“The
The Cardio
Cardio‐Respiratory
Respiratory Human System: Human System:
a simulation study”
“P
“Process System Engineering in Human Physiology”
S t
E i
i i H
Ph i l
”
Elisa Montain, Anibal Blanco, Alberto Bandoni
g
g,
Q (
)
Pilot Plant of Chemical Engineering, PLAPIQUI (UNS‐CONICET)
Bahía Blanca, Argentina
PASI 2011
PASI 2011 Process Modeling and Optimization for Energy and Sustainability
S
Saturday, July 23, 2011, Angra dos Reis, RJ, Brazil
d J l 23 2011 A
d R i RJ B il
PASI 2011 - A. Bandoni
1
Background







The cardiovascular system (CVS) is responsible for supplying oxygen
and nutrients to tissues and organs
organs..
CV diseases are a major cause of death in humans
humans..
Many experimental studies have studied the mechanisms and therapy of
the CV diseases
Together with experimental approaches, mathematical modeling has
become a popular way to analyze the CVS
CVS..
Many models have been published since the preliminary and basic
model of Godins in 1959
Approaches include
include:: hemodynamic models of the vascular system,
distributed impedance and pulmonary arterial stress, lumped parameter
models of the integrated CVS, hemodynamic monitoring models, etc..
etc..
In the last few
fe years
ears there have
ha e been important developments
de elopments in
integrated lumped parameter models of the circulatory and nervous
control systems.
systems.
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2
Motivation
 Assistance
A i t
iin th
the d
decision
i i making
ki off medical
di l practice
ti
 Diagnosis of diseases of the CVS (coronary arteries and heart muscles
dysfunctions, valvular disorders and pulmonary disease.

Comprehend the math. concepts and terms defining how CVS system behaves.

To teach about the complex interactions of the cardiovascular system.
system.

H l tto vascular
Help
l surgeons iin treatment
t t
t planning
l
i and
d tto engineers
i
iin designing
d i i
better medical devices.
devices.

A promising integration strategy involves the personalization of mathematical
models based on biophysical measurements.
measurements.

Analysis of the hemodynamics (blood flow dynamics) of the CVS.

Capacity to locate factors that are not directly observable . Key role in the
measurement of pump efficiency and tissue stress, to assist treatment decisions.
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Motivation
 Anesthesia control and drug delivery control:
control:

Control of patient physiological variables during intensive care is achieved through
drug
g delivery.
delivery
y.

Drug delivery process depends on the value of the physiological variable under
control and on the patient's
patient s condition

Drugs such sodium nitroprusside (SNP) and dopamine (DP) are normally used for
regulation of Media Arterial Pressure (MAP) or Cardiac Output (CO)
(CO)..

Doctors use their discretion to regulate variables that are difficult to quantify in
practice or inferred from other measurements and patient responses to certain
surgical procedures.
procedures.

Currently, the drug infusion is done manually or by programmable pumps.
pumps. The
professional is responsible for monitoring the controlled variable (MAP, CO) and
g delivery
y according
g to the measurement.
the drug
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Objectives
 Development of an integrated distributed parameter model of
the human cardio respiratory system.
 Development of a computational tool to help physicians in the
diagnosis of various heart diseases
diseases.
 Studyy of the drug
g delivery
y ((SNP, DP, etc.))
The developed
p model contain the following
g sub-models:
 Circulatory system
 Baroreceptors
 Respiratory system
 Gas transport and distribution in organs
 Pharmacological effect of drugs on the hemodynamic variables.
variables
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Anatomy and
Ph i l
Physiology
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The Cardiovascular System
The Cardiovascular System:
It consists of:
 The heart, which is a muscular pumping device
 A closed system of vessels (arteries, veins, and
capillaries).
The Heart
 The heart is a hollow muscular pump that provides the force necessary to
circulate
i l t the
th blood
bl d to
t allll the
th tissues
ti
i the
in
th body
b d through
th
h blood
bl d vessels.
l
 The normal adult heart pumps about 5 liters of blood every minute
throughout life.
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Heart Anatomy
Aorta
Superior
vena cava
Pulmonary
truck
Pulmonary
vein
Left
Atrium
Pulmonary
valve
Right
atrium
Mitral
valve
Tricuspid
valve
Aortic
valve
Left
Ventricle
Right
ventricle
Inferior vena
cava
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Functions of the Heart

Generates blood pressure

Routes blood
 Heart separates pulmonary and systemic circulation

Ensures oneone-way blood flow
 Heart valves ensure oneone-way
y flow

Regulates blood supply
 Changes
Ch
in
i contraction
t ti rate
t and
d fforce match
t h bl
blood
dd
delivery
li
tto
changing metabolic needs
 Most healthy
y people
p p can increase cardiac output
p by
y 300–
300–500%

Heart failure is the inability of the heart to provide enough blood flow to
maintain
i t i normall metabolism
t b li
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The Chambers

Separated by
 Interatrial Septum
 Interventricular Septum

Right Atrium
 Blood
Bl d ffrom S
Superior
i and
d iinferior
f i venae cavae and
d th
the coronary sinus
i
 Right Ventricle
 Receives blood from the right atrium via the right AV valve
valve, tricuspid
valve
 Thin wall
 Left Atrium
 Receives blood from R and L Pulmonary Veins
 Left Ventricle
 Receives blood from the Left AV valve
 Thick wall
 Pumps to body via Aortic Semilunar Valve
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The Valves

Two types off valves:
T
l
k
keep
the
h blood
bl d
flowing in the correct direction.

Between atria and ventricles:
called atrioventricular valves (also
called cuspid
p valves))

Bases of the large vessels leaving
the ventricles: called semilunar
valves.

When the ventricles contract,
contract atrioventricular valves close to prevent
blood from flowing back into the atria.

When the ventricles relax,
relax semilunar valves close to prevent blood from
flowing back into the ventricles.

