Assessment of Chaos in a Hemodynamic Model of Sickle
Cell Disease in the Microcirculation
by
Akwasi Asare Apori
B.S. Aeronautics and Astronautics
Massachusetts Institute of Technology, 1998
Submitted to the Department of Aeronautics and Astronautics
in partial fulfillment of the requirements for the degree of
Master of Science in Aeronautics and Astronautics
OF TECHNOLOGY
at the
-SEP 11 2001
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
LIBRAR ES
June 2001
© Massachusetts Institute of Technology, 2001. All Rights Reserved.
.....................
Department of Aeronautics and Astronautics
May 30, 2001
A uthor ................... '.........
Certified by .................
-
Wesley L. Harris
ssor of Aeronautics and Astronautics
Thesis Supervisor
/
Accepted by ......................
...
.
/
AA,
....
Wallace E. Vander Velde
Professor of Aeronautics and Astronautics
Chair, Committee on Graduate Students
2
Assessment of Chaos in a Hemodynamic Model of Sickle
Cell Disease in the Microcirculation
by
Akwasi Asare Apori
Submitted to the Department of Aeronautics and Astronautics on
May 30, 2001, in partial fulfillment of the requirements for the
degree of Master of Science in Aeronautics and Astronautics
Abstract
This thesis explores the mathematics describing both sickle cell blood flow and the nature
of chaotic systems. An Eulerian model describing the manifestation of sickle cell disease
in the capillaries was formulated to study the onset of sickle cell crises. Oxygen concentration, red blood cell velocity, cell stiffness and plasma viscosity were modeled as state variables of the system. The model was solved with respect to a physiological parameter (c2 )
representing the rheology of the unique oxygen-cell stiffness relationship of sickle cell
erythrocytes. The existence of chaos was found in solutions for certain ranges of c2 . These
solutions proved to be deterministic in nature, aperiodic, and sensitive to initial conditions.
The characteristics of the sickle cell flow solutions as they progress from steady state to
chaotic appeared consistent with the change from a healthy state to a crisis state in a sickle
cell patient.
Thesis Supervisor: Wesley Harris
Title: Professor of Aeronautics and Astronautics
Acknowledgements
I would like to thank Professor Harris for proposing this research topic to me and for believing
that I could make a contribution in this field. His guidance and knowledge of the bigger picture
kept me thinking critically and provided constant motivation. I also want to thank Professor Rosales who took the time to explain the finer points of chaos theory to me.
There are also many people that supported me over the past few years who deserve thanks. This
includes my entire family who was always there for me. I especially want to thank my mother,
Florence Apori, who always checked up on me to make sure things were going well. I also want
to thank my sister, Yaa, who helped me to not overcommit and stay focused and my brother Kofi
for helping me relax over breaks. I would also like to thank the Narnor-Harris's; Patience, Vincent and my niece Afia-Grace who is an inspiration just by being here. Finally, I have to acknowledge and thank Erin Matzen and all of my friends who constantly encouraged me when I got
frustrated and reminded me that life after MIT was right around the corner.
This work is dedicatedto the memory of Jaksel, and to the many people whose lives have been
affected by the pain of sickle cell disease.
4
Table of Contents
1 Introduction ....................................................................................................
1.1
13
Sickle Cell as a Chaotic Phenom enon ..........................................................
1.1.1 Classifying D iseases as Chaotic......................................................
1.1.2 Criteria for Chaotic D isease.............................................................
Background of Sickle Cell D isease.............................................................
1.2.1 Cause and Sym ptom s......................................................................
1.2.2 Treatm ent ........................................................................................
Objectives and Outline of Thesis.................................................................
1.3.1 Objective ..........................................................................................
13
13
13
14
14
15
16
16
1.3.2
O utline.............................................................................................
16
2 L iterature R eview ..........................................................................................
17
1.2
1.3
2.1
Previous M odels...........................................................................................
2.1.1 Previous Results...............................................................................
2.1.2 Evolution of Sickle Cell M odeling .................................................
2.1.3 Limitations ......................................................................................
17
17
18
20
2.1.4
Chaos M odeling ...............................................................................
21
Current M odels ............................................................................................
22
3 Fundamentals of Chaos T heory ................................................................
23
2.2
3.1
3.2
3.3
Properties of Chaotic System s ....................................................................
23
3.1.1 Description of Chaotic Systems......................................................
3.1.2 Sensitivity to Initial Conditions ......................................................
3.1.3 Aperiodicity ......................................................................................
3.1.4 Determinism....................................................................................
3.1.5 Additional Properties of Chaotic Systems ......................................
Origins of Chaos in Systems......................................................................
3.2.1 Causes of Chaos...............................................................................
3.2.2 The Equations .................................................................................
3.2.3 Examples of Chaotic Systems.........................................................
Demonstrating Chaos in a System ...............................................................
3.3.1 General M ethod ...............................................................................
3.3.2 Finding Chaos in the Sickle Cell M odel..........................................
23
23
25
29
32
35
35
36
37
40
40
43
4 The Apori-H arris Sickle Cell M odel........................................................
4.1
45
The Physics of Sickle Cell Blood Flow ......................................................
4.1.1 Physiology of Sickle Cell Disease .................................................
4.1.2 Capillary Flow Theory....................................................................
45
45
46
4.1.3
Oxygen Transport ..........................................................................
51
4.1.4
4.1.5
Sickle Cell Rheology ......................................................................
Contributions of Hydrodynamic Sickle Cell M odel.......................
54
55
5
4.2
4.3
4.4
State Equations of the Apori-Harris Sickle Cell Model...............................56
4.2.1 State Variables .................................................................................
56
4.2.2 Oxygen Concentration .....................................................................
57
4.2.3 Blood Velocity .................................................................................
59
4.2.4 Cell Stiffness....................................................................................
60
4.2.5 Plasm a Viscosity.............................................................................
62
Assum ptions and Simplifications ..............................................................
63
4.3.1 Oxygen Concentration ...................................................................
63
4.3.2 Blood Velocity .................................................................................
64
4.3.3 Cell Stiffness....................................................................................
64
4.3.4 Plasm a Viscosity.............................................................................
64
Final Apori-H arris Sickle Cell M odel.........................................................
65
5 Solving the A -H M odel......................................................................................67
5.1
5.2
Solutions of the A-H Equations ...................................................................
5.1.1 Fixed Points ......................................................................................
5.1.2 Linearized Stability Analysis of Fixed Points .................................
Param eter Determination .............................................................................
5.2.1 Dim ensional Analysis ......................................................................
5.2.2 Determination of Control Param eter................................................
5.2.3 Final Param eter Values ....................................................................
67
67
69
75
75
77
80
6 A nalysis of the A -H A ttractor ........................................................................
6.1
6.2
83
Bifurcation Analysis .......................................
6.1.1 Bifurcation Analysis ........................................................................
6.1.2 Route to Chaos..................................................................................85
Phase Space Properties ...............................................................................
6.2.1 Four Dim ensional A-H Attractor....................................................
6.2.2 Basin of Attraction...........................................................................
6.2.3 Long Term Attractor Behavior ............................................................
83
83
95
95
97
98
7 Results of Chaos Analysis on A-H Model...................101
7.1
7.2
7.3
D eterm inism ...................................................................................................
101
7.1.1 D eterminism in A-H Equations .........................................................
101
7.1.2 Poincare Section.................................................................................102
Aperiodicity ...................................................................................................
105
7.2.1 Aperiodic Time Series .......................................................................
105
7.2.2 Fourier Analysis.................................................................................106
Initial Condition Sensitivity...........................................................................110
7.3.1 Sensitivity of the Strange Attractor and Tim e Series.........................110
7.3.2 Lyapunov Exponent ...........................................................................
114
8 C onclusions .........................................................................................................
8.1
Key Findings..................................................................................................117
6
117
8.2
8.3
A -H M odel Limitations..................................................................................122
Future W ork ...................................................................................................
123
A ppendix A Fourier Analysis...................................................................................125
A ppendix B M aple Routines ....................................................................................
127
B.1 Solving OD E's w ith M aple............................................................................127
B.2 Poincare Section.............................................................................................127
B.3 Fourier Pow er Spectrum ................................................................................
129
B .4 Lyapunov Exponent.......................................................................................130
B.5 A dditional M aple W orksheets .......................................................................
133
Appendix C Governing Equations Tested for A-H Sickle Cell Model.........147
C.1 Lagrangian Form ulation of A -H M odel.........................................................147
C.2 Eulerian Form ulation of A -H M odel .............................................................
149
References .................................................................................................................
7
151
8
List of Figures
Figure 2-1:
Figure 3-1:
Figure 3-2:
Figure 3-3:
Figure 3-4:
Figure 3-5:
Figure 3-6:
Figure 3-7:
Figure 3-8:
Figure 3-9:
Figure 4-1:
Figure 4-2:
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
19
Cima et al. Multivalued Solution for Viscosity .......................................
Lorenz Attractor Sensitivity to Initial Conditions....................................25
Fourier Power Spectrum of Periodic and Chaotic Motion.......................27
28
Fourier Power Spectrum for Chaotic Lorenz Attractor ..........................
30
.................................................
Duffing's
Equation
Section
for
Poincare
First Difference Plots for Random Data and Chaotic Data.....................31
Rossler Attractor and Time Series for x(t)...............................................38
39
Lorenz Attractor and Time Series for x(t) ...............................................
Effect of Control Parameter on Stability for Lorenz System...................40
Approach for Analyzing the Sickle Cell Model......................................43
Red Blood Cell Stiffening in Capillary....................................................45
47
Red Blood Cell Rheology ........................................................................
51
4-3: Krogh Cylinder M odel.............................................................................
4-4: Relation Between Oxygen and Viscosity in Capillary for Sickle Cell ........ 58
65
4-5: Block Diagram of A-H Sickle Cell Model.............................................
5-1: Stability of Fixed Point Trace and Determinant ......................................
71
5-2: Trajectories Expelled from Lorenz Attractor...........................................73
6-1: A-H Attractor and Time Series (c 2 x -001 to c2 x -1)--------------...............--87
6-2: A-H Attractor and Time Series (c 2 x 1 to c2 x 100).................................88
6-3: A-H Attractor and Time Series (c2 x 1000).............................................89
6-4: Stability Change With Control Parameter C2 ------------------91
. . . . . . .------..... ........
6-5: Route to Chaos of A-H Attractor (c 2 x 2)...............................................91
92
6-6: Route to Chaos of A-H Attractor (c 2 x 3, c 2 x 3.1, c 2 x 4) -------............
6-7: Route to Chaos of A-H Attractor (c 2 x 5)...............................................93
6-8: 3-D Phase Space Frames for A-H Attractor.............................................95
6-9: Time Series Frames for A-H Attractor ...................................................
97
6-10: Long Term Behavior of A-H Attractor ..................................................
98
7-1: Poincare Sections of A-H Attractor ...........................................................
103
7-2: Aperiodic Time Series for A-H Attractor ..................................................
105
7-3: Time Series of Limit Cycle and A-H Attractor..........................................107
7-4: Fourier Power Spectrum of Limit Cycle and A-H Attractor ..................... 109
7-5: Sensitivity to Initial Conditions of A-H Time Series.................................111
7-6: Sensitivity to Initial Conditions of A-H Chaotic Attractor ........................ 113
7-7: Lyapunov Exponent vs. Control Parameter C 2 ----------------. . . . . . . . . . . . . . . . . . . -114
9
10
List of Tables
72
Table 5-1: Linearized Analysis of Origin ...............................................................
72
Table 5-2: Linearized Analysis of F' and F ...........................................................
Table 5-3: Characteristic Body Constants ..............................................................
75
Table 5-4: Characteristic Values of A-H Parameters ............................................
76
Table 5-5: Key Parameter Values Demonstrating Chaos ........................................
78
Table 5-6: Final A-H Parameter Values for Chaos Analysis...................................81
Table 6-1: Linearized Analysis of Fixed Points .....................................................
84
Table 6-2: Initial Conditions...................................................................................
85
Table B-1: Poincare Plane Bounds ............................................................................
11
128
12
Chapter 1: Introduction
1.1 Sickle Cell as a Chaotic Phenomenon
1.1.1 Classifying Diseases as Chaotic
There are many reasons why one might seek to prove or disprove chaotic behavior of a
disease. Foremost among these are to contribute to the understanding and treatment of the
disease. By determining the nature of outbreaks or crises in sickle cell disease, one could
more accurately predict and subdue these life threatening crises before they ever occur.
Determining the possible chaotic mechanism behind sickle cell crises could enable control
theory to be applied in a preventative way to treat the symptoms of sickle cell crises. It
could also lead to more efficacious delivery and use of medicine to those who currently
suffer from the disease. The goal of this research is to use chaos theory to analyze the
occurrence of sickle cell crises based on a model of the blood dynamics in the microcirculation.
1.1.2 Criteriafor Chaotic Disease
The criteria used for classifying a disease as chaotic are the same for that of classifying
any system as chaotic. The state variables of the system must be determined initially.
Then, the disease must be physically modeled to derive the governing equations. The
solutions to these governing equations are analyzed to see if they exhibit determinism,
aperiodicity, and sensitivity to initial conditions. These criteria are discussed with more
13
detail in chapter three.
1.2 Background of Sickle Cell Disease
1.2.1 Cause and Symptoms
Sickle cell disease is a painful disease caused by a hemoglobin disorder that primarily
affects people of African descent. Sickle cell patients suffer painful attacks, termed crises,
caused by a lack of adequate oxygen reaching the body tissue. Sickle cell disease affects
red blood cells. Sufferers have an abnormal type of hemoglobin (hemoglobin S) which
makes the red blood cell predisposed to taking a sickle shape at low oxygen tension.
These sickle shaped cells clog small blood vessels preventing an adequate supply of blood
from reaching that part of the body. All organ systems are affected including most notably
the spleen, lungs, kidney, and brain. The most common type of crisis is the infarctive
sickle cell crisis (simply referred to as sickle cell crises in this thesis)[5]. In the medical
sense, this crisis is defined as the obstruction of small vessels by sickle cells. In terms of a
hydrodynamic sickle cell model, a crisis is defined as the change of a critical state variable
to a range that is physiologically indicative of a crisis state in the body[1].
The major symptoms of sickles cell disease are like that of chronic anemia. These include
fatigue and decreased exercise tolerance, jaundice and susceptibility to gallstones. Other
manifestations include acute chest syndrome, retinal problems, stroke, renal and pulmonary failure[5]. During crises, symptoms include skeletal pain persisting for several days
or weeks and associated fever. Crises in young children can also include pooling of red
14
blood cells in the blood stream. It is not uncommon for these crises to result in death[5].
1.2.2 Treatment
The approach to staying healthy for patients suffering from sickle cell disease is to avoid
crises and treat them immediately when they do arise. Sickle cell patients are counseled to
avoid high altitudes, maintain an adequate fluid intake, to treat infections promptly, take
dietary supplements, and receive various vaccinations. When a patient is already suffering from a crisis there is little physicians can do. No drugs exist that are effective in preventing crises. Therapy consists of hydration, keeping the patient warm and possibly
using oxygen. Chronic transfusion therapy has also been used to decrease the frequency
of crises[5].
Other treatments include anti-sickling agents. These are given to the more severely
affected patients. Allogenic bone marrow transplantation is the only treatment that can
cure the disease. However, it is a high risk procedure with a 5 to 10 percent mortality rate.
Gene therapy is a possible future treatment and/or cure which has proven difficult to this
point[5].
In general, sickle cell disease is difficult to treat. There is virtually no way to predict and
prevent a crisis before it occurs. Once a crisis has occurred, there is no immediate way to
subdue it. It must be left to run it's course while the patient is at risk of death.
15
1.3 Objectives and Outline of Thesis
1.3.1 Objective
The objective of this thesis is to develop a model for sickle cell blood flow in the microcirculation that captures the complex interaction of physiological parameters that lead to chaotic manifestation of crises in the microcirculation.
1.3.2 Outline
Chapter one of the thesis introduces the problem of sickle cell crises. Chapter two summarizes previous work that has contributed to modeling sickle cell blood flow in the
microcirculation as well as work in diagnosing chaos in physical systems. Chapter three
details the fundamentals of chaos and the tools required to analyze chaotic systems.
Chapter four describes the hemodynamics of sickle cell disease in the microcirculation
and the development of the Apori-Harris (A-H) sickle cell model. Chapter five shows the
process of solving the A-H model and selecting control parameters. Chapter six details
the dynamics of the chaotic attractor. Chapter seven presents the results of the chaotic
analysis for the A-H model. Chapter eight gives general conclusions and suggestions for
further study.
16
Chapter 2: Literature Review
2.1 Previous Models
2.1.1 Previous Results
Several models formulated over the years provide the foundation for this research. The
Lighthill-Fitzgerald (1969) lubrication theory is the model used to describe the flow of red
blood cells through the microcirculation[2]. The L-F lubrication theory extended upon the
solution for a bubble flowing through a tightly fitted circular pipe in which it was found
that a bubble flows through a pipe faster than the average velocity of fluid through the
pipe[4]. Lighthill and Fitzgerald extended this theory to create the first accurate model for
blood cells flowing single file through the capillaries by taking into account the rheology
of the red blood cells. The L-F model established the solution for the total pressure drop
down a capillary as a sum of contributions from each individual cell and the blood plasma
between cells. These contributions were a function of blood velocity, plasma viscosity
and cell stiffness. The description of the dynamics for the flow of red blood cells in the
microcirculation was the major contribution of the L-F model.
The theory for oxygen transport in the microcirculation was developed by August Krogh
in the early 1920's. Krogh's model described how oxygen carried in red blood cells coming from the lungs was dispersed to the body[6]. Starting with the diffusion equation and
convection, Krogh modeled the transfer of oxygen in the capillaries to the surrounding
body tissue as the red blood cells moved through the capillaries. Krogh described the
17
region of tissue supported by one capillary as a cylinder surrounding each capillary
through which no flux of oxygen escapes. The Krogh cylinder model provided a solution
for the oxygen concentration in both the capillary and the surrounding tissue as a function
of the axial and radial position along the capillary.
These two models provide the framework for the investigation done in this thesis. Coupling the solutions of the L-F capillary flow theory with the Krogh oxygen transport model
results in a physiological model for capillary blood flow. In the following section,
research on the unique rheology of sickle cell carrying red blood cells is explained to provide a physiological model describing sickle cell blood flow in the capillaries.
2.1.2 Evolution of Sickle Cell Modeling
Berger and King (1980) developed a mathematical model incorporating capillary flow theory and oxygen transport theory to describe the flow of sickle cell blood in the capillaries[3]. The B-K model included the dependence between sickle-cell erythrocyte rheology
and local oxygen concentration. The introduction of this relationship was key because it
was unique to sickle cell erythrocytes and differentiated the B-K sickle cell model from
existing models of capillary blood flow for normal erythrocytes. Berger and King solved
for oxygen profiles as a function of the distance a cell travels down the capillary. The B-K
model found that depending on initial conditions and physiological parameter values it
was possible for cells to be depleted of oxygen before exiting the capillary. The B-K
model established a possible mathematical representation of conditions that led to red
blood cell sickling in the capillary.
18
Cima, Discher, Tong, and Williams (1994) followed up the B-K model with an improved
sickle cell model of erythrocyte dynamics in the capillaries[l1]. Cima et al. used improved
values for key physiological parameters that affect blood flow. They also improved upon
the equation relating red cell rheology to cellular oxygen tension and the modeling of metabolic oxygen consumption by the tissue. These improvements led to the discovery of a
previously unknown region of multi-valued solutions for capillary blood cell velocity as a
function of blood plasma viscosity (figure 2-1).
0.6
0.4
0.3-
S0.20.1.
0.0
1.4
visacosity
1.6
1.8
2.0
plasma viscosity,A (mPa-s)
Figure 2-1: Cima et al. Multivalued Solution for Viscosity[1]
Cima et al. showed that the multi-valued behavior was caused by the sensitivity of cell
stiffness to local oxygenation for the sickle cell case. They also conjectured that this
multi-valued behavior led to the sudden and abrupt change from normal to crisis state.
Solutions for blood cell velocity would progress down one branch of the curve before
jumping to the lower branch of the curve when entering the multi-valued regime. Cima et
19
al. explained the drop to the lower branch of the curve as a preference for the lower energy
dissipation in that state. The Cima model showed that the abrupt onset of a sickle cell crises could be caused by a catastrophic change in the blood velocity in the multi-valued
regime similar to state bifurcations in catastrophe theory. The Cima model also showed
the extreme dependence of the onset of the multi-valued regime, and thus possible catastrophe induced crises, upon small changes in physiological parameters that vary between
individuals.
2.1.3 Limitations
Past sickle cell models have not been constructed to allow the analysis of long term behavior of sickle cell blood flow. The approach of previous research has been to find the governing equations of sickle cell blood flow on the microvascular level. These equations are
important in describing how sickle cell manifests itself hydrodynamically and more
importantly in describing what conditions within the capillaries may lead to a crisis. However, these models are limited by a lack of autonomy and the Lagrangian formulation of
the governing equations.
The previous models stop short of modeling the entire sickle cell capillary flow system
autonomously. The state variables of the sickle cell model represent physiological properties of the body which are all interdependent. Previous models always kept one variable
(i.e. blood viscosity) independent to see how the other variables varied with respect to the
independent variable. Then, the profile along the capillary over a finite period of time was
solved. In reality, there should be no independent variable for sickle cell blood flow other
than time if the state variables aren't being physically controlled. To generate realistic
20
behavior of the system over time it is necessary to make all of the state variables time
dependent. None of the previous models explored blood flow in the capillary as an autonomous system. This prevented a systematic investigation of both the erratic occurrence of
sickle cell crises and the fundamental causes of this apparently chaotic phenomenon.
The Lagrangian perspective employed for solving the oxygen profile over the length of the
capillary also prevents the analysis of the systems long term behavior. The Cima et al.
model essentially looses validity after the amount of time it takes for one blood cell to
traverse the length of the capillary. An Eulerian model which continuously shows the state
of the flow in the capillary is necessary to demonstrate the long term behavior.
The model presented in this work built upon findings of the previous models. Contributions unique to this thesis include the following: i) The sickle cell problem was formulated in the Eulerian framework. ii) Viscosity was modelled as a state variable. iii) The
erratic onset of sickle cell crises was systematically treated.
2.1.4 Chaos Modeling
There are several fields in which chaos theory has been used to model the behavior of
physical systems. Many of these models have been useful in demonstrating the techniques
for discerning chaos in physical systems. The Logistic Equation is a population model
developed by Verhulst in 1838. It shows how outside influences can make populations of
various animals fluctuate wildly instead of reaching a steady state. The Lorenz Equations
(Lorenz 1963) are a simplified model for convection roles in the atmosphere. This simple
model also exhibits extremely erratic dynamics which demonstrate the difficulties of long
21
term weather prediction. They were also the first equations to launch the study of chaos
theory in it's current form. There are many more recent examples ranging from a variety
of fields of research including biology, lasers, noise, particle mixing, chemical reactions,
magnetic fields, etc[7]. As chaos theory is more utilized it provides the opportunity to
conduct research into physical phenomena previously thought to possess no order or be
indescribable from a mathematical standpoint.
2.2 Current Models
The author conducted an extensive literature search and is not aware of any research on the
chaotic nature of sickle cell disease. The most recent work recent work was that of Cima,
Discher, Tong, and Williams(1994) as discussed above.
22
Chapter 3: Fundamentals of Chaos Theory
3.1 Properties of Chaotic Systems
3.1.1 Descriptionof Chaotic Systems
The term chaos is synonymous with disorder. Chaotic systems are ones that seem disordered on the surface but which have some underlying order underneath. The complexity
of chaotic systems makes predicting long term behavior impossible even though their
exists some underlying order to the system. Population models, weather prediction models, integrated circuits and many other systems show how chaos occurs physically. Not all
systems that appear to be disordered or irregular are necessarily chaotic. Chaos only
occurs in nonlinear, dynamic systems. In the mathematical sense, a chaotic system is generally defined as having the following characteristics: sensitivity to initial conditions, aperiodicity, and deterministic origin. No universal definition of chaos currently exists, but
these three conditions are considered sufficient to demonstrate chaos[9].
3.1.2 Sensitivity to Initial Conditions
Chaotic systems show extreme sensitivity to initial conditions. Nearby trajectories separate exponentially fast. Two trajectories starting very close together will eventually
spread out over the entire attractor. Within the given accuracy of measurement for all systems there are errors introduced that will diverge over each iteration of the solution. This
means that long-term prediction essentially becomes impossible since small uncertainties
23
are amplified enormously fast. This is contrary to non-chaotic dissipative systems in
which trajectories either converge or remain equidistant [7:3 20].
One quantitative measure of the sensitivity of a system to initial condition disturbances is
the Lyapunov exponent. The Lyapunov exponent is a measure of the average rate of convergence or divergence of trajectories in phase space. A positive Lyapunov exponent
shows that neighboring trajectories diverge at a rate proportional to the value of the exponent. A Lyapunov exponent of zero indicates that trajectories separated by a finite distance will maintain the same separation distance over time. A negative Lyapunov
exponent means that neighboring trajectories converge with one another over time. These
three trajectory behaviors are also known as asymptotically unstable, marginally stable,
and asymptotically stable respectively. Chaotic systems posses positive Lyapunov exponents because their trajectories diverge exponentially fast until reaching a maximum separation equal to the diameter of the attractor[9:211, 353].
