# Help me formulate a formuler

## 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
Homework Helper
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.

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
Homework Helper
That's exactly what I said!

EnumaElish
Homework Helper
HallsofIvy said:
... 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 mi - ni (= net growth rate for patient i) through the regression equation:

Pi(t+1) = bi Pi(t) + ui(t)
where bi = mi - ni + 1 and you can test whether mi - ni > 0 by testing whether bi > 1 (that is, whether bi - 1 > 0);

or more generally:

Pi(t+1) = ai + bi Pi(t) + ui(t),
where you can also test the expectation ai = 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).

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