A stochastic markov model of cellular response to radiation

Mathematical modeling in radiotherapy for cancer: a comprehensive narrative review

Mathematical modeling has long been a cornerstone of radiotherapy for cancer, guiding treatment prescription, planning, and delivery through versatile applications. As we enter the era of medical big data, where the integration of molecular, imaging, and clinical data at both the tumor and patient levels could promise more precise and personalized cancer treatment, the role of mathematical modeling has become even more critical. This comprehensive narrative review aims to summarize the main applications of mathematical modeling in radiotherapy, bridging the gap between classical models and the latest advancements. The review covers a wide range of applications, including radiobiology, clinical workflows, stereotactic radiosurgery/stereotactic body radiotherapy (SRS/SBRT), spatially fractionated radiotherapy (SFRT), FLASH radiotherapy (FLASH-RT), immune-radiotherapy, and the emerging concept of radiotherapy digital twins. Each of these areas is explored in depth, with a particular focus on how newer trends and innovations are shaping the future of radiation cancer treatment. By examining these diverse applications, this review provides a comprehensive overview of the current state of mathematical modeling in radiotherapy. It also highlights the growing importance of these models in the context of personalized medicine and multi-scale, multi-modal data integration, offering insights into how they can be leveraged to enhance treatment precision and patient outcomes. As radiotherapy continues to evolve, the insights gained from this review will help guide future research and clinical practice, ensuring that mathematical modeling continues to propel innovations in radiation cancer treatment.

Biophysical Modeling of the Ionizing Radiation Influence on Cells Using the Stochastic (Monte Carlo) and Deterministic (Analytical) Approaches

This review article describes our simplified biophysical model for the response of a group of cells to ionizing radiation. The model, which is a product of 10 years of studies, acts as (a) a comprehensive stochastic approach based on the Monte Carlo simulation with a probability tree and (b) the thereof derived detailed deterministic models describing the selected biophysical and radiobiological phenomena in an analytical manner. Specifically, the presented model describes effects such as the risk of neoplastic transformation of cells relative to the absorbed radiation dose, the dynamics of tumor development, the priming dose effect (also called the Raper-Yonezawa effect) based on the introduced adaptive response approach, and the bystander effect. The model is also modifiable depending on users' potential needs.

Modeling Cell Reactions to Ionizing Radiation: From a Lesion to a Cancer

This article focuses on the analytic modeling of responses of cells in the body to ionizing radiation. The related mechanisms are consecutively taken into account and discussed. A model of the dose- and time-dependent adaptive response is considered for 2 exposure categories: acute and protracted. In case of the latter exposure, we demonstrate that the response plateaus are expected under the modelling assumptions made. The expected total number of cancer cells as a function of time turns out to be perfectly described by the Gompertz function. The transition from a collection of cancer cells into a tumor is discussed at length. Special emphasis is put on the fact that characterizing the growth of a tumor (ie, the increasing mass and volume), the use of differential equations cannot properly capture the key dynamics-formation of the tumor must exhibit properties of the phase transition, including self-organization and even self-organized criticality. As an example, a manageable percolation-type phase transition approach is used to address this problem. Nevertheless, general theory of tumor emergence is difficult to work out mathematically because experimental observations are limited to the relatively large tumors. Hence, determination of the conditions around the critical point is uncertain.

Atomic Bomb Survivors Life-Span Study: Insufficient Statistical Power to Select Radiation Carcinogenesis Model

The atomic bomb survivors life-span study (LSS) is often claimed to support the linear no-threshold hypothesis (LNTH) of radiation carcinogenesis. This paper shows that this claim is baseless. The LSS data are equally or better described by an s-shaped dependence on radiation exposure with a threshold of about 0.3 Sievert (Sv) and saturation level at about 1.5 Sv. A Monte-Carlo simulation of possible LSS outcomes demonstrates that, given the weak statistical power, LSS cannot provide support for LNTH. Even if the LNTH is used at low dose and dose rates, its estimation of excess cancer mortality should be communicated as 2.5% per Sv, i.e., an increase of cancer mortality from about 20% spontaneous mortality to about 22.5% per Sv, which is about half of the usually cited value. The impact of the “neutron discrepancy problem” – the apparent difference between the calculated and measured values of neutron flux in Hiroshima – was studied and found to be marginal. Major revision of the radiation risk assessment paradigm is required.