The Lyapunov exponent is defined as the average of the local slopes of the attractor over a
period of time.
(3.1)
X1=limn-,1/ni=0...n-1 loge I f'(x i),
In (3.1) f' represents the rate of change at point xi. This definition refers to the first or
largest Lyapunov exponent because there are actually n Lyapunov exponents for an ndimensional system.
24
Sensitivity to initial conditions can also be observed qualitatively in the phase space and
temporal space. Time series observations show what trajectories with similar starting
positions will do over a period of time. In figure 3-1 below, x(t) of the Lorenz attractor is
plotted for two slightly different initial conditions. One can observe that when similar trajectories reach their time horizon they diverge and prediction breaks down.
x vs. t for x(O)=-2.62 and x(O)=-2.6400001
0
5
10
15
25
35
Figure 3-1: Lorenz Attractor Sensitivity to Initial Conditions
Sensitivity to initial conditions is what makes the long term behavior of chaotic systems
break down and is a necessary but not sufficient characteristic of chaos[8].
3.1.3 Aperiodicity
Systems that exhibit chaos never settle down into periodic motion. Trajectories will
traverse the phase space in a pattern that never exactly repeats itself, thus making it impossible to predict their future location. Often, a system may appear disordered due to the
interpretation and measurement of the data. The periodic motion of a system with a large
period may go unnoticed without adequate data points while that of a small period may be
unobserved if the data is not sampled often enough. One clear way to determine whether
25
or not a system is periodic is through Fourier analysis[8:130].
Observing a chaotic system in the frequency domain can provide insight not available in
the time domain or phase space. The Fourier analysis allows a time series to be described
in terms of its periodic constituents. The presence or absence of these constituents reveals
how periodic a system is.
Fourier analysis involves breaking down the measured time series of a variable into constituent waves or harmonics. Time series are known as the composite wave and they are
typically decomposed into periodic components like cosine or sine waves. These periodic
components are summed to give Fourier coefficients for each harmonic that add up to the
value of the composite wave at any given time. A detailed description of Fourier analysis
is contained in Appendix A.
For practical purposes Fourier coefficients are usually computed and plotted as a variance
(power). The power spectrum Sh2of harmonic h in terms of Fourier coefficients ax and $ is
defined as:
sh2-(ah2 +
ph2)/ 2
(3.2)
The power spectrum shows the relative importance of the two Fourier coefficients for a
given harmonic. When looking at the power spectrum one can observe periodic motion
from any visible spikes. Spikes at various frequencies show that these frequencies dominate over other frequencies present in the data. Frequency spikes will typically be stron-
26
gest at the frequency of periodic motion and less pronounced for the various harmonics.
The manifestation of periodic motion in the frequency domain can be seen in figure 3-2.
Figure 3-2 shows flow pattern solutions starting with constant flow before moving onto
periodic flow and completely turbulent flow.
Flow pattern
Velocity history
Power spectrum
Time
Frequency
Time
Frequency
Time
Fmquency
Thime
Frequency
Time
Frequency
Flow patterns with varying Reynolds number, velocity history, and power spectra.
(Source: Roads to Chaos by L. Kadanoff. Reproduced with permission of the author.)
Figure 3-2: Fourier Power Spectrum of Periodic and Chaotic Motion [10:202]
Data that is aperiodic does not show a strong peak in power at any one given frequency.
For example, random data or noise essentially has a constant power over the entire frequency spectrum because every frequency contributes to the constituent wave with no
27
preference.
Chaotic data must settle down to aperiodic behavior. By definition, it is required that
exhibiting aperiodic behavior isn't a rare event[8]. This means that the Fourier spectrum
should show some broadening when compared to that of periodic data. There may be
large ranges of several frequencies that are favored over other frequency ranges in the
spectrum but there are never any individual frequencies that dominate the motion of a chaotic system. Figure 3-3 shows an example of the Fourier spectrum for the chaotic Lorenz
attractor. Note the lack of single dominant frequency spikes.
S vs. k for b=8/3 r=28 s=10, [xO,yO,z]=[-2.64,-44.3,17.06]
700
00
500-
400S
300
200-
100
0
0
100
200
300
400
500
k
Figure 3-3: Fourier Power Spectrum for Chaotic Lorenz Attractor
All chaotic systems must have aperiodic motion where trajectories do not settle down to
fixed points, periodic orbits, or quasi-periodic orbits over time. This motion is best demonstrated in the frequency domain. The Fourier power spectrum allows the observation of
periodic constituent waves and their relative impact on the system. Aperiodicity contrib-
28
utes to the inability to make long term predictions and is a necessary characteristic of chaotic systems.
3.1.4 Determinism
All chaotic systems much be deterministic. The irregular behavior of the system must
stem from the nonlinearity of the system's governing equations, not from random or noisy
inputs or parameters of the system. This characteristic is very important in a system for
which data has been measured experimentally and the governing equations are unknown.
Often, noise can mask data and make it appear irregular. The same is true of random data
input to the system. It may produce a very irregular output even if the system is linear[9:324].
If the governing equations of a system are known to be deterministic and there are no
noisy or random inputs or parameters, then by definition the output of the system is deterministic. However, when this is unknown, one tool used for determining whether or not a
system is deterministic is the Poincare section. The Poincare section is simply a cross section taken through the phase space. An example is shown in figure 3-4.
29
0.6
-02
Figure 3-4: Poincare Section for Duffing's Equation[14]
The Poincare section for deterministic data should trace out curves or patterns with points
intersecting the plane in an orderly fashion. If the data is random or noisy then they will
not follow equations governing the dynamics in the phase space. For random data, the
position of a trajectory at any time t would not be correlated to the position of the trajectory at the time t+1. This can also be seen in other return maps like first difference plots.
First difference plots illustrate the same concept as the Poincare section. First difference
plots graph the difference between an observation xt and some later observation xt.
Fig-
ure 3-5 shows first difference plots for random data and the Logistic equation within its
chaotic regime.
30
RANDOM NUMBERS
(b) Difference plot
(a) Time series
I.
N
60
i0Y200
S
.5
Time (iterations
I.e-X
+f
LOGISTIC EQUATION, k - 3.99
(c) Time series
(d) Difference plot
1.0
1
0.5
.00
0La * cm
I
4:
4:
.s-
oS
if
-0.5
0
40
120 160 200
5
5
-1.0
Time (iterations)
-1.0
-0.5
0
0.5
1.0'
X1+I-XI
Figure 3-5: First Difference Plots for Random Data and Chaotic Data[7:262]
As expected, there is a systematic relation on the first difference map for a chaotic system
which points to the underlying determinism. This is in contrast to the random data which
shows considerable scatter. The scatter indicates a lack of relation between successive
points.
Poincare sections and related maps are good geometric tools for showing how data is correlated and whether or not it is deterministic. All chaotic systems must show a pattern in
the Poincare section because the value of a trajectory at any given time will be influenced
31
by it's previous value. This is a property of deterministic systems. Determinism alone is
not sufficient to prove chaos but it is helpful in ruling out irregularity due to random and
noisy input or parameters of the system.
3.1.5 Additional Propertiesof ChaoticSystems
Determinism, sensitivity to initial conditions, and aperiodicity are properties that every
chaotic system must exhibit. Those conditions are sufficient in proving that a system is
chaotic in the mathematical sense. However, there are several other properties that may be
found in chaotic systems. In some cases they can provide additional insight into the
dynamics of the system.
One geometric property of chaotic systems is the strange attractor. A strange attractor
describes the dynamics of the trajectories plotted in the phase plane. It is defined as a
complex surface in the phase space to which trajectories converge asymptotically in time
and on which they move chaotically. Like all attractors a strange attractor has the following properties[7:221]:
*
a set of points in the phase space that the system settles down to over time
" has a geometric shape
e
occupies only certain zones within the bounded phase space
* reproducible
e
invariant probability distribution
32
The main characteristics distinguishing strange attractors are:
* irregular, erratic trajectories
" no crossing trajectories
* divergence of trajectories that start close together (sensitivity to initial conditions)
*
folding due to bounded phase space
*
complex and usually fractal internal structure
*
elaborate or unusual outer geometry
*
non-integer dimension
There are other properties of chaos which can be measured quantitatively. The Kolmogorov-Sinai entropy, mutual information, and redundancy are statistics that provide information about how chaotic a system is.
The K-S entropy represents the average amount of uncertainty in predicting the next n
events. It can also be described as the average rate at which predictability decays with
increasing prediction time. Kolmogorov-Sinai entropy (HKS)is actually an entropy rate
(average entropy per unit time) taken as time increases to infinity and box size decreases
towards zero[7:38 1]:
HKS
(3.3)
=limEM-Olit->o (HAt/time)
=lim--+0 lit->- Eli=1..Nr s ogI/s)]/time
=limF-4olimt-4. (Ht-Ht-1)-
33
In (3.3) Ps is the sequence probabilities of an element taking any one of Nr phase space
routes and Ht is entropy at time t. HKS is zero for deterministic systems that are not chaotic. These systems have no loss of predictability over time. For chaotic systems HKS is a
positive constant indicating the finite window of time in which predictability will break
down. For a random process of uniformly distributed data HKS is infinite.
Mutual information and redundancy are two techniques that take into account the relation
between two or more variables of a system. Knowing information about the state of one
variable at any given moment will provide information about the others. So, the uncertainty of the value of one system is reduced by knowing the value of another. These techniques have potential applications in estimating appropriate lags for attractor
reconstruction, estimating K-S entropy, determining whether a time series is periodic, chaotic or random, and estimating the time envelope for making reliable predictions. These
techniques are complicated and not widely used for practical implementation[7:407].
There are a few additional quantitative measures that provide insight into the chaotic
dynamics of systems. In general, they require additional complex calculations and their
results are sensitive to misapplication. Verification of the deterministic nature of the system, sensitivity to initial conditions, and aperiodicity is the most straightforward way to
show that a system is indeed chaotic. This investigation applies this straightforward
approach to sickle cell disease.
34
3.2 Origins of Chaos in Systems
3.2.1 Causes of Chaos
The causes of chaos in the real world are not well known. There are many systems in
nature thought to be chaotic, but finding conclusive evidence outside of the controlled setting of the laboratory is difficult. When chaos does occur in our physical world a few different reasons are usually proposed[7:13]:
*
A change in the value of a critical control parameter: A system with a stable point
solution may move through zones of periodicity and intermittency on its way to chaotic behavior as the control parameter is increased or decreased. Varying the control
parameter can directly influence the amount of disorder in the system.
e
The nonlinear interaction of two or more individual processes: When two individual
operations are nonlinearly coupled it may lead to behavior much more complex than
that of either operation. An example of this is the problem of the double pendulum
where one pendulum dangles from the end of the lower. Moving the first pendulum in
an orderly way can make the lower pendulum move in a chaotic manner.
" The presence of environmental noise on motion that would be regular otherwise.
Chaos detected in controlled settings can generally be attributed to the first cause stated
above. Changing critical parameters in a small way may drastically change the dynamics
of a system-going from fairly regular behavior to complex chaotic behavior.
35
3.2.2 The Equations
There are certain characteristics found in a system of equations that leads to chaos. Not
every nonlinear equation has the potential to generate chaos. For example, a power law
equation of the form y=axb will never develop chaos. Solutions when iterated will either
become infinitely small if xO is between 0 and 1 or they will grow to infinity if xO is greater
than 1. The same is true for an exponential equation of the form y=abx which becomes
infinitely large upon iteration. Equations or systems of equations that lead to chaos must
be nonmonotonic, noninvertible, and contain at least one unstable fixed point[7:203].
*
Nonmonotonic equations have a slope that changes sign at some point. This means
that there is a switchback or hump in the curve that allows the solution to both
increase and decrease over time.
*
Noninvertible equations are ones in which the value of xt cannot be determined from
simply knowing the value of xt+1 in an iterative equation. The unique value of the
antecedents isn't obtainable from the current state because it may have reached that
point via any number of paths.
*
The solutions to the equations must contain at least one unstable fixed point. In chaotic systems unstable fixed points repel solutions in the phase plane preventing them
from settling down to point attractors or periodic motion.
In general, mathematicians think that most discrete nonlinear equations or systems of differential nonlinear equations have the possibility to generate chaos. The key is to find the
36
- __- -
M.,
correct parameter ranges in the equations over which there is chaos[7].
3.2.3 Examples of Chaotic Systems
There are many examples of systems thought to exhibit chaotic behavior. There are examples of epidemics, pollen production, population models, rainfall, and droughts among
other things that appear chaotic [7:12]. Fields in which chaos has already been studied
include chemical reactions, lasers, biological rhythms, superconducting circuits, turbulence, geology, economics, and sociology just to name a few[9:ix].
One simple three dimensional chaotic system is the Rossler system. Otto Rossler (1976)
was inspired to develop the equations based on the motion of taffy-pulling machines. The
Rossler equations only have one nonlinearity in the xz term[9:434]:
dx_ - -=dt
z
(3.4)
dy = x+ay
(3.5)
dz
I
dt
= b+z(x-c)
(3.6)
When the parameter values are set at a=b=0.2 and c=5.7 numerical integration yields a
strange attractor and chaotic time series. On the attractor in figure 3-6 trajectories appear
to spiral out in the x-y plane until reaching a critical distance from the repelling point.
Then trajectories jump into the z-direction before returning closer to the repelling point in
the x-y plane. The effect of this behavior is also apparent in the accompanying time series.
37
x
vs. y vs. z
x vs. t
1VO1
Figure 3-6 Rossler Attractor and Time Series for x(t)
Another very well studied chaotic system is the Lorenz system. In 1963 Ed Lorenz
derived three equations describing a simplified model for convection rolls in the atmosphere. When he reproduced a previous prediction of the model on his computer he forgot
to specify the initial conditions to the same significant digits. The results of his model's
prediction were drastically different than the prediction of the previous run. The discovery
of the Lorenz equations and their sensitivity to initial conditions launched the field of
chaos theory as known today. The Lorenz equations are as follows:
=
a(y - x)
(3.7)
dt
dy
dt
=rx
=z
-y -xz
(3.8)
xy-bz
(3.9)
38
When the parameters are set to a =10, b=8/3, and r=28 the system exhibits chaos. Figures
3-7 and 3-8 show the Lorenz strange attractor, a sample of the time series in the chaotic
regime, and the affect of varying the control parameter r on the system. The strange
attractor has trajectories that spiral away from the center until reaching a critical distance
at which they jump towards the center of the opposite attracting surface. The time series
for the Lorenz system shows that for the parameter values plotted chaos ensues. The analysis of the parameter space in figure 3-8 shows that the Lorenz system varies from having
a single stable fixed point to multiple fixed points to no fixed points due to a series of
bifurcations. When a and b are held constant, the parameter r controls the stability of the
system and whether or not chaos will appear.
x vs. y vs. z
x vs. t
Figure 3-7: Lorenz Attractor and Time Series for x(t)
39
unstable
limit cycle
~-~
x
I
It
r
--
-
-
-
- -
-
-
- - r
1
I
13.926
stable
origin
I
24.06
stable fixed points C~,
24.74 = ru
C
I
I
transient chaos
transient chaos
I
strange attractor
strange attractor
Figure 3-8: Effect of Control Parameter on Stability for Lorenz System[9:330]
There are many more examples of chaotic systems. The Rossler and Lorenz systems are
well studied because of their relative simplicity and because their governing equations are
known. The Logistic equation and a few other one dimensional iterated maps have also
been well explored and are known to create chaos within various parameter ranges. Other
experimental data has been collected on chaotic systems like cardiac rhythms and brain
activity[1 1]. The study of chaos in real life systems is a complex process and is still progressing.
3.3 Demonstrating Chaos in a System
3.3.1 General Method
One may desire to demonstrate chaos in a variety of systems. The observed system can be
a process occurring in nature or experiments run in the laboratory or on a computer. In
many real world examples, the only information available are measurements of the output
40
with time. In the more ideal case the governing equations of the system are known. There
are numerous tests for chaos though none really provides completely exhaustive proof.
Whether starting from experimental data or a system of equations, one can proceed with
the following steps to detect chaos in a system[10:208].
1. Obtain or generate a time series. Visual inspection of the time series is an important
first step. If the data is obviously linear or if there aren't any fluctuations then there is
no chaos. An erratic time series is a better candidate for chaotic behavior. When
observing the time series it is important to make sure to take into account the scale to
which the data is plotted because this can cause a plot to look linear when it is not.
2. Check for random inputs and parameters. If there is irregular behavior observed in the
time series than the system should be double checked to make sure the behavior is not
due solely to random data going into the system. In general, chaos should be suspected when the amplitude of the output is much larger than that of the input.
3. Perform a Fourier Analysis. The Fourier power spectrum for a chaotic system will
show spectral broadening. The presence of pronounced spikes in the power spectrum
indicates periodic motion which is unlikely to be chaotic.
4. Obtain the autocorrelation function. The calculation of the autocorrelation function is
difficult in practice but can provide valuable information to the cause of spectral
broadening. If the system is chaotic the correlation function will decay. Chaotic data
is only correlated to its very recent past causing long term predictability to break
41
down.
5. Obtain a Poincare section or return map. A Poincare section of the phase space will
show the cross section of the strange attractor cut through with a plane. This provides
insight to the fractal nature of the strange attractor and helps differentiate between
deterministic data and randomly generated data. Both the Poincare section and return
map of a chaotic system will plot as a curve as opposed to points spread out over the
section like random data.
6. Calculate the Lyapunov exponent & Kolmogorov entropy. For a chaotic system both
values will be positive because neighboring trajectories on the attractor will diverge
over time.
7. Plot the solutions. When chaos is suspected to exist, a geometric analysis of the system can provide additional confirmation. Plotting trajectories in the phase space will
display important properties of the attractor such as the bounds of the attractor and the
basin of attraction. A bifurcation plot will show how the system progresses from normal to chaotic dynamics as the control parameter is increased.
When working with only experimental data, the process of detecting chaos is more difficult because the attractor must be reconstructed in the phase space by methods that are not
as reliable as solving the equations. Knowing the governing equations gives extra insight
into the system. They make it easier to see nonlinearity, determine key parameters, and
determine behavior of the inputs among other things.
42
The previous seven steps are a general procedure for demonstrating chaos in a system. In
an ideal case one would satisfy all of the conditions, but in reality the necessary data may
not be available. In general, the more of these conditions that a system satisfies, the more
confident any conclusion that the system is chaotic.
3.3.2 Finding Chaos in the Sickle Cell Model
Determining whether chaos exists in the sickle cell model will be done by following the
steps outlined for detecting chaos in systems. The focus will be on proving the three necessary characteristics of chaos: determinism, aperiodicity, and sensitivity to initial conditions. Figure 3-9 shows the process for analyzing the sickle cell model.
Create simplified model
of Sickle Cell dynamics
Exhausted all methods
of simplifying problem?
N
3 or more state variables?
Nonlinear coupling?
A
Y
<3n
i
Fourier
Analysis
Attractor
Return Map
Lyapunov
Exponent
>3n
Disprove chaos
N
All 3 demonstrate chaos?
Y
Analyze as nonlinear
dynamics problem
Conclusion: System
demonstrates chaos
Figure 3-9: Approach for Analyzing the Sickle Cell Model
43
N
The subsequent chapters of this thesis present the above process in more detail. Chapter
four shows the development of a novel sickle cell model with assumptions and approximations. Chapter five presents the solutions to the A-H sickle cell model. Chapter six details
the dynamics of the chaotic attractor found by solving the A-H sickle cell model. Chapter
seven presents results of the chaos tests performed on the sickle cell model. Chapter eight
gives conclusions and discussion drawn from the results of the analysis.
44
Chapter 4: The Apori-Harris Sickle Cell Model
4.1 The Physics of Sickle Cell Blood Flow
4.1.1 Physiology of Sickle Cell Disease
Patients suffering from sickle cell disease undergo painful attacks, or crises which result in a lack
of adequate oxygen reaching the tissue. The manifestation of this disease in the microcirculation
can be seen as the red blood cells flow through the capillaries. The red blood cells in a person suffering from sickle cell disease stiffen as they give up oxygen. The pathology of this stiffening is
described by the polymerization of hemoglobin inside the cell. The stiffening increases the difficulty of passage of cells through the capillaries (figure 4.1). Eventually the cells may slow to the
point where they are unable to pass through the capillaries altogether, thus causing a crisis.
red cells-
The red blood cells flow through the capillary single file. The capillary walls are
assumed to be rigid. The walls of the red cells are assumed to be inelastic but flexible.
Both the plasma flowing between the cells and the fluid inside the cells are incompressible.
For a person with sickle cell disease the red blood cells stiffen and straighten out as they
flow down the capillaries.
Figure 4-1: Red Blood Cell Stiffening in Capillary[1]
45
From a hydrodynamic point of view, the plasma viscosity plays an important role in the transition
from non-crisis state to crisis. The plasma viscosity can increase for any number of reasons. This
increase leads to a decrease in the velocity of the flow. The decrease in the flow velocity causes
cells to stay in the capillary longer allowing them to release more oxygen. The increase in oxygen
release leads to an additional rise in viscosity continuing a 'viscous cycle' which can ultimately
lead to the sickling (collapsing their shape) and coagulating of cells in the capillaries[1].
After many incidents of sickling, some red cells remain permanently sickled and may obstruct
small vessels until removed from the circulatory system. Numerous crises over a persons life
leads to irreversible damage in the circulatory system causing an early death amongst people who
suffer from the disease.
4.1.2 CapillaryFlow Theory[1,2]
The theory for blood flow through the capillaries stems from Bretherton's bubble problem[4].
Bretherton found the solution of a bubble of air flowing through a circular pipe. It was found that
the bubble flowed through the pipe faster than the average velocity of the fluid in the pipe. The
lubrication layer thickness was a function of the fluid's viscosity and the velocity of the flow. In
1954 Lighthill followed Bretherton's problem by solving for pressure forcing of tightly fitting pellets along fluid filled elastic tubes[ 12]. His results gave equations with characteristics which more
closely resembled circulatory flow. Finally, in 1969 Fitzgerald expounded upon Lighthill's work
providing a solution for the microcirculatory flow which took into account red blood cell rheology. The Lighthill-Fitzgerald lubrication theory will be presented as the basis for modeling capillary blood flow[2].
46
NON
The Lighthill-Fitzgerald lubrication theory solves for the pressure drop across the capillary as a
function of the flow speed and viscosity. Several assumptions are taken in this model. In general
the flow is described by a red blood cell flowing down a capillary with a lubrication layer, h(x),
between the flexible cell wall and the capillary wall at Re (figure 4-2). Since the red blood cell
flows faster than the plasma the relative velocity causes a force on the cell opposite the flow direction. This force is assumed to create a bowing stress that deforms the cell into a uniform parachute shape. The capillary tube and the blood cell are both assumed to be non-elastic and
axisymmetric.
h(x)--lubrication layer
Re- capillary radius
Rf- radius of curvature
R'n--inner radius of curv.
U, p
_ _
..-.
P(g)-- downstream pressure
P(-g)- upstream pressure
S-- resistance to bending
_
-
T-
U-- velocity of flow
p-
viscosity of plasma
R-- radius of blood cell
g --
downstream
-g--
upstream
Figure 4-2: Red Blood Cell Rheology[2]
To find the pressure drop across one blood cell there must be an equation describing the rheology
of the cell. The shape that the cell takes is determined by balancing the surface and bending energies of the cell. For small deformations one can assume that small local deformations depend lin-
47
early on local pressure. The equation for the pressure difference across a curved membrane gives
the pressure difference across the downstream and upstream sides.
upstream side
downstream side
Pinternal- Finternal- P(g) =
P(-g) - Pinternal+ Finternal = S(,-n
7
Combining these two equations gives the pressure drop across the entire cell.
P(-g) - P(g) =
E
(1 +n)
(4.1)
where S is the 'resistance to bending' as measured by Rand & Burton (1964) for normal blood
cells. For sickled cells S is a function of the oxygen concentration.
The equations describing the dynamics of the blood flow are derived from the Navier Stokes equations. The fluid in the capillary is approximated as incompressible with an insignificant pressure
gradient in the transverse direction. Inertial terms are also ignored leading to the 'lubrication
equations'(4.2) and (4.3).
-
ax
r
ap
0
ar
r r -a
(4.2)
(4.3)
ar
where r is the radial coordinate and x the axial coordinate.
48
The flow in the capillary must meet the condition of no slip on the capillary wall. It is convenient
to consider the frame in which the cell is at rest and the walls move with a velocity -U. In that reference frame, the boundary conditions are:
= 0
u =-U
at
r = h(x)
(4.4)
at
r = 0
(4.5)
Integrating the equations twice and satisfying the boundary conditions gives velocity in terms of
the pressure difference.
1
aPr2 - R 2_
u = ~-- 1 ,r
4g ax
2Rh+h 2
-log~
h)
log( 1 + -log(
r
R
Ruog(j
1+
h
(4.6)
In order to solve for a additional equations must be introduced. The total flux passing through
any cross section must equal the fluid leaking back across the lubrication layer by conservation of
mass. The flux leaking back across the lubrication layer is designated as
Q.
