SIMULATION OF BLOOD FLOW WITHIN THE BRAIN Sharif Kurdi

kurdi2@uic.edu

This report is produced under the supervision of BIOE310 instructor Prof. Linninger.

Abstract Cerebrovascular diseases are diseases including strokes that constrict the blood flow within the

brain causing major complications for a person [3]. This stoppage of blood flow can happen in

any artery of the brain, as well as the Circle of Willis. The goal of this project is to estimate blood

flow through a specific network of blood vessels in the brain. This project uses three main

equations: the Hagen Poiseuille equation, Conservation of Flow equation, and Species Transport

Equation. The first two equations mentioned are used to solve for the flows and pressure of a

network while the Species Transport Equation solves for the concentration values. Different

boundary conditions (pressure inlet and pressure outlet values) are chosen and the differences

between concentrations over time graph curves, as well as the difference in residuals, are

compared and analyzed. Overall this paper proposes a model that can help estimate blood flow

within the brain in hopes that it will be useful for doctors and potential cerebrovascular diseased

patients in the near future.

Keywords: Circle of Willis, Dye Concentration, Blood Flow, Cerebrovascular Disease.

1. Introduction

Cerebral vascular disease is known to be the fourth

leading cause of death in the United States; according

to the CDC there are approximately 128,932 deaths

from cerebrovascular diseases a year in the U.S. [1].

A few types of cerebral vascular disease include

strokes, vertebral stenosis, aneurysms, and vascular

malformation to name a few [3]. These diseases

attack and affect the circulation of blood flow within

the brain [3] as well as arteries within the Circle of

Willis, which can be seen in Figure 1.

The Circle of Willis’ main function is to help supply

blood to the brain and is made up of the anterior

cerebral artery (ACA), anterior communicating

artery, internal carotid artery (ICA), posterior

cerebral artery (PCA), and posterior communicating

artery; the basilar artery and middle cerebral artery

are also closely related [2]. Without going into too

much detail, the posterior cerebral arteries (PCA)

streams blood to the back of the brain, while the

anterior and middle cerebral artery supply blood to

parts of the frontal and parietal lobes of the brain [2].

The goal of this project is attempting to estimate

blood flow through a network within the brain. A

current technique that is used for this matter is digital

subtraction angiography (DSA), though this method

is flawed. This method only gives a 2D image of the -

agent over time. In this proposed method a simulation

model to predict blood flow will be created and

compared to an estimation model to see if the flow

values are correct. If this method can surpass the

current one at hand, doctors could be supplied with

the successful models and more effectively and

clearly estimate blood flow within the brain while

ultimately saving a number of lives in the process.

Figure 1 – The Circle of Willis is a group of arteries in the

brain that help supply the brain with blood [1].

2. Methods

2.1 Solving for Flows and Pressures

The first step in creating this model is solving flows

and pressures within a network; in this case the flows

and pressures were solved first within a bifurcation

network, and second in a more complicated Circle of

Willis shaped network. In order to solve the flows

and pressure the following equations were used:

∑ 𝐹𝐼𝑛 = ∑ 𝐹𝑂𝑢𝑡 (1)

This Conservation of Flow equation is simply saying

that the summation of the flow going into the

network must be equal to the summation of flow

going out of the network.

∆𝑃 = 𝛼𝐹; 𝛼 =

8𝜇𝐿

𝜋𝑟4 (2)

This Hagen-Pouiseuille equation is affirming that the

change in pressure (∆P) is equaled to the resistance

(α) multiplied by the flow (F).

It is necessary to note that while using these two

equations to solve for the flows and pressures it is

assumed that the resistance is equaled to one. The

inlet and outlet pressure values (boundary conditions)

are also known values. These equations were

subsequently implemented into and solved using

MATLAB.

2.2 Solving for Concentration Values

After solving for the flows and pressure, the

concentration values through a network over a certain

amount of time were solved for. To complete this a

third equation is needed:

VdC

dt= FInCIn − FOutCOut + Inj (3)

In other words this equation (species-transport

equation) is stating that volume (V) multiplied by the

derivative of concentration over an amount of time

(dC

dt) is equaled to the inflow (FIn) multiplied by

concentration going into a network (CIn), subtracted

by the outflow (FOut) multiplied by the concentration

leaving the network (COut) plus injection rate (Inj). It

is important to note that when carrying out this

equation that volume and injection rate values are

known.

To solve the concentrations using equation (3),

Implicit Euler’s Method was applied in MATLAB.

Implicit Euler’s Method will take the differential

equation and convert it into an algebraic function;

subsequently concentration over time graphs will be

plotted with the new algebraic equations.

