A proliferation saturation index to predict radiation response and personalize radiotherapy fractionation

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.

Predicting and monitoring response to head and neck cancer radiotherapy using multimodality imaging and radiobiological digital twin simulations

Objective.To predict radiotherapy treatment response for head and neck cancer (HNC) using multimodality imaging and personalized radiobiological modeling.Approach.A mechanistic radiobiological model was combined with multi-modality imaging data from diffusion weighted-magnetic resonance imaging and positron emission tomography scans with [18F]Fluorodeoxyglucose (FDG) and [18F]Fluoromisonidazole (FMISO) tracers to develop personalized treatment response models for human papilloma virus associated HNC patients undergoing chemo-radiotherapy. Models were initialized to incorporate patient-specific imaging and updated to reflect longitudinal measurements of nodal gross tumor volume throughout treatment. Prediction accuracy was assessed based on mean absolute error (MAE) of weekly volume predictions and in predicting locoregional recurrence (LRR) following treatment.Main results.Personalized modeling based on pretreatment imaging significantly improved longitudinal volume prediction accuracy and correlation with measurement compared with a generic population model (MAE = 23.4 ± 10.0% vs 24.9 ± 9.0%,p= 0.002 on pairedt-test,R= 0.82 vs 0.72). Adding volume measurements from weeks 1 and 2 further improved prediction accuracy for subsequent weeks (MAE = 12.5 ± 8.1%, 10.7 ± 9.9%). When incorporating feedback with longitudinal measurements, penalizing large deviations from pretreatment model parameters using a variational regularization method was necessary to maintain model stability. Model-predicted volumes based on baseline + week-1 information significantly improved LRR prediction compared with week-1 volume data alone (area under the curve, AUC = 0.83 vs 0.77,p= 0.03) and was similar to prediction using week-3 volume data (AUC = 0.83 vs 0.85,p= non-significant).Significance.The proposed approach, which integrates clinical imaging and radiobiological principles, could be a basis to guide pretreatment prescription personalization as well as on-treatment adaptations.

Identifiability and model selection frameworks for models of high-grade glioma response to chemoradiation

We have developed a family of biology-based mathematical models of high-grade glioma (HGG), capturing the key features of tumour growth and response to chemoradiation. We now seek to quantify the accuracy of parameter estimation and determine, when given a virtual patient cohort, which model was used to generate the tumours. In this way, we systematically test both the parameter and model identifiability. Virtual patients are generated from unique growth parameters whose growth dynamics are determined by the model family. We then assessed the ability to recover model parameters and select the model used to generate the tumour. We then evaluated the accuracy of predictions using the selected model at four weeks post-chemoradiation. We observed median parameter errors from 0.04% to 72.96%. Our model selection framework selected the model that was used to generate the data in 82% of the cases. Finally, we predicted the growth of the virtual tumours using the selected model resulting in low error at the voxel-level (concordance correlation coefficient (CCC) ranged from 0.66 to 0.99) and global level (percentage error in total tumour cellularity ranged from -12.35% to 0.07%). These results demonstrate the reliability of our framework to identify the most appropriate model under noisy conditions expected in the clinical setting.This article is part of the theme issue 'Uncertainty quantification for healthcare and biological systems (Part 2)'.

The temporal response of a glioma cell population to irradiation: modelling the effect of dose and cell density

Time-lapse fluorescence microscopy experiments were performed to track the cell density of F98 glioma cells under varying radiation doses and initial cell densities. Based on these results, a compartmental model characterizing the temporal response of a cancerous cell population to single-dose radiation therapy was developed. This model reproduces very well all the experimental data, with only three free parameters (and four others that are fixed). It allows us to have access and follow the evolution of different cell populations after irradiation, in particular, the senescent and repaired cell populations. From these different cell populations, surviving fractions could also be estimated. Most importantly, our model allows us to analyze and quantify an inhibition effect (or cohort effect) of the dead and senescent cell populations on the regrowth of the repaired one.

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.

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.

Calibrating tumor growth and invasion parameters with spectral spatial analysis of cancer biopsy tissues

The reaction-diffusion equation is widely used in mathematical models of cancer. The calibration of model parameters based on limited clinical data is critical to using reaction-diffusion equation simulations for reliable predictions on a per-patient basis. Here, we focus on cell-level data as routinely available from tissue biopsies used for clinical cancer diagnosis. We analyze the spatial architecture in biopsy tissues stained with multiplex immunofluorescence. We derive a two-point correlation function and the corresponding spatial power spectral distribution. We show that this data-deduced power spectral distribution can fit the power spectrum of the solution of reaction-diffusion equations that can then identify patient-specific tumor growth and invasion rates. This approach allows the measurement of patient-specific critical tumor dynamical properties from routinely available biopsy material at a single snapshot in time.

