Help me formulate a formuler

[Purgum]

Main Question or Discussion Point

If a tumor cell grows with rate m, and dies with rate n (m>n), their population number is P, after tme t, how can set up a mathematical fomuler for growth ? If i also have data from 100 patients, is it useful ?
Thanks

 

[HallsofIvy]

Essentially you are saying that the tumor cell grows with rate m-n so that you have the differential equation P'(t)= (m-n)P(t). Exponential functions have the property that their rate of change is a multiple of their value so P(t)= Ce^(m-n)t.
I'm not sure what data you have but if you have P(t) for at least 3 different times for each patient, you can use the data to determine C, m, and n for each patient. Just put the values for t and P(t) into the equation and you will have three equations to solve for C, m, and n. If you have more than 3 "data points" for a patient, you can check how well those values correspond to P(t) calculated from the formula to see how good that model is. Of course, it would be interesting to see if m and n are at least approximately the same for the different patients.

 

[mustafa]

If Purgum meant that rate is m-n, the differential eqn would be P'(t)=m-n.

However, I think, the data given is incomplete. The rate should be dependent in some way on the number of tumour cells; for example, the growth rate proportional to the no of tumour cells and death rate constant. The question given as such does not seem logical.

 

[HallsofIvy]

That's exactly what I said!

 

[EnumaElish]

... I'm not sure what data you have but if you have P(t) for at least 3 different times for each patient, you can use the data to determine C, m, and n for each patient. ...

If you have lots of data, you can statistically estimate m_i - n_i (= net growth rate for patient i) through the regression equation:

P_i(t+1) = b_i P_i(t) + u_i(t)
where b_i = m_i - n_i + 1 and you can test whether m_i - n_i > 0 by testing whether b_i > 1 (that is, whether b_i - 1 > 0);

or more generally:

P_i(t+1) = a_i + b_i P_i(t) + u_i(t),
where you can also test the expectation a_i = 0.

You need to check for serial autocorrelation; almost surely you will encounter positive autocorrelation (e.g. using Durbin-Watson test); if so, you'll need to correct for it.

You can also test whether m - n is identical across patients and/or over time for each patient, by building slightly more complicated regression equations.

Alternatively you may want to specify percentage growth rates (instead of linear as above).