Can pdepe be used when there is a boundary condition in the centre of the system - Invasion assay

So I am trying to model cells going through an invasion chamber so there are cells in the top of the chamber and food in the bottom of the chamber. The top and bottom chambers are separated by a permeable membrane that the cells can pass through. This means that there are boundary coniditons that apply to the top and bottom of the system (or left and right in the notation that it used for boundary conditions when using pdepe). But there are also boundary conditions that apply inbetween the top and bottom chamber. The boundary conditions at the top and bottom of the chamber are zero flux conditions meaning that the cells or fluid can't leave the chamber. The boundary conditions for the centre of the chamber are Kedem-Katchalsky boundary conditions meaning that the flux through the membrane is proportional to the concentration difference on either side of the membrane.

I have used pdepe before for a model when there was just left and right boundary conditions but I was wondering if I can also be used when there is also a boundary condition on the centre of the system as well as left and right.

Answer 1

[Josh Meyer]

This example shows how to solve a system where there is a material interface:

https://www.mathworks.com/help/matlab/math/solve-pde-with-material-interface.html

You might be able to use the techniques there for your problem if you can express the central boundary conditions as a change in the equations. Another option would be to split your integration up into two calculations: one from the top of the chamber to the membrane, and then another from the membrane to the bottom. You can use the results from the first simulation as the initial conditions of the second to pick up where you left off.

[Zoe Bell]

Thank you for your suggestion. In the paper that I am trying to recreate, they solve the system of two pdes using the finite difference method. I am finding it hard to find any examples of code where a system of two equations have been solved using finite difference. Do you know where I can find an example or how to alter code that solves 1 equation to solve 2?

[Josh Meyer]

The burgersode example file uses finite differences to approximate partial derivatives taken with respect to x, and this effectively converts the PDE into an ODE of one variable so that the solver can solve for the derivatives taken with respect to time. Type "edit burgersode" in MATLAB to see it. I don't think all of the techniques displayed in that example are applicable here, and the system of equations is much larger than 2, but it's an example that uses finite differences nonetheless.

Can you provide a reference link for the paper you're examining?

[Zoe Bell]

Thanks again for the reply. This is the paper that I am trying to recreate the model of, it has the breakdown of the equations that were used for the finite difference method, I am just unsure where to start https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0162553

[Josh Meyer]

This should get you started:

1. Write down the equations assuming all constants are equal to 1 (ignore them for now). Expand the derivatives out explicitly and see if you can get the equations into the form supported by PDEPE. If you can, then you can start writing a program to solve the problem using the constant values outlined in the paper. You would most likely need to account for the central boundary by breaking the integration into two as I outlined above.
2. If the equations can't be put into a form supported by PDEPE, then you'll need to use finite differences to solve the problem, as they did to perform the simulations in the paper. This will make the code longer and it will require more debugging time, so I'd recommend starting with a test problem before scaling up. For that, here is a resource I found with a quick search: Lecture 8: Solving the Heat, Laplace and Wave equations using finite difference methods.

[Zoe Bell]

Thanks again for the suggestions. I have tried to implement it using pedpe but for now I have ignored the central boundary so I am just assuming that the cells are able to easily move through the central boundary. I need to plot time on the x axis against the cell density on the y axis but when I try to plot it, I get a curve for every x value where x is the depth of the well starting from the top. Instead of matab just plotting the value at that one x point, I almost need it to be cumulativite so I get a full picture of what is happening through the depth of the well instead of just at one x position. Is there a way to make this happen? Sorry it is hard for me to fully explain what I mean