Measurable imaging-based changes in enhancement of intrahepatic cholangiocarcinoma after radiotherapy reflect physical mechanisms of response

Background

Although escalated doses of radiation therapy (RT) for intrahepatic cholangiocarcinoma (iCCA) are associated with durable local control (LC) and prolonged survival, uncertainties persist regarding personalized RT based on biological factors. Compounding this knowledge gap, the assessment of RT response using traditional size-based criteria via computed tomography (CT) imaging correlates poorly with outcomes. We hypothesized that quantitative measures of enhancement would more accurately predict clinical outcomes than size-based assessment alone and developed a model to optimize RT.

Methods

Pre-RT and post-RT CT scans of 154 patients with iCCA were analyzed retrospectively for measurements of tumor dimensions (for RECIST) and viable tumor volume using quantitative European Association for Study of Liver (qEASL) measurements. Binary classification and survival analyses were performed to evaluate the ability of qEASL to predict treatment outcomes, and mathematical modeling was performed to identify the mechanistic determinants of treatment outcomes and to predict optimal RT protocols.

Results

Multivariable analysis accounting for traditional prognostic covariates revealed that percentage change in viable volume following RT was significantly associated with OS, outperforming stratification by RECIST. Binary classification identified ≥33% decrease in viable volume to optimally correspond to response to RT. The model-derived, patient-specific tumor enhancement growth rate emerged as the dominant mechanistic determinant of treatment outcome and yielded high accuracy of patient stratification (80.5%), strongly correlating with the qEASL-based classifier.

Conclusion

Following RT for iCCA, changes in viable volume outperformed radiographic size-based assessment using RECIST for OS prediction. CT-derived tumor-specific mathematical parameters may help optimize RT for resistant tumors.

Practical parameter identifiability and handling of censored data with Bayesian inference in mathematical tumour models

Mechanistic mathematical models (MMs) are a powerful tool to help us understand and predict the dynamics of tumour growth under various conditions. In this work, we use 5 MMs with an increasing number of parameters to explore how certain (often overlooked) decisions in estimating parameters from data of experimental tumour growth affect the outcome of the analysis. In particular, we propose a framework for including tumour volume measurements that fall outside the upper and lower limits of detection, which are normally discarded. We demonstrate how excluding censored data results in an overestimation of the initial tumour volume and the MM-predicted tumour volumes prior to the first measurements, and an underestimation of the carrying capacity and the MM-predicted tumour volumes beyond the latest measurable time points. We show in which way the choice of prior for the MM parameters can impact the posterior distributions, and illustrate that reporting the most likely parameters and their 95% credible interval can lead to confusing or misleading interpretations. We hope this work will encourage others to carefully consider choices made in parameter estimation and to adopt the approaches we put forward herein.

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.

To modulate or to skip: De-escalating PARP inhibitor maintenance therapy in ovarian cancer using adaptive therapy

Toxicity and emerging drug resistance pose important challenges in poly-adenosine ribose polymerase inhibitor (PARPi) maintenance therapy of ovarian cancer. We propose that adaptive therapy, which dynamically reduces treatment based on the tumor dynamics, might alleviate both issues. Utilizing in vitro time-lapse microscopy and stepwise model selection, we calibrate and validate a differential equation mathematical model, which we leverage to test different plausible adaptive treatment schedules. Our model indicates that adjusting the dosage, rather than skipping treatments, is more effective at reducing drug use while maintaining efficacy due to a delay in cell kill and a diminishing dose-response relationship. In vivo pilot experiments confirm this conclusion. Although our focus is toxicity mitigation, reducing drug use may also delay resistance. This study enhances our understanding of PARPi treatment scheduling and illustrates the first steps in developing adaptive therapies for new treatment settings. A record of this paper's transparent peer review process is included in the supplemental information.

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.

Intra-promoter switch of transcription initiation sites in proliferation signaling-dependent RNA metabolism

Global changes in transcriptional regulation and RNA metabolism are crucial features of cancer development. However, little is known about the role of the core promoter in defining transcript identity and post-transcriptional fates, a potentially crucial layer of transcriptional regulation in cancer. In this study, we use CAGE-seq analysis to uncover widespread use of dual-initiation promoters in which non-canonical, first-base-cytosine (C) transcription initiation occurs alongside first-base-purine initiation across 59 human cancers and healthy tissues. C-initiation is often followed by a 5' terminal oligopyrimidine (5'TOP) sequence, dramatically increasing the range of genes potentially subjected to 5'TOP-associated post-transcriptional regulation. We show selective, dynamic switching between purine and C-initiation site usage, indicating transcription initiation-level regulation in cancers. We additionally detail global metabolic changes in C-initiation transcripts that mark differentiation status, proliferative capacity, radiosensitivity, and response to irradiation and to PI3K-Akt-mTOR and DNA damage pathway-targeted radiosensitization therapies in colorectal cancer organoids and cancer cell lines and tissues.

