Simulating tumor volume dynamics in response to radiotherapy: Implications of model selection.

From the beginning of the usage of radiotherapy (RT) for cancer treatment, mathematical modeling has been integral to understanding radiobiology and for designing treatment approaches and schedules. There has been extensive modeling of response to RT with the inclusion of various degrees of biological complexity. In this study, we compare three models of tumor volume dynamics: (1) exponential growth with RT directly reducing tumor volume, (2) logistic growth with direct tumor volume reduction, and (3) logistic growth with RT reducing the tumor carrying capacity with the objective of understanding the implications of model selection and informing the process of model calibration and parameterization. For all three models, we: examined the rates of change in tumor volume during and RT treatment course; performed parameter sensitivity and identifiability analyses; and investigated the impact of the parameter sensitivity on the tumor volume trajectories. In examining the tumor volume dynamics trends, we coined a new metric - the point of maximum reduction of tumor volume (MRV) - to quantify the magnitude and timing of the expected largest impact of RT during a treatment course. We found distinct timing differences in MRV, dependent on model selection. The parameter identifiability and sensitivity analyses revealed the interdependence of the different model parameters and that it is only possible to independently identify tumor growth and radiation response parameters if the underlying tumor growth rate is sufficiently large. Ultimately, the results of these analyses help us to better understand the implications of model selection while simultaneously generating falsifiable hypotheses about MRV timing that can be tested on longitudinal measurements of tumor volume from pre-clinical or clinical data with high acquisition frequency. Although, our study only compares three particular models, the results demonstrate that caution is necessary in selecting models of response to RT, given the artifacts imposed by each model.

Predicting Radiotherapy Patient Outcomes with Real-Time Clinical Data Using Mathematical Modelling.

Longitudinal tumour volume data from head-and-neck cancer patients show that tumours of comparable pre-treatment size and stage may respond very differently to the same radiotherapy fractionation protocol. Mathematical models are often proposed to predict treatment outcome in this context, and have the potential to guide clinical decision-making and inform personalised fractionation protocols. Hindering effective use of models in this context is the sparsity of clinical measurements juxtaposed with the model complexity required to produce the full range of possible patient responses. In this work, we present a compartment model of tumour volume and tumour composition, which, despite relative simplicity, is capable of producing a wide range of patient responses. We then develop novel statistical methodology and leverage a cohort of existing clinical data to produce a predictive model of both tumour volume progression and the associated level of uncertainty that evolves throughout a patient's course of treatment. To capture inter-patient variability, all model parameters are patient specific, with a bootstrap particle filter-like Bayesian approach developed to model a set of training data as prior knowledge. We validate our approach against a subset of unseen data, and demonstrate both the predictive ability of our trained model and its limitations.

Classical mathematical models for prediction of response to chemotherapy and immunotherapy.

Classical mathematical models of tumor growth have shaped our understanding of cancer and have broad practical implications for treatment scheduling and dosage. However, even the simplest textbook models have been barely validated in real world-data of human patients. In this study, we fitted a range of differential equation models to tumor volume measurements of patients undergoing chemotherapy or cancer immunotherapy for solid tumors. We used a large dataset of 1472 patients with three or more measurements per target lesion, of which 652 patients had six or more data points. We show that the early treatment response shows only moderate correlation with the final treatment response, demonstrating the need for nuanced models. We then perform a head-to-head comparison of six classical models which are widely used in the field: the Exponential, Logistic, Classic Bertalanffy, General Bertalanffy, Classic Gompertz and General Gompertz model. Several models provide a good fit to tumor volume measurements, with the Gompertz model providing the best balance between goodness of fit and number of parameters. Similarly, when fitting to early treatment data, the general Bertalanffy and Gompertz models yield the lowest mean absolute error to forecasted data, indicating that these models could potentially be effective at predicting treatment outcome. In summary, we provide a quantitative benchmark for classical textbook models and state-of-the art models of human tumor growth. We publicly release an anonymized version of our original data, providing the first benchmark set of human tumor growth data for evaluation of mathematical models.

