Serious Glitches..! Worried Now..

11:36 pm

Have been trying to avoid it, but that's going to be dangerous, gotto face it, and find a way out! Not run away! So here we go!

After reading the "Model of the Main Arterial Tree" part from the paper very carefully I have realised the following things:

The model is based on, firstly, the Mass Conservation Equation given by,

where,

 v is the mean blood flow velocity

S is the arterial cross-section

z is the distance from the aortic valve along the arterial tree

psi is a function describin outflow from the small arteries which are not geometrically depicted

They have mentioned in the paper that they’ve solved this equation using the method of characteristics. Studied it with the help of this video and tried to solve my equation in the following way: (hope it is all understandable)

As you can see, I encounter many problems while trying to solve the mass conservation equation.

Firstly, I do not know if v is a variable which is dependent on the value of z or not. Thinking about it, we realize that the velocity does actually change depending on its distance away from the heart. But that happens only due to the change in pressure with distance which we rae going to accommodate for. If we include this dependence, the equation becomes difficult to be solved using the method of characteristics. Saying this, coz I tried doing that as shown below:

Thus, I decided to assume it independent of z, and continued as shown in the first image.

Secondly, knowing that v is the mean velocity, we try to understand the meaning of the word mean velocity. Considering the fact that in the paper, they have mentioned that the objective of using this method is to be able to compute blood velocity and pressure at any point in the arterial tree as the contribution of the incident and reflected velocity and pressure. Which implies that, by mean velocity, they mean, the resultant of velocity and pressure from maternal side and the fetal side. This makes us think that the v mentioned, needs to be the input to the simulink block that will solve the mass conservation equation, and before entering this block, there needs to exist a simulink block that will calculate this mean velocity. At whichever point in the arterial tree we  need to check the velocity or pressure, we need to do the calculation for that part.

This conclusion gives rise to a controversy. Is this input velocity the same that will be shown as the output velocity? Cannot accurately answer this question right now, coz we haven’t come to that point in the process still, but all that I can guess right now, is that, this input velocity will be acted upon by the mass conservation equation, then the stokes navier equation etc. and get modified. Coming to think of it now, I feel that the input to the system is this mean velocity, but the output might be the pressure wave. This might be needed to be converted into velocity somehow, since none of the equations in this part give velocity as the output.<== Problem

Going into the depth of the above mentioned problem, now we need to

1. Figure out how to calculate the mean velocity
2. Find the expression that describes the relation between velocity and pressure of an incompressible fluid.

Once the mean velocity calculation problem is solved, we need to know how to incorporate this into the mass conservation equation, and design a simulink block that implements the mass conservation equation.

So now, let us talk about this solution of the mass conservation equation, In the equation, they have mentioned psi, whose value hasn’t been mentioned anywhere in the paper. Need to ask what value to input as PSI. Assuming currently that it is zero, I move ahead. As shown in the first image, this gives the solution as

S= f(z_o)=f(z-vt)

This equation says that the cross sectional area for the curve represented by z-vt = z_o is given by the function which is the initial condition at t=0 for that curve.

For all arterial segment we have been given the length of the segment and the cross sectional area at its 2 ends. The table is as follows:

Now as the solution the cross section needs to be a function of z-vt. What function? As the tutorial video suggests, this function is the initial condition given. What is the initial condition given to us? These are just numbers. Thus we need to form relationships, preferably, linear from the given values of cross sections and length. To do this, I used curve fitting tool from MATLAB. Here’s the video of how I did it. Sorry for the delay between video and audio, and the overall bad quality.

Okay, the video refuses to get uploaded, sorry! Here are the equations that I got:

For Arterial segment 2,   

For Arterial segment 3,   

For Arterial segment 4,  

For all the above expressions, y is the respective Cross section and x is the respective position coordinate.

For arterial segment 1 and arterial segment 5, the cross section remains constant. That is it does not change for any value of length. Thus the initial value remains the same for all space coordinates.

Now that we have a relationship that describes the initial condition and relates cross section with the z coordinates, we need to figure out how to incorporate the value of z into the initial condition and how to incorporate the initial condition into the solution.

For each of the above x, we substitute z- vt where z is the distance from the aortic valve along the arterial tree to the beginning / end (?) of the arterial segment, v is the computed mean velocity and t is the simulation time. So now, we have got the logic for doing a part of the Arterial segments modeling.

After this, is the block that computes the arterial vascular elasticity and incorporates that into the model. This part is given as follows:

So, c₁(p) is a polynomial block, and c₂(z) is a waveform to be approximated. these can be implemented on SIMULINK.

All this to be implemented on MATLAB and have to think about the mean velocity block, tomorrow!

ATB!