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.

An adaptive information-theoretic experimental design procedure for high-to-low fidelity calibration of prostate cancer models

The use of mathematical models to make predictions about tumor growth and response to treatment has become increasingly prevalent in the clinical setting. The level of complexity within these models ranges broadly, and the calibration of more complex models requires detailed clinical data. This raises questions about the type and quantity of data that should be collected and when, in order to maximize the information gain about the model behavior while still minimizing the total amount of data used and the time until a model can be calibrated accurately. To address these questions, we propose a Bayesian information-theoretic procedure, using an adaptive score function to determine the optimal data collection times and measurement types. The novel score function introduced in this work eliminates the need for a penalization parameter used in a previous study, while yielding model predictions that are superior to those obtained using two potential pre-determined data collection protocols for two different prostate cancer model scenarios: one in which we fit a simple ODE system to synthetic data generated from a cellular automaton model using radiotherapy as the imposed treatment, and a second scenario in which a more complex ODE system is fit to clinical patient data for patients undergoing intermittent androgen suppression therapy. We also conduct a robust analysis of the calibration results, using both error and uncertainty metrics in combination to determine when additional data acquisition may be terminated.

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.

Investigating the Influence of Growth Arrest Mechanisms on Tumour Responses to Radiotherapy

Cancer is a heterogeneous disease and tumours of the same type can differ greatly at the genetic and phenotypic levels. Understanding how these differences impact sensitivity to treatment is an essential step towards patient-specific treatment design. In this paper, we investigate how two different mechanisms for growth control may affect tumour cell responses to fractionated radiotherapy (RT) by extending an existing ordinary differential equation model of tumour growth. In the absence of treatment, this model distinguishes between growth arrest due to nutrient insufficiency and competition for space and exhibits three growth regimes: nutrient limited, space limited (SL) and bistable (BS), where both mechanisms for growth arrest coexist. We study the effect of RT for tumours in each regime, finding that tumours in the SL regime typically respond best to RT, while tumours in the BS regime typically respond worst to RT. For tumours in each regime, we also identify the biological processes that may explain positive and negative treatment outcomes and the dosing regimen which maximises the reduction in tumour burden.

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.

Multicomponent mathematical model for tumor volume calculation with setup error using single-isocenter stereotactic radiotherapy for multiple brain metastases

We evaluated the tumor residual volumes considering six degrees-of-freedom (6DoF) patient setup errors in stereotactic radiotherapy (SRT) with multicomponent mathematical model using single-isocenter irradiation for brain metastases. Simulated spherical gross tumor volumes (GTVs) with 1.0 (GTV 1), 2.0 (GTV 2), and 3.0 (GTV 3)-cm diameters were used. The distance between the GTV center and isocenter (d) was set at 0-10 cm. The GTV was simultaneously translated within 0-1.0 mm (T) and rotated within 0°-1.0° (R) in the three axis directions using affine transformation. We optimized the tumor growth model parameters using measurements of non-small cell lung cancer cell lines' (A549 and NCI-H460) growth. We calculated the GTV residual volume at the irradiation's end using the physical dose to the GTV when the GTV size, d, and 6DoF setup error varied. The d-values that satisfy tolerance values (10%, 35%, and 50%) of the GTV residual volume rate based on the pre-irradiation GTV volume were determined. The larger the tolerance value set for both cell lines, the longer the distance to satisfy the tolerance value. In GTV residual volume evaluations based on the multicomponent mathematical model on SRT with single-isocenter irradiation, the smaller the GTV size and the larger the distance and 6DoF setup error, the shorter the distance that satisfies the tolerance value might need to be.

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.

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.

Spatial structure impacts adaptive therapy by shaping intra-tumoral competition

Background

Adaptive therapy aims to tackle cancer drug resistance by leveraging resource competition between drug-sensitive and resistant cells. Here, we present a theoretical study of intra-tumoral competition during adaptive therapy, to investigate under which circumstances it will be superior to aggressive treatment.

Methods

We develop and analyse a simple, 2-D, on-lattice, agent-based tumour model in which cells are classified as fully drug-sensitive or resistant. Subsequently, we compare this model to its corresponding non-spatial ordinary differential equation model, and fit it to longitudinal prostate-specific antigen data from 65 prostate cancer patients undergoing intermittent androgen deprivation therapy following biochemical recurrence.

Results

Leveraging the individual-based nature of our model, we explicitly demonstrate competitive suppression of resistance during adaptive therapy, and examine how different factors, such as the initial resistance fraction or resistance costs, alter competition. This not only corroborates our theoretical understanding of adaptive therapy, but also reveals that competition of resistant cells with each other may play a more important role in adaptive therapy in solid tumours than was previously thought. To conclude, we present two case studies, which demonstrate the implications of our work for: (i) mathematical modelling of adaptive therapy, and (ii) the intra-tumoral dynamics in prostate cancer patients during intermittent androgen deprivation treatment, a precursor of adaptive therapy.