Modelling Radiation Cancer Treatment with a Death-Rate Term in Ordinary and Fractional Differential Equations.

Fractional calculus has recently been applied to the mathematical modelling of tumour growth, but its use introduces complexities that may not be warranted. Mathematical modelling with differential equations is a standard approach to study and predict treatment outcomes for population-level and patient-specific responses. Here, we use patient data of radiation-treated tumours to discuss the benefits and limitations of introducing fractional derivatives into three standard models of tumour growth. The fractional derivative introduces a history-dependence into the growth function, which requires a continuous death-rate term for radiation treatment. This newly proposed radiation-induced death-rate term improves computational efficiency in both ordinary and fractional derivative models. This computational speed-up will benefit common simulation tasks such as model parameterization and the construction and running of virtual clinical trials.

The Tumor Invasion Paradox in Cancer Stem Cell-Driven Solid Tumors.

Cancer stem cells (CSCs) are key in understanding tumor growth and tumor progression. A counterintuitive effect of CSCs is the so-called tumor growth paradox: the effect where a tumor with a higher death rate may grow larger than a tumor with a lower death rate. Here we extend the modeling of the tumor growth paradox by including spatial structure and considering cancer invasion. Using agent-based modeling and a corresponding partial differential equation model, we demonstrate and prove mathematically a tumor invasion paradox: a larger cell death rate can lead to a faster invasion speed. We test this result on a generic hypothetical cancer with typical growth rates and typical treatment sensitivities. We find that the tumor invasion paradox may play a role for continuous and intermittent treatments, while it does not seem to be essential in fractionated treatments. It should be noted that no attempt was made to fit the model to a specific cancer, thus, our results are generic and theoretical.

Heterogeneity analysis of MRI T2 maps for measurement of early tumor response to radiotherapy.

External beam radiotherapy (XRT) is a widely used cancer treatment, yet responses vary dramatically among patients. These differences are not accounted for in clinical practice, partly due to a lack of sensitive early response biomarkers. We hypothesize that quantitative magnetic resonance imaging (MRI) measures reflecting tumor heterogeneity can provide a sensitive and robust biomarker of early XRT response. MRI T2 mapping was performed every 72 hours following 10 Gy dose XRT in two models of pancreatic cancer propagated in the hind limb of mice. Interquartile range (IQR) of tumor T2 was presented as a potential biomarker of radiotherapy response compared with tumor growth kinetics, and biological validation was performed through quantitative histology analysis. Quantification of tumor T2 IQR showed sensitivity for detection of XRT-induced tumor changes 72 hours after treatment, outperforming T2-weighted and diffusion-weighted MRI, with very good robustness. Histological comparison revealed that T2 IQR provides a measure of spatial heterogeneity in tumor cell density, related to radiation-induced necrosis. Early IQR changes were found to correlate to subsequent tumor volume changes, indicating promise for treatment response prediction. Our preclinical findings indicate that spatial heterogeneity analysis of T2 MRI can provide a translatable method for early radiotherapy response assessment. We propose that the method may in future be applied for personalization of radiotherapy through adaptive treatment paradigms.

High School Internship Program in Integrated Mathematical Oncology (HIP IMO): Five-Year Experience at Moffitt Cancer Center.

Modern cancer research, and the wealth of data across multiple spatial and temporal scales, has created the need for researchers that are well versed in the life sciences (cancer biology, developmental biology, immunology), medical sciences (oncology) and natural sciences (mathematics, physics, engineering, computer sciences). College undergraduate education traditionally occurs in disciplinary silos, which creates a steep learning curve at the graduate and postdoctoral levels that increasingly bridge multiple disciplines. Numerous colleges have begun to embrace interdisciplinary curricula, but students who double major in mathematics (or other quantitative sciences) and biology (or medicine) remain scarce. We identified the need to educate junior and senior high school students about integrating mathematical and biological skills, through the lens of mathematical oncology, to better prepare students for future careers at the interdisciplinary interface. The High school Internship Program in Integrated Mathematical Oncology (HIP IMO) at Moffitt Cancer Center has so far trained 59 students between 2015 and 2019. We report here on the program structure, training deliverables, curriculum and outcomes. We hope to promote interdisciplinary educational activities early in a student's career.

