Estimating Blood Flow
TO:
FROM:
DATE:
SUBJECT:
1
Professor Andreas Linninger / Chih-Yang Hsu
Ossama Anis
December 2, 2014
Class Project
Dear Prof. Linninger and Chih-Yang Hsu,
The following report will discuss the approach taken in the estimation of blood flow in a
vascular network. I have clearly stated the purpose and significance of this project. Figures have
been generated to simulate the relationship between concentration, pressure, and blood flow.
Known and unknown variables have been established along with boundary conditions. Three
equations that are applied to these variables allow one to gain the unknown in the problem. The
Conservation, Hagen-Poiseuille, and the species transport equations are utilized in this problem.
Thank you,
Ossama Anis
Estimating Blood Flow
2
ESTIMATING BLOOD FLOW IN THE CEREBRAL VASCULATURE
Ossama Anis*
University of Illinois at Chicago*
oanis2@uic.edu
December 2014
This report is produced under the supervision of BIOE310 lecturer Prof. Linninger.
Abstract
One of the leading causes of death in the United States is cerebrovascular disease. The condition of
a person’s blood vessels determines how prone one is to a vascular disease. Hemodynamics studies
the movement of blood through vessels or vascular systems. To properly assess the symptoms of
these diseases, extensive knowledge in hemodynamics is a necessity. Imaging techniques allow
physicians to observe blood flow and perform surgeries simultaneously. However, this technique
only allows physicians to determine blood flow and vessel health by observing the change in image
intensity. The significance of this project is to use dynamic image signals to estimate blood flow in
cerebral blood vessels. In doing so, some information is discovered to help determine the impact of
blood flow on different aspects of vascular health. Furthermore, this study may help improve the
quality of healthcare provided to patients with vascular disease. Simulations are used to predict
blood flow and dye transport. The known variables are initial concentration, time, resistance, and
pressure boundary conditions whereas the unknowns consist of blood flow, pressures, and
concentration profiles. Then, a comparison can be made between the simulated dye concentration
profiles and the parameter estimation model, where flow rates are unknown. Estimated flow rates
will be produced using least square error approach via the known concentration profiles. If the flow
rates obtained are correct, the concentration profile dynamics will be the same.
1. Introduction
Vascular disease is one of the most common forms of
death that occurs in the United States. Cerebrovascular
disease, along with diseases of the heart and arteries,
is one type of vascular disease. When observing
arteries, the diameter of the vessels is of great concern.
Plaque can accumulate along the walls of the artery,
thereby shrinking its diameter. Stenosis, which is the
narrowing of a vessel, can cause a disruption in the
blood flow or even close off the vessel. A narrow or
blocked artery located in the brain can lead to a stroke.
A buildup of plaque can result in damaging the arterial
wall which, in turn, can lead to blood clots. Pieces of
plaque can also breakoff into the bloodstream and get
stuck in smaller branches of the vasculature. Plaque
can also decrease or stop blood flow completely by
building up in a certain place in the artery. When
something of this sort occurs in the brain, certain parts
of the brain will not receive the blood that they need
due to a significant decrease in blood flow. The Circle
of Willis, as seen in Figure 1, is a group of arteries that
are the main suppliers of blood [1]. If these become
clogged, severe damage can occur. The Circle of
Willis consists of the basilar and internal carotid
arteries, both of which are inflowing. The basilar
artery branches off into the posterior cerebral artery,
thus supplying blood to the back of the brain. The
internal carotid artery flows into the anterior and
middle cerebral arteries supplying the front part of the
brain. Furthermore, these elongate out to both sides of
the brain, consisting of the right and left sides of the
anterior cerebral, middle cerebral, and posterior
cerebral arteries. All of these deliver blood to certain
parts of the brain and, if clogged, can cause serious
damage. The flows and pressures differ in direction
and magnitude in each vessel as depicted in Figure 3.
To enhance current procedures concerning vascular
health, simulations are created to demonstrate blood
flow. The data gathered from the observation of blood
flow can be interpreted into potentially valuable
information. This should help to improve the measures
taken when dealing with hemodynamics and health.
Estimating Blood Flow
3
(b) will be comprised of the solutions to the set of
equations as denoted in the matrix equation.
𝐴𝑥 = 𝑏
(3)
Boundary conditions of the system must be known in
order to solve for pressures and flows. Solving for (x)
will conclude values for each pressure and flow
contained within the system. The last equation utilized
was the species transport equation, which represents
the difference in concentration with respect time.
Figure 1. The Circle of Willis is a network of arteries that
supply blood to the brain [5].
𝑉
2. Methods
There are various things to be cautious about when
estimating blood flow through a network of cerebral
blood vessels. To determine blood flow, there are
many crucial concepts to consider. Conservation law
explains how a measureable property in an isolated
physical system does not vary as the system
progresses. This is represented in the conservation
equation which represents the flow going in the system
being equal to the flow going out of the system as
shown in equation 1.
