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

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Zoe Bell on 15 Jan 2021
Commented: Zoe Bell on 28 Jan 2021
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.

Josh Meyer on 18 Jan 2021
This example shows how to solve a system where there is a material interface:
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 on 28 Jan 2021
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