Fractal calculus in tumor growth simulations: The proof is in the pudding.

Mathematical modeling in oncology has a long history. Recently, mathematical models and their predictions have made inroads into prospective clinical trials with encouraging results. The goal of many such modeling efforts is to make predictions, either to clinician's choice therapy or into "optimal" therapy - often for individual patients. The mathematical oncology community rightfully puts great hope into predictive modeling and mechanistic digital twins - but with this great opportunity comes great responsibility. Mathematical models need to be rigorously calibrated and validated, and their predictive performance ascertained, before conclusions about predictions into the unknown can be drawn. The recent article "Modeling tumor growth using fractal calculus: Insights into tumor dynamics" (Golmankhaneh et al., 2023), applied fractal calculus to tumor growth data. In this short commentary, I raise concerns about the study design and interpretation. In its current form, this study is poised to put cancer patients at risk if interpreted as concluded by the authors.

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.

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.

Mathematical modeling of radiotherapy: impact of model selection on estimating minimum radiation dose for tumor control.

Introduction: Radiation therapy (RT) is one of the most common anticancer therapies. Yet, current radiation oncology practice does not adapt RT dose for individual patients, despite wide interpatient variability in radiosensitivity and accompanying treatment response. We have previously shown that mechanistic mathematical modeling of tumor volume dynamics can simulate volumetric response to RT for individual patients and estimation personalized RT dose for optimal tumor volume reduction. However, understanding the implications of the choice of the underlying RT response model is critical when calculating personalized RT dose. Methods: In this study, we evaluate the mathematical implications and biological effects of 2 models of RT response on dose personalization: (1) cytotoxicity to cancer cells that lead to direct tumor volume reduction (DVR) and (2) radiation responses to the tumor microenvironment that lead to tumor carrying capacity reduction (CCR) and subsequent tumor shrinkage. Tumor growth was simulated as logistic growth with pre-treatment dynamics being described in the proliferation saturation index (PSI). The effect of RT was simulated according to each respective model for a standard schedule of fractionated RT with 2 Gy weekday fractions. Parameter sweeps were evaluated for the intrinsic tumor growth rate and the radiosensitivity parameter for both models to observe the qualitative impact of each model parameter. We then calculated the minimum RT dose required for locoregional tumor control (LRC) across all combinations of the full range of radiosensitvity and proliferation saturation values. Results: Both models estimate that patients with higher radiosensitivity will require a lower RT dose to achieve LRC. However, the two models make opposite estimates on the impact of PSI on the minimum RT dose for LRC: the DVR model estimates that tumors with higher PSI values will require a higher RT dose to achieve LRC, while the CCR model estimates that higher PSI values will require a lower RT dose to achieve LRC. Discussion: Ultimately, these results show the importance of understanding which model best describes tumor growth and treatment response in a particular setting, before using any such model to make estimates for personalized treatment recommendations.

Rules-based Volumetric Segmentation of Multiparametric MRI for Response Assessment in Recurrent High-Grade Glioma.

We report domain knowledge-based rules for assigning voxels in brain multiparametric MRI (mpMRI) to distinct tissuetypes based on their appearance on Apparent Diffusion Coefficient of water (ADC) maps, T1-weighted unenhanced and contrast-enhanced, T2-weighted, and Fluid-Attenuated Inversion Recovery images. The development dataset comprised mpMRI of 18 participants with preoperative high-grade glioma (HGG), recurrent HGG (rHGG), and brain metastases. External validation was performed on mpMRI of 235 HGG participants in the BraTS 2020 training dataset. The treatment dataset comprised serial mpMRI of 32 participants (total 231 scan dates) in a clinical trial of immunoradiotherapy in rHGG (NCT02313272). Pixel intensity-based rules for segmenting contrast-enhancing tumor (CE), hemorrhage, Fluid, non-enhancing tumor (Edema1), and leukoaraiosis (Edema2) were identified on calibrated, co-registered mpMRI images in the development dataset. On validation, rule-based CE and High FLAIR (Edema1 + Edema2) volumes were significantly correlated with ground truth volumes of enhancing tumor (R = 0.85;p < 0.001) and peritumoral edema (R = 0.87;p < 0.001), respectively. In the treatment dataset, a model combining time-on-treatment and rule-based volumes of CE and intratumoral Fluid was 82.5% accurate for predicting progression within 30 days of the scan date. An explainable decision tree applied to brain mpMRI yields validated, consistent, intratumoral tissuetype volumes suitable for quantitative response assessment in clinical trials of rHGG.

Logistic tumor-population growth and ghost-points symmetry.