Dynamics of cellular responses to radiation

Understanding the consequences of exposure to low dose ionizing radiation is an important public health concern. While the risk of low dose radiation has been estimated by extrapolation from data at higher doses according to the linear non-threshold model, it has become clear that cellular responses can be very different at low compared to high radiation doses. Important phenomena in this respect include radioadaptive responses as well as low-dose hyper-radiosensitivity (HRS) and increased radioresistance (IRR). With radioadaptive responses, low dose exposure can protect against subsequent challenges, and two mechanisms have been suggested: an intracellular mechanism, inducing cellular changes as a result of the priming radiation, and induction of a protected state by inter-cellular communication. We use mathematical models to examine the effect of these mechanisms on cellular responses to low dose radiation. We find that the intracellular mechanism can account for the occurrence of radioadaptive responses. Interestingly, the same mechanism can also explain the existence of the HRS and IRR phenomena, and successfully describe experimentally observed dose-response relationships for a variety of cell types. This indicates that different, seemingly unrelated, low dose phenomena might be connected and driven by common core processes. With respect to the inter-cellular communication mechanism, we find that it can also account for the occurrence of radioadaptive responses, indicating redundancy in this respect. The model, however, also suggests that the communication mechanism can be vital for the long term survival of cell populations that are continuously exposed to relatively low levels of radiation, which cannot be achieved with the intracellular mechanism in our model. Experimental tests to address our model predictions are proposed.

The effect of low-dose radiation on cells and tissues is a public health concern, because the human population is exposed to low-dose ionizing radiation coming from a variety of sources, such as cosmic rays, soil radioactivity, environmental contaminations, and various medical procedures. At low doses of radiation, phenomena are observed that do not occur at higher doses, such as radioadaptive responses as well as low-dose hyper-radiosensitivity (HRS) and increased radioresistance (IRR), which are so far not fully understood. Each of these phenomena have been investigated separately, and specific mechanisms have been suggested to explain them. Using mathematical models that are successfully fitted to experimental data under a variety of conditions, we show that a set of basic and documented assumptions about cellular responses to low-dose radiation can explain all three low-dose phenomena, indicating that they are inter-related. According to the model, these phenomena are brought about by the multi-factorial interactions that underlie the population dynamics of the cells involved, and this provides a new framework to understand these responses, and to evaluate the risk to human health posed by exposure to low-dose radiation.

Computational modeling of cellular effects post-irradiation with low- and high-let particles and different absorbed doses

The use of computational methods to improve the understanding of biological responses to various types of radiation is an approach where multiple parameters can be modelled and a variety of data is generated. This study compares cellular effects modelled for low absorbed doses against high absorbed doses. The authors hypothesized that low and high absorbed doses would contribute to cell killing via different mechanisms, potentially impacting on targeted tumour radiotherapy outcomes. Cellular kinetics following irradiation with selective low- and high-linear energy transfer (LET) particles were investigated using the Virtual Cell (VC) radiobiology algorithm. Two different cell types were assessed using the VC radiobiology algorithm: human fibroblasts and human crypt cells. The results showed that at lower doses (0.01 to 0.2 Gy), all radiation sources used were equally able to induce cell death (p>0.05, ANOVA). On the other hand, at higher doses (1.0 to 8.0 Gy), the radiation response was LET and dose dependent (p<0.05, ANOVA). The data obtained suggests that the computational methods used might provide some insight into the cellular effects following irradiation. The results also suggest that it may be necessary to re-evaluate cellular radiation-induced effects, particularly at low doses that could affect therapeutic effectiveness.