Modeling Cerebral Hemodynamics
Sam Dreyer, Chih-Yang Hsu, & Andreas Linninger
Bioengineering, University of Illinois at Chicago, Chicago, IL
December 5, 2014
This report is produced under the supervision of BIOE310 instructor Prof. Linninger.
ABSTRACT
Despite recent advances research on the human brain, our understanding of cerebral
vasculature lags behind other areas of the human body. This is due largely to the difficulty of
imaging and performing surgery on live subjects. A better approach is to expand the knowledge
of the scientific community with computational models. The Circle of Willis is a crucial structure
in supplying the brain with blood and it is critical to have a developed model to further medical
understanding.
Keywords: Circle of Willis, Cerebral Vasculature, Computational Hemodynamics
INTRODUCTION
The purpose behind studying cerebral
hemodynamics is to better understand how
the blood flows through different areas of
the brain. This is especially significant
because currently there is not much known
about how this blood flow occurs. If doctors
and physicians had a better idea of how
blood flows in the brain and how it changes
in response to different diseases or injuries,
drug delivery and neurosurgery could be
much more precise. One of the ways that
this can be studied is by releasing a dye at a
single point in the brain and tracing how it
flows over time.
The Circle of Willis (Figure 1) is a
circulatory anastomosis on the inferior
surface of the brain that is vital to supplying
blood to the brain.
A
B
Figure 1A. A graphic representation of the Circle of
Willis [1]. Figure 1B. The network that was used in
the simulation of blood flow and dye concentrations.
Dimensions were chosen based on standard
anatomical measurements [2].
The redundancies within the Circle of Willis
keep a single clot or other deformation such
as a stenosis from blocking blood flow to a
large portion of the brain.
The purpose is to model the blood flow
within the Circle of Willis to better
understand how blood flows in the brain and
how it relates to pressure and flow rates.
During the experimental modeling, the time
and concentration of dye will be measured at
each point throughout the Circle of Willis. It
is intended to use those pieces of
information to solve for the unknown
pressure and flow rates and compare these
values
to
experimentally
derived
measurements.
METHODS
From previous literature, it was possible to
estimate the dimensions of the arteries that
comprise the Circle of Willis in order to
build an accurate model [2]. In the interest
of time, it was assumed that the diameters
throughout the Circle of Willis were
constant as well as the volume of blood.
Artery
Posterior
Cerebral
Posterior
Communicating
Internal Carotid
Anterior Cerebral
Anterior
Communicating
Length
(cm)
0.69
Diameter
(cm)
0.22
1.34
0.15
0.48
1.43
0.25
0.42
0.23
0.19
Furthermore, various equations can be
leveraged to assist in modeling the flow of
any substance and in any network.
Conservation laws must be followed as there
is no generation or destruction of blood in
the brain. From this, it can be extrapolated
that the following equations must hold true
[3].
∑𝐹 = 0
(1)
(2)
Another important aspect to consider is the
resistance of an artery. In the equation
below, μ is the viscosity, L is the length, and
d is the diameter of the artery [3].
𝛼=
128∗𝜇∗𝐿
𝜋∗𝑑4
(3)
One can assume that the change in pressure
along the network will be equal to the
change in flow multiplied by the resistance
for each point, i. This constitutive equation
can be summarily described thusly [3].
𝛼𝑖 𝐹𝑖 = ∆𝑃
∆𝑃 =
8∙𝐿∙𝜇∙𝐹
𝜋∙𝑟 4
(5)
Additionally important is using this model to
simulate the flow of dye through the
network once flows and pressures have been
established.
Table 1. The average lengths and diameters of the
arteries that comprise the Circle of Willis.
∑ 𝐹𝑖𝑛 = ∑ 𝐹𝑜𝑢𝑡
model by solving for the change in pressure.
In this equation, r is the radius of the artery
and the rest of the variables have been
defined in the resistance equation [3].
(4)
Having already solved for the flows and
resistance, the Hagen-Poiseuille equation
can be used to ensure the accuracy of the
If matrix A is set as the matrix containing
the pressures and flows of the network, a
vector, b, which contains the initial value of
the bolus of dye and the location where it
will be injected can also be established. At
this
point,
the
following
simple
mathematical operation is performed for
each time step in the simulation. It is
important to note that b is the x that was
determined from the previous step so that
the simulation relies on previously generated
values to predict the future concentrations.
𝑥 = 𝐴\𝑏
(6)
However, this method of simulation is not
perfect. With this method, cerebral blood
flow is modeled with the boundaries of the
arteries not having any shear stress
interaction with the blood. This model also
assumes the arteries would behave in the
same method as rigid tubes, which is also
not true in application.
Another step taken to improve the model
and keep it as applicable as possible is to
artificially create noise. No biological signal
can currently be measured without noise.
Added random perturbations can distort the
signal enough to mimic the noise that would
be encountered while measure a signal from
a living subject.
RESULTS
Initially, a simple bifurcating network was
modeled to determine the efficacy of
modeling the cerebral hemodynamics using
this particular method.
In the bifurcating network, physiological
parameters were applied to have a standard
inflow pressure of 100 (mmHg) and outflow
pressure of 5 (mmHg). This is displayed
graphically as a concentration vs. time plot
below (Figure 2).
