Modeling age-dependent radiation-induced second cancer risks and estimation of mutation rate: an evolutionary approach

Abstract

Although the survival rate of cancer patients has significantly increased due to advances in anti-cancer therapeutics, one of the major side effects of these therapies, particularly radiotherapy, is the potential manifestation of radiation-induced secondary malignancies. In this work, a novel evolutionary stochastic model is introduced that couples short-term formalism (during radiotherapy) and long-term formalism (post-treatment). This framework is used to estimate the risks of second cancer as a function of spontaneous background and radiation-induced mutation rates of normal and pre-malignant cells. By fitting the model to available clinical data for spontaneous background risk together with data of Hodgkin’s lymphoma survivors (for various organs), the second cancer mutation rate is estimated. The model predicts a significant increase in mutation rate for some cancer types, which may be a sign of genomic instability. Finally, it is shown that the model results are in agreement with the measured results for excess relative risk (ERR) as a function of exposure age and that the model predicts a negative correlation of ERR with increase in attained age. This novel approach can be used to analyze several radiotherapy protocols in current clinical practice and to forecast the second cancer risks over time for individual patients.

Appendices

Appendix 1: Evolutionary dynamics of two-hit process

Consider a population of N cells (inside a niche) governed by the Moran process (types—0, 1, 2 indicating normal, pre-malignant and malignant phenotypes). At each time step, a cell is randomly chosen, based on its fitness, to reproduce and another cell is randomly chosen to die. In the presence of mutations, such as in the model presented in this paper, at each time step, there is an alternative possibility (instead of death–birth) that a normal cell may transform into a pre-malignant cell or alternatively a pre-malignant cell may transform into a malignant cell. The mutation rate for the first hit is $u_{1}$ and the rate for the second hit (pre-malignant into malignant) is $u_{2}$ (or $u_{r}$ after radiation). As discussed in the text, let $f_{0}(t, u_{1},u_{2},r,N)$ be the probability that a first malignant cell has emerged before time t given that the initial population consists of all normal cells in the niche. Similarly, let $f_{1}(t, u_{1}, u_{2}, r, N)$ be the probability that a malignant cell emerges in a niche of size N before time t, beginning with one pre-malignant cell and N − 1 normal cells. It is straightforward to show that the two probabilities satisfy the continuous time limit of the Kolmogorov equations,

$$\begin{aligned} \frac{{\rm d}f_{0}(t)}{{\rm d}t}&= Nu_{1} f_{1}(t)\left(1 - f_{0}(t) \right), \\ \frac{{\rm d}f_{1}(t)}{{\rm d}t}&= ru_{2} - (1-r +2ru_{2})f_{1}(t) - r(1-u_{2})f_{1}^{2}(t), \end{aligned}$$

(9)

with the initial condition $f_{0}(0) = 0$. The solutions of the above set of equations can be directly obtained under a branching process approximation, i.e., independence of two lineages starting from two different cells, one can then obtain an analytical solution for these probabilities,

$$\begin{aligned} f_{1}(t, u_{1},u_{2},r,N)&= \frac{\exp \{( r(1-u_{2})(a + b)t \}- 1}{(1/a)\exp \{ r(1-u_{2})(a + b)t \} +(1/b)}, \\ f_{0}(t, u_{1},u_{2},r,N)&= 1 - \exp \left( -Nu_{1}\left\{ \frac{a + b}{c} \ln \left[ \frac{{\rm e}^{ct} + a/b}{1 + a/b}\right] - bt\right\} \right), \end{aligned}$$

(10)

with a, b, c given as,

$$\begin{aligned} a&= \frac{1}{2(1-u_{2})}\left[ -\left( \frac{1-r}{r} + 2u_{2}\right) + \sqrt{\left( \frac{1-r}{r} + 2u_{2}\right) ^{2} + 4u_{2}}\,\right], \\ b&= \frac{1}{2(1-u_{2})}\left[ \left( \frac{1-r}{r} + 2u_{2}\right) + \sqrt{\left( \frac{1-r}{r} + 2u_{2}\right) ^{2} + 4u_{2}}\,\right], \\ c&= r(1-u_{2})(a + b). \end{aligned}$$