Conclusion

Our work shows that the tumour’s spatial architecture is an important factor in adaptive therapy and provides insights into how adaptive therapy leverages both inter- and intra-specific competition to control resistance.

Cancer therapy traditionally focuses on maximising tumour cell kill with the aim of achieving a cure, but such aggressive treatment can open up space for drug-resistant cells to grow. In contrast, adaptive therapy aims to leverage competition between drug-sensitive and resistant cells by adjusting treatment to maintain the tumour at a tolerable size, whilst preserving drug-sensitive cells. This approach is being tested in trials but is not yet widely used as deeper understanding of cell-cell competition is required. Here, we used a mathematical model to investigate how strongly, and with whom, resistant cells compete during continuous and adaptive therapy, and applied our insights to hormone therapy in prostate cancer where adaptive therapy has recently been successfully trialed. Our results provide new insights into how adaptive therapy works and show that, by shaping cell competition, the tumour’s spatial architecture is important in determining therapy response.

Strobl et al. develop an agent-based spatial model of drug resistance in tumour cells under adaptive therapy. Using this model, they investigate how the tumour’s spatial architecture impacts intratumoural competitive dynamics of drug-sensitive vs. -resistant clones in response to therapy.

Mathematical Modeling to Simulate the Effect of Adding Radiation Therapy to Immunotherapy and Application to Hepatocellular Carcinoma

PURPOSE

To develop a comprehensive framework to simulate the response to immune checkpoint inhibitors (ICIs) in combination with radiation therapy (RT) and to apply the framework for investigating ICI-RT combination regimen in patients with hepatocellular carcinoma (HCC).

METHODS AND MATERIALS

The mechanistic mathematical model is based on dynamic biological interactions between the immune system and the tumor using input data from patient blood samples and outcomes of clinical trials. The cell compartments are described by ordinary differential equations and represent irradiated and nonirradiated tumor cells and lymphocytes. The effect of ICI is modeled using an immune activation term that is based on tumor size changes observed in a phase 1/2 clinical trial for HCC. Simulated combination regimen are based on ongoing ICI-RT trials.

RESULTS

The proposed framework successfully describes tumor volume trajectories observed in early-stage clinical trials of durvalumab monotherapy in patients with HCC. For ICI-RT treatment regimen the irradiated tumor fraction is the most important parameter for the efficacy. For 90% of the tumor cells being irradiated, adding RT to ICI yields an increase in clinical benefit from 33% to 71% in nonirradiated tumor sites. The model agrees with clinical data showing an association of outcome with initial tumor volume and lymphocyte counts. We demonstrate model application in clinical trial design to predict progression-free survival curves, showing that the cohort size to show significant improvement heavily depends on the irradiated tumor fraction.

CONCLUSIONS

We present a framework extending radiation cell kill models to include circulating lymphocytes and the effect of ICIs and enable simulation of combination strategies. The simulations predict that a significant amount of the benefit from RT in combination with ICI stems from the reduction in irradiated tumor burden and associated immune suppression. This aspect needs to be included in the interpretation of outcomes and the design of novel combination trials.

A Multi-Compartment Model of Glioma Response to Fractionated Radiation Therapy Parameterized via Time-Resolved Microscopy Data

PURPOSE

Conventional radiobiology models, including the linear-quadratic model, do not explicitly account for the temporal effects of radiation, thereby making it difficult to make time-resolved predictions of tumor response to fractionated radiation. To overcome this limitation, we propose and validate an experimental-computational approach that predicts the changes in cell number over time in response to fractionated radiation.

METHODS

We irradiated 9L and C6 glioma cells with six different fractionation schemes yielding a total dose of either 16 Gy or 20 Gy, and then observed their response via time-resolved microscopy. Phase-contrast images and Cytotox Red images (to label dead cells) were collected every 4 to 6 hours up to 330 hours post-radiation. Using 75% of the total data (i.e., 262 9L curves and 211 C6 curves), we calibrated a two-species model describing proliferative and senescent cells. We then applied the calibrated parameters to a validation dataset (the remaining 25% of the data, i.e., 91 9L curves and 74 C6 curves) to predict radiation response. Model predictions were compared to the microscopy measurements using the Pearson correlation coefficient (PCC) and the concordance correlation coefficient (CCC).

RESULTS

For the 9L cells, we observed PCCs and CCCs between the model predictions and validation data of (mean ± standard error) 0.96 ± 0.007 and 0.88 ± 0.013, respectively, across all fractionation schemes. For the C6 cells, we observed PCCs and CCCs between model predictions and the validation data were 0.89 ± 0.008 and 0.75 ± 0.017, respectively, across all fractionation schemes.

