CBS 613: COMPUTER LAB II (Invasion Waves)

In this lab we will simulate solutions to a model of tumor invasion into healthy tissue. The model is due to Perumpanani, Sherratt, Norbury and Byrne in Physica D, Vol 126 (1999): 145-159. This particular continuum model is a set of three coupled (nonlinear) partial differential equations which describe the spatio-temporal evolution of:

u(x,t) = tumor cell density at position x, time t

c(x,t) = collagen concentration at position x, time t

p(x,t) = protease concentration at position x, time t

The equations were:

where k_j, j=1,2,... are model parameters and K is natural protease deacy rate (which is very large relative to other rates). In the associated paper, the authors rescale u,c,p as well as x and t by parameter combinations with the same units (a process called nondimensionalization). For example,

Because the protease decay is fast (i.e., k is large), the last of these equations implies that very quickly P "equilibriates" to the product UC. Using this fact, it is sufficient to look at the model equations:

The model we will simulate using the "xtc" software is a slight variant which generalizes this reduced system:

Download XTC code

Click here (look at) to download a file named "invasion.xtc" which has code the program "xtc" runs. The program "xtc" will numerically solve the partial differential equations (using the method of lines) and display the results as it calculates it. After saving the file, in a terminal window type the UNIX command:

xtc invasion.xtc &

The code is preset to duplicate an invasion wave.

1) Click the "Go" button and watch the solution u(x,t) evolve.

This simulation is for d1=1 and d2=0 (no diffusion of tumor cells, just haptotaxis).

At the end of the run click

"Window"

then

"Fit 3d"

This will refresh the run and introduce the axis information. Note that in the program "xtc" (x-space,t-time,c-continuum) every mouse operation has an associated shortcut with keystrokes. The equivalent keystroke for any menu selection is the capital letter which occurs on the menu item. So, for instance, the keystroke you just performed could be accomplished by typing "W" then "F". At the end you should have something that looks like

The "x" or spatial axis is horizontal. The time t axis is vertical with t=0 at the top. Increasing t is in the downward direction. Dark areas represent large u(x,t) values (tumor cell concentrations). White represents low u(x,t) values.

Black and white is boring. To review in 3D do:

2a) Click "3D graphs"

2b) Select "3.SurfCol"

2c) Click "OK"

2d) Click "Window"

2e) Click "Fit 3d"

The "height" is u(x,t). You should have

The figure shows an invasion wave with a rather sharp front.

For the remainder of the lab we will use a 2D color representation of the solution. Do the following:

2a) Click "3D graphs"

2b) Select "0.Color"

2c) Click "OK"

2d) Click "Window"

2e) Click "Fit 3d"

You should have:

Question: Is the invasion propagation speed constant? How could you estimate the speed from the figure above?

Changing Initial Conditions

The previous simulation was for very specific initial conditions, i.e. u(x,0) (tumor cell concentration density at position x and time t=0) and c(x,0). Before they were preset. Let's change them and then redo the simulation:

3a) Click "Init. data"

3b) Click "New"

3c) Change u(x,0) to "1-heav(x-20)" (At the top of the big window)

3d) Hit "Return"

3e) Change c(x,0) to "0.7"

3f) Hit "Return"

3g) Click "Go"

Repeat the steps above changing only c(x,0) to 0.3.

Question: Which initial collagen concentration (c=0.7 or c=0.3) resulted in a quicker tumor invasion? Does this make sense to you?

Changing Parameter Values:

Let's change the model a little. The original model presummed that random cell motion (diffusion) to not contribute to the spatial transport of tumor cells into the tissue. We can add the effect of cell diffusion into the model by changing the parameter d2 from zero to some nonzero value:

4a) Click "Parameter"

4b) Type "d2" then "Return" at the top of the big window

4c) (backspace) replace the "0" with a "1"

4d) Hit "Return" twice

Using the same initial conditions as before, type "Go".

Question: Does the inclusion of random cell movement (diffusive transport) increase or decrease the wave speed? Does this make sense?

Note: d1=1 is unrealistically large but illustrates the point

Simulating localized "Chemotherapy"

I've added an extra term to the model equations which represents an external chemical killing of tumor cells in a localized section of the tissue for a designated duration and strength. The term "a*bx(x)*bt(t)" in the code does this. Stop here and let me explain.

The "treatment" is applied on a spatial interval xa 5a) Change u(x,0)=1-heav(x-2)

5b) Change c(x,0)=0.8

5c) Change the parameter a=0.1

5d) Click "Go"

You can see the effect of the treatment but it wasn't strong enoug to stop the invasion wave. Try:

5e) a=0.1, xa=10, xb=30

for a "wider" treatment zone...then "Go"

That had more of an effect. Try:

5f) a=0.15, xa=10, xb=30

For kicks, try:

5g) a=-0.1, xa=50, xb=60

What's going on here?

That's it for today: Quitting

6a) Click "File"

6b) Click "Quit"

6c) Click "Yes"