Vales
V
l close
l
passively
i l under
d blood
bl d pressure. Responsible
R
ibl for
f the
th heart
h t
sounds.
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Circulatory System
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Circulatory System


Deoxygenated
D
t d bl
blood
d returns
t
to
t
the heart via the superior and
inferior vena cava, enters the
right atrium,
atrium passes into the right
ventricle, and from here it is
ejected to the pulmonary artery.
Oxygenated blood returning from
the lungs enters the left atrium via
the pulmonary veins, passes into
the left ventricle, and is then
ejected to the aorta.
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Blood flow pattern through the heart
1.Blood
1.
Blood enters right atrium via the superior
and inferior venae cavae
2.Passes
2.
Passes tricuspid valve into right ventricle
3 Leaves by passing pulmonary semilunar
3.Leaves
3.
valves into pulmonary trunk and to the lungs
to be oxygenated
4.Returns
4.
Returns from the lung by way of pulmonary
veins into the left atrium
5.From
5.
From left atrium past bicuspid valve into left
ventricle
6.Leaves
6.
Leaves left ventricle past aortic semilunar
valves into aorta
7.Distributed
7.
Distributed to rest of the body
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Blood flow pattern through the heart
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Blood Vessels




Blood vessels are divided into a pulmonary circuit and systemic circuit.
Artery - vessel that carries blood away from the heart. Usually
oxygenated. Exception, pulmonary artery.
Vein - vessel that carries blood towards the heart. Usually
deoxygenated. Exception pulmonary veins
Capillary - a small blood vessel that allow diffusion of gases, nutrients
and wastes between plasma and interstitial fluid.
Systemic vessels
Transport blood through the body part from left ventricle and back to
right
g atrium
Pulmonary vessels
Transport blood from right ventricle through lungs and back to left
atrium
Blood vessels and heart are regulated to ensure blood pressure is
high enough for blood flow to meet metabolic needs of tissues
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The Real Thing
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The Real Thing
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History
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Mathematical Modelling in Physiology

With mathematical models it is possible to simulate almost any kind of
phenomena in nature on a computer.

This is a scientific practice of modern science and engineering
(biology, physiology, medicine, climate research,
research, ecology, physics,
chemistry etc.)
chemistry,
etc )

Mathematical modeling in medicine and biology has become so important
that this type of research now has its own name: in silico

Mathematical modeling undoubtedly will become the paradigm of scientific
and medical research in the twenty‐
twenty‐first century.

In research
research, the ultimate goal is mechanisms‐
mechanisms‐based models
models, but in reality
models are more often used in a detective‐
detective‐like way to investigate the
consequences of different hypotheses.

The mathematics modeling is used as a microscope to unveil information
2011 - A. Bandoni
20
about reality, that is otherwisePASI
inaccessible
Heart and Blood Circulation Research History

Since the dawn of civilization,
civilization man has been concerned with the
understanding of living things.

In
I one off the
th mostt ancient
i t medical
di l ttreatises
ti
((N
(Nei
Nei
N i Jing,
Ji
2697
2697--2597 BC)
BC),
blood is mentioned as originating in the heart and distributed in order to
return to the starting point.

Despite widespread knowledge of the anatomy of blood vessels, Greeks
were unable to find the start of blood circulation by not knowing the
principle of conservation of mass.

The Western world had to wait for William Harvey (1578(1578-1657) to establish
the concept of circulation.
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History

Di
Discovery
off the
th closed
l
d circulation
i l ti off blood
bl d by
b William
Willi
H
Harvey
(1578‐‐1657). "De Motu Cordis" ("On the Motion of the
(1578
Heart and Blood“. Frankfurt, 1628)
Stroke volume is 70 ml. per beat and Heart beats 72 times per minute, therefore Cardiac Output
should be 7.258 liters per day

Before 1628, the Galenic view of the body prevailed and the concept of
blood circulation was not imaginable.

Galen or Galenius (Greek physician, II century AD), spent most of his
lifetime observing the human body and its functioning.

Galen believed that the heart acted not as a pump, but rather that it sucked
blood from the veins, that blood flowed from one ventricle to the other of
the heart through a system of tiny pores of the septum
septum.

Using a simple model, Harvey showed that the amount of blood leaving the
h t iin a minute
heart
i t could
ld nott conceivably
i bl b
be absorbed
b b db
by th
the b
body
d and
d
continually replaced by blood made in the liver from chyle
chyle..
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History

Consequently, this model based evidence established the concept that
blood must constantly move in a closed circuit, otherwise the arteries and
the body would explode under the pressure.

This was discovered about 8 years before the light microscope.

The concept or method of using mathematical modeling, as a tool for
making an inaccessible system accessible or an invisible system visible,
is therefore
f
being coined as “the
“
mathematical microscope” in honor off
William Harvey.
The mathematical microscope
p
Ottesen (2011)
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Th Windkessel
The
Wi dk
l
Effect
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The Windkessel Effect

The windkessel effect is use to describe:
• Load faced by the heart in pumping blood through pulmonary or
systemic arterial system.
system
• Relation between blood pressure and blood flow in the aorta or
pulmonary artery

Characteristic parameters of CVS such us compliance and peripheral
resistance can be described in terms of the Windkessel models, which is
useful
f l iin:
• Quantifying the effects of vasodilator or vasoconstrictor drugs.
• The development and operation of mechanical heart and heart-lung
machines.
hi

Windkessel: a german word that can be translated as air (wind) chamber
Windkessel:
(kessel
kkessel).
l).
l)
)

First description
p
by
y German p
physiologist
y
g Otto Frank in 1899.
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The Windkessel Effect

Heart and systemic arterial system similar to a closed hydraulic circuit
comprised of a water pump connected to a chamber.

The circuit is filled with water except for a pocket of air in the chamber
Arterial
compliance
Peripheral
P
i h l
ressistance

As water is pumped into the chamber, the water both compresses the air in
the pocket and pushes water out of the chamber
chamber. .
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The Windkessel Effect

The compressibility of the air in the pocket simulates the elasticit
elasticity
y and
extensibility of the major artery, as blood is pumped into it by the heart
ventricle.

This effect is commonly referred to as arterial compliance
compliance..

The resistance water encounters while leaving the Windkessel, simulates
the resistance to flow encountered by the blood as it flows through the
arterial tree from the major arteries
arteries, to minor arteries
arteries, to arterioles
arterioles, and to
capillaries, due to decreasing vessel diameter.

Thi resistance
This
i t
to
t flow
fl
is
i commonly
l referred
f
d to
t as peripheral
i h l resistance.
resistance
i t
.
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The Windkessel Effect
Hypotheses:
• Unsteady flow.
• The pressure diff. across the resistance is a linear function of the flow
rate
t
• The working fluid is incompressible (constant air pressure to volume
ratio)
• The flow is constant throughout the ejection phase.
The Windkessel 2-elements considers only
y the arterial compliance
p
((C)) and
the peripheral resistance (R).
Symbols:
P : pressure generated by the heart (N.m-2) [mmHg]
Q : blood flow in the aorta (m3.s-1) [l.mn-1]
R : peripheral resistance (N
(N.s.m
s m-55) [dyne
[dyne. ss.cm
cm-55]
C : arterial or systemic compliance (m5.N-1) [ml.mmHg-1]
t : time [(s)
T : period
i d ((s))
Ts: ejection time (s)
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The Windkessel Effect
Theoretical development of the Windkessel effect
air
P(t)
Q
Ts
V(t)
R
Q(t)
Q1(t)
Pcv
t
T
Schematic representation
of a chamber
Systolic phase:
valve in open
position
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Diastole phase:
valve in close
position
29
The Windkessel Effect
I - Systolic
S t li phase
h
( l iin open position)
(valve
iti ) 0  t  T
s
Conservation of mass:
Thus:
Q  Q1 
dV
dt
Qin  Qout  Qcc
Qcc: flow to the compliance chamber
Pcv : central venus pressure: (Pcv<< P)
(Pcv≅5 mmHg vs. P≅100 mmHg ])
Hyp.4: Q = Cte. throughout the systolic phase, thus:
Therefore:
Then:
Q
P dV P dV dP