The 2019 mathematical oncology roadmap.

Whether the nom de guerre is Mathematical Oncology, Computational or Systems Biology, Theoretical Biology, Evolutionary Oncology, Bioinformatics, or simply Basic Science, there is no denying that mathematics continues to play an increasingly prominent role in cancer research. Mathematical Oncology-defined here simply as the use of mathematics in cancer research-complements and overlaps with a number of other fields that rely on mathematics as a core methodology. As a result, Mathematical Oncology has a broad scope, ranging from theoretical studies to clinical trials designed with mathematical models. This Roadmap differentiates Mathematical Oncology from related fields and demonstrates specific areas of focus within this unique field of research. The dominant theme of this Roadmap is the personalization of medicine through mathematics, modelling, and simulation. This is achieved through the use of patient-specific clinical data to: develop individualized screening strategies to detect cancer earlier; make predictions of response to therapy; design adaptive, patient-specific treatment plans to overcome therapy resistance; and establish domain-specific standards to share model predictions and to make models and simulations reproducible. The cover art for this Roadmap was chosen as an apt metaphor for the beautiful, strange, and evolving relationship between mathematics and cancer.

Mathematical Models of Cancer: When to Predict Novel Therapies, and When Not to.

The number of publications on mathematical modeling of cancer is growing at an exponential rate, according to PubMed records, provided by the US National Library of Medicine and the National Institutes of Health. Seminal papers have initiated and promoted mathematical modeling of cancer and have helped define the field of mathematical oncology (Norton and Simon in J Natl Cancer Inst 58:1735-1741, 1977; Norton in Can Res 48:7067-7071, 1988; Hahnfeldt et al. in Can Res 59:4770-4775, 1999; Anderson et al. in Comput Math Methods Med 2:129-154, 2000. https://doi.org/10.1080/10273660008833042 ; Michor et al. in Nature 435:1267-1270, 2005. https://doi.org/10.1038/nature03669 ; Anderson et al. in Cell 127:905-915, 2006. https://doi.org/10.1016/j.cell.2006.09.042 ; Benzekry et al. in PLoS Comput Biol 10:e1003800, 2014. https://doi.org/10.1371/journal.pcbi.1003800 ). Following the introduction of undergraduate and graduate programs in mathematical biology, we have begun to see curricula developing with specific and exclusive focus on mathematical oncology. In 2018, 218 articles on mathematical modeling of cancer were published in various journals, including not only traditional modeling journals like the Bulletin of Mathematical Biology and the Journal of Theoretical Biology, but also publications in renowned science, biology, and cancer journals with tremendous impact in the cancer field (Cell, Cancer Research, Clinical Cancer Research, Cancer Discovery, Scientific Reports, PNAS, PLoS Biology, Nature Communications, eLife, etc). This shows the breadth of cancer models that are being developed for multiple purposes. While some models are phenomenological in nature following a bottom-up approach, other models are more top-down data-driven. Here, we discuss the emerging trend in mathematical oncology publications to predict novel, optimal, sometimes even patient-specific treatments, and propose a convention when to use a model to predict novel treatments and, probably more importantly, when not to.

Toward early detection of Helicobacter pylori-associated gastric cancer.