The observed time evolution of a population is well approximated by a logistic function in many research fields, including oncology, ecology, chemistry, demography, economy, linguistics, and artificial neural networks. Initial growth is exponential at a constant rate and capped at a limit size, i.e., the carrying capacity. In mathematical oncology, the carrying capacity has been postulated to be co-evolving and thus patient-specific. As the relative tumor-over-carrying capacity ratio may be predictive and prognostic for tumor growth and treatment response dynamics, it is paramount to estimate it from limited clinical data. We show that exploiting the logistic function's rotation symmetry can help estimate the population's growth rate and carry capacity from fewer data points than conventional regression approaches. We test this novel approach against a classic oncology database of logistic tumor growth, achieving a 30% to 40% reduction in the time necessary to correctly estimate the logistic growth rate and carrying capacity. Our results will improve tumor dynamics forecasting and augment the clinical decision-making process.

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.

Predicting Patient-Specific Tumor Dynamics: How Many Measurements Are Necessary?

Acquiring sufficient data is imperative to accurately predict tumor growth dynamics and effectively treat patients. The aim of this study was to investigate the number of volume measurements necessary to predict breast tumor growth dynamics using the logistic growth model. The model was calibrated to tumor volume data from 18 untreated breast cancer patients using a varying number of measurements interpolated at clinically relevant timepoints with different levels of noise (0-20%). Error-to-model parameters and the data were compared to determine the sufficient number of measurements needed to accurately determine growth dynamics. We found that without noise, three tumor volume measurements are necessary and sufficient to estimate patient-specific model parameters. More measurements were required as the level of noise increased. Estimating the tumor growth dynamics was shown to depend on the tumor growth rate, clinical noise level, and acceptable error of the to-be-determined parameters. Understanding the relationship between these factors provides a metric by which clinicians can determine when sufficient data have been collected to confidently predict patient-specific tumor growth dynamics and recommend appropriate treatment options.

Range-Bounded Adaptive Therapy in Metastatic Prostate Cancer.

Adaptive therapy with abiraterone acetate (AA), whereby treatment is cycled on and off, has been presented as an alternative to continuous therapy for metastatic castration resistant prostate cancer (mCRPC). It is hypothesized that cycling through treatment allows sensitive cells to competitively suppress resistant cells, thereby increasing the amount of time that treatment is effective. It has been proposed that there exists a subset of patients for whom this competition can be enhanced through slight modifications. Here, we investigate how adaptive AA can be modified to extend time to progression using a simple mathematical model of stem cell, non-stem cell, and prostate-specific antigen (PSA) dynamics. The model is calibrated to longitudinal PSA data from 16 mCRPC patients undergoing adaptive AA in a pilot clinical study at Moffitt Cancer Center. Model parameters are then used to simulate range-bounded adaptive therapy (RBAT) whereby treatment is modulated to maintain PSA levels between pre-determined patient-specific bounds. Model simulations of RBAT are compared to the clinically applied adaptive therapy and show that RBAT can further extend time to progression, while reducing the cumulative dose patients received in 11/16 patients. Simulations also show that the cumulative dose can be reduced by up to 40% under RBAT. Through small modifications to the conventional adaptive therapy design, our study demonstrates that RBAT offers the opportunity to improve patient care, particularly in those patients who do not respond well to conventional adaptive therapy.

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.

Rethinking the immunotherapy numbers game.

Immunotherapies are a major breakthrough in oncology, yielding unprecedented response rates for some cancers. Especially in combination with conventional treatments or targeted agents, immunotherapeutics offer invaluable tools to improve outcomes for many patients. However, why not all patients have a favorable response remains unclear. There is an increasing appreciation of the contributions of the complex tumor microenvironment, and the tumor-immune ecosystem in particular, to treatment outcome. To date, however, there exists no immune biomarker to explain why two patients with similar clinical stage and molecular profile would have different treatment outcomes. We hypothesize that it is critical to understand both the immune and tumor states to understand how the complex system will respond to treatment. Here, we present how integrated mathematical oncology approaches can help conceptualize the effect of various immunotherapies on a patient's tumor and local immune environment, and how combinations of immunotherapy and cytotoxic therapy may be used to improve tumor response and control and limit toxicity on a per patient basis.

Mathematical modeling of radiotherapy and its impact on tumor interactions with the immune system.

Radiotherapy is a primary therapeutic modality widely utilized with curative intent. Traditionally tumor response was hypothesized to be due to high levels of cell death induced by irreparable DNA damage. However, the immunomodulatory aspect of radiation is now widely accepted. As such, interest into the combination of radiotherapy and immunotherapy is increasing, the synergy of which has the potential to improve tumor regression beyond that observed after either treatment alone. However, questions regarding the timing (sequential vs concurrent) and dose fractionation (hyper-, standard-, or hypo-fractionation) that result in improved anti-tumor immune responses, and thus potentially enhanced tumor inhibition, remain. Here we discuss the biological response to radiotherapy and its immunomodulatory properties before giving an overview of pre-clinical data and clinical trials concerned with answering these questions. Finally, we review published mathematical models of the impact of radiotherapy on tumor-immune interactions. Ranging from considering the impact of properties of the tumor microenvironment on the induction of anti-tumor responses, to the impact of choice of radiation site in the setting of metastatic disease, these models all have an underlying feature in common: the push towards personalized therapy.