 
.
R dt R dP dt
Q (t ) 
P (t )
dP(t )
 C.
R
dt
P  Pcv  R.Q1
Compliance (C)
or
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dP (t ) P(t ) Q (t )


dt
R.C
C
30
The Windkessel Effect
Solution of the differential equation
a) Particular solution (Q = Cte.=0)
P(t )  1. exp( 
t
)
R.C
b) Method of variation of parameter ( α1=α1(t) )
1 
t 
t  Q
d 
(
).
exp(

)
(
).
exp(
) 


t
t


 1

 1
dt 
R.C  R.C 
R.C  C

t
t d1 (t )
t  Q
1
1 

t
.1 (t ). exp( 
)  exp( 
)

(
).
exp(

) 
 1
R.C
R.C
R.C
dt
R.C 
R.C  C
Hence:
d1 (t ) Q
t
 . eexp(
p( 
)
dt
C
R.C
Then: 1 (t )  R.Q. exp( 
PASI 2011 - A. Bandoni
t
)  2
R.C
31
The Windkessel Effect
c) The general solution for systolic phase is
t
t


Ps (t )   R.Q. exp( 
)   2 . exp( 
)
R.C
R.C


To determine α2 we can use initial condition P(t=0)=P0 , then α2 = P0-R.Q
P(t  0)  P0
  2  P0  R.Q
Finally, the pressure waveform for the systolic phase can be written as
t
Ps (t )  R.Q  P0  R.Q . exp( 
)
R.C
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The Windkessel Effect
I – Diastolic
Di t li phase
h
( l iin close
(valve
l
position)
iti )
Ts  t  T
air
Following similar reasoning but with Q=Cte.=0
V(t)
P(t)
dP P Q


dt R.C C
Q1(t)
With initial condition: P(t=Ts)= Ps(Ts), the solution to the differential equation is:
P(t )   3. exp(
p( 
t
)
R.C
where


 3  P0   exp(
p( 
t

)  1. R.Q.
R.C

Fi ll the
Finally,
th pressure waveform
f
for
f the
th di
diastolic
t li phase
h
can be
b written
itt as:
t
t




Pd (t )   P0   exp(
p( 
)  1.R.Q . exp(
p( 
)
R.C
R.C




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The Windkessel Effect
C
Complete
l t model
d l
Systolic Phase
0  t  Ts
air
t
Ps (t )  R.Q  P0  R.Q . exp( 
)
R.C
Diastolic Phase
V(t)
P(t)
Q1(t)
Ts  t  T
air
t
t




Pd (t )   P0   exp( 
)  1.R.Q . exp( 
)
R.C
R.C




Given:
Q , R , C , T , Ts and P 0 ( data ) or
PASI 2011 - A. Bandoni
V(t)
P(t)
Q1(t)
Ts


)  1
 exp(
R.C

P0  R.Q. 
T


)  1
 exp(
R.C

34
The Windkessel Effect
The term
Th
t
R C it iis crucial
R.C
i l in
i the
th 2-W
2 W because
b
it determine
d t
i the
th “speed”
“
d”
of the exponential decay. This product is called the “characteristic
time”, called 
P
P
R.Q
P0
Case:
 0
t
Case:
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 
t
35
Case:
The Windkessel Effect
Hypertension: Ps > 140 mmHg
  0,   
Pd > 90 mmHg
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36
The Windkessel Effect
The electrical circuit equivalence
 Basic equation
q
of a 2-element Winkessel model:
 Electric circuit of 2 passive elements:
I(t)
I3
I2
Q (t ) 
P (t )
dP(t )
 C.
R
dt
I(t) : electrical
l t i l currentt
E(t) : electrical potential
C : capacitance of the capacitor
R : resistance of the resistor
From the Ohm and Kirchhoff laws
E(t)
C
R
I (t ) 
E (t )
dE (t )
 C.
R
dt
I(t) ≡ Q(t) (blood flow)
E(t) ≡ P(t) (arterial blood pressure)
C ≡ C (arterial compliance)
R ≡ R (peripheral resistance)
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37
The Windkessel Effect
The 3-element Windkessel model
I(t)
E(t)
R2
C
R
1
I(t)) ≡ Q(t) (blood
I(
(bl d flow)
fl )
E(t) ≡ P(t) (arterial blood pressure)
C ≡ C (arterial compliance)
R1 ≡ R1 (peripheral resistance
(syst. and pulm.circuits))
R2 ≡ R2 ((resistance of valves
(aortic and pulmonary))

R 
dE (t ) P(t )
dP (t )
1  1 . I (t )  C.R1.

 C.
dt
R2
dt
 R2 
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38
The Windkessel Effect
The 4-element Windkessel model
I(t)
E(t)
E(t)
R2
C
L
R1
I(t) ≡ Q(t) (blood flow)
E(t) ≡ P(t) (arterial blood pressure)
C ≡ C (arterial compliance)
R1 ≡ R1 (peripheral resistance
(syst. and pulm.circuits))
R2 ≡ R2 (resistance of valves
(aortic and p
pulmonary))
lmonar ))
L ≡ L (inertia of the blood circulation)


R1 
L  dE (t )
d 2 E (t ) P (t )
dP (t )
1  . I (t )   R1.C  .
 L.C.


C
.
R2  dt
dt 2
R2
dt
 R2 

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39
Compartment
M d l
Models
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40
Compartment Models

They are used to describe transport material in biological sciences

A compartment model contains a certain number of compartments, each
one with a well mixed material

Compartments exchange material following certain rules

Material can be stored in the boxes and transported between them

Every compartment has a number of connections entering and leaving it.

Material can be added from the outside, can be removed or transported.
Source
Drain
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Compartment Models

Material represent the amount of something that we wish to account for

To account for the material, the models must fulfill certain conservation
laws.

Conservations laws state that the difference between input and output
flows amounts how much will be stored.

A compartment model can also represent:
 Ecological systems (material could be energy and the compartment
different species of animals or plants)
 Physiologic system (material could be oxygen and compartment de
organs)

Compartment can not be thought as independent. Flow in and out may
depend on the compartment volume

Inflow to compartment may depend of outflow of other compartment.
PASI 2011 - A. Bandoni
42
Compartment Models

State variables depend on each other and on the state of the system as a
whole.

The transport in and out is characterized by the flows velocities.