BACKGROUND: Gastric cancer is typically diagnosed at a late stage, leading to poor prognoses. Helicobacter pylori is responsible for 70% of gastric cancers globally, and patients with this bacterial infection often present with early stages of the carcinogenic pathway such as inflammation or gastritis. Although many patients continue to progress to advanced-stage disease after antibacterial treatment, there are no follow-up screening protocols for patients with a history of H. pylori. METHODS: Several biomarkers (Lgr5, CD133, CD44) become upregulated during gastric carcinogenesis. A logistic regression model is developed using clinical data from 59 patients at different stages of the carcinogenic pathway to identify the likelihood of being at an advanced stage of disease for all combinations of age, sex, and marker positivity. Using these likelihood distributions and the observed rate of marker positivity increase, time to high likelihood (probability >0.8) of advanced disease for individual patients is predicted. RESULTS: A strong correlation between marker positivity and disease stage was found for all three markers. Disease stage was accurately classified by the respective regression models for more than 86% of retrospective patients. Highly patient-specific predictions of time to onset of dysplasia were made, allowing the classification of 17 patients initially diagnosed with intestinal metaplasia into high-, intermediate-, or low-risk categories. CONCLUSIONS: We present an approach designed to integrate pathology, mathematics, and statistics for detection of the earliest precancerous, treatable lesion. Given the simplicity and robustness of the framework, such technique has the potential to guide personalized screening schedules to minimize the risk of undetected malignant transformation.

Mathematical Modeling of the Effects of Tumor Heterogeneity on the Efficiency of Radiation Treatment Schedule.

Radiotherapy uses high doses of energy to eradicate cancer cells and control tumors. Various treatment schedules have been developed and tried in clinical trials, yet significant obstacles remain to improving the radiotherapy fractionation. Genetic and non-genetic cellular diversity within tumors can lead to different radiosensitivity among cancer cells that can affect radiation treatment outcome. We propose a minimal mathematical model to study the effect of tumor heterogeneity and repair in different radiation treatment schedules. We perform stochastic and deterministic simulations to estimate model parameters using available experimental data. Our results suggest that gross tumor volume reduction is insufficient to control the disease if a fraction of radioresistant cells survives therapy. If cure cannot be achieved, protocols should balance volume reduction with minimal selection for radioresistant cells. We show that the most efficient treatment schedule is dependent on biology and model parameter values and, therefore, emphasize the need for careful tumor-specific model calibration before clinically actionable conclusions can be drawn.

The Evolution of Tumour Composition During Fractionated Radiotherapy: Implications for Outcome.

Current protocols for delivering radiotherapy are based primarily on tumour stage and nodal and metastases status, even though it is well known that tumours and their microenvironments are highly heterogeneous. It is well established that the local oxygen tension plays an important role in radiation-induced cell death, with hypoxic tumour regions responding poorly to irradiation. Therefore, to improve radiation response, it is important to understand more fully the spatiotemporal distribution of oxygen within a growing tumour before and during fractionated radiation. To this end, we have extended a spatially resolved mathematical model of tumour growth, first proposed by Greenspan (Stud Appl Math 51:317-340, 1972), to investigate the effects of oxygen heterogeneity on radiation-induced cell death. In more detail, cell death due to radiation at each location in the tumour, as determined by the well-known linear-quadratic model, is assumed also to depend on the local oxygen concentration. The oxygen concentration is governed by a reaction-diffusion equation that is coupled to an integro-differential equation that determines the size of the assumed spherically symmetric tumour. We combine numerical and analytical techniques to investigate radiation response of tumours with different intratumoral oxygen distribution profiles. Model simulations reveal a rapid transient increase in hypoxia upon regrowth of the tumour spheroid post-irradiation. We investigate the response to different radiation fractionation schedules and identify a tumour-specific relationship between inter-fraction time and dose per fraction to achieve cure. The rich dynamics exhibited by the model suggest that spatial heterogeneity may be important for predicting tumour response to radiotherapy for clinical applications.

Predicting Patient-Specific Radiotherapy Protocols Based on Mathematical Model Choice for Proliferation Saturation Index.

