Zaider-Minerbo Model Probability Equation
Equation
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
* 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:
$\displaystyle\frac{d}{dt}P_i=(i-1)bP_{i-1}-i[b+d+h(t)]P_i+(i+1)(d+h(t))P_{i+1}$ |
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)
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Generatrix Function
Equation
To solve the equation of the model of Zaider-Minerbo a function generatrix
$A(s,t)=\sum_{i=0}^{\infty}P_i(t)s^i$ |
can be introduced.
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Equation of the Zaider-Minerbo Model
Equation
With the generatrix function
$A(s,t)=\sum_{i=0}^{\infty}P_i(t)s^i$ |
and the derivatives
we can rewrite the equation of Zaider Minerbo
$\displaystyle\frac{d}{dt}P_i=(i-1)bP_{i-1}-i[b+d+h(t)]P_i+(i+1)(d+h(t))P_{i+1}$ |
in which function A must satisfy the following partial differential equation:
$\displaystyle\frac{\partial}{\partial t}A(s,t)=(s-1)[bs-d-h(t)]\displaystyle\frac{\partial}{\partial s}A(s,t)$ |
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Lambda Factor
Equation
Solving 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 Lambda function is defined as
$\Lambda(t)=e^{-\displaystyle\int_0^t[b-d-h(t')]dt'}$ |
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Mortality Function
Equation
The
$h(t)=(\alpha+2\beta D(t))\displaystyle\frac{dD}{dt}$ |
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Cell Dynamics
Equation
At a time
new cells. At the same time
If it is added to this that a fraction
That is, the process is described by equation
$\displaystyle\frac{d}{dt}N=bN-(d+h(t))N$ |
where the
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Zaider Minerbo Model Solution
Equation
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
Because we are looking for a solution for which
it can be shown that this is of the form
with
$\Lambda(t)=e^{-\displaystyle\int_0^t[b-d-h(t')]dt'}$ |
With this it can be shown that the
$TCP(t)=\prod_{i=1}^M\left[1-\displaystyle\frac{1}{\left(\Lambda(t)+b\displaystyle\int_0^t\Lambda(u)du\right)}\right]^{v_i}$ |
The
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Simulator Models Poisson and Zaider Minerbo
Html
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
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Correction to the Zaider Minerbo Model
Equation
The Zerider Minerbo model is based on the population equation
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
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