This gives (4.7)
below:
(R + h)
J
(4.7)
urdr = -RcQ
R
Substituting (4.6) into (4.7) gives the solutions dxP for both the lubrication layer where
-+o and
past the red blood cell where R -+ 0 and h -+ RC. As expected, the pressure gradient between the
cells (4.9) is characteristic of the Poiseuille pressure gradient for simple pipe flow.
49
dp _
-
6gU +12Q
h2 +
Region near Lubrication Layer:
dx
Region between Blood Cells:
dp _
dx
(4.8)
3
8gU + 16gQ
2
3
ro
ro
(4-9)
can be solved in terms of the total pressure across the capillary by
The resulting equations for dP
dx
balancing the pressure forces and the drag forces in the flow. This is equivalent to finding the
equilibrium shape by total energy methods. The resulting force balance yields:
g
TR 2[p(-g)-p(g)]
RTdx
= 27C
(4.10)
-g
where
(4.11)
T =
rR
rr = R
(4.11) is derived by substituting in u from (4.6). Substituting (4.8) and (4.9) into (4.10) yields the
solution for the pressure drops across each cell, the pressure drop across the plasma between the
cells, and the total pressure drop.
1
AP =
15
CI
= p
g) _ p(g)
(4.12)
p(-)pg
Sc)
A* k
AP, 1
=
8pU ')(4.13)
28jL(1 -H )
R cR'
C
(4.14)
Aptotai
=XAPC7,i+
APP 1
50
E*, A*, and k, are parameters derived from power law functions used to complete the description
of the model. H is the hematocrit, or percentage of the capillary filled with cells.
4.1.3 Oxygen Transport[3]
The theory for oxygen transport through the microcirculation was developed by August Krogh in
the early 1920's[6]. Oxygen is carried inside of the red blood cells which flow through the circulatory system until reaching the capillaries. It is in the very narrow capillaries where oxygen
exchange with the body tissue occurs. Oxygen inside of the red blood cells diffuses out through
the capillary walls into the surrounding region of tissue. The region of tissue with which a particular capillary interacts is described by the Krogh cylinder model (figure 4-3).
Re--radius of capillary
No Flux Boundary
Rt-- radius of tissue
geC
RRt
L--- length of capillary
c--- 02 concentration
U-- blood velocity
-- plasma viscosity
L
Figure 4-3: Krogh Cylinder Model[6]
A cross-section of body tissue taken in any particular area would show that the capillaries running
through the tissue are spaced an equal distance apart. Taking this into account, Krogh modeled
the area to which a capillary provides oxygen as a cylinder of constant radius. The Krogh model
assumes that there is oxygen flux only up to the boundary of the cylinder but no oxygen flux
through the boundary. It is also assumed that the capillary wall offers negligible resistance to dif-
51
fusion and that the metabolic rate of consumption by the tissue is constant. The model also only
provides for diffusion in the radial direction neglecting any axial diffusion which is likely to
occur.
The problem is to solve for the oxygen concentration in the capillary, c, as a function of the axial
distance along the capillary, x. The diffusion equation is used to determine the flux of oxygen
within both the tissue region and the capillary region separately. In the tissue region the rate of
consumption of oxygen must balance the diffusion of oxygen into the tissue (4.15). In the capillary region the rate of convection and diffusion out of the capillary must equal the rate of oxygen
production within the capillary (4.16).
2
Dt a C + 1 ac
r ar
2
ubac
-ax X
where
r i R,
= JR
t
(4.15)
2
+ ac I + d(c)
(r2
Kar
0:5 r R
(4.16)
rarc
r,
d(c) = -N
Dt and Db are the radial diffusivity of oxygen in the tissue and blood respectively. N is the total
oxygen binding capacity of blood and s is the fractional saturation.
52
To solve for c, (4.15) and (4.16) must be integrated twice satisfying the boundary conditions
below, (4.17)-(4.21).
tac
ar
Dbac
or
=
br r=R
r = RC
(4.17)
r = Rc
C
(4.18)
ut r = Rc
ac
=0
(4.19)
-0
(4.20)
r =0
ac
r= = R,
ar
dc
dx
or C
at
X
0
(4.21)
x =L
Boundary condition (4.17) requires the continuity of oxygen flux at the capillary-tissue interface.
(4.18) requires that the oxygen partial pressure is continuous across the capillary wall. Equation
(4.19) requires that the axis of the capillary is symmetric in the radial direction and (4.20) states
that there is no flux of oxygen leaving the Krogh tissue cylinder. (4.21) requires that the concentration or flux of oxygen at both ends of the capillary be known.
Integrating equations (4.15) and (4.16) with boundary conditions (4.17) - (4.21) gives the final
profile for oxygen concentration throughout the tissue and the capillary.
53
R
r:5 R
2
C(x,r)i=
r2
R2
-
t
tR2l
2D
c
2
Rt
ttc
ab2 RcD
(4.22)
0! r! Rc
it 2
C(4,r) =
2
-5 (r -Rc) t
where,
it2l
r
Ct't
Rc
2_R_
(Rc - R)
R log{ +
2D
2
ob Rc
x -F(4,c)
+
+
4D
(4.23)
In (4.23) a change to a new independent variable has been made by the Oseen approximation[1].
4.1.4 Sickle Cell Rheology[J]
Once equations for the dynamics of blood flow (4.12)-(4.14) and the oxygen transport by blood
cells (4.23) have been determined, an equation describing how sickle cell rheology depends on
oxygen concentration must be established. For a person who does not suffer from sickle cell disease, the oxygen transport equations and blood flow dynamics are uncoupled. However, as stated
earlier, for someone suffering from sickle cell disease the resistance to bending, S, of a blood cell
will decrease as the cell looses oxygen during flow down the capillary.
54
Cima et al. developed a model for S as a function of oxygen concentration which accurately
matched experimental data[1]. The equation is given below:
~
ref
c
a + (1-
2
c:5 cref
(4.24)
)(r;
cre
where Sref indicates the bending resistance found in a normal red blood cell and cref indicates the
oxygen concentration needed to have Sref in the diseased blood cell.
4.1.5 Contributionsof Hydrodynamic Sickle Cell Model
The equations for blood flow dynamics (4.12)-(4.14), oxygen transport (4.23), and sickle cell rheology (4.24) represent a Lagrangian model of sickle cell flow down the capillary. When solved
iteratively, the equations show the hydrodynamic representation of crisis in the microcirculation
for someone suffering from sickle cell disease.
The previous work of Cima et al. showed the onset of crises represented by an increase of blood
viscosity over 1.7 mPa[1]. This is identical to the mean value of plasma viscosity found by a
study of patients just after their admission to a sickle cell clinic for treatment of crisis (Laogun et
al., 1980)[1]. Through this model the key role that blood viscosity plays in sickle cell crisis is
apparent. However, there are limitations to the model. The reason for changes in viscosity levels
have been left unexplored. Viscosity was modeled as an independent parameter varied over a certain range. The Apori-Harris model presented in this thesis incorporates viscosity as a dependent
state variable for the first time.
55
In addition, the sensitivity of crises to the physiology of different individuals as well as the unpredictable nature of crisis cannot be explained through the previous Lagrangian formulation of the
microcirculation (see Appendix C.1). The Lagrangian model solves the state variables of each
cell at each position along the length of the capillary. The Apori-Harris model is presented in an
Eulerian framework where the capillary is considered an open control volume over time. Values
of the state variables in the capillary are averaged values over the length of the capillary. This
allows the investigation of some previously unexplored characteristics of sickle cell flow.
4.2 State Equations of the Apori-Harris Sickle Cell Model
4.2.1 State Variables
The Apori-Harris (A-H) model for sickle cell blood flow in the capillaries is based upon capillary
flow and oxygen consumption as described in the previous section. This includes consideration of
the equations for capillary flow theory, oxygen transport, and sickle cell rheology developed by
Cima and others. The A-H model will be formulated in the Eulerian frame in order to observe the
state of the system over time. The complex physics of sickle cell flow is converted from the
Lagrangian formulation of previous models.
The state variables of the A-H model are oxygen concentration (c), blood velocity (u), blood cell
stiffness (s), and plasma viscosity (p). These four equations are necessary to describe the complete state of the sickle cell flow in the microcirculation.
56
4.2.2 Oxygen Concentration
The oxygen concentration is the first state variable of the A-H sickle cell model. Oxygen level is
very important to qualify the state of a person suffering from sickle cell at any given moment.
When oxygen concentration falls below a certain level in the body for sustained periods this can
cause irreparable damage to body tissue. Low oxygen level is also a key manifestation of a crises
event. Data taken from patients suffering crises upon admittance to hospitals shows that a
decrease in oxygen levels is consistent with sickle cell crises[1].
(AH. 1) below describes the rate of change of the oxygen concentration (c) in the blood:
dc
dt = a 1c + a2u + a3u
(AH.1)
where a,, a2 , and a3 are parameters, u is blood velocity and p is plasma viscosity.
The first term on the right of (AH.1) is a1 c. This term is a diffusion term. The rate of change of
oxygen concentration depends upon the current oxygen concentration. Oxygen is transferred
from the capillary to the surrounding tissue through radial diffusion. When c is large it can diffuse more rapidly from the capillary into the surrounding tissue.
The second term on the right is a2 up. This term represents an important nonlinear coupling
between velocity and viscosity. This coupling is important to the sickle cell case as described in
Cima's article.
57
100
80
u (mm/s)
0.5
60
-.
S40
~
p(MPa~s
0,08
N.41.2
~1.7
00
20 -
KI
0
Of
0.0
0.2
L
2.4
0.4
0.6
08
I.0
1.2
x/L, dimensionless distance along cdpillary
Figure 4-4: Relation Between Oxygen and Viscosity in Capillary for Sickle Cell[l]
The above graph shows how both u and p are coupled in determining the oxygen concentration.
The state of both must be known to determine which of the multivalued solutions occurs at any
given moment.
The third term a3 u is a convective term. Oxygen is transferred from the blood to the tissue by
axial convection. The rate of oxygen convection to the tissue varies directly with the velocity of
the blood flow. This is described in equation (4.16) of the previous section where
ac
1
=
+
d(c).
58
4.2.3 Blood Velocity
Blood velocity is the second state variable of the A-H model. This includes the averaged velocities of the cells and plasma. The velocity of the blood is critical to defining the state of the system
at a given time. When the velocity of the blood is too low, the blood cells will give up their oxygen before reaching the end of the capillary. In someone with sickle cell disease these oxygen
depleted blood cells may sickle and block flow down the capillary. This is the start of a crises
event as body tissue downstream of the oxygen depleted blood receives no oxygen. Blood velocity is closely coupled to the other variables that indicate a crises and it must be monitored accordingly.
The rate of change of the blood velocity (u) is described by the following equation:
dt
= blu+b2gs+b3s
(AH.2)
where bi, b2 , and b3 are parameters, g is viscosity and s is cell stiffness.
The first term on the right of (AH.2) is biu. This term is a diffusive term for blood velocity. The
rate of change of velocity, or acceleration, is dependent upon the current velocity.
The b2ps term represents the affect of nonlinear coupling between the rheology of the cell and the
viscosity of the plasma. As the blood cell undergoes rheological changes to the stiffness from the
loss of oxygen, the plasma viscosity around the cell is increased by the local oxygenation. This is
59
part of the 'viscous cycle' leading to chaos. The direct relationship between viscosity and stiffness can be seen through the solution of the capillary flow theory presented earlier[1]. The equations for the pressure drop across the cell and the plasma from section 4.1.2 are as follows:
APc =
(4.12)
A*
AP; = 8pjUL(1 - H)
PI R )
APtotal
Atota =
(4.13)
Ac, i + APgM.
(4.14)
From the above equations the coupling of gs is apparent in (4.12) which is the dominant term
when solving the equations for fixed pressure drop APtota-
The last term is b3 s. This represents the resistive affect of the stiffness on velocity. As the stiffness of the blood cells are increased they move down the capillary with more difficulty because of
the increased drag. This resistive force can cause a change to the overall velocity of the blood
flowing through the capillary. This is obvious from their inverse relationship in (4.12) which is
the major contribution to pressure drop along the capillary.
4.2.4 Cell Stiffness
The rheology of the red blood cells in the capillary is important to indicating the onset of sickle
cell crises and is the third state variable in the A-H model. The stiffness of the cell is defined as
its resistance to bending and taking a parachute like shape when flowing through the capillary.
60
~-
-w
The stiffer the blood cell as it flows through the capillary, the more elongated it is in the direction
perpendicular to the capillary. The stiffening that blood cells undergo in the capillary of someone
suffering from sickle cell disease causes the coupling of oxygen transport and capillary flow
unlike the healthy case in which they are independent. Because of the presence of defective hemoglobin in the sickle cell case, blood cells stiffen and loose the flexibility that allows them to easily
flow down the capillary. As the cell stiffens it reduces the lubrication layer and increases drag.
Eventually, the pressure differential over the capillary will not be large enough to keep the cells
moving through the capillary and the start of a crises may ensue.
(AH.3) below describes the rate of change of the stiffness (s):
ds
- = cIs + c 2 c
dt
(AH.3)
where ci and c2 are parameters and c is oxygen concentration.
The first term on the right of (AH.3) is c 1 s. This is a diffusive term showing that the current stiffness influences the rate of change of the stiffness. When the cell is already very stiff the stiffness
will increase slower than when the cell is fairly flexible.
The second term in the equation is c2 c. This term represents the rheological changes that the cell
undergoes in the sickle cell case. From experimental data, Cima et al. established the following
equation for stiffness versus oxygen concentration:
61
1
S = Sref
c
(4.24)
2
where cref is the oxygen concentration upon entering the capillary, Sref is the stiffness at cref and
a is a physiological parameter.
(4.24) shows that the cell stiffness is inversely proportional to the oxygen concentration. This
relationship is captured in the second term of AH.3.
4.2.5 Plasma Viscosity
The fourth state variable in the A-H sickle cell model is plasma viscosity. Physiologically, plasma
viscosity is known to be linked to the onset of sickle cell crises. Experimental data for patients
admitted to hospitals suffering from crises show a correlation to elevated levels of plasma viscosity. As plasma viscosity rises due to increased levels of oxygen the flow slows down. This in turn
leads to a cycle where more oxygen is released increasing the viscosity even more which once
again reduces the velocity. Eventually the increased viscosity may prevent cells from exiting the
capillary before giving off all of their oxygen and sickling. Thus, viscosity level plays an important role in quantifying the crises state of the flow. The equation for the rate of change of viscosity
(g) is the following:
d
dt
- dig + d 2c
(AH.4)
where di and d2 are parameters and c is oxygen concentration.
62
The di term is a diffusive term. The rate of change of viscosity is dependent upon the current
viscosity. The viscosity will increase slower when the current viscosity is high and increase faster
when the current viscosity is low.
The second term in (AH.4) is d2c. This term describes the physiology of the plasma viscosity.
Plasma viscosity is known to increase for a variety of reasons all of which are associated with a
change in oxygen concentration because of the increased total time cells spend in the capillary.
Oxygen concentration is the only state variable upon which viscosity depends other than itself.
This dependency is quantified by the second term. Considering plasma viscosity as a state variable is unique to the A-H sickle cell model.
4.3 Assumptionsand Simplifications
4.3.1 Oxygen Concentration
In assuming an Eulerian perspective, the oxygen concentration is averaged over the length of the
capillary. There is no variation in oxygen concentration with radial or axial distance along the
capillary. The oxygen concentration is also constant over the red blood cells and the plasma
between the cells. The increased local oxygenation around cells is neglected as well as the difference between oxygen levels between the cells and surrounding plasma. The actual dynamics of
oxygen transport in the capillary is suspected to lead to a profile of decreasing oxygen with capil-
63
lary distance x as seen in figure 4-4. All other assumptions of the Krogh model for oxygen transport also hold.
4.3.2 Blood Velocity
The velocity of the blood is averaged over the entire capillary. The blood is assumed to be incompressible and flowing strictly in the radial direction with no pressure gradient. The difference
between the plasma velocity and the red blood cell velocity is neglected. Assumptions of the L-F
capillary flow theory also hold true.
4.3.3 Cell Stiffness
The red blood cell stiffness is averaged over the length of the capillary. Differences in stiffness
due to local oxygen concentration are neglected. The cells are assumed to flow down the capillary
in single file motion with uniform spacing between them. All cells are assumed to have not
undergone sickling regardless of their oxygen concentration or amount of time spent in the capillary. The dynamics of flow for a sickled cell would differ from the solutions of the L-F capillary
flow model upon which the A-H model is based.
4.3.4 Plasma Viscosity
The plasma viscosity is averaged over the length of the capillary. Local viscosity differences are
neglected. The difference in viscosity between the plasma and the blood cells which are considered insoluble to the plasma is also neglected. The viscosity is assumed constant over the cells and
plasma.
64
4.4 Final Apori-Harris Sickle Cell Model
The final governing equations of the A-H sickle cell model are as follows:
6 = aic+a2up+a 3u
(AH.1)
a=
(AH.2)
=
biu+b 2 g s+b 3s
(AH.3)
cis+c2 c
(AH.4)
=di+dc
A block diagram describing the novel A-H sickle cell model can be seen in figure 4-5.
physiological parameters (a1..a3,b1..b3,c1,c2,d1,d2)
C0, U0' S0 , g
d, d, S, A
0
Figure 4-5: Block Diagram of A-H Sickle Cell Model
Blood velocity, oxygen concentration, cell stiffness, and plasma viscosity are the state variables
indicating the crises state in the microcirculation at any given moment. The A-H governing equa-
65
tions model the dynamics of sickle cell within the microcirculation. Starting values for the state
equations are input to the A-H model along with the relevant physiological parameters affecting
the crises state. The model then outputs the solutions to the state variables over time.
The A-H model is an Eulerian description for the manifestation of sickle cell disease in the microcirculation. In the following chapters, this system of equations will be solved numerically and
analyzed to show the important features of the solution and their significance to sickle cell crises.
66
Chapter 5: Solving the A-H Model
5.1 Solutions of the A-H Equations
5.1.1 Fixed Points
Simple properties of the A-H equations can be found by deriving the fixed points of the system.
By setting the left hand side of (AH.1)-(AH.4) to zero one obtains the fixed points c *, u*, s*, and
p~*.
O=aic+a 2 u i+a3 u
(AH.1.1)
O=biu+b 2 gs+b 3 s
(AH.2. 1)
O=cis+c 2 c
(AH.3. 1)
O=dli+d 2 c
(AH.4. 1)
There are two types of fixed points for this system. The first is the origin where (c*,u*,p*,s *)=
(0,0,0,0). This is a fixed point for all parameter values. The origin represents no motion or oxygen transport in the blood stream. In physiological terms, this is not a state that is likely to occur
in a living individual. The stability analysis in the following section shows that the origin is an
unstable fixed point for all parameter values. In fact, the A-H model contains no stable fixed point
solutions for sickle cell patients that are physically possible.
67
The second type of fixed points are only present within specific parameter ranges. These fixed
points appear or disappear via bifurcations when the control parameters are changed. Solving
(AH. 1.1)-(AH.4. 1) yields the additional fixed points.
F+=(u*+,c*+,It*+,s *+) and F= (u*~,c*-*-,s *-):
1 (- a2c3b3 - a3c3b2
ai ci bi + a3C3b3 + -a3a2
1
c d2a2
4)
1
(5.2)
1(- a 2c 3b3 - a3c3b 2 ± JDdi)
c*
2
!(-
-
2
(5.1)
a2 b2 d2c3
a 2c3b 3 - a3c3b 2 i
(5.3)
d )
a2 b2cIdI
1(-a 2 c3b3 - a 3c3b2 ±
[-)
(5.4)
a2 b2c3
where
= a22C322 -2-
2
b33b2+ a32 C322
2
-4a 2b 2aicibIc 3
(5.5)
The two fixed points F+ and F will be present only when (Dis positive for all four state variables.
This requirement for real roots bounds the ranges and signs of the parameters. Within the range
of parameters that satisfy (D> 0, there are solutions analogous to a steady state where the flow in
the capillaries represents that of a healthy individual as well as solutions that lead to the onset of
crises. Otherwise, when (Dis negative there will be imaginary roots for the equation.
68
5.1.2 LinearizedStability Analysis of Fixed Points
The motion near the fixed points can be determined by performing a stability analysis on the linearized system of equations. There are only two nonlinear terms in the A-H equations, ug and ps.
Removing these terms from the equations yields the following linearization at the origin:
6 = a 1 c + a 3u
(5.6)
d = bIu+b 3 s
(5.7)
=
cIs+ c 2 c
(5.8)
=
dli+d
(5.9)
2c
The system can be written in the matrix form dx/dt=Ax where x is the vector of independent variables and A is the matrix of coefficients[13]. Following this method, equations (5.6)-(5.8)
become:
aK a3 0
0
c
0 bi b3 0
u
0 c1 0
s
C2
(5.10)
d 2 0 0 d, ja
The general method for linearizing the system at any fixed point requires taking the Jacobian
matrix of the A-H equations. This Jacobian matrix J is defined by (5.11) for the A-H system.
69
d.
dc
d.
J
d. d. d.
du ds d
d. d. d.
-u -u-u-u
dc du ds d
d . d . d. d.
S-S
-S
dc du ds d
d. d. d. d.
-p -pg -p
p9
_dcduds dg_
(5.11)
Substituting the right hand sides of (AH.1)-(AH.4) gives the following:
al a 2 g* + a3
J
0
bi
C2
0
d2
0
0
a2 u*
(5.12)
b2 g* + b3 b 2 s*
0
c1
0
di
The solution to the linear system dx/dt=Jx is written in terms of the eigenvector v and the eigenvalue ? as x(t)=e v. The eigenvalues provide important information on the stability of the system. By definition, the eigenvalues of a 2x2 matrix A are as follows:
A = a5.1b
]
(.3
_c d
T= trace(A) = a+d =
+X2
(5.14)
(5.15)
A = det(A) = ad -bc =XI2
The value of the trace and the determinant give the stability of the fixed point. When A is negative, the eigenvalues are real and have opposite signs resulting in a saddlepoint. When A is positive, eigenvalues are either real with the same sign, or complex conjugate. Real eigenvalues give
nodes while complex eigenvalues give spirals and centers. The stability is determined by t which
70
represents the real parts of the eigenvalues. Figure 5-1 displays stability characteristics according
to the trace and determinant values[9:137].
T
T2 -4A
= 0
A
saddle points
non-isolated
fixed points /
stars, degenerate nodes
Figure 5-1: Stability of Fixed Point Trace and Determinant[9]
The linearized 4 x 4 A-H system can be broken down into six 2 x 2 systems in order to analyze the
2-D in plane stability dynamics. The six 2 x 2 determinants of the linearized A-H system about
the origin are given below.
set 2
set I
[
_ [a, a3 ci
Laj
0 b u
_
S
al 0
c
C2 CI
s
_ b1 b 3 ul
[
0 cI Is
d_ a, 0
w1
set 5
set 4
d
set 3
A[[bi
d2 dL
set 6
0
u
-10 d,
_t
71
c
Wc
}\0
0 s
d, y
The eigenvalues of each of the 2 x 2 systems are important to describing the stability properties of
the full 4-D system of equations. The fixed points F+ and F were broken down into 2 x 2 subset
matrices following the same process. The trace and determinant of all subsets were calculated in
tables 5-1 and 5-2 below for both the origin and the two fixed points F+ and F.
Table 5-1: Linearized Analysis of Origin
Set #
A
T
-r2 -4A
1
alb,
al+b1
(ai+bl) 2 -4aibi
2
a1 c,
aj+c 1
(ai+ci) 2 -4aic1
3
aidi
ai+d1
(ai+di) 2 -4aidi
4
b 1 c,
bi+c1
(bi+ci) 2 -4bic1
5
b1 di
bi+d1
(bi+d1 )2 -4bidi
6
c1 di
c1+d1
(c1 +di) 2 -4cidi
Table 5-2: Linearized Analysis of F* and F
Set
AT2_4
1
aibi
ai+b1
(ai+bi) 2 -4aibi
2
a 1ci
aj+c 1
(ai+ci) 2 -4aici
3
aidl-a 2u*d 2
al+d 1
(ai+di) 2 -4(aidi-a 2 u*d2 )
4
b 1ci
bi+c1
(bi+ci)2 -4bici
5
b1 di
bi+d1
(bi+di) 2 -4bidi
6
c1 di
c1 +d1
(c 1+di) 2 -4cidi
72
The stability dynamics in the different 2-D planes combine to give a picture of the stability of the
4-D system. In table 5-1, the linearized stability analysis about the origin shows that three determinants are negative unless all of the first term parameters a1 -di are the same sign. The origin
looks like a saddle point in these three directions or planes. The other three directions are either
nodes, spirals, or centers depending on the value of r. In the case where ai-di are the same sign,
all determinants are positive and the fixed points are either stable or unstable nodes, spirals, or
centers with trajectories converging or diverging upon the fixed point in all six directions. All
directions of the fixed point are outgoing if t is positive (unstable) or incoming if T is negative
(stable).
x vs. y vs. z
Figure 5-2: Trajectories Expelled from Lorenz Attractor
Analysis of other chaotic systems show that trajectories must be expelled out of any oscillations in
a two dimensional plane before eventually returning in a different location. This can be seen in
73
the Lorenz system where trajectories spiral in towards one fixed point before being expelled from
the plane to oscillate around the other fixed point (figure 5-2). These dynamics are not possible in
a system that is globally stable or globally unstable in every plane. Saddle points are one way of
providing this trajectory ejection mechanism. The three dimensional Lorenz system is evidence
of that as it has two incoming directions and one outgoing direction. When the Lorenz system is
chaotic one direction is a stable spiral, one a stable node, and one a saddle point according to the
linearized stability analysis.