2.3 Optimization

The next method completed was using different

boundary conditions and plugging them into each

network which will give unique pressure, flow, and

concentration values. The residual values of each

graph compared to the original one were then found

and analyzed; this is more clearly explained in the

results and the discussion sections.

2.4 Noise

The optimal boundary conditions concentration over

time graph for the Circle of Willis Network was

given different levels of noise using MATLAB

commands, and the residual values of each unique set

of boundary conditions were examined.

3. Results

It is critical to mention that in a normal scenario the

boundary conditions (PIn and POut values) would

initially be altered in an attempt to fit the model that

is being dealt with. In other words the values and

curves must be as similar as possible, resulting in a

residual as close to zero as possible. In this project,

however, the ideal boundary conditions are known

beforehand as are shown in Figure 2A and 3A. Figure

2B-2F and Figure 3B-3F represent unique boundary

conditions, and panels H illustrate the bifurcation

network (Figure 2) and Circle of Willis network

(Figure 3) worked with.

3.1 Bifurcation Network

In Figure 2G, the residual over outlet pressure graph,

it displays that the low point of the residual seems to

be at 5mmHg, and the residual continuously increases

the further the outlet pressure is from 5mmHg. The

residual values increase more rapidly when choosing

outlet pressures larger than 5mmhHg than outlet

pressures lower than it.

It can also be observed that a larger pressure outlet

value results in more elongated curves while a lower

pressure outlet values gives off a slightly more

condensed curve. The inlet pressure works in reverse

as the higher the inlet pressure value is, the more

compressed the curves are; and the lower the inlet

pressure is the more drawn-out each curve is.

Figure 2 - Panels A – F represent concentration over time graphs for a bifurcation network given unique boundary conditions;

Panel A displays ideal boundary conditions. Panel G represents the relationship between the outlet pressure boundary condition

and the residual value for the network; from the graph it’s clear that 5 is the optimal outlet pressure, as the residual equals zero.

Panel G illustrates the bifurcation network that is being dealt with including flow direction, and concentration points.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 5 10 15 20 250

0.005

0.01

0.015

0.02

0.025

0.03 Bifurcation Network

F1

F2F3

C1

C2

C3

C4

P1

P2

P3

P4

Time

Time

Time

Time

Time

Time

Co

nce

ntr

atio

n

Co

nce

ntr

atio

n

Co

nce

ntr

atio

n

Co

nce

ntr

atio

n

Co

nce

ntr

atio

n

Optimal Conditions

PIn=100mmHg POut=5mmHg

PIn=100 mmHg POut=1mmHg

PIn=200mmHg POut=5mmHg

PIn=100mmHg POut=50mmHg

PIn=1000mmHg POut=50mmHg

PIn=50mmHg POut=5mmHg D

B

C

A

E F

G H

Outlet Pressure

Figure 3 - Panels A – F represent concentration over time graphs for a Circle of Willis network given unique boundary

conditions; Panel A displays ideal boundary conditions. Panel G represents the relationship between the inlet pressure boundary

condition and the residual value for the network; from the graph it’s clear that 10 is the optimal inlet pressure

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20 250

200

400

600

800

1000

1200

1400Circle of Willis Network

C1

C2

C3 C4

C5

C6

C7

C8

C9

C10

C11

C12

C13

C14 C15 C16

C17

P1

P2

P3

P4

P5

P6

P7

P8

P9P10

P11

P12

P13

P14

P15 P16

P17

Time

Co

nce

ntr

atio

n

Optimal Conditions

PIn =10mmHg POut=1mmHg

Time

Co

nce

ntr

atio

n

PIn=5mmHg POut =1mmHg

Time

Co

nce

ntr

atio

n

PIn =2mmHg POut =1mmHg

Co

nce

ntr

atio

n

Time

PIn =15mmHg POut =1mmHg

Co

nce

ntr

atio

n

PIn =20mmHg POut =1mmHg

Time

PIn =25mmHg POut =1mmHg

Co

nce

ntr

atio

n

A B

B

C D

E F

G

Time

H

Inlet Pressure

The following table, represented as Table 1, shows

the relationship between the outlet pressure and the

residual value within a bifurcation network and

depicts the same data as the graph in Figure 2G. The

outlet pressure value of 5mmHg is ideal, as the

residual is 0 at that point.

POut Value

(mmHg)

PIn Value

(mmHg)

Residual

1 100 0.0016

5 100 0

9 100 0.0019

13 100 0.008

17 100 0.019

21 100 0.03

3.2 Circle of Willis Network

Figure 3A-3F depict different concentration over

time graphs based on their given boundary

conditions; Figure 3A contains ideal boundary

conditions. From Figure 3B and Figure 3C it seems

that a lower inlet pressure value resulted in more

elongated and curves while a higher inlet pressure

value shown in Figure 3D-3F stemmed a more

condensed or compressed curve.