CONCLUSION

By proposing a time-resolved mathematical model of fractionated radiation response that can be experimentally verified in vitro, this study is the first to establish a framework for quantitative characterization and prediction of the dynamic radiobiological response of 9L and C6 gliomas to fractionated radiotherapy.

Improving cancer treatments via dynamical biophysical models

Despite significant advances in oncological research, cancer nowadays remains one of the main causes of mortality and morbidity worldwide. New treatment techniques, as a rule, have limited efficacy, target only a narrow range of oncological diseases, and have limited availability to the general public due their high cost. An important goal in oncology is thus the modification of the types of antitumor therapy and their combinations, that are already introduced into clinical practice, with the goal of increasing the overall treatment efficacy. One option to achieve this goal is optimization of the schedules of drugs administration or performing other medical actions. Several factors complicate such tasks: the adverse effects of treatments on healthy cell populations, which must be kept tolerable; the emergence of drug resistance due to the intrinsic plasticity of heterogeneous cancer cell populations; the interplay between different types of therapies administered simultaneously. Mathematical modeling, in which a tumor and its microenvironment are considered as a single complex system, can address this complexity and can indicate potentially effective protocols, that would require experimental verification. In this review, we consider classical methods, current trends and future prospects in the field of mathematical modeling of tumor growth and treatment. In particular, methods of treatment optimization are discussed with several examples of specific problems related to different types of treatment.

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.

Intermittent radiotherapy as alternative treatment for recurrent high grade glioma: a modeling study based on longitudinal tumor measurements

Recurrent high grade glioma patients face a poor prognosis for which no curative treatment option currently exists. In contrast to prescribing high dose hypofractionated stereotactic radiotherapy (HFSRT, [Formula: see text] Gy [Formula: see text] 5 in daily fractions) with debulking intent, we suggest a personalized treatment strategy to improve tumor control by delivering high dose intermittent radiation treatment (iRT, [Formula: see text] Gy [Formula: see text] 1 every 6 weeks). We performed a simulation analysis to compare HFSRT, iRT and iRT plus boost ([Formula: see text] Gy [Formula: see text] 3 in daily fractions at time of progression) based on a mathematical model of tumor growth, radiation response and patient-specific evolution of resistance to additional treatments (pembrolizumab and bevacizumab). Model parameters were fitted from tumor growth curves of 16 patients enrolled in the phase 1 NCT02313272 trial that combined HFSRT with bevacizumab and pembrolizumab. Then, iRT +/- boost treatments were simulated and compared to HFSRT based on time to tumor regrowth. The modeling results demonstrated that iRT + boost(- boost) treatment was equal or superior to HFSRT in 15(11) out of 16 cases and that patients that remained responsive to pembrolizumab and bevacizumab would benefit most from iRT. Time to progression could be prolonged through the application of additional, intermittently delivered fractions. iRT hence provides a promising treatment option for recurrent high grade glioma patients for prospective clinical evaluation.

Tumor-immune ecosystem dynamics define an individual Radiation Immune Score to predict pan-cancer radiocurability

Radiotherapy efficacy is the result of radiation-mediated cytotoxicity coupled with stimulation of antitumor immune responses. We develop an in silico 3-dimensional agent-based model of diverse tumor-immune ecosystems (TIES) represented as anti- or pro-tumor immune phenotypes. We validate the model in 10,469 patients across 31 tumor types by demonstrating that clinically detected tumors have pro-tumor TIES. We then quantify the likelihood radiation induces antitumor TIES shifts toward immune-mediated tumor elimination by developing the individual Radiation Immune Score (iRIS). We show iRIS distribution across 31 tumor types is consistent with the clinical effectiveness of radiotherapy, and in combination with a molecular radiosensitivity index (RSI) combines to predict pan-cancer radiocurability. We show that iRIS correlates with local control and survival in a separate cohort of 59 lung cancer patients treated with radiation. In combination, iRIS and RSI predict radiation-induced TIES shifts in individual patients and identify candidates for radiation de-escalation and treatment escalation. This is the first clinically and biologically validated computational model to simulate and predict pan-cancer response and outcomes via the perturbation of the TIES by radiotherapy.

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.

