In this repository, I would like provide some useful tools to construct ordinary differential equations (ODE) that can be used for describing the cardiovascular system. Due to the electronic–hydraulic analogy, the cardiovascular system can be represented by an electronic circuit as a so called lumped parameter network (LPN). Focus of this repository is the construction and analysis of LPN.

So far, only linear elements are provided.

Let us say, you want to model a vessel: Pressure and volume flow for inflow and outflow are indicated as P and Q in the figure. The vessel is elastic, meaning that it can change its overall volume. This phenomenon is also defined as windkessel effect and can be modeled by using the following circuit (see chapter "analogy" for more details): In other words, we used a combination of a conductor, coil and resistor.

Now, how to define the ODE for this circuit? How to get the relation between input and output?

In matrix form the task could look like:

where the task now is to find the matrix A_result.

For doing so, we can use the two-port network theory to a set of standard elements. We receive A_result by multiplying the matrices of all used elements in the right order:

$$A_{result} = A_C \cdot A_L \cdot A_R $$

In python, this would look like the following:

# loading cardioLPN from this repository
from cardioLPN import A_R, A_L, A_C
from sympy import symbols

# Defining the parameters R, L and C
R, L, C = symbols('R L C', positive=True)

# Computing A_result; The matrics A_C, A_L and A_R are implemented in cardioLPN
A_result = A_C(C)*A_L(L)*A_R(R)

Result for A_result:

>>> A_result
Matrix([
[  1,              L*s + R],
[C*s, C*L*s**2 + C*R*s + 1]])

A_result contains the relation between U_1, U_2, I_1 and I_2.

Let us assume that $U_1 = U_2$ and $I_1 = I_2$. Then, the ODE in s-domain is computed by

# Defining signals U and I
U, I = symbols('U I', positive=True)
ODE = A_result * x - x

The result for the ODE is

>>> ODE
Matrix([
[                        -I*(L*s + R)],
[C*U*s - I*(C*L*s**2 + C*R*s + 1) + I]])

The first equation is $$ -I(s) \cdot (L \cdot s + R) = 0$$

The second equation (after simplification): $$ U(s) - I(s)\cdot (L\cdot s + R) = 0$$

coming later...

Solving the ODE

# defining a function for U_2 and I_2
U_2 = 1/(s*(1+s*5))
I_2 = 1/(s*(1+s*15))

# defining
x_2 = Matrix([U_2, I_2])

x_1 = A_result * x_2

#
U_1 = x_1[0]
I_1 = x_1[1]

u_1 = inverse_laplace_transform(x_1[0], s, t)

Coming soon.

Example 1 (0D/3D coupling)

E. Kung, G. Pennati, F. Migliavacca, T.-Y. Hsia, R. Figliola, A. Marsden, and A. Giardini, “A Simulation Protocol for Exercise Physiology in Fontan Patients Using a Closed Loop Lumped-Parameter Model,” J. Biomech. Eng., vol. 136, no. 8, p. 81007, 2014.