Limitations of the compartment model
•
Is the system closed.
closed. Equation of conservation of mass is correct
only if all material added or removed is included in the model. There
is some lost of detailed information
information.
•
Homogeneity assumption.
assumption. Not always it is possible to keep this
assumption. Then more compartments are needed but also more
information it is required
required.
•
Accuracy of the balance equation.
equation. In real physiological system
typically some mass balance are know and other are not.
•
R l
Relevance
off th
the mass balance.
balance
b l
. Not
N t allll systems
t
can be
b described
d
ib d
in terms of mass balances.
•
Sensitivity analysis.
analysis. Initial conditions and model parameters are not
always known precisely.
PASI 2011 - A. Bandoni
43
Mathematical
M d l
Models
Cardiovascular, Respiratory
and Pharmacodynamic
PASI 2011 - A. Bandoni
44
Human Circulatory System Model

The historical fascination of the heart has lasted for many centuries and
continues to attract considerable attention both theoretically and clinically.
clinically.

To develop a physiologically founded model of the heart and the
vasculature, it is essential to have a good model of the human short term
press re control represented by
pressure
b the baroreceptor mechanism
mechanism..

Using a lumped parameter compartment model, the entire human
cardiovascular system may be described as a network of compliances,
resistances and inductances not reflecting anatomical properties
properties..

Although strikingly simple, the model gives a very good description of the
input impedance of the arterial system.
system.

Such models are valuable tools for understanding cardiovascular diseases
(hypertension weak and enlarged heart,
(hypertension,
heart hemorrhages,
hemorrhages etc
etc..)
PASI 2011 - A. Bandoni
45
Human Circulatory System Model

Models facilitates getting new insight into cardiovascular functions and the
interaction with other system (central nervous system, respiratory systems,
etc..)
etc

This type of models can be reliable and stable, simply enough to run in real
time..
time
ti

Lumped cardiovascular models are divided into pulsatile and non
non--pulsatile
pulsatile..

In the pulsatile case, the heart functioning is guided by a time
time--varying
elastance function
function..

A lumped pulsatile cardiovascular model that embraces principal features
of the human circulation.
circulation.
PASI 2011 - A. Bandoni
46
Human Circulatory System Model

Lumped cardiovascular models are divided into pulsatile and non
non--pulsatile
pulsatile..

I the
In
th pulsatile
l til case, the
th heart
h t functioning
f
ti i
iis guided
id d by
b a time
ti -varying
timei
elastance function
function..

A lumped pulsatile cardiovascular model that embraces principal features
of the human circulation.
circulation.
PASI 2011 - A. Bandoni
47
Human Circulatory System Model
Pulmonar
circulation
Ap3
Ap2
Vp1
Ap1
Vp2
RV
LA
Heart
LV
RA
Vs2
As1
Vs1
As2
Systemic
circulation
As3
PASI 2011 - A. Bandoni
48
Human Circulatory System Model
Pp3
Vp3
Qp2
Cp2
Rp2
Pp2
Vp2
Cp3
Rp3
Ap3
Qp3
Ap2
Ql1
Qp1
Cp1
Rp1
Lp1
Eminrv
Emaxrv
Erv(t)
Lrv
Pra
Vra
Pp11
P
Vp1
Ap1
Qrv
Era
Rra
Lra
Vp2
PV
TV
Qra
Qla
RA
AV
Qv2
Cv2
Rv2
Lv2
Pv2
Vv2
Cv1
Rv1
Pv1
Vv1
Cl2
Rl2
Ll2
Pla
Vla
Ela
Rla
Lla
Plv
Vlv
Eminlv
Elv(t)
Emaxlv
Llv
MV
LV
Qlv
Pa1
Va1
As1
Vs2
Qa1
Qv1
As2
Vs1
Qa3
As3
Pa3
Va3
Cl2
Rl2
Pl2
Vl2
Ql2
LA
RV
Prv
Vrv
Pl1
Vl1
Vp1
Pa2
Va2
Ca1
Ra1
La1
Ca2
Ra2
Qa2
Ca3
Ra3
PASI 2011 - A. Bandoni
49
Human Circulatory System Model
Model of a typical compartment (chamber) of the hemodynamic system
V0 : volumen at p=0
R : ressistance
L : inertia
C : compliance
li
Blood
input
pi
R
Qin
pi
p0
Qin
Qout
L
Blood
output
p0
C
V0
Qout
PASI 2011 - A. Bandoni
Hemodynamic
y
element of a
blood chamber
Equivalence
with an electric
circuit
50
Circulatory System Model (Ottesen et al., 2003)
• Heart Model
o Heart
H t ititself:
lf 4 chambers
h b
(2 atria
t i and
d 2 ventricles)
ti l )
o Vascular part
 Systemic part: 5 chambers (systemic arteries and veins)
 Pulmonary part: 5 chambers (arteries and veins)
• Baroreceptor Model
o Chronotropic effect (on heart rate)
o Inotropic effect (on the cardiac contractility)
o Vascular
V
l effect
ff t (on
( arteries
t i and
d veins)
i )
Respiratory System Model (Christiansen and Dræby, 1996)
• Lung Model
o Upper respiratory tracks: 1 chamber
o Alveoli: 1 chamber
• Gas Transport in Blood Model (O2, CO2, Anesthesia)
o Vascular
V
l part:
t 5 chambers
h b
o Organs and tissues: 8 compartments
 Organs compartments: one part of tissue and one part of blood (equilibrium
of the substances distributed by the blood on both sides it is assumed)
 It is assumed constant blood (VB) and tissue (VT) volumes.
o Capillaries and alveoli: 1 chamber
Ph
Pharmacodynamic
d
i Model
M d l (Gopinath et al., 1995)
• Drug Effect on Hemodynamic Variables Model
PASI 2011 - A. Bandoni
51
The Cardiovascular Model

The Pumping Heart

Based on an elastance model where the cardiac
contraction properties of the two ventricles are
representing by a pair of time
time--varying elastance
functions..
functions

The inertia of blood movements in the ventricles is considered through
an inductance that introduce a phase shift between the ventricular
pressure and the root aortic pressure.
pressure.