An integrated framework for quantifying immune-tumour interactions in a 3D co-culture model

Investigational in vitro models that reflect the complexity of the interaction between the immune system and tumours are limited and difficult to establish. Herein, we present a platform to study the tumour-immune interaction using a co-culture between cancer spheroids and activated immune cells. An algorithm was developed for analysis of confocal images of the co-culture to evaluate the following quantitatively; immune cell infiltration, spheroid roundness and spheroid growth. As a proof of concept, the effect of the glucocorticoid stress hormone, cortisol was tested on 66CL4 co-culture model. Results were comparable to 66CL4 syngeneic in vivo mouse model undergoing psychological stress. Furthermore, administration of glucocorticoid receptor antagonists demonstrated the use of this model to determine the effect of treatments on the immune-tumour interplay. In conclusion, we provide a method of quantifying the interaction between the immune system and cancer, which can become a screening tool in immunotherapy design.

A Mathematical Study of the Influence of Hypoxia and Acidity on the Evolutionary Dynamics of Cancer

Hypoxia and acidity act as environmental stressors promoting selection for cancer cells with a more aggressive phenotype. As a result, a deeper theoretical understanding of the spatio-temporal processes that drive the adaptation of tumour cells to hypoxic and acidic microenvironments may open up new avenues of research in oncology and cancer treatment. We present a mathematical model to study the influence of hypoxia and acidity on the evolutionary dynamics of cancer cells in vascularised tumours. The model is formulated as a system of partial integro-differential equations that describe the phenotypic evolution of cancer cells in response to dynamic variations in the spatial distribution of three abiotic factors that are key players in tumour metabolism: oxygen, glucose and lactate. The results of numerical simulations of a calibrated version of the model based on real data recapitulate the eco-evolutionary spatial dynamics of tumour cells and their adaptation to hypoxic and acidic microenvironments. Moreover, such results demonstrate how nonlinear interactions between tumour cells and abiotic factors can lead to the formation of environmental gradients which select for cells with phenotypic characteristics that vary with distance from intra-tumour blood vessels, thus promoting the emergence of intra-tumour phenotypic heterogeneity. Finally, our theoretical findings reconcile the conclusions of earlier studies by showing that the order in which resistance to hypoxia and resistance to acidity arise in tumours depend on the ways in which oxygen and lactate act as environmental stressors in the evolutionary dynamics of cancer cells.

Image-based personalization of computational models for predicting response of high-grade glioma to chemoradiation

High-grade gliomas are an aggressive and invasive malignancy which are susceptible to treatment resistance due to heterogeneity in intratumoral properties such as cell proliferation and density and perfusion. Non-invasive imaging approaches can measure these properties, which can then be used to calibrate patient-specific mathematical models of tumor growth and response. We employed multiparametric magnetic resonance imaging (MRI) to identify tumor extent (via contrast-enhanced T1-weighted, and T2-FLAIR) and capture intratumoral heterogeneity in cell density (via diffusion-weighted imaging) to calibrate a family of mathematical models of chemoradiation response in nine patients with unresected or partially resected disease. The calibrated model parameters were used to forecast spatially-mapped individual tumor response at future imaging visits. We then employed the Akaike information criteria to select the most parsimonious member from the family, a novel two-species model describing the enhancing and non-enhancing components of the tumor. Using this model, we achieved low error in predictions of the enhancing volume (median: - 2.5%, interquartile range: 10.0%) and a strong correlation in total cell count (Kendall correlation coefficient 0.79) at 3-months post-treatment. These preliminary results demonstrate the plausibility of using multiparametric MRI data to inform spatially-informative, biologically-based predictive models of tumor response in the setting of clinical high-grade gliomas.

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.

The roles of T cell competition and stochastic extinction events in chimeric antigen receptor T cell therapy

Chimeric antigen receptor (CAR) T cell therapy is a remarkably effective immunotherapy that relies on in vivo expansion of engineered CAR T cells, after lymphodepletion (LD) by chemotherapy. The quantitative laws underlying this expansion and subsequent tumour eradication remain unknown. We develop a mathematical model of T cell-tumour cell interactions and demonstrate that expansion can be explained by immune reconstitution dynamics after LD and competition among T cells. CAR T cells rapidly grow and engage tumour cells but experience an emerging growth rate disadvantage compared to normal T cells. Since tumour eradication is deterministically unstable in our model, we define cure as a stochastic event, which, even when likely, can occur at variable times. However, we show that variability in timing is largely determined by patient variability. While cure events impacted by these fluctuations occur early and are narrowly distributed, progression events occur late and are more widely distributed in time. We parameterized our model using population-level CAR T cell and tumour data over time and compare our predictions with progression-free survival rates. We find that therapy could be improved by optimizing the tumour-killing rate and the CAR T cells' ability to adapt, as quantified by their carrying capacity. Our tumour extinction model can be leveraged to examine why therapy works in some patients but not others, and to better understand the interplay of deterministic and stochastic effects on outcomes. For example, our model implies that LD before a second CAR T injection is necessary.

