Friday, December 30, 2011

Cancer..

Here is a primitive cancer model that quickly struck to mind.. Let me digg..

There are 'T' number of active white blood cells and there are 'c' number of cancer cells. Here goes my assumption

  1. Rate of production T cells is fixed. 
  2. Natural rate of death T is first order. 
  3. Assume a secondary rate of death for T, second order on T and c
  4. There is fixed nucleation of 'c' might be tiny. 
  5. First order growth of 'c' by cell division. 
  6. Second order death of 'c' by interaction with 'T'
  7. and we can assume natural death of 'c' which is first order. 
With all proportionality constants. 

dT/dt = A - B*T -C*T*c
dc/dt = D + E*c - F*T*c -G*c

From the nature of the cancel cells E>>G so let us assume a constant EG = E-G which is a positive
dc/dt = D + EG*c - F*T*c

What other conditions on the co-efficients?
  • C << F, just 'cas in general 'T' cells are effective in killing 'c'
  • For a not cancer situation, D/c + EG < F*T
  • Equilibrium white blood cells count = A/B, assuming 'c' is not there 
  • Steady state 'c' cells count = D /(F*T - EG) 
  • This means, T > EG/F is stable steady state. and T < EG/F is a run way system.
Note: So, by some external decease, if we take T less than EG/F, essentially we are out, assuming our immune system doesn't scale up. That means, A is fixed. One thing I did ignore is the fluctuations that are possible due to C*T*c


ps: I read some info regarding cancer when Steve Jobs died of the same. Above is the summary of the points I picked... 
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