The viscous properties of blood in the two atria are included by
ventricular filling
g resistance
PASI 2011 - A. Bandoni
52
Ql
aR
la
pla
Left
Heart
LA
Right
Heart
Ros
pa1
RA
PA
Ra1
La1
pa2
Qr
aRra
TV
Lra
pv1
a3
prv
Era
Rop
pas
LV
Ra2 p Ra3
Ca2
Ca1
pra
AV
Elv(t)
AA
PV
Pulm
.
Circ.
Lla
Ela
pas
Syst.
Circ.
Ql
M vplv L
lv
V
Ca3
Qrv
Rrv
PV
Rv1
Cv1
pv2
Rv2
Lv2
CVi
Cv2
pap
RV
Erv(t)
pp1
Rp1
Cp1
Lp1
pp2
Rp2
Cp2
PASI 2011 - A. Bandoni
Ra3
pp3
Cp3
Rl1
pl1
Cl1
R
pl2 l2
Cl2
Ll2
CVs
53
The Pumping Heart
dQla pla  plv  Rla .Qla

dt
Lla
if
pla  plv
Qla  0
if
pla  plv
dVla
 Ql 2  Qla
dt
pla  Ela .Vla  Vd ,la 
t
Vlv ,b    * Qlv dt  2ml
t
Elv t   Emin,lv .1   t   Emax,lv . t 
 .t
2. .t

a
.
sin
b
.
sin
, 0  t  tce


 
tce
tce
 t   
0
, tce  t  th

tce   0  1.th
pas  R0 s .Qlv  pas
dQlv plv  pas

d
dt
Llv
Qlv  0
if
plv  pas
if
plv  pas
dVlv
 Qla  Qlv
dt
plv  Elv (t ).
) Vlv  Vd ,lv 
PASI 2011 - A. Bandoni
54
The Pumping Heart
Elastance model
Emin,lv
Emax,lv
tce
th
2. .t
 .t

a
.
sin

b
.
sin
, 0  t  tce
 

tce
tce
 t   
0
, tce  t  th

Elv t   Emin,lv .1   t   Emax,lv . t 
PASI 2011 - A. Bandoni
55
The Circulatory System Model
Single chamber model
pa1
Q 1
Qa1
pa2
Va2
Qa2
dVa 2
 Qa1  Qa 2
dt
V V
pa 2  a 2 un ,a 2
Ca 2
Qa 2 
PASI 2011 - A. Bandoni
pa 2  pa 3
Ra 2
56
The Baroreceptors Model
 Baroreceptors
(BR) are sensors of mean blood pressure that are located in
the blood vessels of several mammals.
 BR
nerves are stretch receptors which responds to changes in blood
pressure.
 BR
can send messages to the CNS to increase or decrease total peripheral
resistance and cardiac output (CO).
 BR
act immediately as part of a negative feedback system called the
baroreflex, returning mean arterial blood pressure (MAP) to a normal level
as soon as there is a change
change.
 BR
detect the amount of stretch of the blood vessel walls, and send the
signal to the CNS system in response to this stretch.
 A hysteresis-like
phenomena is observed: the response to a pressure
increase is different to the response to a pressure-decrease
PASI 2011 - A. Bandoni
57
The Baroreceptors Model
① Increased blood pressure
stretched carotid arteries and aorta
causing the baroreceptor to increase
their basal rate of action potential
generation.
② Action
A ti potential
t ti l are conducted
d t db
by
the glossopharyngeal and the vagus
nerves to the cardioregulatory and
vasomotor
t centers
t
in
i the
th medulla
d ll
oblongata.
③ As a result of increased
stimulation from the baroreceptor, the
cardioregulatory center increased
parasymphatic stimulation to the
heart, which decreases the heart rate.
④ Also, as a result of increased stimulation from the baroreceptor, the
cardiorvascular center decreases sympathetic stimulation to the heart, which
decreases heart rate stroke volume.PASI 2011 - A. Bandoni
58
The Baroreceptors Model
⑤ The vasomotor center decreases sympathetic
stimulation to blood vessels, causing vasodilatation.
The vasodilatation along with the decreased heart rate
and decreased stroke volume bring the elevated blood
pressure back toward normal.
Iff the initial problem were decrease in blood pressure,
the activities and effect of baroreceptors,
cardiovascular center and vasomotor center would be
opposite
it off what
h t was illustrated.
ill t t d
PASI 2011 - A. Bandoni
59
The Baroreceptors Model
Baroreceptor
system
Heart
frequency
Systolic
maximum
elastance
Cardiovascular
System
MAP
Systemic
resistance
arteries
H
Emaxlv, Emaxrv
Ra1, Ra2, Ra3
Compliance
in veins and
arteries
Cv1, Cv2
Unstressed
vol in syst.
vol.
syst
veins
Vunv1, Vunv2
PASI 2011 - A. Bandoni
60
The Baroreceptors Model
Afferent sector
MAP
Efferent sector
Central
Nervous
System
n
Sensors
ns MAP  
n p MAP  
1

 MAP 
1 

  
np
Eferent
pathways
th
xi
 ib MAP   i .ns .MAP   i .n p .MAP    i
dxi t  1
  xi t    ib MAP , i  E
dt
i
1
 MAP 
1 

  
ns

i  E  H , Emax , R ps ,Vun , Cv 
PASI 2011 - A. Bandoni
61
The Respiratory System Model

The respiratory
p
y system
y
is concerned with the transport
p
of oxygen
yg
between atmosphere and the tissue and organs in the body

O
Oxygen
iis continuously
ti
l transported
t
t d by
b the
th llung and
d blood
bl d circuit
circuit.
i it.

Carbon dioxide is a waste product of the oxidative metabolism and is
carried by the blood in the opposite direction
PASI 2011 - A. Bandoni
62
The Respiratory System Model
O2 CO2 Atmosphere
Ventilation
Alveoli O2 CO2
Gas exchange
O2
CO2
Pulmonary circulation
Right
Ri
ht
Heart
Gas transport
Left
L
ft
Heart
Systemic circulation
O2
Cell
CO2
Gas exchange
PASI 2011 - A. Bandoni metabolism
63
The Respiratory System Model

Lung model: pressure
R0
Alveoli
Upper
airway
i
Um (t)
Atmosphere
or
respiratory
i t
mask
■ Connect atmosphere (mask)
with alveoli trought
expressions of gas flow
C0
R1
R2
Ri
C1
C2
Ci
Ut (t)
■ The lung is divided in
compartments
■ In each compartment gas
flows are calculated (O2,
CO2, Anesthesia)
■ The outputs of the model are:
pressure in different sectors,
Muscles
the net volume of air flow,
partial
pa
t a pressure
p essu e o
of e
expired
p ed a
air
and alveoli.
PASI 2011 - A. Bandoni
64
The Respiratory System Model

Distribution of substances in the organs through blood
Alveolus
Capillary
Alveolus
κ.pA
pcp
Central
venous
compartment
Viscere
venous
compartment
pv
Metabolism
s
cvv
Lean venous
compartment
cvl
Adipose
Adi
tissue venous
compartment
cav
pas
Central
C
t l
arterial
compartment
pli
Liver
pki
Kidney
phe
Heart
pbr
Brain
pre
Other
organs
pco
pmu
pad
Connective
tissue
Muscles
Adipose tissue
PASI 2011 - A. Bandoni
Q.cb
Q
κ.p
Vbcb (p)
((1- λ )).Q.c
Q vs
λ.Q.cvs
Metabolism
M-
M+
Vtct (p)
Vbcb (p)
zi.Q(p).cb
zi.Q.cas
65
The Respiratory System Model
Upper
airways p0
R0
C0
f0
Alveoli
pcp
pi
RiCi
fi
U m  p0  R0 .i 1 Ci .
n
dpi
dt
dp0

dt
R0 .C0
dpi p0  pi  U t

, i  1...n A
d
dt
Ri .Ci
Pressure model
 I  U m  p0 
n I   pi  p0 
df0
RT
. f e  f0 
. fi  f 0  



 i 1
2
dt C0 p0  V00 p0 
R0
Ri

 I   p0  U t  pi 

dfi
RT
. f0  fi 



κ
.
p
f
p



cp
i i 
2


dt Ci . pi  V0i . pi 
Ri

0
I  x   
x
x0
x0
Molar fractions model
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66
The Pharmacodynamic Model

Pharmacology: the history, source, physical and chemical properties,
Pharmacology:
biochemical and physiological effect, mechanisms of action, absorption,
distribution, biotransformation and excretion, and therapeutic and other
uses of drugs.
drugs.

