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Analyses and Simulation of Medicine
Carrying Blood Flow in MATLAB
MTH 499
Jill Mercik and Ryan Banci
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Our Research Project
 Simulating
through
MATLAB how blood
flows through the
human circulatory
system.
 1D
simulations
 Advisor: Prof. Yanlai
Chen
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Why are we doing this?
 Simulations
of the natural flow of blood can give us
information on how different medicines are
distributed through the body.
 This
information can help researchers to design
specific treatments that can improve upon patient
care.
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Overview
 Circulatory
System
 Mathematical Techniques
Finite Difference Method
 Navier-Stokes Equation

 Numerical
Scheme
 MATLAB

1D Simulations
 Conclusion

Future Work
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Circulatory System
Made up of:

Arteries

Arterioles

Capillaries

Veins

Heart
Blood is made up of:

Cells

plasma
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Mathematical Techniques
Finite Difference Method

Used to approximate the solutions to differential equations.

Algebraic in form and solutions are related to grid points.
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Finite Difference Method
Example
Consider the ODE:
u' ( x )  3u( x )  2
The Euler method uses the finite difference quotient:

u( x  h )  u( x )
h
 u' ( x )
In order to approximate the differential equation by first substituting

u' ( x ) and applying some algebra to get:
u( x  h )  u( x )  h ( 3u( x )  2)


Solving this last equation can give us an approximate solution to the
differential equation.
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Finite Difference Method
How it is used in our research project

We will use implicit finite difference.

Using the given equations we will create a system of
equations that we will put in matrix form in order to be
solved in MATLAB.

The solutions of this matrix of equations will give us the
values for A(cross sectional area) and q(flow rate) that we
need to simulate the branch for time domain n+1. We can
then apply this solution to solve the next sequential time
domain then repeat until we have reached our requested
simulation time.
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Mathematical Techniques
Navier Stokes Equations
What is it??
Nonlinear PDE’s the describe the motion of fluid substances.

Comes from applying Newton’s 2nd law to fluid motion.

F = ma
The Navier-Stokes Equations are used in a lot of practical
problems such a modeling the weather, ocean current, and
water flow in a pipe.
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Navier-Stokes Equations
General form of the Navier-Stokes equations that are
used to model blood flow are expressed as:
Equation (1) is a form of the momentum equation while equation
(2) is a volume continuity equation that was simplified from the
mass continuity equation.
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Navier-Stokes Equations
The previous slide’s equations can be
written in component form as:
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Numerical Scheme
In order to simulate a branch of the blood stream, we need to find
the flow rate (q) and cross-sectional area (A) for a section of points
along the branch and repeat this for each time step.
The more points (k) we use,
the more accurate the
calculation results will be.
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Numerical Scheme
We need a few equations in order to make a 1D simulation. The
equations for flow and pressure can be expressed through the
conservation of mass and momentum as:
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Numerical Scheme
From those equations, we can derive the equations we need
to find a system of equations that we will put into matrix form
to be solved in MATLAB.
We found diagonal patterns in the matrices that we could
code in a MATLAB script file to generate automatically given
a user input for the sample size k.
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MATLAB
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MATLAB
%Variables;
k=input('The sample size is: ');
dt=input('The change of time (delta t) is: ');
dx=input('The change of position (delta x) is: ');
K =(dt/dx);
E =(dt/(dx^2));
q = 1; %(68*10^-3);
A = 1; %(2*10^-11);
p = 1;
h0 = (6000*10^-9);
R0 = (2209*10^-9);
A0 = (1.5333*10^-11);
v = 1.1;
Estat = (7*10^-6);
a = (8/3)*(q/A);
b = ((-4/3)*(q/A)^2)+((A/p)*((Estat*h0)/(R0*A0)));
d = v;
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MATLAB
%Inlet conditions (B4, B5, B6)
g1=[1 zeros(1,k-1) (K/2) zeros(1,(k-2))];
g2=[0 1 zeros(1,k-2) 0 (K/4) zeros(1,(k-3))];
g3=[(-b*K/4) 0 (b*K/4) zeros(1,k-3) (1+E*d)
(a*K/4)-(E*d/2) zeros(1,(k-3))];
%Outlet conditions (B12, B13)
g4=[zeros(1,k) 1 zeros(1,k-5) (-K/2) 0 (K/2)];
g5=[zeros(1,2*k-2) 1];
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MATLAB
%Middle Matrix Begins (B2, B3)
x1 = ones(1,k-3);
x2 = diag(x1,0);
y1 = (K/4)*ones(1,k-3-1);
y2 = diag(y1,1);
y3 = (-K/4)*ones(1,k-3-1);
y4 = diag(y3,-1);
yout = y2 + y4;
j1 = (b*K/4)*ones(1,k-3-1);
j2 = diag(j1,1);
j3 = (-b*K/4)*ones(1,k-3-1);
j4 = diag(j3,-1);
jout= j2 + j4;
z1 = ((a*K/4)-(E*d/2))*ones(1,k-3);
z2 = diag(z1,0);
z3 = (1+E*d)*ones(1,k-3-1);
z4 = diag(z3,-1);
z5 = ((-a*K/4)-(E*d/2))*ones(1,k-3-2);
z6 = diag(z5,-2);
zout = z2 + z4 + z6;
b1 = zeros(2*k-6,2);
b2 = zeros(2*k-6,3);
%Middle Matrix Ends
%buffer zeros
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MATLAB
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1D Simulations

After solving the matrices with the given boundaries, we are able to
simulate the branch by applying the values of A (Cross Sectional
Area) and q (Flow rate) to each part of the branch for one time
domain. Then using the data from that time domain to solve the next
sequential domain.
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Conclusion
This project ended up being a lot harder than we originally
thought. We had hoped to create some 2D simulations, but were
only able to create 1D simulations. Modeling blood flow is a lot
harder than modeling something like water in a pipe because of
the different pulse flows and internal/external forces.
Future Work
We need to apply the program to a large sample size of 100 or
greater in order to solve for the n+1 time domain. Then we would
analyze the plots for flow rate vs. time and for flow vs. area. Also,
we will use the solutions for the solved n+1 time domain to solve
the next sequential time domains. This would complete our 1D
simulations. Then, the next logical step would be to create 2D
simulations, and ultimately 3D simulations.