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Zaider Minerbo Model Solution
Equation

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The equation of the Zaider-Minerbo model:

$\displaystyle\frac{\partial}{\partial t}A(s,t)=(s-1)[bs-d-h(t)]\displaystyle\frac{\partial}{\partial s}A(s,t)$



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

$\Lambda(t)=e^{-\displaystyle\int_0^t[b-d-h(t')]dt'}$



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

$TCP(t)=\prod_{i=1}^M\left[1-\displaystyle\frac{1}{\left(\Lambda(t)+b\displaystyle\int_0^t\Lambda(u)du\right)}\right]^{v_i}$

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

ID:(4706, 0)


Simulator Models Poisson and Zaider Minerbo
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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:

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Correction to the Zaider Minerbo Model
Equation

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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:

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

Where the f function must be modeled separately.

ID:(4707, 0)