Pharmacokinetics: absorption,
Pharmacokinetics:
absorption distribution,
distribution metabolism and excretion of
drugs..
drugs

Pharmacodynamics: biochemical and physiological effects and their
Pharmacodynamics:
mechanisms of action.
action.
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67
Drug Concentrattion
D
The Pharmacodynamic Model
Time
Concentration of drug in the body as a function of time
PASI 2011 - A. Bandoni
68
The Pharmacokinetic Model
dc 
dp  dc t
 Vt
 Vb b 
dt  dp
dp 
1






0


cO2
M    M O2



c
O2
O2 

caa


M
 aa   c 
aa
aa 

dp  dc b 

 Vb
dt  dp 
1
z i Qc as  c b p   M  p   M  p 
 M CO 2

M   0
 0






Q1   c vs  c b p    p A  p 
dp  dc b 

 Vb
dt  dp 
1
Qc x  c b p 
Pressure balance
i the
in
th organs
Pressure balance in
the capillaries
Q c  Q2 c 2
cx  1 1
Q1  Q2
Pressure balance in
the compartments
p
PASI 2011 - A. Bandoni
69
The Pharmacodynamic Model
Cd
Drug
effect
MAP
Baroreceptors
EmaxBARO
RsisBARO
Emax= EmaxBARO(1±Eff)
R= RsisBARO(1±Eff)
dEff
 k1.CdN .Eff max  Eff   k2 .Eff
dt
 dEffCa 1 PFL 
dCa1

 Ca1BASE .
dt

 dt
Cardiovascular
system


dEff Emax lvDP
dEmax lv dEmax lvBARO

1  Eff Emax lvDP  Emax lvBARO
l BARO
dt
dt
dt
 dEff RsisDP dEff RsisSNP
dRsis dRsisBARO

1  Eff
ff RsisDP  Eff
ff RsisSNP  RsisBARO 

dt
dt
dt
 dt


PASI 2011 - A. Bandoni



70
The Pharmacodynamic Model
Drug (intravenous)
Affected variable
Action
SNP (sodium
nitroprusside)
Peripheral resistance
MAP
DP ((dopamine)
p
)
Peripheral resistance,
systolic maximum elastance
MAP
PFL (propofol)
BIS
MAP
unconsciousness
Systolic maximum elastance
Peripheral resistance
PASI 2011 - A. Bandoni
71
The Pharmacodynamic Model

DP and SNP drugs are chosen to increase ventricular contractility and
reduce
d
th resistance
the
i t
off arteries
t i to
t blood
bl d flow,
fl
respectively.
respectively
ti l .

PFL is chosen to conduct unconsciousness by measured of BIS
parameter..
parameter

DP increases the MAP and CO.
CO. SNP decreases and increases CO
MAP..
MAP

Sceneries are simulated by delivering a step of 1μg/kg/min of SNP, DP
and PFL and registering the dynamic response of the physiological,
pharmacokinetic and pharmacod
pharmacodynamic
namic variables.
ariables
PASI 2011 - A. Bandoni
72
Computational Implementation

Model implemented in Fortran

Diff. Eqs
Eqs.. solved with a 4th order Runge
Runge--Kutta method.

Resolution sequence: (i) the cardiovascular model is solved until to
reach steady state, (ii) the CO obtained from this model is used in the
breathing
g model,, ((iii)) the breathing
g model is solved until to reach steady
y
state.

Th d
The
drug iinjection
j ti iis simulated
i l t d for
f a cycle
l off breathing
b thi (5 sec.).
sec.)). Then
Th the
th
cardiovascular model is fed with the drug concentration Cd to simulate
the 0.8 sec. a heartbeat.
heartbeat. The updated value of CO is fed back to the
breathing model.
PASI 2011 - A. Bandoni
73
Computational Implementation
CO2, O2
Cd (alveoli)
Cd(inhalable)
Cd(inyectable)
Transport and
distribution,
Pharmacokinetics of
drugs
Respiratory
system
Qa3, Qp3
Cd (organs)
Cardiovascular
C
di
l
system
Pharmacodynamics
Ph
d
i
of drugs
Baroreflex
MAP
EmaxBARO
Ra2BARO
Ra3BARO
Emax
Rsis
EffEmax
EffRa2
EffRa3
Control
Action
PASI 2011 - A. Bandoni
74
Computational Implementation
Dimensions of the integrated model
Model
Var./Eqs.
Algebraics
Var./Ecs.
Differenctials
Parameters
Cardiovascular-Respiratory
37
39
53
Respiratory-Pharmacodynamic
60
93
85
Total
97
132
138
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75
Results
PASI 2011 - A. Bandoni
76
Results: cardio vascular system
Wiggers Diagram
PASI 2011 - A. Bandoni
77
Results: cardio vascular system
Left ventricular volume
vs. time
Left ventricle and root aortic
pressure vs. time
PASI 2011 - A. Bandoni
78
Results: cardio vascular system
Outflow of the left ventricle
Left ventricular pressure
Pressure vs. Volume left ventricle
PASI 2011 - A. Bandoni
79
Results: baroreflex system
Heart period vs.
vs time
Resist sect.
Resist.
sect As1 of syst.
syst arteries vs
vs. time
Compliance in sect. Vs1 of sistemic
Unstres. Vol. of sect. Vs1 of
veins vs. time
PASI 2011 - A. Bandoni sistemic veins vs. time
80
Results: baroreflex system
Sistolic max. elastance of left ventr.vs. time
Comparison of CO vs. time in front of 10 %
bleeding, with and without baroreceptor
Comparison of MAP vs. time in front of
10 % bleeding, with and without
baroreceptor
PASI 2011 - A. Bandoni
81
Results: gas transport
Partial pressure of O2 in different
compartments of the body
Partial pressure of CO2 in different
compartments of the body
PASI 2011 - A. Bandoni
82
Results: respiratory system
expiración
inspiración
Volume vs. Pressue diagram in lungs
Partial pres. profile of
CO2 in lung and alveoli.
PASI 2011 - A. Bandoni
Partial pres. profile of O2 in
lung and alveoli.
83
Results: pharmacodymic system
Effect of the SNP action
1µg/kg/min
SNP concentration profile at the central arterial compartment
Mean Arterial Pressure, MAP
PASI 2011 - A. Bandoni
Cardiac Output, CO
84
Results: pharmacodynamic system
Effect of the SNP action
1µg/kg/min
Resistance, Ra3
Resistance, Ra2
PASI 2011 - A. Bandoni
85
Results: pharmacodynamic system
Effect of the DP action
5µg/kg/min
DP concentration profile at the central arterial compartment
Mean Arterial Pressure, MAP
PASI 2011 - A. Bandoni
Cardiac Output, CO
86
Results: pharmacodynamic system
Effect of the DP action
5µg/kg/min
Medial arterial resistances
Elastance
PASI 2011 - A. Bandoni
87
Results: pharmacodynamic system
Effect of the DP action 2, 4, 6, 8 µg/kg/min
Cardiac Index vs. infusion doses (time)
Volume Index vs. infusion doses (time)
Systolic and
diastolic pressure
vs. infusion doses
(time)
PASI 2011 - A. Bandoni
88
Results: pharmacodynamic system
Effect of the DP action 2, 4, 6, 8 µg/kg/min
Systemic Resistance vs. infusion
d
doses
(ti
(time))
Cardiac frequency vs. infusion
doses (time)
PASI 2011 - A. Bandoni
89
Results: pharmacodynamic system
Effect of the PFL action
150µg/kg/min
/ /
Mean Arterial Pressure, MAP
PFL conc. at the central arterial comp.
Cardiac Output, CO
Compliance of sector a1 of systemic arteries
PASI 2011 - A. Bandoni
90
Conclusions

