Proliferation saturation index in an adaptive Bayesian approach to predict patient-specific radiotherapy responses

Predicting Individual Tumor Response Dynamics in Locally Advanced Non-Small Cell Lung Cancer Radiation Therapy: A Mathematical Modelling Study

PURPOSE

To predict individual tumor responses to radiation therapy (RT) in non-small cell lung cancer.

MATERIALS AND METHODS

The proliferation saturation index (PSI) model, which models tumor dynamics in response to RT as an instantaneous reduction in tumor volume, was fit to n = 162 patients with 4 distinct dose fractionation schedules (30-32 fractions × 2 Gy, 23-24 fractions × 2.75 Gy, 32-42 fractions × 1.8 Gy, and 30 fractions × 1.5 Gy Bidaily, followed by 5-12 fractions × 2 Gy daily). Following initial training, the predictive power of the model was tested using only the first 3 tumor volume measurements as measured on daily imaging. The remainder of tumor volume regression during RT was simulated using the PSI model. Comparisons of the measured to the simulated volumes were made using scatter plots, coefficient of determination (R2), and Pearson correlation coefficient values.

RESULTS

The PSI model predicted tumor volume regression during RT with a high degree of accuracy. Comparison of the measured versus predicted volumes resulted in R2 values of 0.968, 0.954, 0.968, and 0.937, and Pearson correlation coefficient values of 0.984, 0.977, 0.984, and 0.968 in the 2 Gy, 1.8 Gy, 2.75 Gy, and Bidaily groups, respectively.

CONCLUSIONS

The proliferation saturation model can predict, with a high degree of accuracy, non-small cell lung cancer tumor volume regression in response to RT in 4 distinct dose fractionation schedules.

Optimizing fractionation schedules for de-escalation radiotherapy in head and neck cancers using deep reinforcement learning

PURPOSE

Patients with locally-advanced head and neck squamous cell carcinomas (HNSCCs), particularly those related to human papillomavirus (HPV), often achieve good locoregional control (LRC), yet they suffer significant toxicities from standard chemoradiotherapy. This study aims to optimize the daily dose fractionation based on individual responses to radiotherapy (RT), minimizing toxicity while maintaining a low risk of LRC failure.

METHOD

A virtual environment was developed to simulate tumor dynamics under RT for optimizing dose schedules. Patients predicted to maintain LRC were selected for de-escalation experiments. The proliferation saturation index (PSI) and linear-quadratic model were used to predict responses. A deep reinforcement learning (DRL) agent optimized fractionation schemes by interacting with the simulation environment, aiming to reduce the OAR's biologically effective dose (BED) while preserving LRC. The impact of model uncertainty was analyzed and a support vector machine (SVM) model was used to segment parameter space and identify patients more robust to noise.

RESULTS

Personalized de-escalation plans were compared to conventional RT in a cohort of 5000 virtual patients. Personalized fractionation reduced the tumor dose and OAR's BED by 29%, with an average OAR BED reduction of 5.61 ± 2.96 Gy. Prognostic outcomes were nearly identical, with 99.80% of patients in the low-risk LRC failure group. Model uncertainty impacted dosimetric indicators and prognosis, but the high-BED benefit group showed greater robustness to noise. SVM decision boundaries defined parameters range for patient selection.

CONCLUSION

Optimizing fractionated doses based on patient responses minimizes toxicity while maintaining LRC in HNSCCs. Stratifying patients can mitigate model uncertainty and reduce treatment risks.

Harnessing Flex Point Symmetry to Estimate Logistic Tumor Population Growth

The observed time evolution of a population is well approximated by a logistic growth function in many research fields, including oncology, ecology, chemistry, demography, economy, linguistics, and artificial neural networks. Initial growth is exponential, then decelerates as the population approaches its limit size, i.e., the carrying capacity. In mathematical oncology, the tumor carrying capacity has been postulated to be dynamically evolving as the tumor overcomes several evolutionary bottlenecks and, thus, to be 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 published pan-cancer animal and human breast cancer data, achieving a 30% to 40% reduction in the time at which subsequent data collection is necessary to estimate the logistic growth rate and carrying capacity correctly. These results could improve tumor dynamics forecasting and augment the clinical decision-making process.

