I am not familiar with this much-advanced probability and not even this much-advanced biology. I am confused about what does this equation finally comes to. I am not able to derive the value of
[MATH]f_n(\tau ) = \int f_n ( \tau , \lambda) * \gamma( \lambda \: | \: k , \theta) \: d\lambda [/MATH]
where [MATH]F_n(s) = 1- e^{- \frac{\mu_n}{\xi +1} * S(\xi +1)}[/MATH]
FYI: [MATH]\lambda[/MATH] is the growth rate related to the tumor doubling time. The growth rate follows a gamma distribution, with shape and scale parameters [MATH] K \text{ and } \theta.[/MATH]
We define the tumor cell’s activity [MATH]\alpha[/MATH], to which the growth rate [MATH]\lambda[/MATH] is proportional, [MATH]\lambda = \epsilon_1 * \alpha[/MATH] (where [MATH]\epsilon_1 [/MATH] is a constant). The cell detachment rate, [MATH]\beta[/MATH], is also proportional to [MATH]\alpha[/MATH], [MATH]\beta = \epsilon_2 * \alpha[/MATH] (where [MATH]\epsilon_2[/MATH] is another constant). Thus, [MATH]\beta = \frac{\epsilon_2}{\epsilon_1} \lambda[/MATH], where [MATH]\frac{\epsilon_2}{\epsilon_1} = \xi[/MATH] is a parameter representing the relationship between [MATH]\beta \text{ and } \lambda[/MATH]. If the tumor with volume S grows exponentially, [MATH]S=e^{\lambda * t}[/MATH] , the total number of detached cells before time [MATH]\tau_0[/MATH] is [MATH]e^{\xi \lambda \tau_0} = S_0^{\xi}[/MATH]; we assume [MATH]0<\xi<1[/MATH], the interpretation of which is that cells always detach from the primary tumor but not all tumor cells
will detach.
For more information, you can refer to this article (pg 2)
[MATH]f_n(\tau ) = \int f_n ( \tau , \lambda) * \gamma( \lambda \: | \: k , \theta) \: d\lambda [/MATH]
where [MATH]F_n(s) = 1- e^{- \frac{\mu_n}{\xi +1} * S(\xi +1)}[/MATH]
FYI: [MATH]\lambda[/MATH] is the growth rate related to the tumor doubling time. The growth rate follows a gamma distribution, with shape and scale parameters [MATH] K \text{ and } \theta.[/MATH]
We define the tumor cell’s activity [MATH]\alpha[/MATH], to which the growth rate [MATH]\lambda[/MATH] is proportional, [MATH]\lambda = \epsilon_1 * \alpha[/MATH] (where [MATH]\epsilon_1 [/MATH] is a constant). The cell detachment rate, [MATH]\beta[/MATH], is also proportional to [MATH]\alpha[/MATH], [MATH]\beta = \epsilon_2 * \alpha[/MATH] (where [MATH]\epsilon_2[/MATH] is another constant). Thus, [MATH]\beta = \frac{\epsilon_2}{\epsilon_1} \lambda[/MATH], where [MATH]\frac{\epsilon_2}{\epsilon_1} = \xi[/MATH] is a parameter representing the relationship between [MATH]\beta \text{ and } \lambda[/MATH]. If the tumor with volume S grows exponentially, [MATH]S=e^{\lambda * t}[/MATH] , the total number of detached cells before time [MATH]\tau_0[/MATH] is [MATH]e^{\xi \lambda \tau_0} = S_0^{\xi}[/MATH]; we assume [MATH]0<\xi<1[/MATH], the interpretation of which is that cells always detach from the primary tumor but not all tumor cells
will detach.
For more information, you can refer to this article (pg 2)
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