Carcinogenesis and cancer progression are often modeled using population dynamics equations for a diverse somatic cell population undergoing mutations or other alterations that alter the fitness of a cell and its progeny. Usually it is then assumed, paralleling standard mathematical approaches to evolution, that such alterations are slow compared to selection, i.e., compared to subpopulation frequency changes induced by unequal subpopulation proliferation rates. However, the alterations can be rapid in some cases. For example, results in our lab on in vitro analogues of transformation and progression in carcinogenesis suggest there could be periods where rapid alterations triggered by horizontal intercellular transfer of genetic material occur and quickly result in marked changes of cell population structure. We here initiate a mathematical study of situations where alterations are rapid compared to selection. A classic selection-mutation formalism is generalized to obtain a “proliferation-alteration” system of ordinary differential equations, which we analyze using a rapid-alteration approximation. A system-theoretical estimate of the total-population net growth rate emerges. This rate characterizes the diverse, interacting cell population acting as a single system; it is a weighted average of subpopulation rates, the weights being components of the Perron–Frobenius eigenvector for an ergodic Markov-process matrix that describes alterations by themselves. We give a detailed numerical example to illustrate the rapid-alteration approximation, suggest a possible interpretation of the fact that average aneuploidy during cancer progression often appears to be comparatively stable in time, and briefly discuss possible generalizations as well as weaknesses of our approach.

Purpose

Optimal treatment planning for fractionated external beam radiation therapy requires inputs from radiobiology based on recent thinking about the "five Rs" (repopulation, radiosensitivity, reoxygenation, redistribution, and repair). The need is especially acute for the newer, often individualized, protocols made feasible by progress in image guided radiation therapy and dose conformity. Current stochastic tumor control probability (TCP) models incorporating tumor repopulation effects consider "stem-like cancer cells" (SLCC) to be independent, but the authors here propose that SLCC-SLCC interactions may be significant. The authors present a new stochastic TCP model for repopulating SLCC interacting within microenvironmental niches. Our approach is meant mainly for comparing similar protocols. It aims at practical generalizations of previous mathematical models.

Methods

The authors consider protocols with complete sublethal damage repair between fractions. The authors use customized open-source software and recent mathematical approaches from stochastic process theory for calculating the time-dependent SLCC number and thereby estimating SLCC eradication probabilities. As specific numerical examples, the authors consider predicted TCP results for a 2 Gy per fraction, 60 Gy protocol compared to 64 Gy protocols involving early or late boosts in a limited volume to some fractions.

Results

In sample calculations with linear quadratic parameters α = 0.3 per Gy, α∕β = 10 Gy, boosting is predicted to raise TCP from a dismal 14.5% observed in some older protocols for advanced NSCLC to above 70%. This prediction is robust as regards: (a) the assumed values of parameters other than α and (b) the choice of models for intraniche SLCC-SLCC interactions. However, α = 0.03 per Gy leads to a prediction of almost no improvement when boosting.

Conclusions

The predicted efficacy of moderate boosts depends sensitively on α. Presumably, the larger values of α are the ones appropriate for individualized treatment protocols, with the smaller values relevant only to protocols for a heterogeneous patient population. On that assumption, boosting is predicted to be highly effective. Front boosting, apart from practical advantages and a possible advantage as regards iatrogenic second cancers, also probably gives a slightly higher TCP than back boosting. If the total number of SLCC at the start of treatment can be measured even roughly, it will provide a highly sensitive way of discriminating between various models and parameter choices. Updated mathematical methods for calculating repopulation allow credible generalizations of earlier results.