In Figure 3G, the residual vs. inlet pressure graph, it

displays that the low point of the residual seems to be

at 10, and the residual continuously increases from

that point whether the inlet pressure increases or

decreases; this is will be explained in the discussion

section.

The following table, represented as Table 2, shows

the relationship between the inlet pressure and the

residual value within a Circle of Willis network and

depicts the same data as Figure 3G.

POut Value

(mmHg) PIn Value

(mmHg)

Residual

(mmHg)

1 2 1264.8

1 5 161.5

1 10 0

1 15 29.7

1 20 68.3

1 25 97.7

3.3 Circle of Willis Network with Noise

Figure 4 – Optimal Boundary Conditions with a Signal to

Noise Ratio of 5dB.

To test the effect that adding noise to a concentration

over time graph has on the residual for each graph,

noise was added to the Circle of Willis optimal

boundary conditions as expressed in Figure 3A.

Figure 4 and Figures 6A, 6C, 6E, and 6F shows noise

added with signal to noise ratios of 5dB, 10dB, 15dB,

20dB, and 25dB respectively.

Figure 5 – Residual over Inlet Pressure graph comparing

each set of boundary conditions’ residuals with the optimal

set of boundary conditions with added noise.

Figure 5 and Figures 6B, 6D, 6F, 6H all show the

relationship between the residual and the inlet

pressure when a signal to noise ratio of 5dB, 10dB,

15dB, 20dB, and 25dB respectively is added. The

graphs are all very identical with one another.

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20 250

200

400

600

800

1000

1200

1400

PIn and Residual Relationship

POut and Residual Relationship

Co

nce

ntr

atio

n

Res

idual

Time

Inlet Pressure

Signal to Noise Ratio=5dB

Figure 6 – Panels A,C,E,G represent concentration over time graphs with varied amounts of noise, while panels B,D,F,H

represent the relationship between the residual and inlet pressure at each amount of noise.

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20 250

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20 250

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20 250

200

400

600

800

1000

1200

1400

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20 250

200

400

600

800

1000

1200

1400

Res

idual

R

esid

ual

R

esid

ual

R

esid

ual

Inlet Pressure

Inlet Pressure

Inlet Pressure

Inlet Pressure

Time

Time

Time

Time

Co

nce

ntr

atio

n

Co

nce

ntr

atio

n

Co

nce

ntr

atio

n

Co

nce

ntr

atio

n

A

G H

F

D

E

C

B Signal to Noise Ratio=10dB

Signal to Noise Ratio=15dB

Signal to Noise Ratio=20dB

Signal to Noise Ratio=25dB

4. Discussion

4.1 Bifurcation Network

As shown in Figure 2B the POut value was reduced to

1mmHg while the PIn value was unchanged at

100mmHg. It’s clear that there isn’t much of a

difference between this graph and the optimal graph

in Figure 2A. This is furthermore supported from the

fact that the residual is the extremely low value of

0.0016. From this graph it can be deduced that

merely lowering the POut value by 4, to the largest

whole number, has a very insignificant effect on the

concentration values, though it does very slightly

condense the graph’s curves.

As represented in Figure 2C the next set of boundary

conditions uses the POut values of 50mmHg, and PIn

value of 100mmHg. From raising the POut value

tenfold and leaving the PIn value the same, it is clear

that the curves reach a higher concentration value

than the original’s, and subside at a later time. This

proves that a larger POut value will give a larger

concentration value at each node, and will take the

concentration a longer amount of time to leave the

network. This makes sense, because a larger POut

value will result in smaller flow values in bifurcation

networks. Smaller flow values will subsequently keep

the dye in each node for a longer amount of time. The

residual value is at 0.5764 for this graph which

makes sense, as there visually is a higher amount of

error from the ideal graph in panel A than there was

in panel B.

In the case of Figure 2D the POut is kept at 5mmHg

(from the original model), and instead the PIn value is

halved to 50mmHg. These curves look very similar to

the previous graph’s curves, because the

concentration values are larger at each node, along

with the time elapsed for the dye to leave the network

being longer. This makes complete sense: since the

PIn value is lesser than the ideal, it takes a longer time

for the dye to enter the network resulting in a longer

amount of time for the dye to leave the network. The

larger elapsed time can also be explained by the

smaller flow values that are obtained by the lower

pressure inlet values. After analyzing these last few

graphs it’s clear that a rise in the POut value OR a

drop in the PIn value give off the same result: higher

concentration values and more time being elapsed

before the dye completely leaves the network. The

residual of the graph is 0.8193 which signifies that

we are getting farther from the solution compared to

the last few.