Multi-modality imaging parameters that predict rapid tumor regression in head and neck radiotherapy

Background and purpose

Volume regression during radiotherapy can indicate patient-specific treatment response. We aimed to identify pre-treatment multimodality imaging (MMI) metrics from positron emission tomography (PET), magnetic resonance imaging (MRI), and computed tomography (CT) that predict rapid tumor regression during radiotherapy in human papilloma virus (HPV) associated oropharyngeal carcinoma.

Materials and methods

Pre-treatment FDG PET-CT, diffusion-weighted MRI (DW-MRI), and intra-treatment (at 1, 2, and 3 weeks) MRI were acquired in 72 patients undergoing chemoradiation therapy for HPV+ oropharyngeal carcinoma. Nodal gross tumor volumes were delineated on longitudinal images to measure intra-treatment volume changes. Pre-treatment PET standardized uptake value (SUV), CT Hounsfield Unit (HU), and non-gaussian intravoxel incoherent motion DW-MRI metrics were computed and correlated with volume changes. Intercorrelations between MMI metrics were also assessed using network analysis. Validation was carried out on a separate cohort (N = 64) for FDG PET-CT.

Results

Significant correlations with volume loss were observed for baseline FDG SUVmean (Spearman ρ = 0.46, p < 0.001), CT HUmean (ρ = 0.38, p = 0.001), and DW-MRI diffusion coefficient, Dmean (ρ = -0.39, p < 0.001). Network analysis revealed 41 intercorrelations between MMI and volume loss metrics, but SUVmean remained a statistically significant predictor of volume loss in multivariate linear regression (p = 0.01). Significant correlations were also observed for SUVmean in the validation cohort in both primary (ρ = 0.30, p = 0.02) and nodal (ρ = 0.31, p = 0.02) tumors.

Conclusions

Multiple pre-treatment imaging metrics were correlated with rapid nodal gross tumor volume loss during radiotherapy. FDG-PET SUV in particular exhibited significant correlations with volume regression across the two cohorts and in multivariate analysis.

Comparison of 3-year local control using DAHANCA radiotherapy guidelines before and after implementation of five millimetres geometrical GTV to high-dose CTV margin

INTRODUCTION

Treatment planning using a five-millimetre geometrical margin from GTV to high-dose CTV (CTV1) has been used in DAHANCA treatment centres since 2013. We aimed to evaluate changes in CTV1 volumes, local control (LC), and recurrence pattern after the implementation of five-millimetre geometrical margins nationally.

MATERIALS AND METHODS

1,948 patients with pharyngeal, and laryngeal squamous cell carcinomas completed definitive IMRT-based treatment in 2010-2012 and 2013-2015 in three centres. The patient-specific margin was calculated as median surface distance from primary tumour GTV (GTV-T) to CTV1. Radiologically verified local recurrences were analysed using a centre of mass (COM) of the delineated recurrence volume, measuring the shortest distance between COM to GTV-T and CTV1 boundaries.

RESULTS

Median GTV-CTV1 was 0.9 (0.0-0.97) and 0.47 cm (0.4-0.5) for 2010-2012 and 2013-2015, respectively. Median CTV1 changed in three centres from 76, 28, 42 cm3 to 61, 53, 62 cm3 for 2010-2012 and 2013-2015, respectively. Local failures occurred at 247 patients during first three years after radiotherapy. The 3-year LC rate for 2010-2012 and 2013-2015 was 0.84 and 0.87 (p = 0.06). Out of 146 radiology-verified analysable local recurrences, 102 (69.9%) were inside the CTV1. In 74.6% and 91% of cases, the LRs were covered by 95% isodose in 2010-2012 and 2013-2015, respectively.

CONCLUSION

DAHANCA radiotherapy guidelines based on a geometrically generated isotropic CTV1 margin led to less variation in treatment volumes and between centres than previous guidelines. The transition towards consensus GTV-CTV1 margins did not influence local tumour control. The majority of local recurrences were inside CTV1 and covered by the prescription dose.

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.

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.

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.