Development of an integrated cardiovascular,
baroreceptor, respiratory,
pharmacokinetic and
pharmacodynamic model
model..

The effect of certain drugs on hemodynamic variables
was studied
studied..
PASI 2011 - A. Bandoni
91
Future Works

General model validation with real patient data.
data. Collaboration
with a research group formed by doctors (Favaloro University,
Bs..As
Bs
As.. – Español Hospital,
Hospital B. Blanca,
Blanca Arg
Arg..)

Model validation for inhalable anesthesia effects
effects..

Model validation for simultaneously drugs administration
administration..

Development of a control model for handling dose of drug
administration..
administration

Development of a teaching simulation model of the
cardiovascular system (Instituto Nacional de Tecnología
Industrial, INTI, Bs.
Bs.As
As.., Arg
Arg..)
PASI 2011 - A. Bandoni
92
Basic References:
Cardiovascular Model:
 Ottesen J., Olufsen M. and Larsen J. Applied Mathematical Models in
Human Physiology . SIAM, Philadelphia. (2004)
Pharmacodynamic Model:
 Gopinath R., Bequette B., Roy R. and Kaufman H. Issues in the Design
of a Multirate Model- based Controller for a Nonlinear Drug Infusion
System Biotechnol.
System.
Biotechnol Prog.
Prog 11 (3)
(3), pp 318
318–32.
32 (1995)
Respiratory Model:
 Christiansen
Ch i ti
T and
T.
dD
Dræby
b C
C. Modeling
M d li the
th R
Respiratory
i t
S
System
t
Technical. Report IMFUFA, Roskilde University Denmark Text No. 318.
(1996)
PASI 2011 - A. Bandoni
93
Other References:
 Dua P and Pistikopulos E.
E Modelling and control of drug delivery systems
systems. Comp.
Comp
Chem. Eng. 29 pp. 2290-96. (2005)
 Montain M,
M Bandoni J y Blanco A . Modelado del sistema cardiorespiratorio
humano: un estudio de simulación. VI CAIQ (Congreso Argentino de Ing. Química)
Mar del Plata 26 al 29 de septiembre (2010).
 Rao R, Bequette B and Roy R. Simultaneous regulation of hemodynamic and
anesthetic states: a simulation study; Annals of Biomedical Engineering, 28 pp. 71(
)
84. (2000)
 Dua P, Dua V and Pistikopoulos E. Modelling and mult-parametric control for
delivery of anaesthetic agents. Med. Biol. Eng. Comput. 48 543-53. (2010).
 Massoud T., G. Eorge, J. Hademenos, W. Young , E. Gao, J. Pile-Spellman and F.
Uela. Principles and philosophy of modeling in biomedical research.The FASEB
Journal, vol. 12 no. 3, pp.275-285, March 1, 1998.
 Ottesen J.T. The Mathematical Microscope ‐ Making the inaccessible accessible.
Bi
Biomedical
di l and
d Lif
Life Sciences
S i
S
Systems
t
Bi l
Biology
‐ Volume
V l
2
2, 2011.
2011
PASI 2011 - A. Bandoni
94
“With growing emphasis being placed
d on the information
processing aspects of biomedical investigation, theoretical and
experimental studies assume increasing importance.
importance In many
instances, however, there are questions that appear to be
present experimental
p
techniques;
q ; in such cases,,
unanswerable byy p
models can usefully augment direct scientific experimentation.
The essential
Th
i l ingredient
i
di
off the
h scientific
i ifi method
h d is
i the
h use off
models. Good modeling is more likely to be achieved by following
the rules of good thinking.
thinking However,
However the ideal model cannot be
achieved. Partial models, imperfect as they may be, are the only
means developed
p byy and available to scientists for understanding
g
the universe”
Principles and philosophy of modeling in biomedical research.
T. Massoud, G. Eorge, J. Hademenos, W. Young , E. Gao, J. Pile-Spellman and F. Uela
(University of California at Los Angeles, Columbia University, University of Dallas)
The FASEB Journal, vol. 12 no. 3, pp.275-285, March 1, 1998
PASI 2011 - A. Bandoni
95
PASI 2011 - A. Bandoni
96
Muchas gracias
PASI 2011 - A. Bandoni
97
Cámara izquierda del corazón
p  plv  Rla Qla
dQla
 la
Lla
dt
Qla  0
si
Qlv  0
si

pla  Ela Vla  Vd ,la
si
Vlv,b  

t
*
dQa1
p  p a 2  Ra1Qa1
 a1
dt
La1
p a1 

Va1  Vun, a1
C a1
dVa 2
 Qa1  Qa 2
dt
plv  p as
plv  pas
dVlv
 Qla  Qlv
dt
t
pla  plv
pla  plv
dVla
 Ql 2  Qla
dt
dQlv
p  p as
 lv
dt
Llv
si
Circulación sistémica
Qa 3 