∑ 𝐹𝐼𝑛 = ∑ 𝐹𝑂𝑢𝑡
(1)
In this circumstance, the blood flow entering a node is
equal to the blood flow exiting that node. Conservation
balances will be applied to each face, which represents
a vessel or connections between each node. Pressure
values will be assigned to each node while flows are
assigned to each face. Once the conservation balances
are derived, constitutive equations may be formulated
using the Hagen-Poiseuille law as shown below.
∆𝑃 = 𝛼𝐹; 𝛼 =
8𝜇𝐿
𝜋𝑟 4
(2)
In fluid dynamics, this law declares that the change in
pressure is equal to the flow rate between the two
nodes multiplied by the resistance. The pressures
coming in and out of the system as well as the viscosity
of the blood must be established in order to consider
the two concepts mentioned above. Both conservation
and constitutive equations will be put into a matrix
(A), while the vector (x) consists of all the pressures at
each node and the flows in between them. The vector
𝑑𝐶
= 𝐹𝐼𝑛 𝐶𝐼𝑛 − 𝐹𝑂𝑢𝑡 𝐶𝑂𝑢𝑡 + 𝐼𝑛𝑗
𝑑𝑡
(4)
Diameters and lengths of the vessels are known and
needed in order to determine the volume of each
vessel. Once volumes and viscosities have been
established, it will be used to calculate the
concentration profiles at each node. These
observations allow us to approximate volumetric flow
rates, pressure drops, and other steady-state properties
at each vessel in the network.
Utilizing the fundamental concepts described above
allows one to determine the concentration profiles in
each node. Implicit Euler methods are applied to
observe the difference in concentration with respect to
time. After simulation of concentrations, one can use
these representations to interpret the flow. Simulating
with different boundary conditions will provide
altered concentrations and, in turn, yield various
residuals. This way can be tedious and allows for
alternative methods to be of use. Integral
approximation, a method that will be discussed and
elaborated on upon further reading, may work more
efficiently and can be utilized as a source of validation.
These methods have been done on a simple bifurcated
network and the results coincide. This can be applied
to any specific network, such as the Circle of Willis in
this case, to determine the blood flow specific to that
system.
3. Results
In Figure 3, the Circle of Willis is displayed in a basic
format that represents the network. This diagram has
been broken down and labeled at each pressure point
and flow terminal. The direction of the flows are
observed either flowing in or out of the node. The
communicating arteries consist of a bidirectional flow
that can supply blood to either side of the brain in case
of insufficient flow. This diagram can easily be used
to derive a matrix for both faces and nodes, by utilizing
Estimating Blood Flow
4
conservation and constitutive laws which will, in turn,
determine flows and pressures in this network. The
inputs in this network will be the left and right internal
carotid arteries, along with the two vertebral arteries
that join to become the basilar artery. The outputs are
the left and right middle cerebral arteries, anterior
cerebral arteries, and posterior cerebral arteries. The
equations have been formulated and implemented with
a simple bifurcated network before advancing on to a
more complicated model such as the Circle of Willis.
From conducting this simulation, the results obtained
allowed for the comparison of different aspects within
the network. The Circle of Willis is observed after dye
was injected. The observations determined that there
was an initial increase in concentration, then peaked to
Figure 2. The simple bifurcated network with three
flows connecting the four pressure points. Assume the
network is injected with dye at pressure point one (P1).
a maximum, followed by a decrease. As observed in
Figure 4, the concentration is initially very high
because the time begins when injection of dye occurs.
Once the dye completely exits the vessel, the
concentration profile will be zero. Figure 4 also
depicts that various initial pressures only changes the
duration of the concentration profiles. Various initial
pressures were applied in order to compare their
residuals, displayed in Figure 5. The results conclude
that if there is a higher pressure coming into the
network, the concentration profiles will be shorter in
duration. Simulating with different initial pressures
and comparing their residuals will allow for
interpretation of flows.
Figure 3. Illustrates a network of the Circle of Willis. This diagram displays
pressure points and the directions of the flows throughout the system.
Estimating Blood Flow
5
Concentration Profiles (pIn = 75)
1
0.9
0.9
0.8
0.8
0.7
0.7
Concentrations
Concentrations
Control for Concentration Profiles (pIn = 100)
1
0.6
0.5
0.4
0.6
0.5
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
0.05
0.1
0.15
0.2
Time
0.25
0.3
0.35
0
0.4
0
0.05
0.1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.5
0.4
0.2
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0.1
0.1
0.15
0.2
Time
0.25
0.3
0.35
0
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0
0.05
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0.9
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0.7
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Time
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Time
0.5
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0.15
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0
0.4
Concentration Profiles (pIn = 200)
1
Concentrations
Concentrations
Concentration Profiles (pIn = 175)
1
0
0.35
0.4
0.3
0.05
0.3
0.5
0.2
0
0.25
0.6
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0
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Time
Concentration Profiles (pIn = 150)
1
Concentrations
Concentrations
Concentration Profiles (pIn = 125)
0.15
0.3
0.35
0.4
0
0
0.05
0.1
0.15
0.2
Time
0.25
Figure 4. The graphs above display the concentration of dye injected into a network with various initial pressures applied. Each
graph represents the dye concentration profiles with respect to time. It is easy to depict that a higher inlet pressure will result in a
shorter duration of the concentration profile throughout the system.