The critical components for effective adaptive radiotherapy in patients with unresectable non-small-cell lung cancer: who, when and how

Adaptive radiotherapy (ART) is a new radiotherapy technology based on image-guided radiation therapy technology, used to avoid radiation overexposure to residual tumors and the surrounding normal tissues. Tumors undergoing the same radiation doses and modes can occur unequal shrinkage due to the variation of response times to radiation doses in different patients. To perform ART effectively, eligible patients with a high probability of benefits from ART need to be identified. Confirming the precise timetable for ART in every patient is another urgent problem to be resolved. Moreover, the outcomes of ART are different depending on the various image guidance used. This review discusses 'who, when and how' as the three key factors involved in the most effective implementation for the management of ART.

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.

Integrating mechanism-based modeling with biomedical imaging to build practical digital twins for clinical oncology

Digital twins employ mathematical and computational models to virtually represent a physical object (e.g., planes and human organs), predict the behavior of the object, and enable decision-making to optimize the future behavior of the object. While digital twins have been widely used in engineering for decades, their applications to oncology are only just emerging. Due to advances in experimental techniques quantitatively characterizing cancer, as well as advances in the mathematical and computational sciences, the notion of building and applying digital twins to understand tumor dynamics and personalize the care of cancer patients has been increasingly appreciated. In this review, we present the opportunities and challenges of applying digital twins in clinical oncology, with a particular focus on integrating medical imaging with mechanism-based, tissue-scale mathematical modeling. Specifically, we first introduce the general digital twin framework and then illustrate existing applications of image-guided digital twins in healthcare. Next, we detail both the imaging and modeling techniques that provide practical opportunities to build patient-specific digital twins for oncology. We then describe the current challenges and limitations in developing image-guided, mechanism-based digital twins for oncology along with potential solutions. We conclude by outlining five fundamental questions that can serve as a roadmap when designing and building a practical digital twin for oncology and attempt to provide answers for a specific application to brain cancer. We hope that this contribution provides motivation for the imaging science, oncology, and computational communities to develop practical digital twin technologies to improve the care of patients battling cancer.

From Fitting the Average to Fitting the Individual: A Cautionary Tale for Mathematical Modelers

An outstanding challenge in the clinical care of cancer is moving from a one-size-fits-all approach that relies on population-level statistics towards personalized therapeutic design. Mathematical modeling is a powerful tool in treatment personalization, as it allows for the incorporation of patient-specific data so that treatment can be tailor-designed to the individual. Herein, we work with a mathematical model of murine cancer immunotherapy that has been previously-validated against the average of an experimental dataset. We ask the question: what happens if we try to use this same model to perform personalized fits, and therefore make individualized treatment recommendations? Typically, this would be done by choosing a single fitting methodology, and a single cost function, identifying the individualized best-fit parameters, and extrapolating from there to make personalized treatment recommendations. Our analyses show the potentially problematic nature of this approach, as predicted personalized treatment response proved to be sensitive to the fitting methodology utilized. We also demonstrate how a small amount of the right additional experimental measurements could go a long way to improve consistency in personalized fits. Finally, we show how quantifying the robustness of the average response could also help improve confidence in personalized treatment recommendations.

Dynamics-Adapted Radiotherapy Dose (DARD) for Head and Neck Cancer Radiotherapy Dose Personalization

Standard of care radiotherapy (RT) doses have been developed as a one-size-fits all approach designed to maximize tumor control rates across a population. Although this has led to high control rates for head and neck cancer with 66-70 Gy, this is done without considering patient heterogeneity. We present a framework to estimate a personalized RT dose for individual patients, based on pre- and early on-treatment tumor volume dynamics-a dynamics-adapted radiotherapy dose (D_DARD). We also present the results of an in silico trial of this dose personalization using retrospective data from a combined cohort of n = 39 head and neck cancer patients from the Moffitt and MD Anderson Cancer Centers that received 66-70 Gy RT in 2-2.12 Gy weekday fractions. This trial was repeated constraining D_DARD between (54, 82) Gy to test more moderate dose adjustment. D_DARD was estimated to range from 8 to 186 Gy, and our in silico trial estimated that 77% of patients treated with standard of care were overdosed by an average dose of 39 Gy, and 23% underdosed by an average dose of 32 Gy. The in silico trial with constrained dose adjustment estimated that locoregional control could be improved by >10%. We demonstrated the feasibility of using early treatment tumor volume dynamics to inform dose personalization and stratification for dose escalation and de-escalation. These results demonstrate the potential to both de-escalate most patients, while still improving population-level control rates.