We consider dosing regimens designed to cure patients by eradicating colony forming units (CFU) such as bacteria. In the field of "population" pharmaco-kinetics/dynamics (PK/PD), inter-individual variability (IIV) of patients is estimated using model parameter statistical distributions. We consider a more probabilistic approach to IIV called stochastic process theory, motivated by the fact that tumor treatment planning uses both approaches. Stochastic process PD can supply additional insights and suggest different dosing regimens due to its emphasis on the probability of complete CFU eradication and its predictions on "pure chance" fluctuations of CFU number per patient when treatment has reduced this integer to less than ~100. To exemplify the contrast between stochastic process PD models and standard deterministic PD models, which track only average CFU number, we analyze, neglecting immune responses, neonatal intravenous gentamicin dosing regimens directed against Escherichia coli. Our stochastic calculations predict that the first dose is crucial for CFU eradication. For example, a single 6 mg/kg dose is predicted to have a higher eradication probability than four daily 4 mg/kg doses. We conclude: (1) neonatal gentamicin dosing regimens with larger first doses but smaller total doses deserve investigation; (2) in general, if standard PK/PD models predict average CFU number drops substantially below 100, the models should be modified to incorporate stochastic effects more accurately, and will then usually make more favorable, or less unfavorable, predictions for front boosting ("hit hard early"). Various caveats against over-interpreting the calculations are given.

Age and physiologic status, such as menopause, are risk factors for breast cancer. Less clear is what factors influence the diversity of breast cancer. In this study, we investigated the effect of host age on the distribution of tumor subtypes in mouse mammary chimera consisting of wild-type hosts and Trp53 nullizygous epithelium, which undergoes a high rate of neoplastic transformation. Wild-type mammary glands cleared of endogenous epithelium at 3 weeks of age were subsequently transplanted during puberty (5 weeks) or at maturation (10 weeks) with syngeneic Trp53-null mammary tissue fragments and monitored for one year. Tumors arose sooner from adult hosts (AH) compared with juvenile hosts (JH). However, compared with AH tumors, JH tumors grew several times faster, were more perfused, exhibited a two-fold higher mitotic index, and were more highly positive for insulin-like growth factor receptor phosphorylation. Most tumors in each setting were estrogen receptor (ER)-positive (80% JH vs. 70% AH), but JH tumors were significantly more ER-immunoreactive (P = 0.0001) than AH tumors. A differential expression signature (JvA) of juvenile versus adult tumors revealed a luminal transcriptional program. Centroids of the human homologs of JvA genes showed that JH tumors were more like luminal A tumors and AH tumors were more like luminal B tumors. Hierarchical clustering with the JvA human ortholog gene list segregated luminal A and luminal B breast cancers across datasets. These data support the notion that age-associated host physiology greatly influences the intrinsic subtype of breast cancer.

As the number of cancer survivors grows, prediction of radiotherapy-induced second cancer risks becomes increasingly important. Because the latency period for solid tumors is long, the risks of recently introduced radiotherapy protocols are not yet directly measurable. In the accompanying article, we presented a new biologically based mathematical model, which, in principle, can estimate second cancer risks for any protocol. The novelty of the model is that it integrates, into a single formalism, mechanistic analyses of pre-malignant cell dynamics on two different time scales: short-term during radiotherapy and recovery; long-term during the entire life span. Here, we apply the model to nine solid cancer types (stomach, lung, colon, rectal, pancreatic, bladder, breast, central nervous system, and thyroid) using data on radiotherapy-induced second malignancies, on Japanese atomic bomb survivors, and on background US cancer incidence. Potentially, the model can be incorporated into radiotherapy treatment planning algorithms, adding second cancer risk as an optimization criterion.

Mathematical models of radiation carcinogenesis are important for understanding mechanisms and for interpreting or extrapolating risk. There are two classes of such models: (1) long-term formalisms that track pre-malignant cell numbers throughout an entire lifetime but treat initial radiation dose–response simplistically and (2) short-term formalisms that provide a detailed initial dose–response even for complicated radiation protocols, but address its modulation during the subsequent cancer latency period only indirectly. We argue that integrating short- and long-term models is needed. As an example of this novel approach, we integrate a stochastic short-term initiation/inactivation/repopulation model with a deterministic two-stage long-term model. Within this new formalism, the following assumptions are implemented: radiation initiates, promotes, or kills pre-malignant cells; a pre-malignant cell generates a clone, which, if it survives, quickly reaches a size limitation; the clone subsequently grows more slowly and can eventually generate a malignant cell; the carcinogenic potential of pre-malignant cells decreases with age.