Estimating Blood Flow
6
Concentration Profiles with Noise (10 dB)
Pressures and Corresponding Residuals (10 dB)
40
1
0.9
35
0.8
30
0.6
Residuals
Concentrations
0.7
0.5
0.4
25
0.3
20
0.2
0.1
0
0
0.05
0.1
0.15
0.2
Time
0.25
0.3
0.35
15
60
0.4
80
Concentration Profiles with Noise (20 dB)
100
120
140
Pressure (mmHg)
160
180
200
180
200
180
200
Pressures and Corresponding Residuals (20 dB)
18
1
16
0.8
14
0.7
12
0.6
Residuals
Concentrations
0.9
0.5
10
8
0.4
6
0.3
4
0.2
2
0.1
0
0
0.05
0.1
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0.2
Time
0.25
0.3
0.35
0
60
0.4
80
Concentration Profiles with Noise (30 dB)
100
120
140
Pressure (mmHg)
160
Pressures and Corresponding Residuals (30 dB)
1
16
0.9
14
0.8
12
10
0.6
Residuals
Concentrations
0.7
0.5
0.4
8
6
0.3
4
0.2
2
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0
0
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Time
0.25
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60
80
100
120
140
Pressure (mmHg)
160
Figure 6. The graphs on the left represent the concentration of dye with respect to time, injected into the Circle of Willis, with an
initial pressure of 100 mmHg. Noise is taken into account and corresponding signal-to-noise ratios (SNR) are calculated in
decibels. The higher the SNR, the less noise there is interrupting the signal. The graphs on the right validate that noise and residuals
are directly proportional. The more noise contained within the signal, the larger the residual will be.
Estimating Blood Flow
7
potentially life-threatening. The medical world can
benefit from this correlation by using it to find
efficient drug delivery systems. Hemodynamics is a
very crucial concept that, if understood completely,
could be a very valuable tool. If correlations can be
confirmed between blood flow and other health
concepts, these types of simulations would prove to be
useful. These correlations still need to be explored and
could possibly consist of preventative measures for
strokes. Furthermore, if these simulations could mirror
actual patient’s brains, the benefits would be
exponential.
Pressures and Corresponding Residuals
15
Residuals
10
5
6. Intellectual Property
0
60
80
100
120
140
Pressure (mmHg)
160
180
200
Figure 5. Displayed above are the residuals corresponding
to various boundary conditions. The residual is zero when
the inlet pressure is 100 mmHg.
4. Discussion
Blood flow rates can be estimated using a few different
methods. One of these methods includes the use of
inversing data. The least-squares approach or matrix
pseudo-inverse is utilized to determine flow rates from
experimentally gained concentration profiles.
Integration of the concentration profiles is necessary,
therefore, it is logical to suggest the trapezoidal
method. Although it is an easy procedure to carry out,
it brings with it numerical and experimental error.
Limitations include aerodynamic properties of the
blood flow along with the fact that inlet pressures into
the internal carotid and vertebral arteries may not be
the same.
Biological and physiological data and some modeling
procedures provided to you from Dr. Linninger’s lab
are subject to IRB review procedures and Intellectual
property procedures.
Therefore, the use of these data and procedures are
limited to the coursework only. Publications need to
be approved and require joint authorship with staff of
Dr. Linninger’s lab.
7. References
1.
Bhara, A., D. Espino Barros Perez. Collateral
Blood Flow in Diseased State. Chicago:
University of Illinois at Chicago, Bioengineering
Department.
2.
Hsu, C., & A. Linninger. Modeling the Whole
Brain Vascular Network for Hemodynamics
Computation. Chicago: Laboratory for Product
and Process Design. 2014.
3.
Kulkarni, K. Mathematical Modeling, Problem
Inversion and Design of Distributed Chemical and
Biological Systems. Chicago. (2008)
4.
McEneaney, J., & M. Dytso. Simulation of
Capillary Blood Flow Rates after Occlusion and
Vasodilation of Left Cerebral Arteries. Chicago.
5.
""Brainstem Circle of Willis"" Brain Anatomy.
WordPress, 2014. Web. 27 Nov. 2014.
<http://brainanatomy.tk/tag/brainstem-circle-ofwillis>.
5. Conclusion
In conducting these simulations, it has been found that
estimating blood flow in the cerebral vasculature is,
indeed, possible. Pressures and flows can be
determined for each vessel by taking the correct
measures. Concentration profiles can then be
determined using fundamental concepts, such as the
Euler method. We can infer blood flow rates from the
concentration profiles by creating simulations with
different boundary conditions and comparing their
residuals or by integral approximation. With this
knowledge, it is inferred that the initial pressure of the
dye injected is relative to the blood flow rate. This
concept can be applied to real-life situations that are