Early response dynamics predict treatment failure in patients with recurrent and/or metastatic head and neck squamous cell carcinoma treated with cetuximab and nivolumab.

OBJECTIVES: Recurrent and/or metastatic (R/M) head and neck squamous cell carcinoma (HNSCC) is currently an incurable disease. To improve treatment strategies, combinations of cetuximab plus nivolumab or pembrolizumab were evaluated for efficacy and safety for incurable R/M HNSCC. While some patients had a significant clinical benefit with complete or partial response, most patients had stable or progressive disease (PD). To identify patients with a high likelihood of treatment failure and prevent futile treatments, we developed a mathematical model of early response dynamics as an early biomarker of treatment failure. MATERIALS AND METHODS: Demographics, RECIST assessment, and outcome were obtained from patients who were treated with combination of cetuximab and nivolumab on a previously published phase I/II clinical trial. We trained a tumor growth inhibition (TGI) ordinary differential equation (ODE) model describing patient-specific pre-treatment growth rate and uniform initial treatment sensitivity and rate of evolution of resistance. In a leave-one-out approach, we forecasted tumor burden and predicted time to progression (TTP) and PD. RESULTS: The TGI model accurately represented tumor burden dynamics (R2=0.98; RMSE=0.57 cm) and predicted PD with accuracy=0.71,sensitivity=1.00, and specificity=0.69 after three serial response assessment scans. Patient-specific pre-treatment growth rate correlated negatively with TTP (Spearman's ρ=-0.67,p=5.7e-05). CONCLUSION: The TGI model can identify patients with high likelihood of PD based on early dynamics. Further studies including prospective validation are warranted.

The Radiosensitivity Index Gene Signature Identifies Distinct Tumor Immune Microenvironment Characteristics Associated With Susceptibility to Radiation Therapy.

PURPOSE: Radiation therapy (RT) is a mainstay of cancer care, and accumulating evidence suggests the potential for synergism with components of the immune response. However, few data describe the tumor immune contexture in relation to RT sensitivity. To address this challenge, we used the radiation sensitivity index (RSI) gene signature to estimate the RT sensitivity of >10,000 primary tumors and characterized their immune microenvironments in relation to the RSI. METHODS AND MATERIALS: We analyzed gene expression profiles of 10,469 primary tumors (31 types) within a prospective tissue collection protocol. The RT sensitivity of each tumor was estimated by the RSI and respective distributions were characterized. The tumor biology measured by the RSI was evaluated by differentially expressed genes combined with single sample gene set enrichment analysis. Differences in the expression of immune regulatory molecules were assessed and deconvolution algorithms were used to estimate immune cell infiltrates in relation to the RSI. A subset (n = 2368) of tumors underwent DNA sequencing for mutational frequency characterization. RESULTS: We identified a wide range of RSI values within and across various tumor types, with several demonstrating nonunimodal distributions (eg, colon, renal, lung, prostate, esophagus, pancreas, and PAM50 breast subtypes; P < .05). Across all tumor types, stratifying RSI at a tumor type-specific median identified 7148 differentially expressed genes, of which 146 were coordinate in direction. Network topology analysis demonstrates RSI measures a coordinated STAT1, IRF1, and CCL4/MIP-1ß transcriptional network. Tumors with an estimated high sensitivity to RT demonstrated distinct enrichment of interferon-associated signaling pathways and immune cell infiltrates (eg, CD8+ T cells, activated natural killer cells, M1-macrophages; q < 0.05), which was in the context of diverse expression patterns of various immunoregulatory molecules. CONCLUSIONS: This analysis describes the immune microenvironments of patient tumors in relation to the RSI gene expression signature.

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.

Predictive Radiation Oncology - A New NCI-DOE Scientific Space and Community.

With a widely attended virtual kickoff event on January 29, 2021, the National Cancer Institute (NCI) and the Department of Energy (DOE) launched a series of 4 interactive, interdisciplinary workshops-and a final concluding "World Café" on March 29, 2021-focused on advancing computational approaches for predictive oncology in the clinical and research domains of radiation oncology. These events reflect 3,870 human hours of virtual engagement with representation from 8 DOE national laboratories and the Frederick National Laboratory for Cancer Research (FNL), 4 research institutes, 5 cancer centers, 17 medical schools and teaching hospitals, 5 companies, 5 federal agencies, 3 research centers, and 27 universities. Here we summarize the workshops by first describing the background for the workshops. Participants identified twelve key questions-and collaborative parallel ideas-as the focus of work going forward to advance the field. These were then used to define short-term and longer-term "Blue Sky" goals. In addition, the group determined key success factors for predictive oncology in the context of radiation oncology, if not the future of all of medicine. These are: cross-discipline collaboration, targeted talent development, development of mechanistic mathematical and computational models and tools, and access to high-quality multiscale data that bridges mechanisms to phenotype. The workshop participants reported feeling energized and highly motivated to pursue next steps together to address the unmet needs in radiation oncology specifically and in cancer research generally and that NCI and DOE project goals align at the convergence of radiation therapy and advanced computing.