The linearization of the origin for the A-H equations shows that chaos is likely to occur only when
ai-dl do not all have the same sign. In this case there are three saddle point directions. Another
requirement is that T # 0. Otherwise, the origin will be a center. This also means that
a, # -b, ...cl # -dl as seen in Table 5-2.
The linearization of the F+ and F fixed points is done in the same manner as that of the origin.
The dynamics in all planes are identical to that of the origin except for in plane three which represents the c-p plane. In this plane the dynamics depend upon the u* value of the critical point.
Since u* is a function of the parameters ai-d 2 , this means that the stability of F+ and F will be
dependent upon the parameter values. Only one of the six directions is influenced by the critical
point u*, so the parameter ranges for chaos and having saddle point directions are similar to those
described above for the origin.
74
5.2 Parameter Determination
5.2.1 DimensionalAnalysis
The physiological parameter ranges are key in determining the dynamics of the system. In an
experimental setup, parameter value ranges could be measured from known physical quantities in
the body. For the A-H system, parameters have been derived in order to normalize each term of
(AH. 1)-(AH.4). This allows consistency in units within each equation. These parameters are
assumed to correspond to physiological parameters that can be measured in the body.
Ranges of all parameters must be defined consistently. The first step taken in defining the parameters for the A-H equations was to define a set of characteristic values for length, time, stiffness,
viscosity, and oxygen concentration. These values represent reference values likely to be found in
the body. They are presented in table 5-3 below.
Table 5-3: Characteristic Body Constants
Length
556 x 10~6 m
Time
1.112 sec
Cell Stiffness
Viscosity
Oxygen
Concentration
185 x 10-6 dynes/m
1.4 x 10-3 Pa-s
8.3 x 10-6 mol/L
The characteristic length was taken as the average length of the capillary as defined by Cima et
al.[1]. This is the capillary length over which oxygen is contributed to the surrounding body tissue as described by the Krogh model[6]. The characteristic time was defined as the time it took
for a cell to traverse the capillary with a healthy, non-crises viscosity of p = 1.4 Pa-s. This time is
an average for a healthy individual found in the results of the Cima model[1]. It does not reflect
75
the extended periods that cells are stuck in the capillary during a crisis. The characteristic stiffness was taken as Sref which is the stiffness for healthy blood cells as defined by Cima et al. This
stiffness was interpolated from data measured in experiments where red blood cells were passed
through filters[1]. The characteristic viscosity is the value for normal blood and sickle blood in a
non-crises state. The value of the viscosity comes from patient data[1]. The characteristic value
for oxygen concentration is the average starting capillary inlet concentration for people with
sickle cell blood[1]. This is the average concentration of fully oxygenated red cells that have left
the lungs and travelled through the microvascular system to the capillaries.
In the next step, all parameters had the characteristic values substituted to get the correct units and
order of magnitude for each term. Table 5-4 displays the units for each parameter and the initial
parameter values found by substituting the characteristic values for the required units.
Table 5-4: Characteristic Values of A-H Parameters
Parameter
Units
Value
ai
1/s
.89928
a2
(s/m)(1/Pa-s)(mol/L)(1/s)
10.66
a3
(s/m)(mol/L)(1/s)
.014928
bi
1/s
.88928
b2
(1/Pa-s)(m/dynes)(m/s 2 )
1736
b3
(L/mol)(dynes/m)(1/s)
2.43
ci
1/s
.88928
C2
(L/mol)(dynes/m)(1/s)
20.44
d
1/s
.88928
d2
(L/mol)(Pa-s)/s
151.7
76
The values of these parameters make all terms of (AH. 1)-(AH.4) the same order of magnitude. In
the physiological sense the parameters correspond to levels of substances in the body or individual biological properties that vary from individual to individual.
This could include pH levels,
oxygen carrying capacity of red blood cells, or environmental effects like stress or lactose
buildup. The body is a complex system with numerous parameters affecting its operation. However, quantifying the affect of parameters is a very difficult process that cannot be achieved with
complete accuracy. Defining the parameters and their interactions as best as possible is important
to creating an effective model.
5.2.2 Determinationof Control Parameter
The process for choosing a control parameter required testing the sensitivity of the system to
changes in each parameter. The complexity of the roots shown in (5.1)-(5.5) makes it difficult to
clearly visualize the impact of each parameter and bifurcations that may occur. Therefore, all of
the parameters were checked through trial and error to see if they could drive the solutions from a
steady state to a chaotic one.
Each parameter was varied individually while all others were held at their original characteristic
values shown in table 5-4. Varying each parameter individually while all others were held constant did not produce chaos in the system. Each parameter was varied from a value 1/100th to 100
times the original value. Solutions were plotted in the phase space and temporal domain in order
to view the dynamics of the system.
77
The next step was to vary different combinations of parameters together. There were a total of 10!
combinations of parameters. The complete spectrum of parameter combinations was not
explored. The focus was given to those seen to have the greatest impact on the geometry of the
attractor in the phase space. This included parameters that stretched trajectories, folded trajectories upon themselves, dispersed trajectories, etc. Parameters that affected the phase space in the
most noticeable way were the ones likely to create the complex dynamics of chaos in the system.
Upon exploring different combinations in the parameter space, the key parameters were found to
be a,, a2 , b2 , ci, c2 , and dl. Table 5-5 describes the ranges over which these selected parameters
generated solutions that appeared chaotic.
Table 5-5: Key Parameter Values Demonstrating Chaos
Parameter
Value
Set #1
> characteristic
< characteristic
a,, c,
Set #2
Set #3
c 2 , d,
a2 , di
a,
ci
Set #4
a2 , c 2 , di
Set #5
Set #6
di
c2 , di
b2 , c1
b2
In table 5-5, all parameters other than those specified were held at their characteristic parameter
values. Specified parameters were either multiplied by a constant to obtain a greater than (>)
characteristic value or divided by a constant to obtain a less than (<) characteristic value. The
combinations of parameters displayed in table 5-5 were the minimal combination sets required to
cause chaos. Varying additional parameters in conjunction with having the minimal set would
also likely lead to chaos in the model.
78
The key to the dynamics of the attractor proved to be the parameters ai, a2 , bi, ci, c2 , di and the
various combinations made between them.
c = aic+ a2 up+ a3u
(AH.1)
Q = blu+b 2 gs+b 3s
(AH.2)
=
cIs+c 2 c
(AH.3)
=
dig +d 2 c
(AH.4)
As seen in the A-H equations above, a1 is the diffusion term for oxygen concentration. In terms of
the dynamics of the attractor, a1 was found to change the speed at which trajectories travel in the
phase space. This is equated with varying the strength of attraction or repulsion of the fixed
points. A2 is the coefficient to the nonlinear coupling between velocity and viscosity. The a2 term
controlled spacing of trajectories in the phase space and their convergence and divergence from
limit cycles. B 2 is the coefficient to the nonlinear coupling between viscosity and stiffness. The
B2 term also caused trajectories to converge or diverge from a spiral or a chaotic attractor in the
phase space. The ci term is the diffusive term for stiffness. This term changed the strength of the
attractor in one direction by causing trajectories to converge upon or diverge from the manifold of
the attractor. It also had the property of lengthening or shortening the attractor. The c 2 term is the
dependence of cell stiffness on oxygen concentration. The c2 term controlled the spiraling
strength about the fixed points by changing the strength of attraction or repulsion of the attractor.
Finally, the d, diffusive viscosity term controlled the attraction of trajectories towards the two
79
dimensional spiral plane in the phase space as well as their convergence or divergence upon a
limit cycle.
The previous analysis demonstrates that there were at least six key parameters that could have
been selected as control parameters. The decision was made to use c 2 as the primary control
parameter during this investigation.
C2 is the coefficient of the oxygen concentration term in the cell stiffness equation AH.3
(
= cIs + c2c). In terms of physiology, this parameter represents the complex relationship
between the stiffness of the cell and oxygen concentration. The dependence of cell stiffness on
oxygen concentration is a phenomenon known to be the key difference between sickle cell blood
and normal blood. A control parameter that emphasizes this key relationship provides valuable
insight. C2 could represent anything from changes based on genetic makeup to changes due to the
function of environment or stress introduced to the body. The biological representation of c2 is an
important aspect to be further explored.
Because of the large number of parameters in the system, there is more than one that leads to
chaos. However, for simplicity only c2 is varied as a control parameter in further analysis. All
calculations in the following chapter hold the remaining parameters constant.
5.2.3 Final ParameterValues
The final parameter values analyzed in the subsequent chapters are listed in table 5-6. They were
chosen from set #6 of the parameter combinations that caused chaos. Set #6 consisted of varying
80
b2 , c2 , and di. This combination of parameters was chosen because it produced chaos over a large
value of the control parameter c2 . Parameters ai, bi, ci and d, were each varied by 10-5 off their
characteristic values in order to satisfy t # 0 or a, # -bI ... cI # -dI . This prevents the fixed
points from being a center as described in table 5-1 and table 5-2 of the linearized analysis results.
Table 5-6: Final A-H Parameter Values for Chaos Analysis
Value
Parameter
a,
.088928
a2
-10.66
a3
-.014928
b,
-.89929
b2
1736 x.35
b3
2.43
ci
-.89930
c2
varied (characteristic value
is 20.44)
d,
-.89931 x 3
d2
151.7
Assigning signs to each term was also critical to the stability dynamics of the solutions. In order
to give the fixed point dynamics of a saddle point, ai-dl were not given the same sign. A positive
value was assigned to a, while bi, c1 , and d, were assigned negative values. Determining signs
for the initial terms was also a result of exploring the stability dynamics of the system over different parameter ranges as shown in table 5-5.
81
Though the first term determines the eigenvalues, the subsequent terms in each equation also
impact the dynamics of the system. They are not present in the linearized analysis but the long
term behavior away from the fixed points is dependent upon the additional terms which include
the nonlinearities of the system. All parameters of any one equation (i.e. a,, a2 and a 3 ) cannot
have the same sign or the state variable will grow to positive infinity or negative infinity over time.
All equations must have a switchback mechanism or hump so that they can both increase or
decrease. In the mathematical sense this means that the derivative of any of (AH. 1)-(AH.4) must
pass through zero, thus making the equation non-monotonic. For this reason, parameters following ai,bi,ci, and di were assigned opposite signs. In the case of (AH.1) and (AH.2) there are two
terms after the initial term. The sign of the nonlinear terms of (AH.1) and (AH.2) were chosen
based on their impact on the stability dynamics of the system. The nonlinear terms of both
(AH.1) and (AH.2) led to chaotic behavior when their signs were opposite the first term.
All other parameter coefficients were determined by their impact on the attractor. Each of the key
parameters of table 5-5 was varied over a large range. From this process the impact of each on the
stability of the attractor was observed. The final physiological parameter values of Table 5-6
reflect the optimal conditions for developing a chaotic attractor in the phase space.
82
Chapter 6: Analysis of the A-H Attractor
6.1 BifurcationAnalysis
6.1.1 BifurcationAnalysis
The A-H model for sickle cell blood flow has many solutions. They vary with the change of physiological parameters and/or initial conditions. These changes, which may occur at any moment,
greatly impact the final state of the blood flow. The fixed points of the state variables defining
sickle cell blood flow appear and disappear, move throughout the phase space, and change stability characteristics as the selected control parameter c2 is varied.
The linearized analysis from 5.1.1 shows that there is a fixed point at the origin along with two
other points called F+ and F. The values of these fixed points in terms of the control parameter
are shown below.
c* = -0.961x10 -6(38.86C2 i V167.6c2 2 + 26917.6C2)
(6.1)
C2
u* .= 1.5x10--3 - 1.87x10-5C2 ±1.44x10-6)V167.6c22 + 26917.6C2
(6.2)
s* = - 4.15x10~5 C2 i 1.0686x10- V167.6c2 2+ 26917.6C2
(6.3)
c2 2 + 26917.6c 2 )
*= -5.4Cx10(-5(38.86c 2 ±V167.6c2
83
(6.1)
The results of the linearization about the three fixed points along with their predicted stability
dynamics are summarized in the table 6-1.
Table 6-1: Linearized Analysis of Fixed Points
Set #
T2-4A
A
Stability
1
-.8087
-.00001
3.235
saddle point
2
-.8087
-.00002
3.235
saddle point
3 Origin
-2.426
-1.799
12.94
saddle point
3 F+ and F-
-2.426 + 1617u*
-1.799
3.235 + 4(2.426 - 1617u*)
function of c2
4
.8087
-1.799
0
center
5
2.426
-3.597
3.235
stable spiral
6
2.426
-3.597
3.235
stable spiral
According to the linearization results, the origin has three saddle point directions, one center, and
two stable spirals when observed in the six 2-D planes. The dynamics of the fixed points F+ and
F- vary with c 2 . They both have five directions identical to that of the origin. Notably, there are at
least two saddle point directions, one center, and two stable spirals. The final direction of F+ and
F~ is determined by A of set 3. When c 2 is negative, A3 +> 0 and A3~ <0. For F-, the final direction
is a saddle making it identical to the origin. For F+, A3 + > 0 and t < 0 means that this direction is
either a stable spiral or stable node depending on the value of T 2 - 4A. Solving 12 - 4A = 0 shows
that for c2 < 6.67 this point is a stable node and for c 2 > 6.67 it gives a stable spiral.
In general, linearized dynamics are only accurate close to the fixed points and in many cases linearization may even incorrectly predict the stability dynamics of a fixed point. The key insight
84
taken from the linearized analysis of the A-H fixed points is the prediction of numerous saddle
direction for all the fixed points. These saddle directions are present throughout the control
parameter range when c2 > 0. The linearization results predict that all fixed points in the 4-D
phase space are unstable and will lead to complex dynamics throughout the control parameter
ranges. At least one unstable fixed point in the phase space is a requirement for chaotic systems,
so the fixed points of the A-H system are consistent with those leading to chaotic behavior.
6.1.2 Route to Chaos
The phase space for the four dimensional A-H system contains exotic limit cycles, strange attractors, spiraling trajectories and other solutions in multiple dimensions. If all parameters are varied
simultaneously, the phase space would be extremely complex and difficult to visualize. In this
section, the attractor analysis was limited to varying the control parameter c2 while holding all
others constant.
The first step to finding the A-H solution in the phase space was to decide on a set of initial conditions. (AH. 1)-(AH.4) were solved with respect to the chosen set of initial conditions and parameter values. The results gave the state variables c, u, s, and pt as a function of time. (AH. 1)-(AH.4)
were solved numerically using Maple 6 computational software (Appendix B. 1). Table 6-2 displays the value of the initial conditions used.
Table 6-2: Initial Conditions
Oxygen.
Concentration
Velocity
8.3 x 10-6 mol/L
5 x 10-4 m/s
Stiffness
1.85 x 10-4 dynes/m
85
Viscosity
1.4 x 10-3 Pa*s
These initial conditions were taken from those found in previous investigations of sickle cell
blood flow. Many of them are the same as the characteristic values used to define the physiological parameters in section 5.2. The co value is the initial average capillary inlet oxygen concentration as measured for sickle cell patients[3]. Uo is the starting velocity for capillary flow in the
non-sickle cell case calculated using the average normal starting viscosity of 1.4 x 10-3 Pa-s [1].
SO is the reference stiffness of both a healthy cell or fully oxygenated sickle cell [1]. The
so value
is the normal viscosity for healthy blood and sickle cell blood in the non-crises case[1]. Small
variations of these initial conditions can greatly affect the final state of the solutions. This phenomenon is explored further in chapter seven.
The solution to the A-H model goes from stable to chaotic as the control parameter is increased.
As c2 is varied from zero to infinity the solution starts as an unstable fixed point, then becomes a
limit cycle, and finally a strange attractor. The solutions are plotted in the phase space and temporal space in figures 6-1, 6-2, and 6-3 as c2 is varied from c2 x 10~3 to c2 x 103 .
86
c2 x .001
c2 x .001
I
A
I
*1
Multi-
r
Limit
Cycle
c2 x.c2 x.-1
-2
Multi-
Limit
*
Cycle
C2 X
.1
C2
X .1
Single
Limit
Cycle
Figure 6-1: A-H Attractor and Time Series (c2 x .001 to c2 x -1)
87
C2 X 1
C2 x 1
4
Single
Limit
Cycle
C2
X
10
c2 x 10
5
Chaos
IN
c2 x 100
c2 x 100
-L
I. I i . I
6
11
HyperChaos
H1 II'! U1
Ij
I
Figure 6-2: A-H Attractor and Time Series (c2 x 1 to c2 x 100)
88
C2 x 1000
C2 X 1000
7
HyperLimit
Cycle
-am
Figure 6-3: A-H Attractor and Time Series (C2 x 1000)
At c2 = 0 there is a single saddle point in the phase space at the origin. All trajectories in the
phase space regardless of where they start are expelled out to infinity. As c2 is increased beyond
zero, two more fixed points (F+ and F) are created in the phase space. Both of these points are
unstable saddle points. They repel trajectories out in a spiral towards a limit cycle shaped somewhat like a figure eight as seen for c2 X .001 and c2 x .01 in plots 1 and 2 of figure 6-1. The plots
of these limit cycles are labeled multi-limit cycle because they encircle multiple fixed points. The
limit cycles are actually saddle cycles, attracting in-plane trajectories while repelling trajectories
in the direction perpendicular to the plane. As c2 is increased, the fixed points F+ and F move in
towards the origin from infinity causing the limit cycle to shrink. The portion of the limit cycle
around F+ grows with respect to that around F as c2 increases. Trajectories spend more time circling only F+ while escaping only briefly to circle F. Eventually, at c 2 x .0441 the limit cycle
about F completely disappears and trajectories only circle F. At this point any trajectories starting close to F would spiral outward until they eventually are attracted to the limit cycle around
89
F+. Trajectories starting close to F+ would also spiral outwards to the limit cycle around F+. Any
other trajectories started in the phase space either escape to infinity or are also attracted to the
limit cycle around F+. The limit cycle around F+ is present for c 2 x .1 in plot 3 of figure 6-1. It is
labeled a single limit cycle because it only encircles one point.
As c2 is further increased the fixed points F+ and F~ continue to approach the origin. The limit
cycle around F+ decreases in size and is deformed as it approaches the stable and unstable manifold of the saddle point at the origin. At c2 x 3.1 the limit cycle is so deformed by approaching the
other fixed points that it disappears and the set becomes a strange attractor. This strange attractor
can be seen in plot 5 of figure 6-2 for c 2 x 10.
The trajectories for the strange attractor trace a path around both F+ and F- while also being influenced by the saddle point at the origin. All trajectories starting within the envelope of attraction
go to the strange attractor over time tracing out a path on the attractor that never repeats. Trajectories outside of the basin of attraction escape to infinity. Increasing c2 beyond this point causes
trajectories to move on the attractor even faster. This is called hyper-chaos because of the high
frequency. Mathematically, hyperchaos is defined as having more than one positive Lyapunov
exponent[16]. Hyperchaos is visible for c 2 x 100 in plot 6. The attractor also increases in size
along its various manifolds as the attraction and repulsion strength are increased. When c is
2
increased to very large values, it is possible for regions of periodic motion and chaotic motion to
exist depending on the value of c 2 . For example, plot 7 of c 2 x 1000 shows a complex hyper-limit
cycle upon which trajectories move so fast that the time series appears to be dense over the range
90
plotted. Figure 6-4 summarizes the various fixed points and solution states in the phase space as
the control parameter is varied. Plots 1-7 from figures 6-1, 6-2, and 6-3 are placed in their corresponding solution state ranges in figure 6-4.
Figure 6-4: Stability Change With Control Parameter C2
Closer analysis of the phase space as c2 approaches 3.1 shows the route to chaos in further detail.
The solutions in the phase space leading up to the appearance of the strange attractor are shown in
figures 6-5, 6-6, and 6-7.
C2 x 2
C2 x 2
Weak
Saddle
Cycle
M
40
M
s
Figure 6-5: Route to Chaos of A-H Attractor (c2 x 2)
91
U
C2 x 3
C2 x 3
2
Strong
Saddle
Cycle
c2 x 3.1
c2 x 3.1
3
Onset
of Chaos
-022t
c2 x 4
C2 x 4
Weak
Chaos
Figure 6-6: Route to Chaos of A-H Attractor (c2 x 3, 02 x 3.1, c2 x 4)
92
C2 x 5
C2 x 5
Weakl
Moderate
Chaos
t
00
Figure 6-7: Route to Chaos of A-H Attractor (c2 x 5)
At c2 x 1 a weak saddle cycle is present. When c 2 is increased to c 2 x 2 the fixed point F+ in the
center of the limit cycle increases its strength of attraction in the perpendicular direction. This is
due in part to the approach of the F~ fixed point in that area. Trajectories begin to spiral out
towards the limit cycle and then shoot back towards the center where the fixed point lies. Time
series 1 of figure 6-5 shows a period seven limit cycle-spiral type attractor. At c2 x 3, the trajectories become more attracted to the limit cycle as the strength of attraction of F+ in the perpendicular direction increases. Time series 2 of figure 6-6 shows that the limit cycle is still quasi-periodic
repeating every fourth oscillation. The change from periodicity to chaos occurs around c2 x 3.1 of
plot 3. Further increasing to c2 x 4 in plot 4 causes the frequency of oscillation to start increasing
again. The attractor begins to look like a very complex form in the phase space with one surface
that contains a spiral. Trajectories travel on the spiral surface until they get too far away from the
center and jump off. After leaving the spiral they traverse a path around the second fixed point
before returning closer to the spiral's center. At this point the attractor is already weakly chaotic.
93
At C2 x 5 and above, the fixed points continue to approach each other making the various attractor
manifolds fold and collide even further. This increases the erraticness of the time series making it
look less and less like any periodic behavior existed.
The route to chaos is important in the physiological sense. C2 is the driving parameter in the
march towards chaos. C2 represents the key individual dynamics of the blood cell rheology for a
sickle cell patient. If the relation between cell stiffness and oxygen concentration is changed for
any reason, an individual can be sent from a periodic state for the solution to capillary blood flow
to one in which blood flow becomes chaotic. This may cause state variables to fall below acceptable critical limits in the body resulting in a crises.
The relationship of c2 to the actual physiology of the body is key to understanding the change
from a periodic solution to a chaotic solution when c2 or other physiological parameters are varied. However, the A-H model is only valid up to the point it predicts this change to chaotic behavior which may be linked with the onset of a crisis. Reaction of the body to reversing the onset of
a crises and returning to a steady blood flow solution is beyond the scope of the A-H model.
94
6.2 Phase Space Properties
6.2.1 FourDimensionalA-H Attractor
The full phase space for the A-H system is four dimensional. Visualizing the full 4-D phase space
is difficult, so only 3-D phase space frames are shown at any given time. There is a total of four
possible different 3-D phase space frames. These are the c-u-s, c-u-p, c-s-g, and u-s-g phase
spaces. Figure 6-8 shows the strange attractor for c2 x 10 in all four frames.
c-s-
c-u-pg
M
C-U-S
U-S-p
Figure 6-8: 3-D Phase Space Frames for A-H Attractor
95
The stability dynamics do not vary between the different 3-D phase space frames. A fixed point,
attractor, or other solution exists regardless of the frame chosen for plotting. Each merely presents a different view into what trajectories are doing in the overall 4-D phase space. For the purpose of attractor analysis in this thesis, trajectories are primarily plotted in the c-u- plane. The
assumption was made that the other phase spaces do not contain additional unseen dynamics
unless otherwise noted.
There are also four possible time series plots. One for each state variable. Each state variable has
the same stability dynamics over time. If c has a periodic solution the same will be true for u, g,
and s since all variables are interdependent. For simplicity, in this thesis the time series of only
one variable is plotted. The assumption is made that other variables vary with time in the same
manner. Figure 6-9 shows the chaotic time series for all the state variables with an identical solution.
96
u
VS.t
vs. t
c vs. t
s vs. t
0.01,
-0.01,
C
S
-0.01
0
O
lDo
0
0
2D
Figure 6-9: Time Series Frames for A-H Attractor
6.2.2 Basin of Attraction
The basin of attraction for the A-H attractor is the total set of points in the phase space within
which starting trajectories end up on the attractor. The initial conditions used to solve the A-H
system all fall within the basin of attraction. When these initial conditions are varied by amounts
that are within the physiological limits of the body, they are still within the basin of attraction.
The size of the basin of attraction is determined by parameter values. For most values of the control parameter c2 , the initial conditions that are outside of the basin of attraction are assumed to be
97
M
physiologically impossible. However, varying c2 changes the basin of attraction because of its
effect on the phase space. When the control parameter has a value of c2 x 10, the limits of the
basin of attraction are approximately c0 =3 x 10-3, u0 =6 x 10-1, s=1.1, and m0 =12 x 10-2. To find
these limits each initial condition was varied independently while the others were held at their
standard values from table 6-2. Solving with initial conditions greater than these limits generates
trajectories that are expelled to infinity over time even when the control parameter c2 is within the
chaotic regime.
