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Brief Review of the Zaider-Minerbo Model (ZMM)

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Brief Review of the Zaider-Minerbo Model (ZMM)

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Equations

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

dP_i/dt=(i-1)bP_i-1 - i[b+d+h(t)]P_i + (i+1)(d+h(t))P_i+1

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

dN/dt=bN-(d+h(t))N

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

h(t)=(alpha+2beta D)dD/dt

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

Lambda(t)=exp(-int(0,t,b-d-h))

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

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

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

Examples

Zaider-Minerbo Model Probability 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 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)

Generatrix Function

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

equation

can be introduced.

Equation of the Zaider-Minerbo Model

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

Lambda Factor

Solving the equation of the Zaider-Minerbo model

equation=8810

the Lambda function is defined as

equation

Mortality Function

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

equation

Cell Dynamics

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.

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