The key to Zaider Minerbo's model is the introduction and solution of a differential equation that allows us to determine how the probability of having a population of i cancer cells in time P_i (t) varies. To do this, it introduces the factors of probability of birth of a cell b, of natural death d and of death by effect of treatment h. With this, the probability varies according to cells that reach the universe of i cells by:

* Birth of a cell in the population $P_{i-1}$
* By death of a cell in the population $P {i + 1}$

It also considers that the number is reduced to the extent that:

* A cell dies by increasing the population of $P{i-1}$
* A new one is born by increasing the population of $P {i + 1}$

In this way the resulting equation is:

equation

For more details see the original paper at:

Tumor control probability: a formulation applicable to any temporal protocol of dose delivery
M.Zaider and G.N.Minerbo

[Phys. Med. Biol. 45 (2000) 279-293] (http://downloads.gphysics.net/papers/ZaiderMinerbo2000.pdf)

To solve the equation of the model of Zaider-Minerbo a function generatrix

equation

can be introduced.

With the generatrix function

equation=8809

and the derivatives

P_i(t)=\displaystyle\frac{1}{i!}[\displaystyle\frac{\partial^i}{\partial s^i}A]_{s=0}

we can rewrite the equation of Zaider Minerbo

equation=4705

in which function A must satisfy the following partial differential equation:

equation

Solving the equation of the Zaider-Minerbo model

equation=8810

the Lambda function is defined as

equation

The h that is used to calculate the Lambda of the Zaider-Minerbo model is calculated by the equation:

equation

At a time dt and the N cells will multiply at a rate b so there will be a total of

bNdt

new cells. At the same time dt of the N cells died by natural causes a fraction d so they will be lost

dNdt

If it is added to this that a fraction h dies by effect of the radiation it is necessary that the total number will vary in

dN=bNdt - (d+h)Ndt

That is, the process is described by equation

equation

where the h function may vary over time.

The equation of the Zaider-Minerbo model:

equation=8810

The solution of this equation will allow us to calculate the TCP(t) since

TCP(t)=A(s=0,t)

Because we are looking for a solution for which

A(s,0)=s^n

it can be shown that this is of the form

A(s,t)=\left[1-\displaystyle\frac{1}{\left(\displaystyle\frac{\Lambda(t)}{1-s}+b\displaystyle\int_0^t\Lambda(t')dt'\right)}\right]

with

equation=8808

With this it can be shown that the TCP function is of the form:

equation

The h function can be modeled with the L-Q model for the applicable dose history.

The following in a simulator that allows to calculate the TCP both under Poisson and Zaider Minerbo assuming two types of cells (birth rate, death, factors \alpha and \beta) dose and number of treatments:

The Zerider Minerbo model is based on the population equation

\displaystyle\frac{d}{dt}N=(b-d+h(t))N

however the births can be conditioned by what the generalization of the model can be based on the more general equation:

equation

Where the f function must be modeled separately.