Forecasting Individual Patient Response to Radiation Therapy in Head and Neck Cancer With a Dynamic Carrying Capacity Model

Purpose:

To model and predict individual patient responses to radiation therapy.

Methods and Materials:

We modeled tumor dynamics as logistic growth and the effect of radiation as a reduction in the tumor carrying capacity, motivated by the effect of radiation on the tumor microenvironment. The model was assessed on weekly tumor volume data collected for 2 independent cohorts of patients with head and neck cancer from the H. Lee Moffitt Cancer Center (MCC) and the MD Anderson Cancer Center (MDACC) who received 66 to 70 Gy in standard daily fractions or with accelerated fractionation. To predict response to radiation therapy for individual patients, we developed a new forecasting framework that combined the learned tumor growth rate and carrying capacity reduction fraction (δ) distribution with weekly measurements of tumor volume reduction for a given test patient to estimate δ, which was used to predict patient-specific outcomes.

Results:

The model fit data from MCC with high accuracy with patient-specific δ and a fixed tumor growth rate across all patients. The model fit data from an independent cohort from MDACC with comparable accuracy using the tumor growth rate learned from the MCC cohort, showing transferability of the growth rate. The forecasting framework predicted patient-specific outcomes with 76% sensitivity and 83% specificity for locoregional control and 68% sensitivity and 85% specificity for disease-free survival with the inclusion of 4 on-treatment tumor volume measurements.

Conclusions:

These results demonstrate that our simple mathematical model can describe a variety of tumor volume dynamics. Furthermore, combining historically observed patient responses with a few patient-specific tumor volume measurements allowed for the accurate prediction of patient outcomes, which may inform treatment adaptation and personalization.

Kinetic modeling and numerical simulations to predict patient-specific responses to radiotherapy

Recent research indicates that quiescent tumor cells are significantly less radiosensitive with a greater repair capacity than proliferative cells. In order to better predict patient-specific responses to radiotherapy, we develop a mathematical model with treatment terms to describe dynamical behaviors of tumor growth. The global stabilities of the tumor-free equilibrium, the tumor-present equilibrium and the corresponding sufficient criteria are obtained. In addition, we simulate volumetric imaging data from 12 head-and-neck cancer patients and estimate the patient-specific responses to radiotherapy. Results indicate that radiosensitivity of proliferative cells is a critical factor that determines a successful radiotherapy. By comparison with previous simulation results, we find that the model presented in this paper is more suitable to describe the radiotherapy procedure of head-and-neck cancer. Finally, we discuss the influences of different radiotherapy strategies on therapeutic effect. The results show that treatment strategies with large dose or short treatment cycle can obtain better treatment effect.

Towards an Image-Informed Mathematical Model of In Vivo Response to Fractionated Radiation Therapy

Simple Summary

Using medical imaging data and computational models, we develop a modeling framework to provide personalized treatment response forecasts to fractionated radiation therapy for individual tumors. We evaluate this approach in an animal model of brain cancer and forecast changes in tumor cellularity and vasculature.

Abstract

Fractionated radiation therapy is central to the treatment of numerous malignancies, including high-grade gliomas where complete surgical resection is often impractical due to its highly invasive nature. Development of approaches to forecast response to fractionated radiation therapy may provide the ability to optimize or adapt treatment plans for radiotherapy. Towards this end, we have developed a family of 18 biologically-based mathematical models describing the response of both tumor and vasculature to fractionated radiation therapy. Importantly, these models can be personalized for individual tumors via quantitative imaging measurements. To evaluate this family of models, rats (n = 7) with U-87 glioblastomas were imaged with magnetic resonance imaging (MRI) before, during, and after treatment with fractionated radiotherapy (with doses of either 2 Gy/day or 4 Gy/day for up to 10 days). Estimates of tumor and blood volume fractions, provided by diffusion-weighted MRI and dynamic contrast-enhanced MRI, respectively, were used to calibrate tumor-specific model parameters. The Akaike Information Criterion was employed to select the most parsimonious model and determine an ensemble averaged model, and the resulting forecasts were evaluated at the global and local level. At the global level, the selected model’s forecast resulted in less than 16.2% error in tumor volume estimates. At the local (voxel) level, the median Pearson correlation coefficient across all prediction time points ranged from 0.57 to 0.87 for all animals. While the ensemble average forecast resulted in increased error (ranging from 4.0% to 1063%) in tumor volume predictions over the selected model, it increased the voxel wise correlation (by greater than 12.3%) for three of the animals. This study demonstrates the feasibility of calibrating a model of response by serial quantitative MRI data collected during fractionated radiotherapy to predict response at the conclusion of treatment.

