07:52 pm
Last 2 Sundays have been spent in preparing the rough sketch of discrete parts of the model. To begin with, I divided the model into:
1. Fetal Heart (Pressure Source 1)
2. Maternal Uterine Arteries (Pressure Source 2)
3. Peripheral areas:
(PA.1) Brain
(PA.2) Kidneys
(PA.3) Peripherals at Primitive Iliac Artery Bifurcation
(PA.4) Peripherals at External Iliac Artery Bifurcation
(PA.5) Peripherals at Internal Iliac Artery Bifurcation
4. Arterial Segments:
(Seg.1) Ascending Aorta (Ductus Arteriosus)
(Seg.2) Thoracic Aorta
(Seg.3) Abdominal Aorta
(Seg.4) Primitive Iliac Artery
(Seg.5) Internal Iliac Artery
(Seg.6) Umbilical Arteries
(Seg.7) Uterine Arteries
5. Bifurcations:
(Bif.1) Cerebral Arteries
(Bif.2). Renal Arteries
(Bif.3) Primitive Iliac Arteries
(Bif.4) External Iliac Arteries
(Bif.5) Internal Iliac Arteries
6. Placenta on the fetal side of the feto-maternal circulation
7. Placenta on the maternal side of the feto-maternal circulation
1. Modeling of the fetal heart:Fetal Heart parameters, ventricular pressure and ventricular volume follow the relation,
p(t) = E(t)[V(t) - V0]
where,
p(t) => ventricular pressure in mmHg
V(t) => ventricular volume in ml
V0 => Reference Volume
E(t) => Ventricle Elastance
In this relation, p(t) and V(t) are unknowns. Thus, I had to create a pressure wave that approximates the one given in the paper. I used the following values:
time axis => [0, 0.049, 0.054, 0.059, 0.07, 0.08, 0.1, 0.149, 0.173, 0.198, 0.208, 0.217, 0.225, 0.251, 0.4]
pressure axis => [0, 0, 1.09, 1.49, 1.79, 1.93, 1.99, 1.99164, 2, 1.96, 1.9, 1.82, 1.65, 0, 0]
To find E(t), there exists an equation
E(t) = Emax En(t)
where En(t) = 5.412t^6 - 20.066t^5 + 25.542t^4 - 13.71t^3 + 2.714t^2 + 1.08t + 0.029
Here t is taken as the simulation time. Thus, to create the block, now we need to have the values of Emax and V0.
After the analysis of fetal lamb and dog heart and approximately equating it to human fetus, we obtain, Emax = 6 mmHg per ml and V0 = -8ml.
Error: The output produced shows pulsetile increase in pressure, but it does not reduce to the baseline!2. Modeling of the maternal uterine arteries:
I approximated the above shown waveform which was the only information provided about the uterine pressure. The values that I used are:
Pressure values => [78, 77, 78, 80, 90, 100, 110, 118, 122, 130, 131, 130, 122, 118, 112, 108, 103, 100, 90, 87, 84, 82, 84, 84, 80, 79, 78]
Time Values => [0, 0.05, 0.1, 0.12, 0.14, 0.16, 0.18, 0.2, 0.22, 0.24, 0.25, 0.26, 0.28, 0.3, 0.32, 0.34, 0.36, ,0.38, 0.4, 0.45, 0.5, 0.55, 0.6, 0.65, 0.7, 0.75, 0.8]
3. Peripheral Areas
Q(t) = {[p(t) - pc] / R} + {C dp(t)/dt}
where
Q(t) => Blood flow
R => Resistance of PA
C => Compliance of PA
pc => Critical Pressure of PA
With the above mentioned values from the paper and the equation mentioned, I created the PA basic subsystem in Simulink as shown:
and varied the pc, R, and C values for every PA.
Arterial Segments and Bifurcations : Main arterial tree
1. In the paper that we are referring to, they have specified the geometrical parameters on the fetal side of feto-maternal arterial tree for each of the segment and bifurcation for fetus at term. BUT, In reference to the blood flow in these segments they have NOT given any values for the parameters like
- the density of blood
- viscosity
which are necessary in the formation of the Navier-Stokes equation. This equation, together with mass and momentum conservation equationsform the constitutive equations of the model.
Suggested Solution: We could use values from any other reference.
2. This highly mathematical part of the model in the paper, which requires the solution to the Navier - Stokes Equation as solved using Method of Characteristics, has not been provided.
Suggested Solution: We could use the pdetool in MATLAB to solve the Navier- Stokes Equation for incompressible fluid. For this, I found this attached document on the internet. Asked my External Guide to go through it and see if we could use it, He happily showed the green signal. Now gotta move onto the next question.Since we see that in the above mentioned paper, twhy use the PDETool, to solve the fluid flow equations, and I am using Simulink, I need to somehow interface these two. For this I just found
this paper very useful.It gives an example in which they have attempted an interface between, Simulink and PDETool. Will be studying and implementing the model from these sources!
ATB!