6.2.3 Long Term AttractorBehavior
A chaotic system must show erratic behavior over the long term. Solutions to a non-chaotic system may engage in apparently chaotic behavior for some time before settling down to a steady
state or being repelled to infinity. Figure 6-10 shows the A-H attractor plotted to t = 1000s for
C2 x 3.1 where chaos just begins and c 2 x 10 where the system is moderately chaotic.
c2 x 10, t = 1..1
c2 x 3.1, t = 1..10 0 0
Figure 6-10: Long Term Behavior of A-H Attractor
98
000
Now---
----
Both solutions settle down to chaotic behavior over the long term as the attractors are filled in
with trajectories. The trajectories never leave the attractors. They simply trace new paths on the
attractor surfaces causing them to look more and more dense. This means the chaotic solutions
for the blood flow of sickle cell patients described by the A-H model exist as long as the physiological parameters are in the chaotic ranges. A patient whose physiological parameters are in the
chaotic regime can not have steady blood flow. The state of a sickle cell patient's blood flow
would remain chaotic indefinitely barring any input from body mechanisms to return the blood
flow to normal.
99
100
Chapter 7: Results of Chaos Analysis on A-H Model
7.1 Determinism
7.1.1 Determinism in A-H Equations
For chaos to exist, the data observed must be generated by a deterministic process. The apparent
disorder must be caused by the complexity of the governing equations and not by random or noisy
inputs to the system. Differentiating between random and chaotic data may be difficult for a system occurring in nature. It requires attractor reconstruction among other mathematical tools to
prove that the data is correlated. In the case of experimentally generated chaos, proving the system deterministic simply requires knowledge of the governing equations and all inputs to the system.
Solutions to the A-H model were found by solving (AH. 1)-(AH.4) below.
c = aic+a 2 ug+a 3 u
(AH.1)
u = blu+b 2 g s+b 3s
(AH.2)
(AH.3)
= c 1s+C 2C
=
(AH.4)
djt+d2 c
(AH.1)-(AH.4) are all ordinary differential equations. When solved without any random inputs
from parameters or initial conditions, by definition their solutions are deterministic. Throughout
101
the analysis of the A-H model parameters a,, a2 , a 3 , bi, b2 , b3 , ci, di, and d2 were kept constant.
Different solutions were found by varying c2 and the initial conditions but no parameter or initial
condition was varied during the iterations to produce a numerical solution.
The chaos found in the solution of the A-H model was strictly a function of the complex nonlinearity of the governing equations. However, the actual physiological parameters of the body may
change over time. The physiological parameters are unlikely to change randomly or chaotically
themselves. What is more likely to happen is that the control parameter of the A-H system
changes values over some range until reaching a critical limit that leads to an inherently chaotic
solution for the blood flow.
7.1.2 PoincareSection
The Poincare section is a plot that show a cross section of the attractor within a plane. From this
cross section the form of the attractor can be seen. A Poincare section is useful for determining if
and how trajectories are correlated in the phase space. The Poincare section of a period one limit
cycle will only consist of two points. One where the trajectories are going through the plane and
one where they are coming back through the plane. Since all trajectories are on the limit cycle the
cross section will not show any other points. The Poincare section for a chaotic attractor's trajectories will resemble a curve or pattern. No trajectory will ever intersect the plane at the exact
same point but since a strange attractor does have surfaces neighboring trajectories will intersect
the cross section plane as if on the surface of the strange attractor. This curve differs from the
Poincare section of random data. The Poincare section of random data would have points spread
102
out over the entire cross section. Since the solutions at different times are not correlated the trajectories of a random system would have no order in the phase space.
Poincare sections for the A-H attractor at c2 x .001 and c2 x 50 are plotted in figure 7-1. The plots
were generated after all initial transients died down by a method that approximated the intersection between trajectories and the cross section plane (see Appendix B.2).
u vs. mu, C2
50
u vs. mu, c2 x
.001,
X
0.W
aw
*
-W +
+
+++
.
4
+ ++
.-0DM
-J
-aol
a
0
-.0.01
01
001
U
CVS. S,C2 X50
c vs. s, c2 x .001
0*
-aol-
9_C.Cz
-004~
U,,,,
.0041
+
44,
*'~~
,#
4,,
*
4,~**
-Owl
n~i -%4m
-4-0
-59.-04
Figure 7-1: Poincare Sections of A-H Attractor
103
-40
+
-40
For the c2 x 50 plots, the concentration of the points on the attractor surface provides confirmation
of the deterministic and chaotic nature of the solutions. The g-u Poincare section shows that the
attractor intersects the plane in a pattern of lines spread out in a fan-like shape. For the c-s
Poincare section, the attractor has a crescent shaped cross section. As solutions are plotted over a
long period of time, the points in the cross section will become more dense as trajectories continue to intersect the plane in different locations along the pattern. Chaotic trajectories don't
repeat themselves or intersect so they will remain on the surface of the attractor though they can
be found anywhere on the surface. These non-repeating intersections of trajectories with the
plane are further proof that the attractor is chaotic at c2 x 50.
In contrast, there are only two points on the Poincare section for c2 x .001 when the attractor generates a limit cycle. This is because all trajectories converge on the limit cycle and trace out an
identical path in the phase space. This causes them to intersect the plane in the exact same location as they oscillate around the limit cycle.
The verification of the deterministic nature of chaos has important meaning in describing sickle
cell crises. This supports the fact that crises can occur without drastic and erratic changes to physiological parameters. A patient suffering from sickle cell would not have to go through extreme
changes or stressors to the body for the chaotic state of sickle cell blood flow to occur. Chaos
could be induced when physiological body parameters vary slightly for normal everyday reasons
without a shock to the system or a chaotic perturbation to the body.
104
7.2 Aperiodicity
7.2.1 Aperiodic Time Series
The A-H model at c2 x 50 exhibits aperiodic behavior. The state variables vary erratically over
time in a pattern that doesn't repeat itself. As seen in figure 7-2, the oxygen concentration oscillates in a pattern of increasing and decreasing amplitudes. The periods of oscillation also vary
over time. The oxygen concentration in the blood for somebody with parameter values within the
chaotic range would vary between high and low values never becoming stable. This instability
would be indicative of a crisis in a sickle cell patient.
c vs. t for c2 x 50
D
5
0
55
70
75
Figure 7-2: Aperiodic Time Series for A-H Attractor
The cause of the aperiodicity in oxygen concentration over time is the movement of trajectories
from one part of the attractor to another. The trajectories may spiral around one fixed point of the
attractor until moving too far away from the center and being captured into the path around a different fixed point. This process explains the pattern of oscillations increasing to a maximum
amplitude near the edges of the attractor before decreasing again. The time it takes for this cycle
105
to occur is governed by the trajectories exact location in the attractor. This extreme sensitivity to
the precise trajectory location and the lack of a stable fixed point is why the trajectories remain
aperiodic.
7.2.2 FourierAnalysis
Performing a Fourier analysis on the A-H equations displays the solutions in the frequency
domain. This provides a different interpretation of the solutions and shows characteristics not visible in the time domain or phase space. Plotting the Fourier power spectrum shows the relative
strength of frequencies present in the solution. Abrupt spikes in the power spectrum indicates
dominant frequencies. Dominant frequencies induce periodic motion about those frequencies.
When there are no dominant frequencies the power spectrum is spread out over a range of frequencies. This is the case for a chaotic power spectrum which shows broadening over the range
of frequencies present in the attractor. The size of these ranges varies with the dynamics of the
fixed points and the shape of the attractor in the phase space. Random data also has a broad power
spectrum. However, random data shows a constant spectral broadening across the entire range of
frequencies since every frequency has the same chance of occurring.
The plots of figure 7-3 show the time series for each state variable. They were plotted on the left
with the control parameter producing a limit cycle (c 2 x .01) and on the right with a chaotic attractor (c 2 x 50). For c 2 x .01 all of the state variables oscillate in a single period limit cycle except
for u which oscillates in a period two limit cycle. When the control parameter is increased to c x
2
50 any observable regularity in the time series disappears and the solutions for all state variables
become chaotic.
106
Chaotic Attractor, C2 x 50
Limit Cycle, C2 X .01
I
am
0
w
illil
Ale
41M
m
m
/Yf
~
~
a
m
Figure 7-3: Time Series of Limit Cycle and A-H Attractor
107
IVu
In figure 7-4 are the corresponding Fourier power spectrums for the plots in 7-3. A fast Fourier
transform was calculated with Maple 6.0 to generate the plots in figure 7-4 (Appendix B.3).
Along the left are plotted solutions for c2 x .01 and on the right are the solutions for c2 x 50. The
power spectrum shows large frequency spikes along with their harmonics for the limit cycle solution (c2 x .01). The largest spikes represent the frequency of oscillation while the smaller harmonics show other frequencies influencing the motion. In the plot for velocity the harmonics are
almost as large as the dominant frequency. The velocity is greatly affected by both frequencies.
This explains why it oscillates in a period two limit cycle.
The power spectrum plots on the right for the A-H attractor show results characteristic of chaotic
systems. There are no frequency spikes since no one frequency dominates the motion on the
attractor. There are however ranges of frequencies that are prevalent. This is because trajectories
tend to oscillate around fixed points in the attractor but they never return to trace out the same
path. This causes trajectories to always oscillate at a slightly different frequency. The frequency
ranges that show up in the chaotic domain correspond to the motion of the chaotic time series
plotted in figure 7-3. The time series for c, s, and g are much more chaotic than that of u. The
power spectrum shows that this is due to the larger ranges of frequencies over which the solution
varies for c, s, and p.
108
Limit Cycle, C2 X -01
Chaotic Attractor, c2 x 50
C
U
S
s
s
S.(
10-
S 5.
p
0~
0
200
400
k
600
600
800
860
1000
1000
Figure 7-4: Fourier Power Spectrum of Limit Cycle and A-H Attractor
109
The Fourier power spectrum for the A-H attractor shows that at c2 x 50 solutions generated are not
periodic in nature. All state variables oscillate in an erratic and unpredictable way. Since there
are no dominant frequencies, solutions don't repeat themselves even if iterated for a long period
of time. Physiologically, this may indicate that once a sickle cell patient enters the crisis state,
levels of oxygen, viscosity, cell stiffness, and blood velocity are impossible to predict. This lack
of predictability could what makes it difficult to effectively treat patients who are already having a
crisis. Regulating values of the state variables so they don't fall to critical and potentially life
threatening levels during a crises would be difficult when the state variables are changing in a chaotic manner.
7.3 Initial Condition Sensitivity
7.3.1 Sensitivity of the Strange Attractorand Time Series
All chaotic systems exhibit extreme sensitivity to initial conditions. This is another cause of their
unpredictable nature. An infinitesimally small change to an initial condition can cause a drastic
difference in the final state of the solution over time. This is because trajectories starting at
nearby points separate exponentially fast as they spread out over the attractor.
Five different solutions are plotted in figure 7-5. The solutions are for c 2 x 50 which lies in the
chaotic regime. The first solution plotted contains the baseline values of the initial condition from
Table 6-2. In each of the additional plots one initial condition has been increased on the order of
110
10-6 with respect to the initial condition. The exact values of the initial conditions that were
changed are presented in figure 7-5.
c vs. t of 5 Nearly Identical Initial Conditions
Initial
Conditions
0.0042
c0 , u0 , so, g0
-0.0002
-0.04
-oOD6
c + 10-12
c vs. t Time Horizon of 5 Nearly Identical Initial Conditions
co, u0 +10-10, so go
-
o_-
c , u0 , s 0 + 10 1 0 , o
0
-0.0002--
c0 , u0 , s , 0+10-9
14
16
24
18
Figure 7-5: Sensitivity to Initial Conditions of A-H Time Series
Figure 7-5 shows to what degree the trajectories diverge on the A-H attractor for the state variable
c. Initially, the solutions are and the difference in starting conditions is unnoticeable. However,
111
by the end of the plot at t = 70 it is impossible to tell the value of each individual because they are
all drastically different. The small initial difference has been magnified and the trajectories in the
phase space are spread out over various ranges. The lower graph in figure 7-5 shows a close-up of
the region t = 13 to 25 where the solutions separate. The solutions are virtually identical until
approximately t = 17 at which point they begin to diverge. This is the time horizon for which
accurate predictions can be made about the A-H system for c 2 x 50. Beyond this time, all predictability breaks down and solutions that were once similar become entirely different. For a sickle
cell patient, the time horizon represents the amount of time after which the physiological control
parameter passes into the chaotic regime that the state of their blood flow could be predicted and
possibly treated. This time threshold may be important to understanding effective methods of
treating and avoiding crises.
The divergence of trajectories is also visible in the phase space. Figure 7-6 shows the same solution for c2 x 50 plotted on the strange attractor. The trajectories spread out over the entire attractor
in a dense pattern filling in the space on the attractor surfaces. The lower left plot of the five solutions from t = 1 to 10 shows that they initially follow the same path in the phase space. However,
the plot of t = 60 to 70 on the right shows that the trajectories end up on different parts of the
attractor as the magnification of the initial perturbation has grown exponentially to the borders of
the attractor.
112
Phase Space with 5 Nearly Identical Initial Conditions for c 2 x 50
mu
02D6
t = 0..10
t = 60..70
Figure 7-6: Sensitivity to Initial Conditions of A-H Chaotic Attractor
113
7.3.2 Lyapunov Exponent
The Lyapunov exponent is a quantitative measure of a system's sensitivity to initial conditions. It
gives the average rate of convergence or divergence of the attractor in the phase space. A positive
Lyapunov exponent indicates divergence of neighboring trajectories while a negative Lyapunov
exponent indicates convergence of neighboring trajectories. Chaotic attractors have a positive
Lyapunov exponent. Maple was used to calculate an approximation of the Lyapunov exponents
for the A-H attractor (see Appendix B.4). A plot of Lyapunov exponent vs. c2 for all four state
variables is shown in figure 7-7.
oxygen concentration (c)
velocity (u)
4'
2*
C2
cell stiffness (s)
plasma viscosity (m)
Figure 7-7: Lyapunov Exponent vs. Control Parameter C2
114
There is a Lyapunov exponent for the principle axis described by each of the four state variables.
The dynamics of the system are controlled by the largest Lyapunov exponent for any given value
of c2 . For the A-H attractor, which Lyapunov exponent is largest depends on the value of c2 - Xs
reaches the maximum value of any of the four Lyapunov exponents.
Figure 7-7 shows that most of the Lyapunov exponents go negative from 0 to 60, which is approximately c2 x 3.1. This is consistent with the regime for limit cycles in the phase space shown in
figure 6-6. As c2 increases past c2 x 3.1, the Lyapunov exponents stay positive. This is where the
limit cycles disappear and a strange attractor is born corresponding to chaos in the sickle cell
blood flow solutions. Trajectories switch from asymptotically stable to asymptotically unstable at
C2 x 3.1 and begin to repel each other. As c2 is increased further there is a large region where the
Lyapunov exponent is positive. Within this positive region, there are a few pockets where the
erratic drops in the value of the Lyapunov exponent will cause it to become negative again for
short periods of time. These short pockets of stability are evidence of the small windows of periodic motion known to be present within chaotic control parameter ranges.
The extreme sensitivity of the A-H attractor to changes in initial conditions is further proof of the
presence of chaos for specific ranges of the control parameter c2 . This is another characteristic
contributing to the break down of predicting the long term behavior of chaotic systems. In physiological terms, this means that there may be various behaviors of the blood flow for different
patients who each have a unique physiological background. There may also be different reactions
for one individual at different times of the day or in different environments that affect the initial
conditions of the blood upon entry to the capillary. All of these physiological and environmental
115
factors can have an affect on the initial conditions of the blood flow and the ensuing chaotic
behavior. The sensitivity to initial conditions also means that the treatment of any individual
patient in crisis or approaching crisis would have to be approached carefully. Reactions to different treatments could vary greatly from person to person because various drugs or other remedies
would also affect the initial conditions leading to the chaotic behavior of sickle cell flow.
116
Chapter 8: Conclusions
8.1 Key Findings
This thesis developed a novel model describing the manifestation of sickle cell in the microcirculation. The Apori-Harris (A-H) model for sickle cell blood flow was developed based upon the
framework of several well explored Lagrangian models of sickle cell blood blow. The A-H model
took the unique approach of formulating sickle cell blood flow as an autonomous system in the
Eulerian framework. For the first time, this allowed blood plasma viscosity to be modelled as a
state variable instead of a parameter. In addition, a systematic mathematical treatment of the
erratic onset of sickle cell crisis was performed using chaos theory. The analysis led to several
new and important conclusions.
-
The steady state solution for non-crises blood flow in sickle cell patients is periodic.
The governing equations (AH. 1)-(AH.4) of the A-H model were solved with respect to initial conditions and physiological parameters to give oxygen concentration, blood velocity, cell stiffness,
and plasma viscosity as a function of time. A key finding of the solutions to the A-H model was
the lack of stable fixed points for all selected characteristic parameter values. Only stable limit
cycles existed for steady state solutions. Assuming an accurate designation of characteristic
parameter values, this implies that the normal steady state blood flow solution is periodic for
117
sickle cell patients. The red blood cell velocity and other state variable values would oscillate in
the capillary. An explanation of this result may be some transient effect of the pumping of blood
through the heart. Though most models have assumed capillary blood to flow at constant velocity,
there may be an effective pulsing distinguishable in the microcirculation. Another possible explanation could be the flow dynamics associated with the stiffening of red blood cells as they loose
oxygen in the sickle cell case. This phenomenon should be further explored.
-
Chaotic solutions for sickle cell blood flow are not uncommon.
Another key result was the confirmation of the existence of several chaotic solutions to the A-H
sickle cell model. The regimes for chaotic solutions appeared and disappeared as a function of the
various physiological parameters. As shown in Table 5-5, there were at least six different combinations of parameters that were found to lead to chaos when varied together. The key parameters
were a,, a 2 , b2 , c1 , c2 , and di. Chaos was achieved for the parameter combinations starting with
initial conditions of normal and sickle cell patients measured experimentally. This underscores
the fact that chaotic solutions are not simply possible but they are not uncommon for the A-H
sickle cell model. Knowing this, in addition to the fact that sickle cell crises are unpredictable but
not uncommon events, it appears that the onset of a crisis results directly from the sickle cell
blood flow solution going from steady state to chaotic. For the physiological control parameter
c 2 chosen in this thesis, sickle cell crisis would occur for c2 > 3.1 when the A-H model solutions
initially become chaotic.
-
The onset of crises is caused by physiological parameters reaching a critical value.
118
The physiological parameters were found to play an important role in going from chaos to steady
solutions. C2 was used to control the onset of chaos in the A-H model. C2 represents the difference in the rheology between a red blood cell of a sickle cell patient and a healthy individual.
Physiological parameters like stress, pH levels, temperature, environmental factors, and others are
known to impact when a crisis occurs[5]. It is probable that c2 represents one of these key crises
influencing parameters because its ability to cause the onset of crises in the A-H model has been
demonstrated. The theory presented by Cima[1] stated that crises were caused by plasma viscosity, which Cima et al. modeled as a parameter, reaching a critical value. Though the A-H equations modeled plasma viscosity as a state variable, the analysis presented in this thesis confirms
the importance of physiological parameters. The results of chapter six and seven show that sickle
cell crises are apparently caused by key physiological parameters reaching critical values at which
point the state of sickle cell blood flow becomes chaotic. This may be important to the field of
sickle cell research because currently intra-cellular and extra-cellular physiological parameter
changes associated with the onset of crises were not known to be a cause or an effect of crises[17].
This finding shows that some physiological parameter changes may indeed be a cause of crises.
Developing drugs that maintain key physiological parameters at an acceptable level may be a new
approach effective in treating sickle cell patients.
-
The u-g and p-s relationships are key to understanding sickle cell crises.
The nonlinear terms also had a key impact on chaos. In (AH. 1)-(AH.4) the nonlinearities only
occur in a2 up and b2 gs. These nonlinear terms affect the dynamics of the solution's long term
behavior. The key to understanding the onset of sickle cell crises may be in understanding the
119
complexities of the g-u and ji-s relationship and their respective parameters a 2 and b2 . The viscosity coupling with velocity and cell stiffness are the terms that allow chaos to occur in the system as demonstrated in the analysis of chapter seven. Additionally, plasma viscosity can
potentially be managed clinically, so understanding its effect on crises could lead to improved
methods of treatment[1].
e
A crisis can occur from steady changes in the body over time.
The results of the chaos tests provided many additional physical implications for the sickle cell
patient. The determinism of the Poincare attractor showed that the onset of chaos was not due to
random or chaotic changes in physiological parameters, but to any change of a control parameter
to a critical value. Patients suffering from sickle cell do not have to be exposed to extreme conditions inside or outside of the body to be sent into a crisis. This has been found to be true for most
patients who believe that the crises they suffer are not in response to any stimulus. Doctors have
been unable to identify any agent causing the onset of crises, though it is believed that there is
indeed some underlying physical cause[17]. This physical cause may be explained by the findings of chapter six which show that a crisis can occur from steady changes to physiological
parameters in the body over time.
e
Lack of predictability is the major problem with effectively treating patients in the crisis
state.
The results of the aperiodicity found in the A-H model solutions are also important. The long
term aperiodicity of the state variables in the chaotic regime indicate that once a sickle cell patient
does enter a crises state their oxygen concentration, plasma viscosity, cell stiffness and blood
120
velocity are impossible to predict until the body reacts to control their levels. This lack of predictability could be the major problem with effectively treating patients in the crises state. A patient
can not have steady blood flow until the body is able to bring the parameter levels out of the chaotic regime because the chaotic solutions for blood flow exist as long as the physiological parameters are at critical levels.
-
Sensitivity to initial conditions causes each sickle cell patient to have drastically different
reactions to crises and a lack of forewarning of the onset of crises.
The sensitivity to initial conditions found in the chaotic regime indicates that each sickle cell
patient may have drastically different reactions to both the onset of crises and the treatment of crises. An identical perturbation may cause one individual to have a crisis while another does not.
The early treatment of any individual approaching crisis would have to be approached carefully
since the conditions causing the onset of crisis and the reaction of the patient to treatment could
vary widely between individuals or for the same individual in different situations. The key in
treatment of crises would be taking proper care at the onset to avoid state variables falling to critical levels where they cause the most severe symptoms. Most sickle cell patients come in for treatment after their vessels have already been blocked by sickle cells and they are suffering tissue
damage and other painful attacks of a crisis. If crises could be forewarned by only a matter of
hours, patients could be hydrated and oxygenated to prevent the crisis from occurring completely
or to reduce the intensity and shorten the duration of the crisis[17].
121
8.2 A-H Model Limitations
During the process of solving the A-H model, some limitations became apparent. State variables
often dropped down to negative values which are not physically possible. This indicates a problem of scaling the equations. This was not taken into account when determining the characteristic
values of the parameters. This problem can be easily corrected by an axis shift or reduction in
amplitude of the governing equations. For example, the system t = f(x) would give the following:
(8.1)
= f(x)+6
x
=
f(ax)
(8.2)
(8.1) shows an axis shift of 8 and (8.2) shows an amplitude coefficient of x. These constants must
be carefully chosen as they would represent physiological parameters in the body. Because this
data was not readily available at the time the research was performed, the equations were not corrected.
Another limitation of the A-H model is the accuracy of predictions in the chaotic regime when
variables fall below critical levels for the onset of a crisis. The A-H model is only valid up to the
point of the onset of crises but not beyond this. The reaction of the body and the mechanism it
employs to reverse the onset of a crisis and return to a steady flow solution is beyond the scope of
the A-H model.
122
83 Future Work
The goal of studying the chaotic dynamics of sickle cell disease is to contribute to the understanding and treatment. Accurately modeling the nature of crises in sickle cell disease can lead to better prediction of potential crisis events. Knowing just a few hours in advance that the state of
blood flow is approaching the chaotic regime would allow preventative measures to be taken to
avoid crises. This could also improve the methods of treating critical symptoms of crises when
they do occur. In support of this goal, future research should focus on refining the accuracy and
understanding of the A-H model.
The parameter space of the A-H model should be further explored. Though the c2 physiological
parameter was the focus of this analysis, the discovery of chaos was not limited to varying c2 There are a variety of parameter combinations that can lead to chaos all of which might contribute
additional insight on the manifestation of sickle cell disease in the microcirculation.
The results of the A-H model should also be validated against empirical data. The focus should
be on a thorough exploration of well understood and measurable physiological parameters that are
known to affect sickle cell crises. The parameters of the A-H equations should be checked against
those existing in the body to verify their feasibility and correspondence to real physiological
parameters. In addition, the accuracy of the A-H model in capturing the known behavior of these
parameters in steady state and crises conditions must be validated.
123
Future efforts should also work towards producing an experimental setup to monitor sickle cell
capillary blood flow. Measuring the capillary flow of patients prior to crises and during crises
could provide additional insight into the physics of sickle cell blood flow and further validate the
A-H model. This would strengthen the case for the existence of chaotic solutions in sickle cell
blood flow.