WAM to SeeSaw model for cancer therapy - overcoming LQM difficulties

PURPOSE

The assessment of biological effects caused by radiation exposure has been currently carried out with the linear-quadratic (LQ) model as an extension of the linear non-threshold (LNT) model. In this study, we suggest a new mathematical model named as SeaSaw (SS) model, which describes proliferation and cell death effects by taking account of Bergonie-Tribondeau's law in terms of a differential equation in time. We show how this model overcomes the long-standing difficulties of the LQ model.

MATERIALS AND METHODS

We construct the SS model as an extended Wack-A-Mole (WAM) model by using a differential equation with respect to time in order to express the dynamics of the proliferation effect. A large number of accumulated data of such parameters as α and β in the LQ based models provide us with valuable pieces of information on the corresponding parameter b 1 and the maximum volume V m of the SS model. The dose rate b 1 and the notion of active cell can explain the present data without introduction of β, which is obtained by comparing the SS model with not only the cancer therapy data but also with in vitro experimental data. Numerical calculations are presented to grasp the global features of the SS model.

RESULTS

The SS model predicts the time dependence of the number of active- and inactive-cells. The SS model clarifies how the effect of radiation depends on the cancer stage at the starting time in the treatment. Further, the time dependence of the tumor volume is calculated by changing individual dose strength, which results in the change of the irradiation duration for the same effect. We can consider continuous irradiation in the SS model with interesting outcome on the time dependence of the tumor volume for various dose rates. Especially by choosing the value of the dose rate to be balanced with the total growth rate, the tumor volume is kept constant.

CONCLUSIONS

The SS model gives a simple equation to study the situation of clinical radiation therapy and risk estimation of radiation. The radiation parameter extracted from the cancer therapy is close to the value obtained from animal experiment in vitro and in vivo. We expect the SS model leads us to a unified description of radiation therapy and protection and provides a great development in cancer-therapy clinical-planning.

Bayesian Information-Theoretic Calibration of Radiotherapy Sensitivity Parameters for Informing Effective Scanning Protocols in Cancer

With new advancements in technology, it is now possible to collect data for a variety of different metrics describing tumor growth, including tumor volume, composition, and vascularity, among others. For any proposed model of tumor growth and treatment, we observe large variability among individual patients’ parameter values, particularly those relating to treatment response; thus, exploiting the use of these various metrics for model calibration can be helpful to infer such patient-specific parameters both accurately and early, so that treatment protocols can be adjusted mid-course for maximum efficacy. However, taking measurements can be costly and invasive, limiting clinicians to a sparse collection schedule. As such, the determination of optimal times and metrics for which to collect data in order to best inform proper treatment protocols could be of great assistance to clinicians. In this investigation, we employ a Bayesian information-theoretic calibration protocol for experimental design in order to identify the optimal times at which to collect data for informing treatment parameters. Within this procedure, data collection times are chosen sequentially to maximize the reduction in parameter uncertainty with each added measurement, ensuring that a budget of n high-fidelity experimental measurements results in maximum information gain about the low-fidelity model parameter values. In addition to investigating the optimal temporal pattern for data collection, we also develop a framework for deciding which metrics should be utilized at each data collection point. We illustrate this framework with a variety of toy examples, each utilizing a radiotherapy treatment regimen. For each scenario, we analyze the dependence of the predictive power of the low-fidelity model upon the measurement budget.

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.

