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Zaider-Minerbo Model Probability Equation
Equation

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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:

$\displaystyle\frac{d}{dt}P_i=(i-1)bP_{i-1}-i[b+d+h(t)]P_i+(i+1)(d+h(t))P_{i+1}$

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)

ID:(4705, 0)


Generatrix Function
Equation

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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.

ID:(8809, 0)


Equation of the Zaider-Minerbo Model
Equation

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With the generatrix function

$A(s,t)=\sum_{i=0}^{\infty}P_i(t)s^i$



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

$\displaystyle\frac{d}{dt}P_i=(i-1)bP_{i-1}-i[b+d+h(t)]P_i+(i+1)(d+h(t))P_{i+1}$



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

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

ID:(8810, 0)


Lambda Factor
Equation

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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

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

ID:(8808, 0)


Mortality Function
Equation

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

$h(t)=(\alpha+2\beta D(t))\displaystyle\frac{dD}{dt}$

ID:(8807, 0)


Cell Dynamics
Equation

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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

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

where the h function may vary over time.

ID:(8747, 0)