Radiation is commonly used in cancer treatment. Over 50% of all cancer patients will undergo radiotherapy (RT) as part of cancer care. Scientific advances in RT have primarily focused on the physical characteristics of treatment including beam quality and delivery. Only recently have inroads been made into utilizing tumor biology and radiobiology to design more appropriate RT protocols. Tumors are composites of proliferating and growth-arrested cells, and overall response depends on their respective proportions at irradiation. Prokopiou et al. (Radiat Oncol 10:159, 2015) developed the concept of the proliferation saturation index (PSI) to augment the clinical decision process associated with RT. This framework is based on the application of the logistic equation to pre-treatment imaging data in order to estimate a patient-specific tumor carrying capacity, which is then used to recommend a specific RT protocol. It is unclear, however, how dependent clinical recommendations are on the underlying tumor growth law. We discuss a PSI framework with a generalized logistic equation that can capture kinetics of different well-known growth laws including logistic and Gompertzian growth. Estimation of model parameters on the basis of clinical data revealed that the generalized logistic model can describe data equally well for a wide range of the generalized logistic exponent value. Clinical recommendations based on the calculated PSI, however, are strongly dependent on the specific growth law assumed. Our analysis suggests that the PSI framework may best be utilized in clinical practice when the underlying tumor growth law is known, or when sufficiently many tumor growth models suggest similar fractionation protocols.

The Optimal Radiation Dose to Induce Robust Systemic Anti-Tumor Immunity.

The synergy of radiation and the immune system is currently receiving significant attention in oncology as numerous studies have shown that cancer irradiation can induce strong anti-tumor immune responses. It remains unclear, however, what are the best radiation fractionation protocols to maximize the therapeutic benefits of this synergy. Here, we present a novel mathematical model that can be used to predict and dissect the complexity of the immune-mediated response at multiple tumor sites after applying focal irradiation and systemic immunotherapy. We successfully calibrate the proposed framework with published experimental data, in which two tumors were grown in mice at two spatially-separated sites from which only one was irradiated using various radiation fractionation protocols with and without concurrent systemic immunotherapy. The proposed model is calibrated to fit the temporal dynamics of tumor volume at both sites and can predict changes in immune infiltration in the non-irradiated tumors. The model was then used to investigate additional radiation fractionation protocols. Model simulations suggest that the optimal radiation doses per fraction to maximize anti-tumor immunity are between 10 and 13 Gy, at least for the experimental setting used for model calibration. This work provides the framework for evaluating radiation fractionation protocols for radiation-induced immune-mediated systemic anti-tumor responses.

The future of personalised radiotherapy for head and neck cancer.

Radiotherapy has long been the mainstay of treatment for patients with head and neck cancer and has traditionally involved a stage-dependent strategy whereby all patients with the same TNM stage receive the same therapy. We believe there is a substantial opportunity to improve radiotherapy delivery beyond just technological and anatomical precision. In this Series paper, we explore several new ideas that could improve understanding of the phenotypic and genotypic differences that exist between patients and their tumours. We discuss how exploiting these differences and taking advantage of precision medicine tools-such as genomics, radiomics, and mathematical modelling-could open new doors to personalised radiotherapy adaptation and treatment. We propose a new treatment shift that moves away from an era of empirical dosing and fractionation to an era focused on the development of evidence to guide personalisation and biological adaptation of radiotherapy. We believe these approaches offer the potential to improve outcomes and reduce toxicity.

Neoadjuvant radiotherapy of early-stage breast cancer and long-term disease-free survival.

BACKGROUND: Compared with surgery alone, postoperative adjuvant radiotherapy (RT) improves relapse-free survival of patients with early-stage breast cancer. We evaluated the long-term overall and disease-free survival rates of neoadjuvant (presurgical) versus adjuvant RT in early-stage breast cancer patients. METHODS: We used the Surveillance, Epidemiology, and End Results (SEER) database provided by the National Institutes of Health to derive an analytic dataset of 250,195 female patients with early-stage breast cancer who received RT before (n = 2554; 1.02%) or after (n = 247,641; 98.98%) surgery. Disease-free survival, defined as time to diagnosis of a second primary tumor at any location, was calculated from automated patient identification matching of all SEER records. RESULTS: Partial and complete mastectomies were performed in 94.4% and 5.6% of patients, respectively. In the largest cohort of estrogen receptor-positive women who underwent partial mastectomy, the HR of developing a second primary tumor after neoadjuvant compared with adjuvant RT was 0.64 (95% CI 0.55-0.75; P < 0.0001). Overall survival was independent of radiation sequence (HR 1; P = 0.95). Neoadjuvant RT also resulted in a lower HR for second primary cancer among estrogen receptor-positive patients who underwent mastectomy compared with those who received adjuvant RT (HR 0.48, 95% CI 0.26-0.87; P = 0.0162). CONCLUSIONS: Neoadjuvant RT may significantly improve disease-free survival without reducing overall survival, especially for estrogen receptor-positive patients with early-stage breast cancer. This finding warrants further exploration of potential long-term benefits of neoadjuvant radiotherapy for early-stage breast cancer in a controlled, prospective clinical trial setting, with correlative studies done to identify potential mechanisms of superiority.