To test quantitatively whether there are systematic chromosome-chromosome associations within human interphase nuclei, interchanges between all possible heterologous pairs of chromosomes were measured with 24-color whole-chromosome painting (multiplex FISH), after damage to interphase lymphocytes by sparsely ionizing radiation in vitro. An excess of interchanges for a specific chromosome pair would indicate spatial proximity between the chromosomes comprising that pair. The experimental design was such that quite small deviations from randomness (extra pairwise interchanges within a group of chromosomes) would be detectable. The only statistically significant chromosome cluster was a group of five chromosomes previously observed to be preferentially located near the center of the nucleus. However, quantitatively, the overall deviation from randomness within the whole genome was small. Thus, whereas some chromosome-chromosome associations are clearly present, at the whole-chromosomal level, the predominant overall pattern appears to be spatially random.

Cancer is an important long-term risk for astronauts exposed to protons and high-energy charged particles during travel and residence on asteroids, the moon, and other planets. NASA's Biomedical Critical Path Roadmap defines the carcinogenic risks of radiation exposure as one of four type I risks. A type I risk represents a demonstrated, serious problem with no countermeasure concepts, and may be a potential "show-stopper" for long duration spaceflight. Estimating the carcinogenic risks for humans who will be exposed to heavy ions during deep space exploration has very large uncertainties at present. There are no human data that address risk from extended exposure to complex radiation fields. The overarching goal in this area to improve risk modeling is to provide biological insight and mechanistic analysis of radiation quality effects on carcinogenesis. Understanding mechanisms will provide routes to modeling and predicting risk and designing countermeasures. This white paper reviews broad issues related to experimental models and concepts in space radiation carcinogenesis as well as the current state of the field to place into context recent findings and concepts derived from the NASA Space Radiation Program.

Robust predictive models are essential to manage the risk of radiation-induced carcinogenesis. Chronic exposure to cosmic rays in the context of the complex deep space environment may place astronauts at high cancer risk. To estimate this risk, it is critical to understand how radiation-induced cellular stress impacts cell fate decisions and how this in turn alters the risk of carcinogenesis. Exposure to the heavy ion component of cosmic rays triggers a multitude of cellular changes, depending on the rate of exposure, the type of damage incurred and individual susceptibility. Heterogeneity in dose, dose rate, radiation quality, energy and particle flux contribute to the complexity of risk assessment. To unravel the impact of each of these factors, it is critical to identify sensitive biomarkers that can serve as inputs for robust modeling of individual risk of cancer or other long-term health consequences of exposure. Limitations in sensitivity of biomarkers to dose and dose rate, and the complexity of longitudinal monitoring, are some of the factors that increase uncertainties in the output from risk prediction models. Here, we critically evaluate candidate early and late biomarkers of radiation exposure and discuss their usefulness in predicting cell fate decisions. Some of the biomarkers we have reviewed include complex clustered DNA damage, persistent DNA repair foci, reactive oxygen species, chromosome aberrations and inflammation. Other biomarkers discussed, often assayed for at longer points post exposure, include mutations, chromosome aberrations, reactive oxygen species and telomere length changes. We discuss the relationship of biomarkers to different potential cell fates, including proliferation, apoptosis, senescence, and loss of stemness, which can propagate genomic instability and alter tissue composition and the underlying mRNA signatures that contribute to cell fate decisions. Our goal is to highlight factors that are important in choosing biomarkers and to evaluate the potential for biomarkers to inform models of post exposure cancer risk. Because cellular stress response pathways to space radiation and environmental carcinogens share common nodes, biomarker-driven risk models may be broadly applicable for estimating risks for other carcinogens.