(11)

As discussed in the main body of the paper, both the cumulative age incidence of different cancer types and excess relative risk of cancer (second cancer in this case) is expressed in terms of functions $f_{0}(t,u_{1},u_{2},r,N)$ and $f_{1}(t,u_{1},u_{2},r,N)$, which lead to a simple form for the ERR. As is obvious, there is no dependence on the total number of niches $\tilde{N}$, and the dependence on the niche size, N, is very weak. However, for more realistic fits, the value of the proliferation strength r is not extremely close to unity and thus the above approximation might not be appropriate. However, it is easy to see from the above approximate results that any quantity of interest (i.e., number of background age incidences or ERR) is represented by very weak functions of the niche size, N. Thus, the present rough estimates for the value of N do not change any of the important conclusions resulting from this work.

Appendix 2: Initiation–inactivation–repopulation model

In this section, the initiation–inactivation–repopulation (Sachs and Brenner 2005) formalism is briefly reviewed to estimate the number of pre-malignant cells at the end of radiation treatment. The effect of radiation is simplified into two mechanism: (1) cell killing which is modeled by the linear–quadratic approximation and gives the number of either normal or pre-malignant cells killed due to radiation dose. (2) The cell initiation which is assumed to be a linear function of the dose with a small constant mutation rate per dose rate. Two populations of normal stem cells and pre-malignant stem cells are assumed. Normal stem cells grow logistically while they can die during the radiation exposure times, where cell death is given by linear–quadratic formula. They can also transform into pre-malignant cells during the exposure time. The population of pre-malignant cells has a similar growth form while its proliferation is regulated by the population of normal cells and has the same cell death rate due to radiation and also a positive rate due to normal stem cell initiation. The above can be written in the form of the following coupled system of ordinary differential equations.

$$\begin{aligned} \frac{{\rm d}n(t)}{{\rm d}t}&= r_{0}n(t)\left( 1 - \frac{n(t)}{K}\right) - (\alpha + \beta D)\cdot \frac{{\rm d}D}{{\rm d}t}n(t) - \gamma \frac{{\rm d}D}{{\rm d}t}n(t), \\ \frac{{\rm d}m(t)}{{\rm d}t}&= r_{0}\lambda m(t)\left( 1 - \frac{n(t)}{K}\right) - (\alpha + \beta D) \cdot \frac{{\rm d}D}{{\rm d}t}m(t) + \gamma \frac{{\rm d}D}{{\rm d}t}n(t), \end{aligned}$$

(12)

where $r_{0}$ is the normal cell repopulation rate while $r_{0}\lambda$ is the pre-malignant repopulation rate. ${\rm d}D/{\rm d}t$ is the dose delivery rate, $\lambda$ represents the relative proliferation rate of pre-malignant cells to the normal cells and K is the carrying capacity of normal cells. $\alpha$ and $\beta$ are the cell killing rates from linear–quadratic dose dependence (LQ). The number of pre-malignant cells after the total radiation dose has been applied and is the quantity of importance here. This can be analytically calculated for the simplified case of an acute dose, ${\rm d}D/{\rm d}t = {{\rm const}}.$ applied in a finite interval of time

$$\begin{aligned} M&= m(\infty ) = N(\exp (\gamma D) - 1), \\&\sim N\gamma D, \end{aligned}$$

(13)

where D is the total dose. The total number of the cells $N_{{\rm tot}}$, is given by the number of niche times the niches size, $N\tilde{N}$. For the values of the radiation-induced initiation rate $\gamma$, this will result in $M/N_{{\rm tot}} \sim 10^{-6}$ (per Gy). Since we are interested on an estimate for number of radiation-induced pre-malignant cells, the details of cell killing mechanism in LQ formula does not appear in Eq. (13).

The above formalism can be expanded to incorporate fractionation therapy, i.e., non-constant dose–delivery rate which has been discussed in the literature. In the present work, there is no need to deal with these details. However, as a future direction of research, all the details of radiotherapy (fractionation protocol and the dose–volume histogram for each patient) can be used as input to estimate the ERR as a function of exposure age and attained age.