plv  Elv t  Vlv  Vd ,lv

p as  pv1
Ra 3
Va3  Vun, a3
dVa1
 Qlv  Qa1
dt
Qa 2 
p a 2  p a3
Ra 2
pa2 
Va 2  Vun, a 2
Ca2
dVa3
 Qa 2  Qa 3
dt
Qv1 
pv1  pv 2
Rv1
dVv1
 Qa3  Qv1
dt
pv1 
Vv1  Vun, v1
Cv1
dVv 2
 Qv1  Qv 2
dt
dQv 2
p  p ra  Rv 2Qv 2
 v2
dt
Lv 2
p a3 
C a3
Qlv dt  2ml
Elv t   E min,lv 1   t   E max,lv t 
t
2t

 b sin
a sin
t ce
t ce
 t   
0

tce   0  1t h
0  t  t ce
t ce  t  t h
pas  R0 sQlv  pa1
pv 2 
Vv 2  Vun,v 2
Cv 2
PASI 2011 - A. Bandoni
98
Cámara derecha del corazón
Circulación pulmonar
dQra
p  prv  Rra Qra
 ra
dt
Lra
dQ p1
Qra  0
si
p p1 

pra  Era Vra  Vd , ra


Q p3 
prv  pap

prv  Erv t  Vrv  Vd , rv
dVrv
 Qra  Qrv
dt
t
Vrv,b   Qrv dt  2ml
t*
p p3 

pl 2 
pap  R0 pQrv  p p1
 Qrv  Q p1
 Q p1  Q p 2
p p2 
V p 2  Vun, p 2
C p2
p ps  pl1
R p3
dV p3
dt
V p3  Vun, p3
 Q p 2  Q p3
p  pl 2
Ql1  l1
Rl1
Vl1  Vun,l1
pl1 
Cl1
C p3
dQl 2 pl 2  pla  Rl 2Ql 2

dt
Ll 2
dVl 2
 Ql1  Ql 2
dt
Erv t   Emin,rv 1   t   Emax,rv t 
dt
p p 2  p p3
R p2
dVl1
 Q p3  Ql1
dt

dV p1
Q p2 
C p1
dV p 2
prv  pap
si
p p1  p p 2  R p1Q p1
L p1
V p1  Vun, p1
dt
dQrv prv  pap

dt
Lrv
sii
dt
pra  prv
dVra
 Qv 2  Qra
dt
Qrv  0
pra  prv
si
Vl 2  Vun,l 2
Cl 2
Modelo respiratorio (fracción molar)
df 0
RT

dt
C 0 p 02  V 00 p 0
 I U
m  p 0 f e  f 0  
 

R0

df i
RT

dt
C i p i2  V 0 i p i

I   p i  p 0 f i  f 0  

Ri
i 1

n
 I   p 0  U t  p i f 0  f i 

 κ p cp  p i f i

R
i

PASI 2011 - A. Bandoni



0
I  x   
x
x0
x0
99
Barorreceptores
n s MAP  
n p MAP  
1

 MAP 

1  
  
1
 MAP 

1  
  

 ib MAP   i ns MAP   i n p MAP   i
dxi t  1

dt
i

i  E  H , E max , R ps , Vun , C v
Q c  Q2 c 2
cx  1 1
Q1  Q2

0







cO2 
M    M O2

O2  cO2 

caa 

 M aa   c 
aa
aa 

  x t    b MAP, i  E
 i

i


Modelo respiratorio (presión)
dp0

dt
U m  p0  R0

n
i 1
Ci
R0C0
0.27273  1.96364 t

0.66943  0.53554 t
Um  
17.00005  8.50909 t
0.57034  0.05904 t

dpi
dt
dpi p0  pi  U t

dt
Ri Ci
dp  dc b 

  Vb
dt  dp 
1
1
H 
 3
 
 a2 H 
2
1
Qc x  c b p 
M 
 CO2 
M   0 
 0 


 
 a1 H   a0  0
0  x  0.278
0.278  x  1.806
a 2  K a ,Pr  NaOH 0  K a ,CO
1.806  x  1.904
1.904  x  5
a1  K a ,CO N
NaOH
OH 0  cCO2  K a ,Pr K a ,CO  NaOH
N OH 0  H Pr
P 0 

Modelo de transporte de gases en sangre
dc 
dp  dc t
 Vt
 Vb b 
dt  dp
dp 
dp  dc b 

 Vb
dt  dp 
z i Qc as  c b p   M  p   M  p 
Q1   c vs  c b p    p A  p 


a 0  K a ,CO K a ,Pr NaOH 0  H Pr 0  cCO2
Ery
p, pH 
c bCO2 p, pH   c CO
2

c Hb
c Hb
Pla




c
p
,
pH
1

CO2
Ery
 c Ery
c Hb
Hb


1  10 
Ery
Ery
p, pH    CO
c CO
p cCO 2 1  10  pH
2
2
Ery
Ery
p, pH    CO
c CO
p cCO 2
2
2
PASI 2011 - A. Bandoni

Ery



p , pH  pK Ery p , pH 
pH Ery p , pH  pK Ery p , pH
100




Pla
Pla
p, pH    CO
c CO
p cCO 2 1  10  pH  pK
2
2

pK Ery p, pH   6.125  log 10 1  10
Pla
 pH
p , pH 
Ery

p , pH  7.84  0.06 sO2 p , pH 


pH Ery p, pH   7.19  0.77 pH  7.4  0.035 1  sO2 p, pH 

pK
K Pla p, pH
H   6.125  log
l 10 1  10  pH 8.7 

1
1  e  y p 
hp   3.5  ap 

xp   log p O2 / kPa

t
p    t p aa
c aa
 pCO2 
mmol 

  0.07 0.03xHbf cdpg /
5 
ap  0.72 pH cCO2 p  7.4  0.09log
l


 5.33kPa
0.386xHbCO 0.174xHi  0.28xHbf
 

b
p    b p aa
c aa
y p   1.875  xp   x0 p   hp  tanh 0.5343xp   x0 p 
c bO2 p, pH    O2 pO2  c Hb sO2
s O2 

 T

 37 
x0 p   1.946  a p   0.055

 ºC


t
Pla
l
Hb

p, pH   cCO
p, pH 1  cEry
cCO
2
2

 c Hb 

cOt 2 p    O2 pO2
Modelo farmacodinámico
 dEffCa1PFL
dCa1
 Ca1BASE 

dt
dt

dEff
 k1C N Eff max  Eff   k 2 Eff
d
dt






 dEff RsisDP dEff RsisSNP
dRsis dRsisBARO
1  Eff RsisDP  Eff RsisSNP  RsisBARO 



dt
dt
dt
dt







dEff
d
ff E max lvDP
dEmax lv dEmax lvBARO

1  Eff E max lvDP  E max lvBARO
dt
dt
dt
PASI 2011 - A. Bandoni
101