Figure 3. A visualization of dye concentration at each
point in the network over time. P1, P2, P3, P4, P5,
and P6 refer to the different discernible points in the
network. It can be seen how the point of injection
decays after a peak concentration and the points
along the Circle of Willis have a peak at a time
corresponding to their distance from the initial point
and at a lower concentration. For this simulation the
inflow pressure was 10 (mmHg) and the outflow was
1 (mmHg). The artificial noise introduced in this
simulation was 0.1% of the actual values.
Figure 2. A visualization of dye concentration at each
point in the network over time. P1, P2, P3, and P4
refer to the different discernible points in the
network. It can be seen how the point of injection
decays after a peak concentration and the points
along the bifurcation network have a peak at a time
corresponding to their distance from the initial point
and at a lower concentration.
Due to this establishing the efficacy of
simulating with this type of model, it was
then possible to extend the scope of the
testing to model an entire Circle of Willis.
In the model of blood flow in the Circle of
Willis, it is important to use physiological
measurements to obtain as accurate a
simulation as possible. For this model, there
is a standard inflow pressure of 10 mmHg
and outflow pressure of 1 mmHg. This is
displayed graphically as a concentration vs.
time plot (Figure 3).
One method of analyzing flows from this
data is by running more simulations with
different boundary conditions. To quantify
how well the new simulations compare to
the standard model is to calculate the
residual error for each point in the network
at each point in time. This was done for four
different sets of boundary conditions as
shown in Figure 4 and summarized in Table
2. Beyond the four simulations shown in
Figure 4, additional simulations were run at
even more boundary conditions to better
understand how these changes affected the
residual error. The residual errors of all the
different
boundary
conditions
are
summarized in Table 3 and displayed in
Figure 5.
Figure 4. A visualization of dye concentration at each point in the network over time. As in Figure 2; P1, P2, P3, P4,
P5, and P6 refer to the different discernible points in the network. It can be seen how the point of injection decays
after a peak concentration and the points along the Circle of Willis have a peak at a time corresponding to their
distance from the initial point and at a lower concentration. The inflow pressures used for these simulations are
displayed in Table 2. For experimental accuracy, these simulations were run at 0.1% artificial noise, the same as the
standard model displayed in Figure 3.
Simulation Inflow and Outflow Pressures
Outflow
Simulation
Inflow
(mmHg)
(mmHg)
A
2
1
B
15
1
C
50
1
D
5
1
Table 2. The values of inflow and outflow pressure of
the simulations displayed in Figure 3.
Residual Error between Standard Model and
Simulations of Inflow Pressure Changes
Simulation
Residual Error
A
126.49
B
29.46
C
163.39
D
160.99
Table 3. The residual errors between the standard
model in Figure 2 the different inflow pressures
shown in Table 2. and a sampling of different levels
of artificial noise for the same signal. All values
tested is displayed in Figure 5.
Figure 5. A residual error plot for simulations with
different inflow pressures. For these simulations, the
level of artificial noise was kept constant at 0.1% and
only the inflow pressure was varied. It can be seen
that there is a dramatic increase in error as the inflow
pressure approaches zero and a limit as it approaches
infinity.
Another way to test the efficacy of this
model is to measure the residual error
between the standard model and the same
simulation but with different levels of
artificial noise injected into the signal. For
the sake of maintaining experimental
integrity, the inflow and outflow pressures
were kept constant at 10 mmHg and 1
mmHg, respectively. Only the level of
artificial noise was varied. This is shown
below in Figure 6 and quantified in Table 4.
One phenomena of note that was observed
was when there was an inflow pressure 5
mmHg less than the standard model causing
the concentration curves to appear starkly
different. Concurrently, with an inflow
pressure 5 mmHg greater than the standard
model, the concentration curves appear quite
similar.
Future research will examine this strange
trend and attempt to explain why this change
arises.
Figure 6. A residual error plot for 10 simulations of
varying percentages of artificial noise added to the
signal. It can be seen that the more noise in the
signal, the larger the error becomes.
In regards to changing the amount of
artificial noise in the system, an exponential
increase in the residual error corresponds to
an increase in the percentage of noise in the
system. This was expected and verifies the
efficacy of the model for simulating blood
flow.
REFERENCES
Residual Error between Standard Model and
Levels of Artificial Noise
Artificial Noise (%)
Residual Error
0.25
0.014
1
0.499
2
2.223
3
5.178
Table 4. The residual errors between the standard
model in Figure 2 and a sampling of different levels
of artificial noise for the same signal. All values
tested are displayed in Figure 6.
DISCUSSION:
The efficacy of modeling cerebral
hemodynamics has now been shown. It is
possible to determine the flows and
pressures of a system based off of the
change in dye concentration at discreet
points in the brain over time. Furthermore, it
can be observed that a change in inflow
pressure dramatically influences the rate of
change in concentration at each point in the
model of the Circle of Willis.
Intellectual Property
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.
1. Kayembe, K. et al. (1984). Cerebral
Aneurysms and Variations in the Circle of
Willis. Stroke. 15, 5.
2. Kamath, S. (1981). Observations on the
length and diameter of vessels forming the
circle of Willis. Journal of Anatomy. 133, 3.
3. McEneaney, J. et al. (2012). Simulation of
capillary blood flow rates after occlusion
and vasodilation of left cerebral arteries.
4. Kulkarni, K. (2008). Mathematical
modeling, problem inversion and design of
distributed chemical and biological systems.