Validating an accurate model for sickle cell blood flow has important implications for the treatment of sickle cell patients. An accurate model for sickle cell blood flow and the onset of crises
may help in creating new therapies or drugs for treating sickle cell crises. Currently, most treatment revolves around hydrating and oxygenating red blood cells or using drugs to inhibit the
polymerization of hemoglobin so that conditions in the cell are not conducive to sickling. There
are also new therapies which involve temporarily concentrating hemoglobin in the plasma
between the cells. This causes oxygen to be carried in the plasma so that the overall oxygen concentration in the blood is increased and cells are less likely to sickle before leaving the capillary[17]. An accurate sickle cell model should be used to simulate how these current and future
therapies effect the onset of sickle cell crises. These investigations could improve upon the methods used to prevent sickle cell patients from having crises in the future.
124
Appendix A: Fourier Analysis
Below are the basic equations used in calculating the Fourier series[7]:
Yh = ahcos[hO]+Phsin[hO],
ah = Ahsin~h and $h = Ahcos~h
(A. 1)
where Yh is the value of the harmonic h wave, Ah is the amplitude, $h is the phase angle and 0 is
the angle associated with the fundamental wavelength.
In general, the number of harmonics included in the series to reproduce a wave is equal to half of
the number of observations in the data set. Any inclusion of harmonics beyond N/2 doesn't contribute additional information. The definition of the Fourier series is as follows:
y = Xh=O...N/2(Qhcos[hO]+$hsin[hO])
(A.2)
In terms of time:
y = Eh=0... N/2(hcosh[2nt/T]+ hsinh[[2nt/T])
(A.3)
The goal of Fourier analysis is to find the values of the coefficients that represent the relative contributions of the constituent waves. From the above equations the following expression for the
coefficients is derived:
ach = 2/N
$h = 2/N
t=0 ...N-1 Yncos(2nhtn/N)
(A.4)
Et=O.. .N-1 ynsin(2htn/N)
(A.5)
except in the special case where N is even and h=N/2 when
ch = 1/N
Et=O ...N-1 yncos(2nhtn/N)
(A.6)
where ah and $h are the Fourier coefficients for harmonic number h, and yn is the data value
observed at time tn.
For discrete systems, the Discrete Fourier transform (DFT) is used. It is also known as a Fast
Fourier Transform (FFT). Like the continuous Fourier transform, it transforms a series of equally
spaced observations taken in a finite amount of time into a discrete frequency-domain spectrum.
It can be thought of as a discrete approximation of the Fourier integral in which each discrete variance stands for a narrow band of frequencies. The DFT is a series of complex numbers defined as
follows:
DFT = ah +jh
(A.7)
125
For practical purposes Fourier coefficients are usually computed and plotted as a variance
(power). The power sh 2of harmonic h in terms of the Fourier coefficients is:
sh2 = (cXh 2 +
Ph2 )/2
(A.8)
126
Appendix B: Maple Routines
B.1 Solving ODE's with Maple
The A-H equations were solved numerically using maple 6 software from Waterloo Maple[15].
In this thesis, the two Maple functions used to solve ordinary differential equations were
DEplot3d and dsolve. DEplot3d employs the fourth-order classical Runge-Kutta method to generate a numerical solution that is plotted in the phase plane. This method uses a default static
stepsize of .005 or takes 3 steps over the time of integration, whichever is smaller. There is no
error correction used with the fourth-order classical Runge-Kutta method. The default method
used by dsolve is a Fehlberg fourth-fifth order Runge-Kutta method. For dsolve, the sizes of the
error are a function of the number of digits Maple uses in calculating software floating point numbers for a solution.
absolute error = 1 x
relative error = 1 x
1 0 (2-Digits)
(B.1)
(2-Digits)
(B.2)
1 0 (1-Digits)
(B.3)
10
minimum error = 1 x
The default value for the absolute error and the relative error is 1 x 10-8 while the default minimum error is 1 x 10-9. The default errors proved to be adequate for the calculations used in the
thesis. The borderline cases are most sensitive to computer errors and can give inaccurate solutions if the numerical methods are inadequate. To verify whether the methods were adequate, the
Lorenz attractor was solved near the borderline chaos case when the control parameter r = 24.74.
Both DEplot3d and dsolve were found to correctly predict the dynamics of the Lorenz attractor
within a range of r + 1 for the borderline chaos case.
B.2 Poincare Section
An approximation of the Poincare section was created by taking a slice of the phase space and
plotting any points on the attractor that fell within the given region. For example, the Poincare
section of the u- plane would be calculated by plotting solutions with the following constraints:
a < s(t) < b or c < c(t) < d, where a and b are constants. As Ia - b Iand Ic - dIgo to zero the
bounding region becomes a plane and a true Poincare section is present. The limits of the bounding regions for the Poincare sections plotted in figure 7-1 are shown in table B-1. The approximation of the Poincare section was used because solving c(t) = 0 and s(t) = 0 explicitly was difficult
to implement. A program that is able to numerically solve the state variables for zero is recommended for a true Poincare section. Algorithms for calculating a Poincare section can be found in
Parker and Chua's book on chaos[15].
127
Table B-1: Poincare Plane Bounds
c 2 x 50
u-p plane
c-s plane
-5 x 10-5 < c < 5 x 10-5
-0.03
< s < -0.025
0.009
-0.004
< u < 0.012
< g < -0.002
> restart:
> with(DEtools):with(plots):
# Poincare Section for A-H attractor
# by Akwasi Apori adapted from MF11 .mws of Enns and McGuire[ 14]
#
#
#
#
This program calculates the Poincare section by finding solutions of
the attractor that fall in a small region of one of the state
variables. As this region is reduced to zero it represents
intersections of the attractor with a plane.
# Poincare section routine
> sys:=diff(c(t),t)=al *c(t)+a2*u(t)*mu(t)+a3*u(t),#start by specifying the system of equations
> diff(u(t),t)=bl*u(t)+b2*mu(t)*s(t)+b3*s(t),
> diff(s(t),t)=c 1*s(t)+c2*c(t),
> diff(mu(t),t)=d1 *mu(t)+d2*c(t);#
> al:=.89928: a2:=-10.66: a3:=-.014928: bl:=-.89929: b2:=1736*.5:#parameter values
> b3:=2.43: cl:=-.89930: d1:=-.89931*3: d2:=151.7:
> c2:=20.044*.001; # the control parameter value
> ic:=c(0)=8.3e-6,u(0)=5e-4,s(0)=1.85e-4,mu(0)=1.4e-3;
#the initial conditions#
> sys;# shows value of system of equations
> vars:=c(t),u(t),s(t),mu(t);
> sol:=dsolve({sys,ic}, {vars},type=numeric,output=listprocedure, maxfun=-1, maxkop=-1);# solves
>#equations as procedures, be sure to set maxfun and maxkop to negative numbers so that there is no limit
to >#the number of total function calculations of calculations of points with the same solution
> assign(sol);
> c:=c(t);
> u:=u(t);
> s:=s(t);
> mu:=mu(t);
> number:=0:
>sart:=1;
>finish:=100;
>increment:=.03# be careful not to use too small a step size or the calculation will take extremely long,
>also risk to plot many solutions within the bounding plane if the step size is too small
> for i from start to finish by increment do
> uplot:=u(i):
> muplot:=mu(i):
128
> if (c(i) > -le-6 and c(i) < le-6) and (s(i) > -0.03 and s(i) < -0.025) then #slice of phase space is made by
>#constraining the solution in two state variables to fall in the bounded region, the slice is much smaller
>#than the attractor diameter
> number:=number+1
> fi:
> if (c(i) > -le-6 and c(i) < le-6) and (s(i) > -0.03 and s(i) < -0.025) then
> g|jnumber:=plot({ [uplot,muplot] },color=black,style=point)
> else glinumber:=plot({ [0,0]} ,color=black,style=point)
> fi:
> od:
> od:
> number;
> display([g||(2..number)],insequence=false,color=black,axes=boxed,symbo
> =CROSS,view=[-0.016..0.016,-0.003..0.0016],color=blue,labels=["u",""mu
> "],tickmarks=[4,41,symbol=point,title="A-H Poincare
> Section,5e-6>c>-5e-6,13pts,t=5..1 10,c2*.001");
#can name plots and then display them all at the end on one graph
> Aplot:=display([g||(10..number)],insequence=false,color=black,axes=box
> ed,symbol=CROSS,color=blue,
> labels=["c","mu"],tickmarks=[4,4],symbol=CROSS,title="Poincare
> Section: Apori-Harris #Equations"):
# The file is done.
B.3 Fourier Power Spectrum
The Fourier power spectrum calculation was rather straightforward. It was calculated using a Fast
Fourier Transform as described in Appendix A. The FFT calculation sampled 1024 points at a
frequency of Fs = .5 for the limit cycle solution and Fs = 10 for the chaotic solution.
> restart:with(plots):
# Power Spectrum of A-H Equations
# by Akwasi Apori adapted from MF32.mws of Enns and McGuire[ 14]
# This program calculates the power spectrum using a Fast Fourier
# Transform of 1024 points
# Calculation of power spectrum
> numpts:=1024;#number of points over which transform is calculated. must be a power of 2.
> m:=(ilog[2](numpts));
> Digits:=16-m;
> sys:=diff(c(t),t)=al*c(t)+a2*u(t)*mu(t)+a3*u(t),#the system of equations being solved
> diff(u(t),t)=bl*u(t)+b2*mu(t)*s(t)+b3*s(t),
> diff(s(t),t)=c 1*s(t)+c2*c(t),
129
> diff(mu(t),t)=d 1*mu(t)+d2*c(t);#
> al:=.89928: a2:=-10.66: a3:=-.014928: bl:=-.89929: b2:=1736*.5:#parameters
> b3:=2.43: cl:=-.89930: d1:=-.89931*3: d2:=151.7:;
> c2:=20.044*50; # control parameter
> ic:=c(O)=8.3e-6,u(0)=5e-4,s(0)=1.85e-4,mu(O)=1.4e-3; #the initial conditions#
> vars:=c(t),u(t),s(t),mu(t);#defining variables
> sol:=dsolve({ sys,ic }, {vars },type=numeric,output=listprocedure);# numerically solving and output to
#procedure
> assign(sol);
> C:=c(t);
> U:=u(t);
> S:=s(t);
> Mu:=mu(t);
> Fs:=8;#frequency of sampling for the fourier transform. there must be two points sampled per wave
#length. see chapter 3.
> x:=array([seq(U(50+i/Fs),i=O..numpts-1)]):#fourier spectrum plot of state variable u
> y:=array([seq(0,i=1..numpts)]):
> FFT(m,x,y):#maple procedure FFT takes array of values and array of zeros as arguments and outputs
#array of real and imaginary numbers
> for i from 1 to numpts do
> pt[1]:=0,0;
> pt[i]:=i-1,(((x[i]A2 + y[i]A2).5));#for the power spectrum the results of FFT are squared
> od:
> pts:=[seq([pt[z]],z=1..numpts)]:
# Plot of the power spectrum
> plot(pts,font=[TIMES,ROMAN, 12],tickmarks=[4,3],abels=["k","S"],title=
> "Power Spectrum of U with C2=50 and Fs=10");
# The file is done.
BA Lyapunov Exponent
The Lyapunov Exponent was calculated according to the following approximation[9:321]:
11 (t)||~||Sollext
(B.4)
This describes the exponential growth of an initial separation over time between two trajectories.
The initial separation is S0. The separation as a function of time is 8(t), and X is the Lyapunov
exponent. (B.4) is an approximation for continuous systems that is simple to implement on the
computer. However, some anomalies were noted in the results. The program appeared to work
well with positive X values but negative X values only stayed negative briefly. Values of X known
to be negative for the A-H attractor straddled the X = 0 axis between positive and negative values.
The cause of this anomaly was unknown. The author suspects that (B.4) approximates X in a way
that automatically incorporates the affect of the largest Lyapunov exponent into the calculation of
each individual Lyapunov exponent. Since divergences on the attractor are controlled by the larg-
130
est Lyapunov exponent, measuring this divergence for each state variable may just be presenting
the effect of the largest Lyapunov exponent as viewed from four different axes. This method of
approximation was thought to show the general way that X varied with c2 , particularly for positive
values of X. Algorithms for calculating Lyapunov exponents can be found in [15].
> restart:with(plots):
# Lyapunov Exponent for the A-H Equations
# by Akwasi Apori adapted from MF38.mws of Enns and Mcguire[14]
# This code calculates the Lyapunov exponent vs the control parameter c2
# based on the definition on pg. 321 of Strogatz's book [9].
# Calculating the Lyapunov exponent
> Digits:=7:
> h:=O;total:=O;
> startgraph:=.001; endgraph:=1;frequency:=100;# the starting point, ending point, and time step for vary#ing the control parameter
> al:=.89928: a2:=-10.66: a3:=-.014928: bl:=-.89929: b2:=1736*.5:#parameter values
> b3:=2.43: cl:=-.89930: d1:=-.89931*3: d2:=151.7:
> for c2 from startgraph to endgraph by
" ((endgraph-startgraph)/frequency) do
> unassign(c,u,s,mu,co,uo,so,muo,cdelta,udelta,sdelta,mudelta);
#the first loop of solving the system of equations just generates the initial condition values for t=20 (when
#transients have settled down to be used in the second loop. The divergence must be calculated over the
#attractor so that is why initial transients are removed. In the fist loop the tiny separation, delta, of initial
#conditions on the attractor at t=20 is also calculated. The values must be recalculated every time since
#they change as c2 is varied in the second loop
> syso:=diff(co(t),t)=al*co(t)+a2*uo(t)*muo(t)+a3*uo(t),#solve system of equations
> diff(uo(t),t)=bl*uo(t)+b2*muo(t)*so(t)+b3*so(t),
> diff(so(t),t)=c 1*so(t)+c2*co(t),
> diff(muo(t),t)=d1 *muo(t)+d2*co(t);
> ico:=co(O)=8.3e-6,uo(O)=5e-4,so(O)=1.85e-4,muo(O)=1.4e-3; #the initial conditions
> varso:=co(t),uo(t),so(t),muo(t);
> solo: =dsolve({ syso,ico 1, { varso },numeric,output=listprocedure);
> assign(solo);
> co:=co(t);
> uo:=uo(t);
> so:=so(t);
> muo:=muo(t);
> cdeltao:=co(20)+co(20)*10e-6;#solution procedures for delta, which is a small perturbation of lOe-6
> udeltao:=uo(20)+uo(20)* 1Oe-6;
> sdeltao:=so(20)+so(20)*10e-6;
> mudeltao:=muo(20)+muo(20)* 1Oe-6;
131
#there is a second loop of solving the equations for calculating the lyapunov exponent using i.c.'s for t=20
> sys:=diff(c(t),t)=al*c(t)+a2*u(t)*mu(t)+a3*u(t),
> diff(u(t),t)=bl*u(t)+b2*mu(t)*s(t)+b3*s(t),
> diff(s(t),t)=c1*s(t)+c2*c(t),
> diff(mu(t),t)=d1 *mu(t)+d2*c(t);
>
>
>
>
>
>
>
>
ic:=c(20)=co(20),u(20)=uo(20),s(20)=so(20),mu(20)=muo(20); #the initial conditions
vars:=c(t),u(t),s(t),mu(t);
sol:=dsolve({ sys,ic 1, { vars },numeric,output=listprocedure);
assign(sol);
>
>
>
>
sysdelta:=diff(cdelta(t),t)=al*cdelta(t)+a2*udelta(t)*mudelta(t)+a3*udelta(t),#solving
diff(udelta(t),t)=bl*udelta(t)+b2*mudelta(t)*sdelta(t)+b3*sdelta(t),
diff(sdelta(t),t)=c1*sdelta(t)+c2*cdelta(t),
diff(mudelta(t),t)=dl*mudelta(t)+d2*cdelta(t);
c:=c(t);
u:=u(t);
s:=s(t);
mu:=mu(t);
> icdelta:=cdelta(20)=cdeltao,udelta(20)=udeltao,sdelta(20)=sdeltao,mude#the initial conditions of system
# with delta
> lta(20)=mudeltao; #the initial conditions of delta
> varsdelta:=cdelta(t),udelta(t),sdelta(t),mudelta(t);
> soldelta:=dsolve({ sysdelta,icdelta} ,{ varsdelta} ,numeric,output=listprocedure);#solving system + delta
> assign(soldelta);
> cdelta:=cdelta(t);
> udelta:=udelta(t);
> sdelta:=sdelta(t);
> mudelta:=mudelta(t);
> total:=0:
> for j from 20 to 50 do
> d:=abs((mudelta(j)-mu(j))/(mudeltao-muo(20)));#implementing equation B.4
> if d<>0 then f:=ln(d) else f=0 fi;
> total:=total+f;
> od:
> h:=h+1;
> ptsllh:=[c2,total/(j-21)]:#implementing equation B.4
> od:
# The plot
> plot([seq(pts||i,i=1 ..frequency)],style=line,tickmarks=[3,3],color=blu
> e);#
132
B.5 AdditionalMaple Worksheets
The additional Maple worksheets included in this section were either used to make the Maple
generated figures or they were used for other aspects of the analysis contained in the thesis.
# Rossler Attractor and Time Series Figure 3-6
# by Akwasi Apori
> restart:
> with(DEtools):
> f:=[diff(x(t),t)=-y-z,diff(y(t),t)=x+a*y,diff(z(t),t)=b+z*(x-c)];
> b:=.2: a:=.2: c:=5.7:
# Rossler equations are solved and plotted in phase space and time
# series
> DEplot3d(f,[x,y,z],t=0.. 100,{ [0,1,1,1] },scene=[x,y,z],stepsize=.05);#
> DEplot3d(f,[x,y,z],t=0.. 150,{ [0,1,1,1] },scene=[x,t,x],stepsize=.05);#
# Lorenz Attractor, Time Series, and Fourier Spectrum Figures 3-1, 3-3,
# 3-7, 5-2
# Attractor/Time Series by Akwasi Apori. Fourier Spectrum Adapted from
# MF32.mws by Enns and Mcguire[14].
> restart:
> with(DEtools): with(plots):
> f:=[diff(x(t),t)=s*(y-x),diff(y(t),t)=-x*z+r*x-y,diff(z(t),t)=x*y-b*z]
> b:=8/3: r:=28: s:=10:
# Plot of Lorenz time series with different initial conditions on the
# same plot
> Aplot:=DEplot3d(f,[x,y,z],t=0..35,{[0,-2.64,-44.3,17.06]} ,scene=[x,t,x
> ],stepsize=.05):
> Bplot:=DEplot3d(f,[x,y,z],t=0..35,{[0,-2.6400001,-44.3,17.06] },scene=[
> x,t,x],stepsize=.05,linecolor=red):
> display([Aplot,Bplot],title="Lorenz Attractor Time Series for
> x(0)=-2.64 & -2.6400001");#
> display([Aplot]);#
# plot of Lorenz Attractor
> DEplot3d(f,[x,y,z],t=0..50,{ [0,-2.64,-44.3,17.06]),scene=[x,yz],steps
> ize=.01,title);#
# Calculation of power spectrum
> numpts:=512;#
> m:=(ilog[2](numpts));#
> Digits:=16-m;
> sys:=diff(x(t),t)=s*(y-x),diff(y(t),t)=-x*z+r*x-y,diff(z(t),t)=x*y-b*z
> ;#
>
>
>
>
b:=8/3: r:=28: s:=10: #chaotic solution
ic:=x(O)=-2.64,y(0)=-44.3,z(0)=17.06; #the initial conditions
vars:=x(t),y(t),z(t);
sol:=dsolve({sys,ic}, {vars},type=numeric,output=listprocedure);
133
> assign(sol);
> X:=x(t);
> Y:=y(t);
> Z:=z(t);
> Fs:=8;#frequency of sampling for the fourier transform
> p:=array([seq(X(20+i/Fs),i=O..numpts-1)]):
> q:=array([seq(O,i=1..numpts)]):
> FFT(m,p,q):
> for i from 1 to numpts do
> pt[1]:=O,O;
> pt[i]:=i-1,(((p[i]A2 + q[i]A2)A.5));
> od:
> pts:=[seq([pt[g]],g=1..numpts)]:
# Plot of the power spectrum
> plot(pts,font=[TIMES,ROMAN,12],labels=["k","S"],title="Power
> for X on Lorenz Attractor, b=8/3 r=28 s=10,
> [xo,yo,zo]=[-2.64,-44.3,17.06]");
Spectrum
# Solution of Fixed Points and Linearized Analysis Tables 5-1, 5-2, and
# 6-1
# by Akwasi Apori
> restart:
> with(DEtools):with(plots):
> f:=[diff(c(t),t)=al*c+a2*u*mu+a3*u,diff(u(t),t)=bl*u+b2*mu*s+b3*s,diff
> (s(t),t)=c1*s+c2*c, diff(mu(t),t)=dl*mu+d2*c];#
# The fixed points F+ and F- of the system are solved
> solve({al*c+a2*u*mu+a3*u=0,bl*u+b2*mu*s+b3*s=0,c1*s+c3*c=0,dl*mu+d2*c=
> 01,{c,u,s,mu});#
> allvalues(u =
> dl*(al*cl*bl+a3*c3*b3+a3*b2*RootOf(_ZA2*a2*b2+al*c1*bl*c3+a3*c3A2*b3+(
> a2*c3*b3+a3*c3*b2)*_Z,label = _L3))/(c1*d2*a2*bl));#
> allvalues(s =
> RootOf(_ZA2*a2*b2+al*c1*bl*c3+a3*c3A2*b3+(a2*c3*b3+a3*c3*b2)*_Z,label= _L3)*dl/
> (c 1*d2));#
> allvalues(c =-RootOf(_.ZA2*a2*b2+a1*cl*bl*c3+a3*c3A2*b3+(a2*c3*b3+a3*c3*b2)*_Z,abe
>= _L3)*dl/(d2*c3));#
> allvalues(mu =
> RootOf( ZA2*a2*b2+al*c1*bl*c3+a3*c3A2*b3+(a2*c3*b3+a3*c3*b2)*_Z,1abel= .L3)/c3);#
> al:=.89928*.6: a2:=-12.44: a3:=-.014928: bl:=-.89929: b2:=2025.4:
> b3:=2.43: c1:=-.89930: c2:=-16703.3: c3:=20.044: d1:=-.89931*.5:
> d2:=130.016*6:
> solve({al*c+a2*u*mu+a3*u=,bl*u+b2*mu*s+b3*s=0,c1*s+c3*c=O,dl*mu+d2*c=
> O},{c,u,s,mu});#
> al:=.89928: a2:=-10.66: a3:=-.014928: bl:=-.89929: b2:=1736*.5:
> b3:=2.43: cl:=-.89930: c2:=20.044*100: d1:=-.89931*3: d2:=151.7:
> solve({al*c+a2*u*mu+a3*u=0,bl*u+b2*mu*s+b3*s=0,c1*s+c2*c=0,dl*mu+d2*c=
> O},{c,u,s,mu});#
# linearization around origin for extremely chaotic attractor when
# c2*100
> al:=.89928: a2:=-10.66: a3:=-.014928: bl:=-.89929: b2:=1736*.5:
134
>
>
>
>
b3:=2.43: cl:=-.89930: c2:=20.044*100: d1:=-.89931*3: d2:=151.7:
deto1: =a 1*bl1;
deto2:=a1*c1;
deto3:=a1*d1;
> deto4:=b1*c1;
> deto5:=b1*d1;
> deto6:=c1*d1;
> traceO4:=bl+c1;
> trace05:=bl+dl;
> trace06:=c1+d1;
> traceO4A2-4*deto4;
> trace05A2-4*deto5;
> trace06A2-4*deto6;
# origin has 3 saddle points, one center, and two stable spirals
# linearization about fixed point has one different determinant
> detFplus3:=al*d1-a2*.2973000597e-2*d2;
> detFminus3:=al*dl-a2*(-.7493725359e-1)*d2;
> trace3Fplus:=al+dl;
> trace3FplusA2-4*detFplus3;
# so at linearization when c2=100* F- is same type of fixed point as
# origin with stronger saddle and F+ has 2 saddle points, one center and
# 2 stable spirals and one stable node.
# Following is analysis of c2*.001 which displays figure 8 limit cycle
> al:=.89928: a2:=-10.66: a3:=-.014928: bl:=-.89929: b2:=1736*.5:
> b3:=2.43: cl:=-.89930: c2:=20.044*.001: d1:=-.89931*3: d2:=151.7:
> solve({al*c+a2*u*mu+a3*u=0,bl*u+b2*mu*s+b3*s=0,c1*s+c2*c=0,dl*mu+d2*c=
> 0},{c,u,s,mu});
> detFplus3:=a1*d1-a2*. 1533494782e-2*d2;
> detFminus3:=al*d1-a2*.1466388163e-2*d2;
> trace3Fplus:=al+dl;
> trace3FplusA2-4*detFplus3;