Turnover Modulates the Need for a Cost of Resistance in Adaptive Therapy

Adaptive therapy seeks to exploit intratumoral competition to avoid, or at least delay, the emergence of therapy resistance in cancer. Motivated by promising results in prostate cancer, there is growing interest in extending this approach to other neoplasms. As such, it is urgent to understand the characteristics of a cancer that determine whether or not it will respond well to adaptive therapy. A plausible candidate for such a selection criterion is the fitness cost of resistance. In this article, we study a general, but simple, mathematical model to investigate whether the presence of a cost is necessary for adaptive therapy to extend the time to progression beyond that of a standard-of-care continuous therapy. Tumor cells were divided into sensitive and resistant populations and we model their competition using a system of two ordinary differential equations based on the Lotka-Volterra model. For tumors close to their environmental carrying capacity, a cost was not required. However, for tumors growing far from carrying capacity, a cost may be required to see meaningful gains. Notably, it is important to consider cell turnover in the tumor, and we discuss its role in modulating the impact of a resistance cost. To conclude, we present evidence for the predicted cost-turnover interplay in data from 67 patients with prostate cancer undergoing intermittent androgen deprivation therapy. Our work helps to clarify under which circumstances adaptive therapy may be beneficial and suggests that turnover may play an unexpectedly important role in the decision-making process. SIGNIFICANCE: Tumor cell turnover modulates the speed of selection against drug resistance by amplifying the effects of competition and resistance costs; as such, turnover is an important factor in resistance management via adaptive therapy.See related commentary by Strobl et al., p. 811.

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.

Integrating transcriptomics and bulk time course data into a mathematical framework to describe and predict therapeutic resistance in cancer

A significant challenge in the field of biomedicine is the development of methods to integrate the multitude of dispersed data sets into comprehensive frameworks to be used to generate optimal clinical decisions. Recent technological advances in single cell analysis allow for high-dimensional molecular characterization of cells and populations, but to date, few mathematical models have attempted to integrate measurements from the single cell scale with other types of longitudinal data. Here, we present a framework that actionizes static outputs from a machine learning model and leverages these as measurements of state variables in a dynamic model of treatment response. We apply this framework to breast cancer cells to integrate single cell transcriptomic data with longitudinal bulk cell population (bulk time course) data. We demonstrate that the explicit inclusion of the phenotypic composition estimate, derived from single cell RNA-sequencing data (scRNA-seq), improves accuracy in the prediction of new treatments with a concordance correlation coefficient (CCC) of 0.92 compared to a prediction accuracy of CCC = 0.64 when fitting on longitudinal bulk cell population data alone. To our knowledge, this is the first work that explicitly integrates single cell clonally-resolved transcriptome datasets with bulk time-course data to jointly calibrate a mathematical model of drug resistance dynamics. We anticipate this approach to be a first step that demonstrates the feasibility of incorporating multiple data types into mathematical models to develop optimized treatment regimens from data.

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.

Optimization of Dose Fractionation for Radiotherapy of a Solid Tumor with Account of Oxygen Effect and Proliferative Heterogeneity

A spatially-distributed continuous mathematical model of solid tumor growth and treatment by fractionated radiotherapy is presented. The model explicitly accounts for three time and space-dependent factors that influence the efficiency of radiotherapy fractionation schemes—tumor cell repopulation, reoxygenation and redistribution of proliferative states. A special algorithm is developed, aimed at finding the fractionation schemes that provide increased tumor cure probability under the constraints of maximum normal tissue damage and maximum fractional dose. The optimization procedure is performed for varied radiosensitivity of tumor cells under the values of model parameters, corresponding to different degrees of tumor malignancy. The resulting optimized schemes consist of two stages. The first stages are aimed to increase the radiosensitivity of the tumor cells, remaining after their end, sparing the caused normal tissue damage. This allows to increase the doses during the second stages and thus take advantage of the obtained increased radiosensitivity. Such method leads to significant expansions in the curative ranges of the values of tumor radiosensitivity parameters. Overall, the results of this study represent the theoretical proof of concept that non-uniform radiotherapy fractionation schemes may be considerably more effective that uniform ones, due to the time and space-dependent effects.

Immunologic Consequences of Sequencing Cancer Radiotherapy and Surgery

PURPOSE

Early-stage cancers are routinely treated with surgery followed by radiotherapy (SR). Radiotherapy before surgery (RS) has been widely ignored for some cancers. We evaluate overall survival (OS) and disease-free survival (DFS) with SR and RS for different cancer types and simulate the plausibility of RS- and SR-induced antitumor immunity contributing to outcomes.

MATERIALS AND METHODS

We analyzed a SEER data set of early-stage cancers treated with SR or RS. OS and DFS were calculated for cancers with sufficient numbers for statistical power (cancers of lung and bronchus, esophagus, rectum, cervix uteri, corpus uteri, and breast). We simulated the immunologic consequences of SR, RS, and radiotherapy alone in a mathematical model of tumor-immune interactions.

