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)
ID:(4705, 0)
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.
ID:(8809, 0)
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:
$\displaystyle\frac{\partial}{\partial t}A(s,t)=(s-1)[bs-d-h(t)]\displaystyle\frac{\partial}{\partial s}A(s,t)$ |
ID:(8810, 0)
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
$\Lambda(t)=e^{-\displaystyle\int_0^t[b-d-h(t')]dt'}$ |
ID:(8808, 0)
Mortality Function
Equation
The
$h(t)=(\alpha+2\beta D(t))\displaystyle\frac{dD}{dt}$ |
ID:(8807, 0)
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
ID:(8747, 0)