# so at linearization when c2*.001 F- is same type of fixed point as
# origin and F+ has 2 saddle points, one center and 3 stable spirals.
# linearization for c2*.O1 still figure eight
> al:=.89928: a2:=-10.66: a3:=-.014928: bl:=-.89929: b2:=1736*.5:
> b3:=2.43: cl:=-.89930: c2:=20.044*.O1: d1:=-.89931*3: d2:=151.7:
> solve({al*c+a2*u*mu+a3*u=0,bl*u+b2*mu*s+b3*s=0,c1*s+c2*c=0,dl*mu+d2*c=
> 0},{c,u,s,mu});
> detFplus3:=a1 *d1-a2*. 1602732502e-2*d2;
> detFminus3:=a1*d1-a2*. 1390403603e-2*d2;
> trace3Fplus:=al+dl;
> trace3FplusA2-4*detFplus3;
# so at linearization when c2*.O1 F- is same type of fixed point as
# origin and F+ is same as c2*.001 with 2 saddle points, one center and
# 3 stable spirals.
# linearization for c2*. 1 oval limit cycle
> al:=.89928: a2:=-10.66: a3:=-.014928: bl:=-.89929: b2:=1736*.5:
> b3:=2.43: cl:=-.89930: c2:=20.044*.1: d1:=-.89931*3: d2:=151.7:
135
>
>
>
>
solve({al*c+a2*u*mu+a3*u=0,bl*u+b2*mu*s+b3*s=0,c1*s+c2*c=0,dl*mu+d2*c=
0},{c,u,s,mu});
detFplus3:=a1*d1-a2*. 1800433236e-2*d2;
detFminus3:=al*d1-a2*. 1125234472e-2*d2;
> trace3Fplus:=al+dl;
> trace3FplusA2-4*detFplus3;
# so at linearization when c2*. 1 F- is same type of fixed point as
# origin and F+ is same as c2*.01 with 2 saddle points, one center and 3
# stable spirals.
# linearization for c2* 1 oval limit cycle
> al:=.89928: a2:=-10.66: a3:=-.014928: bl:=-.89929: b2:=1736*.5:
> b3:=2.43: cl:=-.89930: c2:=20.044: d1:=-.89931*3: d2:=151.7:
> solve({al*c+a2*u*mu+a3*u=0,bl*u+b2*mu*s+b3*s=0,c1*s+c2*c=0,dl*mu+d2*c=
> 0},{c,u,s,mu});
> detFplus3:=al*d1-a2*.2250737372e-2*d2;
> detFminus3:=al*dl-a2*.2463654679e-6*d2;
> trace3Fplus:=al+dl;
> trace3FplusA2-4*detFplus3;
# so at linearization when c2* 1 F- is same type of fixed point as
# origin and F+ has 2 saddle points, one center, 2 stable spirals and
# one stable node
# linearization for c2* 10 chaotic attractor
> c2:=20.044* 10:
> solve({al*c+a2*u*mu+a3*u=0,bl*u+b2*mu*s+b3*s=0,c1*s+c2*c=0,dl*mu+d2*c=
> 0},{c,u,s,mu});
> detFplus3:=al*dl-a2*.2782596452e-2*d2;
> detFminus3:=al*d1-a2*(-.7278452417e-2)*d2;
> trace3Fplus:=al+dl;
> trace3FplusA2-4*detFplus3;
# so at linearization when c2* 10 F- is same type of fixed point as
# origin and F+ is like c2* 1 with 2 saddle points, one center, 2 stable
# spirals and one stable node
# linearization for c2*1000 goes to infinity (too much for maple to
# handle)
> c2:=20.044* 1000:
> solve({al*c+a2*u*mu+a3*u=0,bl*u+b2*mu*s+b3*s=0,c1*s+c2*c=0,dl*mu+d2*c=
> 0},{c,u,s,mu});
> detFplus3:=al*dl-a2*.2998935580e-2*d2;
> detFminus3:=a1*d1-a2*(-.7496471588)*d2;
> trace3Fplus:=al+dl;
> trace3FplusA2-4*detFplus3;
# so at linearization when c2* 1000 F- is same type of fixed point as
# origin and F+ is like c2*100 with 2 saddle points, one center, 2
# stable spirals and one stable node
# solve where traceA2-4Delta equals 0 as a function of c2 to see where
136
# dynamics change for the linearized fixed points giving them the extra
# stable node present for most of chaotic domain
> restart:
> al:=.89928: a2:=-10.66: a3:=-.014928: bl:=-.89929: b2:=1736*.5:
>b3:=2.43: c1:=-.89930: d1:=-.89931*3: d2:=151.7:
> trace3Fplus:=al+d 1;
>
>
>
>
>
uplus:=dl*(al*c1*bl+a3*c2*b3+1/2*a3*(-a2*c2*b3-a3*c2*b2+srt(a2A2*c2A2
*b3A2-2*a2*c2A2*b3*a3*b2+a3A2*c22*b2^2-4*a2*b2*al*c1*bl*c2))/a2)/(cl*d2*a2*bl);#
uminus:=d1*(al*c1*bl+a3*c2*b3+1/2*a3*(-a2*c2*b3-a3*c2*b2-sqrt(a2A2*c2A
2*b3A2-2*a2*c2A2*b3*a3*b2+a3A2*c2A2*b2A2-4*a2*b2*a1*c1*bl*c2))/a2)/(c1*d2*a2*bl);#
detFplus3:=al*dl-a2*uplus*d2;#
, { c2 1);
# This is a bifurcation point in the system where the dynamics of the
# attractor change
# Solving the fixed points in terms of the attractors
> restart:
> al:=.89928: a2:=-10.66: a3:=-.014928: bl:=-.89929: b2:=1736*.5:
> b3:=2.43: c1:=-.89930: d1:=-.89931*3: d2:=151.7:
> solve({al*c+a2*u*mu+a3*u=0,bl*u+b2*mu*s+b3*s=0,c1*s+c2*c=0,dl*mu+d2*c=0},{c,u,s,mu});#
> solve({al*c+a2*u*mu+a3*u=0,bl*u+b2*mu*s+b3*s=0,c1*s+c2*c=0,dl*mu+d2*c=0},{c,u,s,mu});#
> allvalues(u =.2445295318e-14*c2*RootOf(-31313274832343+3731*_ZA2*c2+161922100*c2*_Z
> solve({trace3FplusA2-4*detFplus3=0}
>+1561842000000*c2,label=_L26)*(73.10520000*RootOf(31313274832343+3731*_ZA2*c2+16192210
>0*c2*_Z+1561842000000*c2,label = _L26)+2114829.));#
> allvalues(u =dl*(al*c1*bl+a3*c2*b3+a3*b2*RootOf(a2*b2*_ZA2+al*c1*bl*c2+a3*c2A2*b3+(
> a2*b3*c2+a3*c2*b2)*_Z,abel = _L3))/(a2*d2*c1*bl));#
> uplus:=dl*(al*c1*bl+a3*c2*b3+1/2*a3*(-a2*b3*c2-a3*c2*b2+sqrt(a22*b3A2*c2A2-2
" *a2*b3*c2A2*a3*b2+a3A2*c2A2*b22-4*a2*b2*al*cl*bl*c2))/a2)/(a2*d2*c1*bl);#
> uplus:=d1*(al*c1*bl+a3*c2*b3+1/2*a3*(-a2*b3*c2-a3*c2*b2+srt(a2A2*b3A2*c2A2-2
" *a2*b3*c2A2*a3*b2+a3A2*c2A2*b22-4*a2*b2*al*cl*bl*c2))/a2)/(a2*d2*cl*bl);#
> allvalues(mu =RootOf(a2*b2*_ZA2+al*c1*bl*c2+a3*c2A2*b3+(a2*b3*c2+a3*c2*b2)*_Z,label
>= _L3)/c2);#
> muplus:=(1/2*(-a2*b3*c2-a3*c2*b2+sqrt(a2A2*b3A2*c2A2-2*a2*b3*c2A2*a3*b
> 2+a3A2*c2A2*b2A2-4*a2*b2*a1*c1*bl*c2))/(a2*b2*c2));#
> allvalues(s =RootOf(a2*b2*_ZA2+al*c1*bl*c2+a3*c2A2*b3+(a2*b3*c2+a3*c2*b2)*_Z,abel=
>_L3)*dl/>(c1*d2));#
> splus:=(1/2*(-a2*b3*c2-a3*c2*b2+sqrt(a22*b3A2*c2A2-2*a2*b3*c2A2*a3*b2
> +a3A2*c2A2*b2A2-4*a2*b2*a1*cl*bl*c2))*dl/(a2*b2*c1*d2));#
> allvalues(c =-RootOf(a2*b2*_ZA2+al*cl*bl*c2+a3*c2A2*b3+(a2*b3*c2+a3*c2*b2)*_Z,abel
>= _L3)*dl/(d2*c2));#
> cplus:=(-1/2*(-a2*b3*c2-a3*c2*b2+sqrt(a2A2*b3A2*c2A2-2*a2*b3*c2A2*a3*b
> 2+a3A2*c2A2*b2A2-4*a2*b2*a1 *c1*bl*c2))*dl/(a2*b2*d2*c2));#
> uplus:=.3276997210e-18*(-161922100*c2+2*sqrt(727459115102500*c2A2+1168
> 29828399471733*c2))*(.9796998124e-2*(-161922100*c2+2*sqrt(727459115102
> 500*c2^2+116829828399471733*c2))/c2+2114829.);#
> trace3Fplus:=al+dl;
> detFplus3:=al *d-a2*uplus*d2;
> detFplus3:=al*d1-a2*uplus*d2;#
> solve({trace3FplusA2-4*detFplus3=0}
> solve({ trace3FplusA2-4*detFplus3=0}
> c2:=6.678314989;
,{ c2 });#
, { c2 });
137
> uplus:=
> .1500316297e-2-.1870008120e-4*c2+.1444434858e-5*sqrt(167.6065801 *c2A2+
> 26917.59246*c2);
> c2:= -.3799614627e12;#
> uplus:=
>.1500316297e-2-.1870008120e-4*c2+.1444434858e-5*sqrt(167.6065801*c2A2+26917.59246*c2);#
> c2:= -.3799614627e12;#
> trace3FplusA2-4*detFplus3;
> c2:=6.678314989;
> trace3FplusA2-4*detFplus3;
> ans:=trace3FplusA2-4*detFplus3;#
> uplus
> ans;
> uplus
> ans;#
.2000455095e-2;
14210620.42;#
> al*bl;
> al+bl;
> (al+b1)A2-4*a1*b1;
> al*c1;
> al+c1;
> (al+c1)A2-4*al*c1;
> al*dl;
> al+dl;
> (al+d1)A2-4*al*d1;
> bl*cl;
> bl+c1;
> (bl+c1)A2-4*bl*c1;
> bl*dl;
> bl+dl;
> (b1+d1)A2-4*bl*d1;
> cl*dl;
> cl+dl;
> (c1+d1)A2-4*c1*d1;
> a2*d2;
>
(al+d1)^2;
> al*dl;
# Chapter 6 AttractorsFigures 6-1 to 6-3
# by Akwasi Apori
> restart:
> with(DEtools):with(plots):
> f:=[diff(c(t),t)=al*c+a2*u*mu+a3*u,diff(u(t),t)=bl*u+b2*mu*s+b3*s,diff
> (s(t),t)=c1*s+c2*c, diff(mu(t),t)=dl*mu+d2*c];
> al:=.89928: a2:=-10.66: a3:=-.014928: bl:=-.89929: b2:=1736*.5:
> b3:=2.43: cl:=-.89930: c2:=20.044*.001: d1:=-.89931*3: d2:=151.7:
# The phase space and time series of the A-H equations are solved for a
# variety of c2 values.
> DEplot3d(f,[c,u,s,mu],t=0.. 100, { [0,8.3e-6,5e-4,.000185,1.4e-3] },scene=
>[c,u,mu],stepsize=. 1,title="c2 x .001");#
> DEplot3d(f,[c,u,s,mu],t=0..100, { [0,8.3e-6,5e-4,.000185,1.4e-3] },scene=
138
> [mu,t,c],stepsize=.1,title="c2 x .001");#
> c2:=20.044*.01:
> DEplot3d(f,[c,u,s,mu],t=0..100,{ [0,8.3e-6,5e-4,.000185,1.4e-3] },scene=
> [c,u,mu],stepsize=. 1,title="c2 x .01");#
> DEplot3d(f,[c,u,s,mu],t=0..100, { [0,8.3e-6,5e-4,.000185,1.4e-3] },scene=
> [mu,t,c],stepsize=. 1,title="c2 x .01");#
> c2:=20.044*.1:
> DEplot3d(f,[c,u,s,mu],t=O.. 100, { [0,8.3e-6,5e-4,.000185,1.4e-3] },scene=
> [u,c,mu],stepsize=.1,title="c2 x .1");#
> DEplot3d(f,[c,u,s,mu],t=0.. 100, { [0,8.3e-6,5e-4,.000185,1.4e-3] },scene=
> [mu,t,c],stepsize=. 1,title="c2 x .1");#
> c2:=20.044:
> DEplot3d(f,[c,u,s,mu],t=O.. 100,{ [0,8.3e-6,5e-4,.000185,1.4e-3] },scene=
> [u,c,mu],stepsize=. 1,title="c2 x 1");#
> DEplot3d(f,[c,u,s,mu],t=0.. 100, { [0,8.3e-6,5e-4,.000185,1.4e-3] },scene=
> [mu,t,c],stepsize=.1,title="c2 x 1");#
> c2:=20.044* 10:
> DEplot3d(f,[c,u,s,mu],t=0.. 100, { [0,8.3e-6,5e-4,.000185,1.4e-3] },scene=
> [c,u,mu],stepsize=.05,title="c2 x 10");#
> DEplot3d(f,[c,u,s,mu],t=0.. 100, { [0,8.3e-6,5e-4,.000185,1.4e-3]} ,scene=
> [mu,t,c],stepsize=.05,title="c2 x 10"); #
> c2:=20.044* 100:
> DEplot3d(f,[c,u,s,mu],t=0.. 100, { [0,8.3e-6,5e-4,.000185,1.4e-3] },scene=
> [c,u,mu],stepsize=.01,title="c2 x 100");#
> DEplot3d(f,[c,u,s,mu],t=0..100,{ [0,8.3e-6,5e-4,.000185,1.4e-3] },scene=
> [mu,t,c],stepsize=.01,title="c2 x 100");#
> c2:=20.044* 1000:
> DEplot3d(f,[c,u,s,mu],t=0..100,{[0,8.3e-6,5e-4,.000185,1.4e-3] },scene=
> [c,u,mu],stepsize=.005,title="c2 x 1000");
> DEplot3d(f,[c,u,s,mu],t=0.. 100,{ [0,8.3e-6,5e-4,.000185,1.4e-3] },scene=
> [mu,t,c],stepsize=.005,title="c2 x 1000");
# Route to Chaos Figures 6-5 to 6-7
# by Akwasi Apori
>
>
>
>
restart:
with(DEtools):with(plots):
f:=[diff(c(t),t)=al*c+a2*u*mu+a3*u,diff(u(t),t)=bl*u+b2*mu*s+b3*s,diff
(s(t),t)=c1*s+c2*c, diff(mu(t),t)=d1*mu+d2*c];#
> al:=.89928: a2:=-10.66: a3:=-.014928: bl:=-.89929: b2:=1736*.5:
> b3:=2.43: cl:=-.89930: d1:=-.89931*3: d2:=151.7:
> c2:=20.044*2:
# The phase space and time series of the A-H equations are solved for
# various values of c2
> DEplot3d(f,[c,u,s,mu],t=0..100,{ [0,8.3e-6,5e-4,.000185,1.4e-3] },scene=
> [c,u,mu],stepsize=. 1,title="c2 x 2");#
> DEplot3d(f,[c,u,s,mu],t=0.. 100, { [0,8.3e-6,5e-4,.000185,1.4e-3] },scene=
> [mu,t,c],stepsize=. 1,title="c2 x 2");#
> c2:=20.044*3:
> DEplot3d(f,[c,u,s,mu],t=0..100,{ [0,8.3e-6,5e-4,.000185,1.4e-3] },scene=
> [c,u,mu],stepsize=. 1,title="c2 x 3");#
139
> DEplot3d(f,[c,u,s,mu],t=0..100, {[0,8.3e-6,5e-4,.000185,1.4e-3] },scene=
> [mu,t,c],stepsize=. 1,title="c2 x 3");#
> c2:=20.044*4:
> DEplot3d(f,[c,u,s,mu],t=0..100, {[0,8.3e-6,5e-4,.000185,1.4e-3] },scene=
> [c,u,mu],stepsize=. 1,title="c2 x 4");#
> DEplot3d(f,[c,u,s,mu],t=0..100, { [0,8.3e-6,5e-4,.000185,1.4e-3] },scene=
> [mu,t,c],stepsize=. 1,title="c2 x 4");#
> c2:=20.044*5:
> DEplot3d(f,[c,u,s,mu],t=0..100,{ [0,8.3e-6,5e-4,.000185,1.4e-3] },scene=
> [c,u,mu],stepsize=. 1,title="c2 x 5");#
> DEplot3d(f,[c,u,s,mu],t=0.. 100, { [0,8.3e-6,5e-4,.000185,1.4e-3] },scene=
> [mu,t,c],stepsize=.1,title="c2 x 5");#
# Analysis of Attractor Near Predicted Linearized Bifurcations
# by Akwasi Apori
> restart:with(DEtools):with(plots):
> f:=[diff(c(t),t)=al*c+a2*u*mu+a3*u,diff(u(t),t)=bl*u+b2*mu*s+b3*s,diff
> (s(t),t)=c1*s+c2*c, diff(mu(t),t)=dl*mu+d2*c];#
> al:=.89928: a2:=-10.66: a3:=-.014928: bl:=-.89929: b2:=1736*.5:
> b3:=2.43: cl:=-.89930: c2:=20.044*.001: d1:=-.89931*3: d2:=151.7:
# Solutions in the phase space are plotted for a variety of c2 values
# near potential bifurcations
> DEplot3d(f,[c,u,s,mu],t=0.. 100,1 [0,-. 1051105867e-2,.1533494782e-2,-.25
> 65636161e-4,-.6472471858e-1] },scene=[c,u,mu],stepsize=.1);#
> DEplot3d(f,[c,u,s,mu],t=0.. 100,1 [0,. 1176411898e-2,.1466388163e-2,.2399
> 154907e-4,.6052480417e-1] },scene=[c,u,mu],stepsize=.1);#
> DEplot3d(f,[c,u,s,mu],t=0.. 100,1 [-0.0001,-0.0001,0,0,0] },scene=[c,u,mu],stepsize=.1);#
> c2:=20.044*.05:
> DEplot3d(f,[c,u,s,mu],t=0..100,1 [0.0001,-0.0001,0,0,0]},scene=[c,u,mu],stepsize=.1);#