The graph in Figure 2E was created using the POut

value of 5mmHg (unchanged from original) and a PIn

of 200mmHg (doubled from the original). Compared

to the original this graph has more condensed curves;

the curves not only have overall smaller

concentration values than our original, but a smaller

amount of time passed for all of the dye to leave the

network. This is the effect drastically raising the PIn

value gives which is logical: lowering the PIn value

lowers flow, but raising PIn raises the flow value; this

results in the dye being pushed through the network

more quickly. The residual value of this graph is

0.3068 which is closer to 0 than our last. This

signifies that doubling our PIn gives is a more correct

solution than halving our PIn value.

The last concentration over dye graph for this section,

labeled as Figure 2F, had PIn and POut values 10x

larger than our optimal boundary condition values.

This graph has the lowest concentration and least

amount of time elapsed thus far by a large amount as

well as the greatest residual (1.0209). This graph

helps to understand the relationship between

boundary conditions well in the bifurcation network:

the farther apart the boundary conditions are from

one another, the smaller the concentration values and

time elapsed will be for each graph (assuming POut is

lower than PIn). The residual values will also be

greater as the distance between the two initial

conditions differ assuming PIn > 100mmHg (the

optimal output value).

The first table describes the relationship between the

outlet pressure value and the residual value; the ideal

outlet pressure value appears to be 5mmHg as already

known; this is signified by the residual equaling zero

at that boundary condition. It is also clear that the

farther away the inlet pressure is from the ideal inlet

value, the larger the residual will be; in other words

the farther away the boundary conditions are from

each other the higher the percent error will be. The

same data can be looked at visually within Figure 2G.

4.2 Circle of Willis Network

Concerning the Circle of Willis network the inlet

pressures were the focus and subsequently were the

only boundary condition changed. The main point to

be taken out of these concentration over time graphs

for the Circle of Willis is simple: the larger the inlet

pressure boundary condition becomes, the higher the

flow will be. On the other side of things the lower the

inlet pressure value becomes, the lower the flow will

become. Since flow has a major effect on the time it

take for all of the dye to completely leave the

network, it is able to be said that the more flow a

network has, the less time it takes to flow out of the

network; and the smaller the flow values are the

longer it takes to leave the network. This can be

proven by Figures 3A-3F. It can also be deduced from

these graphs that merely changing inlet pressure (at

the values used) does not affect the value of the peak

concentration. The peak point looks to be the same

for each graph at each individual node.

Figure 3G and table 2 show the relationship between

the inlet pressure boundary condition and the

residual. From looking at either demonstration it is

confirmed that an inlet pressure of 10mmHg is an

ideal boundary condition as that is when the residual

reaches 0. It is also shown by the graph and table that

the farther the inlet pressure is from the perfect

pressure, the larger the residual will become.

Therefore it can be deduced that the farther away the

inlet pressure is from 10mmHg, the higher the error.

4.3 Circle of Willis Network with Noise

After adding noise to the optimal boundary

conditions of Circle of Willis, one thing is clear: the

more noise you add to a signal, the worse the residual

value is all around. In other words the lower the

signal to noise ratio is, the higher the residual values

will be. Noise is generally unwanted data which

worsens and interferes with the results [4] so it is

logical that the more noise present would result in a

larger residual for every single set of boundary

conditions tested.

5. Conclusion

To conclude the goal was to simulate blood flow

within the brain, specifically the Circle of Willis to

help supply doctors with a more advanced and

efficient way to predict blood flow. The Conservation

of Flow Equation, Hagen Poiseuille Equation, and the

Species transport equation were used to find the

Flows, Pressures, and Concentration values

respectively within a bifurcation network as well as a

Circle of Willis network. Pressure boundary

conditions then must be continuously altered in an

attempt to find the most accurate set of boundary

conditions; in other words to find the boundary

conditions that give the closest residual value to zero.

The residual values were found and analyzed in

conditions with and without noise. It was deduced

that the more noise present, the higher the residual

value would be.

Acknowledgement

The author would like to acknowledge Professor

Andreas Linninger and especially Teaching Assistant

Chih-Yang Hsu for helping and guiding him through

the project.

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.

7. References

1. CDC. "Cerebrovascular Disease or

Stroke." Centers for Disease Control and Prevention.

Centers for Disease Control and Prevention, 14 July

2014. Web. 25 Nov. 2014.

2. "Circle of Willis." Wikipedia. Wikimedia

Foundation, 24 Nov. 2014. Web. 25 Nov. 2014.

3. "What Is Cerebrovascular Disease? What Causes

Cerebrovascula Disease?" Medical News Today.

MediLexicon International, 6 Apr. 2010. Web. 28

Nov. 2014.

4. "Noise." Wikipedia. Wikimedia Foundation, 12

Nov. 2014. Web. 11 Dec. 2014.