RESULTS

RS improved OS for cancers with low 20-year survival rates (lung: hazard ratio [HR], 0.88; P = .046) and improved DFS for cancers with higher survival (breast: HR = 0.64; P < .001). For rectal cancer, with intermediate 20-year survival, RS improved both OS (HR = 0.89; P = .006) and DFS (HR = 0.86; P = .04). Model simulations suggested that RS could increase OS by eliminating cancer for a broader range of model parameters and radiotherapy-induced antitumor immunity compared with SR for selected parameter combinations. This could create an immune memory that may explain increased DFS after RS for certain cancers.

CONCLUSION

Study results suggest plausibility that radiation to the bulk of the tumor could induce a more robust immune response and better harness the synergy of radiotherapy and antitumor immunity than postsurgical radiation to the tumor bed. This exploratory study provides motivation for prospective evaluation of immune activation of RS versus SR in controlled clinical studies.

Personalizing Gastric Cancer Screening With Predictive Modeling of Disease Progression Biomarkers

Gastric cancer (GC) remains the third most common cause of cancer-related death worldwide. Infection with Helicobacter pylori is responsible for over 70% of GC incidence; colonization induces chronic inflammation, which can facilitate progression to intestinal metaplasia, dysplasia, and GC (Correa pathway). Although H. pylori eradication is a necessary first step in GC prevention, some patients continue to progress to advanced stage disease if substantial tissue damage has occurred or inflammation persists. This progression is often asymptomatic until cancer reaches stage IV, yet efficient, cost-effective screening protocols for patients who present with early stages of the Correa pathway do not exist. Given the high interpatient heterogeneity in progression time through this pathway, such screening protocols must necessarily be personalized. This requires the identification of reliable and longitudinally assessable biomarkers of patient-specific progression. Several gastric stem cell (GSC) markers including CD44, CD133, and Lgr5 are upregulated in GC. Here we show a significant stepwise increase in immunohistochemical staining for these markers in biopsies at different stages of the Correa pathway, suggesting GSC fraction to be a promising candidate biomarker for early detection of malignant transformation. We present a mathematical model capable of both simulating clinically observed increases in GSC fraction in longitudinal biopsy samples of individual patients, and forecasting patient-specific disease progression trajectories based only on characteristics identified from immunohistochemistry at initial presentation. From these forecasts, personalized screening schedules may be identified that would allow early stratification of high-risk patients, and potentially earlier detection of dysplasia or early-stage GC.

A Perspective on Expanding Our Understanding of Cancer Treatments by Integrating Approaches from the Biological and Physical Sciences

Multicellular systems such as cancer suffer from immense complexity. It is imperative to capture the heterogeneity of these systems across scales to achieve a deeper understanding of the underlying biology and develop effective treatment strategies. In this perspective article, we will discuss how recent technologies and approaches from the biological and physical sciences have transformed traditional ways of measuring, interpreting, and treating cancer. During the SLAS 2019 Annual Meeting, SBI² hosted a Special Interest Group (SIG) on this topic. Academic and industry leaders engaged in discussions surrounding what biological model systems are appropriate to study cancer complexity, what assays are necessary to interrogate this complexity, and how physical sciences approaches may be useful to detangle this complexity. In particular, we examined the utility of mathematical models in predicting cancer progression and treatment response when tightly integrated with reproducible, quantitative, and dynamic biological measurements achieved using high-content imaging and analysis. The dialogue centered around the impetus for convergent biosciences, bringing new perspectives to cancer research to further understand this complex adaptive system and successfully intervene therapeutically.

Mathematical oncology and it's application in non melanoma skin cancer - A primer for radiation oncology professionals

Cancers of the skin (the majority of which are basal and squamous cell skin carcinomas, but also include the rarer Merkel cell carcinoma) are overwhelmingly the most common of all types of cancer. Most of these are treated surgically, with radiation reserved for those patients with high risk features or anatomical locations less suitable for surgery. Given the high incidence of both basal and squamous cell carcinomas, as well as the relatively poor outcome for Merkel cell carcinoma, it is useful to investigate the role of other disciplines regarding their diagnosis, staging and treatment. Mathematical modelling is one such area of investigation. The use of mathematical modelling is a relatively recent addition to the armamentarium of cancer treatment. It has long been recognised that tumour growth and treatment response is a complex, non-linear biological phenomenon with many mechanisms yet to be understood. Despite decades of research, including clinical, population and basic science approaches, we continue to be challenged by the complexity, heterogeneity and adaptability of tumours, both in individual patients in the oncology clinic and across wider patient populations. Prospective clinical trials predominantly focus on average outcome, with little understanding as to why individual patients may or may not respond. The use of mathematical models may lead to a greater understanding of tumour initiation, growth dynamics and treatment response.

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.