> DEplot3d(f,[c,u,s,mu],t=0.. 100, { [0,. 1176411898e-2,.1466388163e-2,.2399
> 154907e-4,.6052480417e-1]},scene=[c,u,mu],stepsize=.1);#
> c2:=20.044*.35;#
> DEplot3d(f,[c,u,s,mu],t=0..100,1 [0,.01 176411898e-2,.01466388163e-2,.02
> 399154907e-4,.06052480417e-1] },scene=[c,u,mu],stepsize=.1);#
# where do we go from limit cycle encompassing two points to a limit
# cycle encompassing one point?
> c2:=20.044*.03;
> DEplot3d(f,[c,u,s,mu],t=0..100,1 [0,-.0004,.0015,0,-.02] },scene=[c,u,mu],stepsize=. 1);#
> DEplot3d(f,[c,u,s,mu],t=0.. 150,1[0,-.0004,.0015,0,-.02]},scene=[c,t,mu],stepsize=. 1);#
> c2:=20.044*.04;
> DEplot3d(f,[c,u,s,mu],t=0.. 150,1[0,-.0004,.0015,0,-.02]} ,scene=[c,t,mu],stepsize=. 1);#
> c2:=20.044*.043;
> DEplot3d(f,[c,u,s,mu],t=0..200,{ [0,-.0004,.0015,0,-.02] },scene=[c,t,mu],stepsize=.1);#
> c2:=20.044*.044;
> DEplot3d(f,[c,u,s,mu],t=0..200,{[0,-.0004,.0015,0,-.02] },scene=[c,u,mu],stepsize=.1);#
# what happens around the change in stability predicted by linearized
# analysis at c2 x .0441
> DEplot3d(f,[c,u,s,mu],t=0..200,{[0,-.0004,.0015,0,-.02] },scene=[c,t,mu],stepsize=.1);#
> c2:=20.044*.0441;
140
> DEplot3d(f,[c,u,s,mu],t=0..300,{[0,-.0004,.0015,0,-.02]} ,scene=[c,u,mu],stepsize=. 1);#
> DEplot3d(f,[c,u,s,mu],t=0..300,{[0,-.0004,.0015,0,-.02]} ,scene=[c,t,mu],stepsize=. 1);#
> c2:=20.044*.0442;
> DEplot3d(f,[c,u,s,mu],t=0..300,{[0,-.0004,.0015,0,-.02] },scene=[c,t,mu],stepsize=.1);#
> c2:=20.044*.0443;
> DEplot3d(f,[c,u,s,mu],t=0..300,{[0,-.0004,.0015,0,-.02] },scene=[c,t,mu],stepsize=. 1);#
> c2:=20.044*.0445;
> DEplot3d(f,[c,u,s,mu],t=0..300,{[0,-.0004,.0015,0,-.02] },scene=[c,t,mu],stepsize=. 1);#
> c2:=20.044*.045;
> DEplot3d(f,[c,u,s,mu],t=0..100,{ [0,-.0004,.0015,0,-.02] },scene=[c,u,mu],stepsize=. 1);#
> DEplot3d(f,[c,u,s,mu],t=0..250,{[0,-.0004,.0015,0,-.02]} ,scene=[c,t,mu],stepsize=. 1);#
> c2:=20.044*.05;
> DEplot3d(f,[c,u,s,mu],t=0..200,{[0,-.0004,.0015,0,-.02] },scene=[c,t,mu],stepsize=. 1);
# Pinpointing Onset of Chaos of A-H Attractor
# by Akwasi Apori
> restart:
> with(DEtools):with(plots):
> f:=[diff(c(t),t)=al*c+a2*u*mu+a3*u,diff(u(t),t)=bl*u+b2*mu*s+b3*s,diff
> (s(t),t)=c1*s+c2*c, diff(mu(t),t)=dl*mu+d2*c];#
> al:=.89928: a2:=-10.66: a3:=-.014928: bl:=-.89929: b2:=1736*.5:
> b3:=2.43: cl:=-.89930: d1:=-.89931*3: d2:=151.7:
# Time series and phase space for A-H attractor are plotted at c2 x 3.1
# where chaos first starts
> c2:=20.044*3.1:
> DEplot3d(f,[c,u,s,mu],t=0.. 100, { [0,8.3e-6,5e-4,.000185,1.4e-3] },scene=
> [c,u,mu],stepsize=. 1,title="c2 x 3. 1");#
> DEplot3d(f,[c,u,s,mu],t=0.. 100, { [0,8.3e-6,5e-4,.000185,1.4e-3]},scene=
> [c,t,c],stepsize=. 1,title="c2 x 3. 1");#
# General Exploration of C2 Parameter Dynamics
# by Akwasi Apori
> restart:
> with(DEtools):with(plots):
> f:=[diff(c(t),t)=al*c+a2*u*mu+a3*u,diff(u(t),t)=bl*u+b2*mu*s+b3*s,diff
> (s(t),t)=c 1*s+c2*c, diff(mu(t),t)=d1 *mu+d2*c];#
> al:=.89928: a2:=-10.66: a3:=-.014928: bl:=-.89929: b2:=1736*.5:
>b3:=2.43: cl:=-.89930: c2:=20.044*.001: d1:=-.89931*3: d2:=151.7:
# Solutions of the A-H equations are solved for a variety of c2 values
> DEplot3d(f,[c,u,s,mu],t=0..100,{ [0,8.3e-6,5e-4,.000185,1.6e-3]},scene=[c,u,mu],stepsize=. 1);#
> al:=.89928: a2:=-10.66: a3:=-.014928: bl:=-.89929: b2:=1736*.5:
> b3:=2.43: cl:=-.89930: c2:=20.044*.01: d1:=-.89931*3: d2:=151.7:
> DEplot3d(f,[c,u,s,mu],t=0.. 100,{ [0,8.3e-6,5e-4,.000185,1.6e-3] },scene=[c,u,mu],stepsize=. 1);#
> al:=.89928: a2:=-10.66: a3:=-.014928: bl:=-.89929: b2:=1736*.5:
> b3:=2.43: cl:=-.89930: c2:=20.044*.1: d1:=-.89931*3: d2:=151.7:
> DEplot3d(f,[c,u,s,mu],t=0..100,{[0,8.3e-6,5e-4,.000185,1.6e-3] },scene=[c,u,mu],stepsize=. 1);#
> al:=.89928: a2:=-10.66: a3:=-.014928: bl:=-.89929: b2:=1736*.5:
> b3:=2.43: cl:=-.89930: c2:=20.044: d1:=-.89931*3: d2:=151.7:
> DEplot3d(f,[c,u,s,mu],t=0..100,{[0,8.3e-6,5e-4,.000185,1.6e-3] },scene=[c,u,mu],stepsize=.1);#
> c2:=20.044* 10:
141
> DEplot3d(f,[c,u,s,mu],t=0.. 100, { [0,8.3e-6,5e-4,.000185,1.6e-3]
},scene=[c,u,mu],stepsize=.1);#
> c2:=20.044*100:
> DEplot3d(f,[c,u,s,mu],t=0..100,{ [0,8.3e-6,5e-4,.000185,1.6e-3] },scene=[c,u,mu],stepsize=.1);#
> c2:=20.044*1000:
> DEplot3d(f,[c,u,s,mu],t=0..2.5,{ [0,8.3e-6,5e-4,.000185,1.6e-3] },scene=[c,u,mu],stepsize=.1);#
> c2:=20.044*2:
> DEplot3d(f,[c,u,s,mu],t=0..100,{ [0,8.3e-6,5e-4,.000185,1.6e-3] },scene=[c,u,mu],stepsize=.1);#
> DEplot3d(f,[c,u,s,mu],t=0..100,{[0,8.3e-6,5e-4,.000185,1.6e-3] },scene=[c,t,mu],stepsize=.1);#
> c2:=20.044*3:
> DEplot3d(f,[c,u,s,mu],t=0..100,{ [0,8.3e-6,5e-4,.000185,1.6e-3] },scene=[c,u,mu],stepsize=.1);#
> DEplot3d(f,[c,u,s,mu],t=0..100,{ [0,8.3e-6,5e-4,.000185,1.6e-3] },scene=[c,t,mu],stepsize=.1);#
> c2:=20.044*4:
> DEplot3d(f,[c,u,s,mu],t=0..100,{[0,8.3e-6,5e-4,.000185,1.6e-3] },scene=[c,u,mu],stepsize=.1);#
> DEplot3d(f,[c,u,s,mu],t=0.. 100, { [0,8.3e-6,5e-4,.000185,1.6e-3] },scene=[c,t,mu],stepsize=. 1);#
> c2:=20.044*5:
> DEplot3d(f,[c,u,s,mu],t=0.. 100, { [0,8.3e-6,5e-4,.000185,1.6e-3] ,scene=[c,u,mu],stepsize=. 1);#
> DEplot3d(f,[c,u,s,mu],t=0.. 100, { [0,8.3e-6,5e-4,.000185,1.6e-3] ,scene= [c,t,mu],stepsize=. 1);#
> c2:=20.044*6:
> DEplot3d(f,[c,u,s,mu],t=0.. 100,{ [0,8.3e-6,5e-4,.000185,1.6e-3] },scene=[c,u,mu],stepsize=.1);#
> DEplot3d(f,[c,u,s,mu],t=0.. 100, { [0,8.3e-6,5e-4,.000185,1.6e-3] },scene=[c,t,mu],stepsize=. 1);#
> c2:=20.044*7:
> DEplot3d(f,[c,u,s,mu],t=0..100,{ [0,8.3e-6,5e-4,.000185,1.6e-3] },scene=[c,u,mu],stepsize=.1);#
> DEplot3d(f,[c,u,s,mu],t=0.. 100, { [0,8.3e-6,5e-4,.000185,1.6e-3]} ,scene=[c,t,mu],stepsize=. 1);#
> c2:=20.044*8:
> DEplot3d(f,[c,u,s,mu],t=0..100, { [0,8.3e-6,5e-4,.000185,1.6e-3] },scene=[c,u,mu],stepsize=. 1);#
> DEplot3d(f,[c,u,s,mu],t=0.. 100,{ [0,8.3e-6,5e-4,.000185,1.6e-3] },scene=[c,t,mu],stepsize=. 1);#
> c2:=20.044*9:
> DEplot3d(f,[c,u,s,mu],t=0.. 100,{ [0,8.3e-6,5e-4,.000185,1.6e-3] },scene=[c,u,mu],stepsize=.1);#
> DEplot3d(f,[c,u,s,mu],t=0..100,{ [0,8.3e-6,5e-4,.000185,1.6e-3] },scene=[c,t,mu],stepsize=. 1);
# Presentation of the 4 different 3-D phase Space Frames
# by Akwasi Apori
> restart:
> with(DEtools):with(plots):
> f:=[diff(c(t),t)=al*c+a2*u*mu+a3*u,diff(u(t),t)=bl*u+b2*mu*s+b3*s,diff
> (s(t),t)=c1*s+c2*c, diff(mu(t),t)=d1*mu+d2*c];
> al:=.89928: a2:=-10.66: a3:=-.014928: bl:=-.89929: b2:=1736*.5:
> b3:=2.43: cl:=-.89930: d1:=-.89931*3: d2:=151.7:
> c2:=20.044* 10:
# use the commands below to plot the different phase space frames by
# changing the 3 variables included in scene
> DEplot3d(f,[c,u,s,mu],t=0.. 100, { [0,8.3e-6,5e-4,.000185,1.4e-3]},scene=
>[c,u,s],stepsize=.05,title="c2 x 10");
> DEplot3d(f, [c,u,s,mu],t=0.. 100, { [0,8.3e-6,5e-4,.000185,1.4e-3]},scene=
> [c,mu,s],stepsize=.05,title="c2 x 10");
> DEplot3d(f,[c,u,s,mu],t=0.. 100,{ [0,8.3e-6,5e-4,.000185,1.4e-3]},scene=
> [u,mu,s],stepsize=.05,title="c2 x 10");
142
# substitute t into the scene to get different views of the time series
> DEplot3d(f,[c,u,s,mu],t=0.. 100,1 [0,8.3e-6,5e-4,.000185,1.4e-3] },scene=
" [u,t,s],stepsize=.05,title="c2 x 10");
" DEplot3d(f,[c,u,s,mu],t=0.. 100,1 [0,8.3e-6,5e-4,.000185,1.4e-3] },scene=
" [c,t,mu],stepsize=.1,title="c2 x 10");
# Long Term Behavior of A-H Attractor Figure 6-10
# by Akwasi Apori
> restart:with(DEtools):with(plots):
> f:=[diff(c(t),t)=al*c+a2*u*mu+a3*u,diff(u(t),t)=bl*u+b2*mu*s+b3*s,diff
" (s(t),t)=c1*s+c2*c, diff(mu(t),t)=dl*mu+d2*c];#
> al:=.89928: a2:=-10.66: a3:=-.014928: bl:=-.89929: b2:=1736*.5:
> b3:=2.43: c1:=-.89930: d1:=-.89931*3: d2:=151.7:
> c2:=20.044* 10:
# The solutions to the A-H equations are plotted over a very long period
# of time on the attractor for the borderline chaos case of c2 x 3.1 and
# the more chaotic case of c x 10
" DEplot3d(f,[c,u,s,mu],t=1.. 1000,{ [0,8.3e-6,5e-4,.000185,1.4e-3]},scene
"
"
"
"
=[c,u,mu],stepsize=.O1,title="C2=10 and t=1 to 1,000");
restart:with(DEtools):with(plots):
f:=[diff(c(t),t)=al*c+a2*u*mu+a3*u,diff(u(t),t)=bl*u+b2*mu*s+b3*s,diff
(s(t),t)=c 1*s+c2*c, diff(mu(t),t)=d1 *mu+d2*c];#
" al:=.89928: a2:=-10.66: a3:=-.014928: bl:=-.89929: b2:=1736*.5:
>b3:=2.43: cl:=-.89930: d1:=-.89931*3: d2:=151.7:
" c2:=20.044*3.1:
" DEplot3d(f,[c,u,s,mu],t=1..1000,1 [0,8.3e-6,5e-4,.000185,1.4e-3]} ,scene
" =[c,u,mu],stepsize=.01,title="C2=3.1 and t=1 to 1,000");#
# A-H Attractor Time Series for c2 x .01 and c2 x 50 Figure 7-2 and 7-3
# by Akwasi Apori
>
>
>
>
restart:
with(DEtools):with(plots):
f:=[diff(c(t),t)=al*c+a2*u*mu+a3*u,diff(u(t),t)=bl*u+b2*mu*s+b3*s,diff
(s(t),t)=c1*s+c2*c, diff(mu(t),t)=dl*mu+d2*c];#
> al:=.89928: a2:=-10.66: a3:=-.014928: bl:=-.89929: b2:=1736*.5:
> b3:=2.43: c1:=-.89930: d1:=-.89931*3: d2:=151.7:
> c2:=20.044*50:
# time series are plotted for c2 x 50 and c2 x .001 by varying the state
# variables in scene
> DEplot3d(f,[c,u,s,mu],t=50..150,1[0,8.3e-6,5e-4,.000185,1.4e-3] },scene
> =[mu,t,mu],stepsize=. 1,title="Time Series for Mu with c2=50");#
> c2:=20.044*.01:
> DEplot3d(f,[c,u,s,mu],t=50..150,1 [0,8.3e-6,5e-4,.000185,1.4e-3]} ,scene
> =[mu,t,mu],stepsize=.1,title="Time Series for Mu with c2=.01");#
> c2:=20.044*50:
> DEplot3d(f,[c,u,s,mu],t=50..83,1[0,8.3e-6,5e-4,.000185,1.4e-3] },scene=
143
> [c,t,c],stepsize=.1,title="Close-Up Time Series for C with c2=50"); #
# Sensitivity of A-H Time Series to Initial Conditions Figure 7-5
# by Akwasi Apori
> restart:with(DEtools):with(plots):
> f:=[diff(c(t),t)=al*c+a2*u*mu+a3*u,diff(u(t),t)=bl*u+b2*mu*s+b3*s,diff
> (s(t),t)=c1*s+c2*c, diff(mu(t),t)=d1*mu+d2*c];#
> al:=.89928: a2:=-10.66: a3:=-.014928: bl:=-.89929: b2:=1736*.5:
> b3:=2.43: cl:=-.89930: d1:=-.89931*3: d2:=151.7:
> c2:=20.044*50:
# Several time series are plotted using different initial conditions
# then they are all displayed on the same graph using display. This is
# done twice, one for a long time period and once for a close up of the
# time scale when trajectories begin to diverge
> Aplot:=DEplot3d(f,[c,u,s,mu],t=13..25,{[0,8.3e-6,5e-4,.000185,1.4e-3]}
>,scene=[mu,t,c],stepsize=.01):
> Bplot:=DEplot3d(f,[c,u,s,mu],t=13..25,{[0,8.3e-6+10e-12,5e-4,.000185,1
> .4e-3] },scene=[mu,t,c],stepsize=.O1,linecolor=red):
> Cplot:=DEplot3d(f,[c,u,s,mu],t=13..25,{[0,8.3e-6,5e-4+10e-10,.000185,1
> .4e-3] },scene=[mu,t,c],stepsize=.O1,linecolor=blue):
> Dplot:=DEplot3d(f,[c,u,s,mu],t=13..25,{[0,8.3e-6,5e-4,.000185+10e-10,1
> .4e-3] },scene=[mu,t,c],stepsize=.O1,linecolor=green):
> Eplot:=DEplot3d(f,[c,u,s,mu],t=13..25,{[0,8.3e-6,5e-4,.000185,1.4e-3+1
> Oe-9] },scene=[mu,t,c],stepsize=.O1,linecolor=grey):
> display([Aplot,Bplot,Cplot,Dplot,Eplot],title="Close-Up of Chaotic
> Time Series of C for C2=50 containing 5 Trajectories Starting with
> Nearly Identical Initial Conditions");
## Sensitivity of A-H Attractor to Initial Conditions Figure 7-6
# by Akwasi Apori
>
>
>
>
restart:
with(DEtools):with(plots):
f:=[diff(c(t),t)=al*c+a2*u*mu+a3*u,diff(u(t),t)=bl*u+b2*mu*s+b3*s,diff
(s(t),t)=c1*s+c2*c, diff(mu(t),t)=dl*mu+d2*c];#
> al:=.89928: a2:=-10.66: a3:=-.014928: bl:=-.89929: b2:=1736*.5:
> b3:=2.43: cl:=-.89930: d1:=-.89931*3: d2:=151.7:
> c2:=20.044*50:
# A baseline attractor is plotted than 5 4 other attractors with
# slightly different initial conditions are plotted in different colors.
# They are all displayed on the same plot.
> Aplot:=DEplot3d(f,[c,u,s,mu],t=0..70,{[0,8.3e-6,5e-4,.000185,1.4e-3] },
> scene=[c,u,mu],stepsize=.01):
> Bplot:=DEplot3d(f,[c,u,s,mu],t=0..70,{[0,8.3e-6+10e-12,5e-4,.000185,1.
> 4e-3] },scene=[c,u,mu],stepsize=.01,linecolor=red):
> Cplot:=DEplot3d(f,[c,u,s,mu],t=O..70,{[0,8.3e-6,5e-4+10e-10,.000185,1.
> 4e-3] },scene=[c,u,mu],stepsize=.O1,linecolor=blue):
> Dplot:=DEplot3d(f,[c,u,s,mu],t=O..70,{ [0,8.3e-6,5e-4,.000185+Oe- 10,1.
> 4e-3] },scene=[c,u,mu],stepsize=.01 ,linecolor=green):
> Eplot:=DEplot3d(f,[c,u,s,mu],t=0..70,{[0,8.3e-6,5e-4,.000185,1.4e-3+10
144
>
>
>
>
e-9]} ,scene=[c,u,mu],stepsize=.O1 ,linecolor=grey):
display([Aplot,Bplot,Cplot,Dplot,Eplot],title="Chaotic Attractor for
C2=50 containing 5 Trajectories Starting with Nearly Identical Initial
Conditions");
# The first 5 and last 5 time steps are plotted on the attractor for the
# different initial conditions
" Fplot:=DEplot3d(f,[c,u,s,mu],t=O..5,{[0,8.3e-6,5e-4,.000185,1.4e-3] },s
" cene=[c,u,mu],stepsize=.01,view=[-0.0005..0.0003,-0.02..0.08,-0.007..O.006]):
" Gplot:=DEplot3d(f,[c,u,s,mu],t=0..5,{[0,8.3e-6+10e-12,5e-4,.000185,1.4e-3] },scene=[c,u,mu],step
"
"
"
"
"
"
"
size=.01,linecolor=red,view=[-0.0005..0.0003,-0.02..0.08,-0.007..0.006]):
Hplot:=DEplot3d(f,[c,u,s,mu],t=0..5,{[0,8.3e-6,5e-4+10e-10,.000185,1.4
e-3] },scene=[c,u,mu],stepsize=.O1,linecolor=blue,view=[-0.0005..0.0003,-O.02..0.08,-0.007..O.006]):
Iplot:=DEplot3d(f,[c,u,s,mu],t=O..5,{[0,8.3e-6,5e-4,.000185+10e-10,1.4
e-3] },scene=[c,u,mu],stepsize=.01,linecolor=green,view=[-0.0005..0.0003,-0.02..0.08,-0.007..0.006]):
Jplot:=DEplot3d(f,[c,u,s,mu],t=0..5,{[0,8.3e-6,5e-4,.000185,1.4e-3+10e
-9]},scene= [c,u,mu],stepsize=.O1 ,linecolor=grey,view=[-0.0005..0.0003,-0.02..0.08,-0.007..0.006]):
" Kplot:=DEplot3d(f,[c,u,s,mu],t=65..70,{[0,8.3e-6,5e-4,.000185,1.4e-3] },scene=[c,u,mu],stepsize=.01):
" Lplot:=DEplot3d(f,[c,u,s,mu],t=65..70,{[0,8.3e-6+10e-12,5e-4,.000185,1
S.4e-3] },scene=[c,u,mu],stepsize=.01 ,linecolor=red):
" Mplot:=DEplot3d(f,[c,u,s,mu],t=65..70,{[0,8.3e-6,5e-4+10e-10,.000185,1
S.4e-3] },scene=[c,u,mu],stepsize=.01 ,linecolor=blue):
" Nplot:=DEplot3d(f,[c,u,s,mu],t=65..70,{[0,8.3e-6,5e-4,.000185+10e-10,1
S.4e-3] },scene=[c,u,mu],stepsize=.01 ,linecolor=green):
" Oplot:=DEplot3d(f,[c,u,s,mu],t=65..70,{[0,8.3e-6,5e-4,.000185,1.4e-3+1
" Oe-9]} ,scene=[c,u,mu],stepsize=.01 ,linecolor=grey):
" display([Fplot,Gplot,Hplot,Iplot,Jplot],title="Startpoints of
Trajectories for Chaotic Attractor when C2=50");
" display([Fplot,Gplot,Hplot,Iplot,Jplot],title="Startpoints of
Trajectories for Chaotic Attractor when C2=50");
145
146
Appendix C: Governing Equations Tested for A-H
Sickle Cell Model
C. 1 Lagrangian Formulation of A-H Model
The original analysis performed by the author for sickle cell flow and crisis was in the Lagrangian
frame. The flow in the capillary was modeled as a function of distance along the capillary x. The
Lagrangian model never proved to be consistent with the obvious changes in blood flow that occur
over time when a sickle cell patient has a crisis. The Lagrangian formulation of the problem was
limited by the inability to model the change of the solution over time. The Lagrangian sickle cell
model is presented in the equations below.
Substituting and integrating Cima equations (3)-(8) gives[1]:
dc _
e___(C._1)
dx
n-1
u(c) 1+A
c
2
(1+ Bc" .
where
B = k'
and
A = nNk'
(C.2)
(C.3)
A and B are the known Hill constants. To solve for Jc, the oxygen flux in the tissue, several
approaches were taken. First, the oxygen concentration ct was assumed to decrease exponentially
with the radius of the Krogh cylinder.
c,=
-(rR)(C.4)
t=
ce-ez(r-Re)
This approach proved not to be as accurate as desired. In another attempt, equation (12) from
Berger and King was used to solve for ct in an iterative process that required substituting into (7)
of Cima[3,1]. For this method the approximation of ct was described by the following equation:
c = A (r2 - R)
--
R In f
+
c -A
- 1
(C.5)
where Dt is the diffusivity of oxygen in tissue, Rt and R, are the radius of the capillary and Krogh
tissue cylinder and at and ab are the oxygen solubility constants in tissue and blood. (C.5) was
very complex and did not accurately represent ct.
147
The final approximation used for the capillary oxygen concentration was a sinusoidal decay. It
proved to model ct accurately with a simple equation.
(r - Rccs
c, = c(x)cos (R,-Re
)
E
Z + 2)+
2
(C.6)
This led to the final expression for dc/dx below.
dc
dx
EFc
_
(C.7)
Cn-1
1+A
c
(1+Bc")
D
RC t
and
where
E =-
F
=2(R,-R)
(C.8) (C.9)
Obtaining (C.7) allowed the problem to be formulated as a system of three equations based upon
Cima equations (1), (2a), (2b), and (3)[1].
du
d _ p1 (1 PIR \kA1
A(l - k)S (-k)d
dxp5
dx
E;j
(C. 10)
ds dx
(C.11)
dx
EFc
u(x+ P)
Q=
1
dc
_
(C.12)
where
1-a
and
Z 1-
(C.13) (C.14)
c is an empirical constant used to describe the oxyrheology of sickle cells at low c. P is a constant
found by approximating (C.7) with a simple function that appropriately described oxygen concentration levels in the capillary.
148
This was the final set of equations found for the Lagrangian model of sickle cell disease. The
equations were accurate in describing certain aspects of sickle cell flow. They produced profiles
for flow down the capillary with decreasing velocity and oxygen concentration similar to those
found by Cima et al. However, in the suspected case of sickle cell crises, the solutions to these
equations showed no change in dynamics.
Dimensional analysis was used in an attempt to simplify (C. 10)-(C. 12). The equations were too
complex and it did not provide any significant results. (C.10)-(C.12) modeled the sickle cell problem in a very limited way. In order to better study the dynamics of crises, the state variables
needed to be described in a manner that accurately represented the physics of microcirculatory
flow over time. Therefore, the next step was to take averaged values of state variables over the
length of the capillary and proceed with an Eulerian frame for modeling sickle cell flow in the
microcirculation.
C.2 Eulerian Formulation of A-H Model
There were several Eulerian formulations for the governing equations. Equations were developed
and tested to see if they accurately described the physics of sickle cell flown as presented in
Cima's Lagrangian model[1]. Listed below are the various equations explored for modeling
sickle cell blood flow.
1
1
a
= aic + a2u + a3 s
(C.15)
= blus-I + b 2 g~1
(C.16)
a
= bIus-I + b29~1
(C.19)
(C.17)
i
= CIS + c2uc
(C.20)
= CIS + c2uc
=
ac + a2 u + a3s
(C.21)
p = dIsu
1
1
a c + a 2 u + a 3s
u= blu + b 3As-b2
=
cIs+ c 2uc
=di p + d2c
(C.18)
2
+ a 4p
=
(C.22)
a
(C.23)
ac + a2pu
= bju+b
3gsb2
(C.26)
(C.27)
(C.24)
= CIS +c 2uc
(C.28)
(C.25)
p= di pL + d2c
(C.29)
149
d = aic+a 2 g u+a
a
=
3pg
biu+b 3psb2
(C.30)
6 = a 1 c+a
(C.31)
a
=
3
(C.34)
9
blu+b 3 gsb2
(C.32)
= cis+c2uc 2
dip +nd 2 c
(C.33)
=
d = aic+a2 ug + a 9
3
(C.38)
S= a 1 c + a 2u
(C.39)
d = blu + b 2 gs + b 3 s
S= cis + C UC
2
=
a
=
S =
biu + b 2gsb2
cis + C 2UC
di
(C.36)
(C.37)
(C.41)
(C.42)
(C.43)
2c
(C.44)
+ a 3g
(C.45)
e = a c + a39
(C.49)
= blu + b 2 gs + b 3 s
(C.46)
a
(C.50)
e = a 1c + a2 u
s = cIs + c 2 c
+ c 3c
= dig + d 2 c
6 = a 1 c+a
a
+ a 3 9i
= c 1 s+C 2C
(C.40)
A = d 1 R+d
a
+nd 2c
(C.35)
3 9+a
4 cu
blu + b 2 gs + b 3 s
= cis + C 2 c
=
(C.48)
= blu + b 2 gs + b 3 s
S = CIS + C C
2
(C.47)
=
di +d2c
(C.51)
(C.52)
(C.53)
(C.54)
(C.55)
The majority of these equations were ruled out because they did not accurately show a change in
state variables when parameters were varied to simulate the onset of crises. Many of the equations had solutions that were attracting or repelling fixed points and spirals for all parameter
ranges. In addition to (C.41)-(C.44) which were analyzed in this thesis, (C.45)-(C.48) also exhibited chaos as parameters were varied. Though chaos was not found in any of the other equations,
it remains inconclusive whether the existence of chaos is possible in those equations.
150
References
[1]
Cima, L.G., Discher, D.E., Tong, J., & Williams, M.C. (1994). A Hydrodynamic
Interpretationof Crisis in Sickle Cell Anemia. Microvascular Research 47, 41-54.
[2]
Fitz-Gerald, J. M. (1969). Mechanics of red-cell motion through very narrowcapillaries.
Proc. R. Soc. Lond. B. 174, 193-227.
[3]
Berger, S. A., & King, W. S. (1980). The flow of sickle cell blood in the capillaries.
Biophysical J. 29, 119-148.
[4]
Bretherton, F.P. (1961). J. Fluid Mech. 10, 166.
[5]
Beutler, Ernest. Disordersof Hemoglobin. Harrison's Principles of Internal Medicine.
14th ed. Vol. I. 107. McGraw-Hill. 1998.
[6]
Hoofd, L. Updating the Krogh Model-Assumptions and Extensions. Oxygen Transport
In Biological Systems: Modelling of pathways from Environment to Cell. ed: Egginton,
S., Ross, H.F. Cambridge University Press. New York, NY. 1992.
[7]
Williams, Garnett P. Chaos Theory Tamed. Joseph Henry Press. Washington, DC.
1997.
[8]
Discussion with Professor Rodolfo Rosales, MIT Dept. of Applied Mathematics. 5/11/01.
[9]
Strogatz, Steven H. Nonlinear Dynamics and Chaos. Addison-Wesley Publishing
Company. Reading, MA. 1998.
[10]
Cambel, A. B. Applied Chaos Theory: A Paradigm for complexity. Academic Press.
San Diego, CA. 1993.
[11]
Herbert, D.H., Croft, P., Silver, D.S., Williams, S.G., & Woodall, M. Chaos and the
Changing Nature of Science and Medicine. American Institute of Physics Press.
Woodbury, New York. 1996.
[12]
Lighthill, M.J. (1958). Pressure-forcingof tightly fitting pellets alongfluid-filled elastic
tubes. J. Fluid Mech. 34, 113-143.
[13]
Bretscher, Otto. Linear Algebra With Applications. Prentice Hall. Upper Saddle River,
NJ. 1997.
[14]
Enns, R. H., Mcguire, G. C. Nonlinear Physics with Maple for Scientists and Engineers.
Birkhauser. Boston, MA. 2000.
151
[15]
Maple V. 6.0 Software. Help Sections. Waterloo Maple Inc. 2000.
[16]
Parker, T.S., Chua, L.O. Practical Numerical Algorithms for Chaotic Systems. SpringerVerlag. New York, N.Y. 1989.
[17]
Discussion with Dr. Kwaku Ohene-Frempong, Director of the Comprehensive Sickle Cell
Center at the Children's Hospital of Philadelphia. 6/4/01.
152