A stochastic markov model of cellular response to radiation

Abstract

A stochastic model based on the Markov Chain Monte Carlo process is used to describe responses to ionizing radiation in a group of cells. The results show that where multiple relationships linearly depending on the dose are introduced, the overall reaction shows a threshold, and, generally, a non-linear response. Such phenomena have been observed and reported in a number of papers. The present model permits the inclusion of adaptive responses and bystander effects that can lead to hormetic effects. In addition, the model allows for incorporating various time-dependent phenomena. Essentially, all known biological effects can be reproduced using the proposed model.

Keywords: Markov process; Monte Carlo; dose response model; hormesis; low-level radiation; radiation threshold.

FIG. 1.

General idea of the model: two loops over the steps and the cells.

FIG. 2.

First part of the stochastic tree for each cell. See text and Figs 3 to 8 for more details.

FIG. 3.

A portion of the stochastic tree from Fig. 2 – a healthy cell has been hit.

FIG. 4.

A portion of the stochastic tree from Fig. 2 - a healthy cell has not been hit.

FIG. 5.

A portion of the stochastic tree from Fig. 2 – a mutated cell has been hit.

FIG. 6.

A portion of the stochastic tree from Fig. 2 - a mutated cell has not been hit.

FIG. 7.

A portion of the stochastic tree from Fig. 2 - a cancer cell has been hit.

FIG. 8.

A portion of the stochastic tree from Fig. 2 - a cancer cell has not been hit.

FIG. 9.

Four exemplary models (B1, B2, B3, B4) of the probability distribution of the bystander effect (P B) described by up to the third-order polynomials. The labels “D2” and “D3” in the figure’s legend stand for “D²” and “D³”, respectively, where D is the dose per one state (step) of cells

FIG. 10.

Four exemplary models (A1, A2, A3, A4) of the probability distribution of the adaptive response (P_A), described by up to the third-order polynomials. “D2” and “D3” in the figure’s legend stand for “D²” and “D³”, respectively, where D is the dose per one state (step) of cells.

FIG. 11.

Results obtained with use of the set of parameters from Tab. 2. The fraction is shown of mutated (solid line) and cancer cells (dotted line).

FIG. 12.

Results obtained with use of the set of parameters from Tab. 2. The fraction is shown of mutated (solid line) and cancer cells (dotted line). The dose is limited to 0.4 mGy per step.

FIG. 13.

Results obtained with use of the set of parameters from Tab. 2. The total number is shown of mutations in all cells (solid line), along with the number of cancer cells (dotted line).

FIG. 14.

Results obtained with use of the set of parameters from Tab. 2 including the adaptive response and bystander effect. The fraction is shown of the mutated (solid line) and cancer cells (dotted line).

FIG. 15.

Results obtained with use of the set of parameters from Tab. 2 including the adaptive response and bystander effect. The fraction is shown of the mutated (solid line) and cancer cells (dotted line).