Editing Logistic function (section)

=== In medicine: modeling of growth of tumors ===
{{See also|Gompertz curve#Growth of tumors}}
Another application of logistic curve is in medicine, where the logistic differential equation can be used to model the growth of [[neoplasm|tumors]]. This application can be considered an extension of the above-mentioned use in the framework of ecology (see also the [[Generalized logistic curve]], allowing for more parameters). Denoting with <math>X(t)</math> the size of the tumor at time <math>t</math>, its dynamics are governed by

<math display="block">X' = r\left(1 - \frac X K \right)X,</math>

which is of the type

<math display="block">X' = F(X)X, \quad F'(X) \le 0,</math>

where <math>F(X)</math> is the proliferation rate of the tumor.

If a course of [[chemotherapy]] is started with a log-kill effect, the equation may be revised to be

<math display="block">X' = r\left(1 - \frac X K \right)X - c(t) X,</math>

where <math>c(t)</math> is the therapy-induced death rate. In the idealized case of very long therapy, <math>c(t)</math> can be modeled as a periodic function (of period <math>T</math>) or (in case of continuous infusion therapy) as a constant function, and one has that

<math display="block">\frac 1 T \int_0^T c(t)\, dt > r \to \lim_{t \to +\infty} x(t) = 0,</math>

i.e. if the average therapy-induced death rate is greater than the baseline proliferation rate, then there is the eradication of the disease. Of course, this is an oversimplified model of both the growth and the therapy. For example, it does not take into account the evolution of clonal resistance, or the side-effects of the therapy on the patient. These factors can result in the eventual failure of chemotherapy, or its discontinuation.{{citation needed|date=February 2025}}