Evolution and phenotypic selection of cancer stem cells.

Cells of different organs at different ages have an intrinsic set of kinetics that dictates their behavior. Transformation into cancer cells will inherit these kinetics that determine initial cell and tumor population progression dynamics. Subject to genetic mutation and epigenetic alterations, cancer cell kinetics can change, and favorable alterations that increase cellular fitness will manifest themselves and accelerate tumor progression. We set out to investigate the emerging intratumoral heterogeneity and to determine the evolutionary trajectories of the combination of cell-intrinsic kinetics that yield aggressive tumor growth. We develop a cellular automaton model that tracks the temporal evolution of the malignant subpopulation of so-called cancer stem cells(CSC), as these cells are exclusively able to initiate and sustain tumors. We explore orthogonal cell traits, including cell migration to facilitate invasion, spontaneous cell death due to genetic drift after accumulation of irreversible deleterious mutations, symmetric cancer stem cell division that increases the cancer stem cell pool, and telomere length and erosion as a mitotic counter for inherited non-stem cancer cell proliferation potential. Our study suggests that cell proliferation potential is the strongest modulator of tumor growth. Early increase in proliferation potential yields larger populations of non-stem cancer cells(CC) that compete with CSC and thus inhibit CSC division while a reduction in proliferation potential loosens such inhibition and facilitates frequent CSC division. The sub-population of cancer stem cells in itself becomes highly heterogeneous dictating population level dynamics that vary from long-term dormancy to aggressive progression. Our study suggests that the clonal diversity that is captured in single tumor biopsy samples represents only a small proportion of the total number of phenotypes.

TNF-TNFR2/p75 signaling inhibits early and increases delayed nontargeted effects in bone marrow-derived endothelial progenitor cells.

TNF-α, a pro-inflammatory cytokine, is highly expressed after being irradiated (IR) and is implicated in mediating radiobiological bystander responses (RBRs). Little is known about specific TNF receptors in regulating TNF-induced RBR in bone marrow-derived endothelial progenitor cells (BM-EPCs). Full body γ-IR WT BM-EPCs showed a biphasic response: slow decay of p-H2AX foci during the initial 24 h and increase between 24 h and 7 days post-IR, indicating a significant RBR in BM-EPCs in vivo. Individual TNF receptor (TNFR) signaling in RBR was evaluated in BM-EPCs from WT, TNFR1/p55KO, and TNFR2/p75KO mice, in vitro. Compared with WT, early RBR (1-5 h) were inhibited in p55KO and p75KO EPCs, whereas delayed RBR (3-5 days) were amplified in p55KO EPCs, suggesting a possible role for TNFR2/p75 signaling in delayed RBR. Neutralizing TNF in γ-IR conditioned media (CM) of WT and p55KO BM-EPCs largely abolished RBR in both cell types. ELISA protein profiling of WT and p55KO EPC γ-IR-CM over 5 days showed significant increases in several pro-inflammatory cytokines, including TNF-α, IL-1α (Interleukin-1 alpha), RANTES (regulated on activation, normal T cell expressed and secreted), and MCP-1. In vitro treatments with murine recombinant (rm) TNF-α and rmIL-1α, but not rmMCP-1 or rmRANTES, increased the formation of p-H2AX foci in nonirradiated p55KO EPCs. We conclude that TNF-TNFR2 signaling may induce RBR in naïve BM-EPCs and that blocking TNF-TNFR2 signaling may prevent delayed RBR in BM-EPCs, conceivably, in bone marrow milieu in general.