Forecasting tumor and vasculature response dynamics to radiation therapy via image based mathematical modeling

BACKGROUND

Intra-and inter-tumoral heterogeneity in growth dynamics and vascularity influence tumor response to radiation therapy. Quantitative imaging techniques capture these dynamics non-invasively, and these data can initialize and constrain predictive models of response on an individual basis.

METHODS

We have developed a family of 10 biologically-based mathematical models describing the spatiotemporal dynamics of tumor volume fraction, blood volume fraction, and response to radiation therapy. To evaluate this family of models, rats (n = 13) with C6 gliomas were imaged with magnetic resonance imaging (MRI) three times before, and four times following a single fraction of 20 Gy or 40 Gy whole brain irradiation. The first five 3D time series data of tumor volume fraction, estimated from diffusion-weighted (DW-) MRI, and blood volume fraction, estimated from dynamic contrast-enhanced (DCE-) MRI, were used to calibrate tumor-specific model parameters. The most parsimonious and well calibrated of the 10 models, selected using the Akaike information criterion, was then utilized to predict future growth and response at the final two imaging time points. Model predictions were compared at the global level (percent error in tumor volume, and Dice coefficient) as well as at the local or voxel level (concordance correlation coefficient).

RESULT

The selected model resulted in < 12% error in tumor volume predictions, strong spatial agreement between predicted and observed tumor volumes (Dice coefficient > 0.74), and high level of agreement at the voxel level between the predicted and observed tumor volume fraction and blood volume fraction (concordance correlation coefficient > 0.77 and > 0.65, respectively).

CONCLUSIONS

This study demonstrates that serial quantitative MRI data collected before and following radiation therapy can be used to accurately predict tumor and vasculature response with a biologically-based mathematical model that is calibrated on an individual basis. To the best of our knowledge, this is the first effort to characterize the tumor and vasculature response to radiation therapy temporally and spatially using imaging-driven mathematical models.

Hybrid modeling frameworks of tumor development and treatment

Tumors are complex multicellular heterogeneous systems comprised of components that interact with and modify one another. Tumor development depends on multiple factors: intrinsic, such as genetic mutations, altered signaling pathways, or variable receptor expression; and extrinsic, such as differences in nutrient supply, crosstalk with stromal or immune cells, or variable composition of the surrounding extracellular matrix. Tumors are also characterized by high cellular heterogeneity and dynamically changing tumor microenvironments. The complexity increases when this multiscale, multicomponent system is perturbed by anticancer treatments. Modeling such complex systems and predicting how tumors will respond to therapies require mathematical models that can handle various types of information and combine diverse theoretical methods on multiple temporal and spatial scales, that is, hybrid models. In this update, we discuss the progress that has been achieved during the last 10 years in the area of the hybrid modeling of tumors. The classical definition of hybrid models refers to the coupling of discrete descriptions of cells with continuous descriptions of microenvironmental factors. To reflect on the direction that the modeling field has taken, we propose extending the definition of hybrid models to include of coupling two or more different mathematical frameworks. Thus, in addition to discussing recent advances in discrete/continuous modeling, we also discuss how these two mathematical descriptions can be coupled with theoretical frameworks of optimal control, optimization, fluid dynamics, game theory, and machine learning. All these methods will be illustrated with applications to tumor development and various anticancer treatments.

This article is characterized under:

Analytical and Computational Methods > Computational Methods

Translational, Genomic, and Systems Medicine > Therapeutic Methods

Models of Systems Properties and Processes > Organ, Tissue, and Physiological Models

Integrating Mathematical Modeling into the Roadmap for Personalized Adaptive Radiation Therapy

In current radiation oncology practice, treatment protocols are prescribed based on the average outcomes of large clinical trials, with limited personalization and without adaptations of dose or dose fractionation to individual patients based on their individual clinical responses. Predicting tumor responses to radiation and comparing predictions against observed responses offers an opportunity for novel treatment evaluation. These analyses can lead to protocol adaptation aimed at the improvement of patient outcomes with better therapeutic ratios. We foresee the integration of mathematical models into radiation oncology to simulate individual patient tumor growth and predict treatment response as dynamic biomarkers for personalized adaptive radiation therapy (RT).