A patient-specific computational model of hypoxia-modulated radiation resistance in glioblastoma using 18F-FMISO-PET

A mathematical model for predicting the spatiotemporal response of breast cancer cells treated with doxorubicin

Tumor heterogeneity contributes significantly to chemoresistance, a leading cause of treatment failure. To better personalize therapies, it is essential to develop tools capable of identifying and predicting intra- and inter-tumor heterogeneities. Biology-inspired mathematical models are capable of attacking this problem, but tumor heterogeneity is often overlooked in in-vivo modeling studies, while phenotypic considerations capturing spatial dynamics are not typically included in in-vitro modeling studies. We present a data assimilation-prediction pipeline with a two-phenotype model that includes a spatiotemporal component to characterize and predict the evolution of in-vitro breast cancer cells and their heterogeneous response to chemotherapy. Our model assumes that the cells can be divided into two subpopulations: surviving cells unaffected by the treatment, and irreversibly damaged cells undergoing treatment-induced death. MCF7 breast cancer cells were previously cultivated in wells for up to 1000 hours, treated with various concentrations of doxorubicin and imaged with time-resolved microscopy to record spatiotemporally-resolved cell count data. Images were used to generate cell density maps. Treatment response predictions were initialized by a training set and updated by weekly measurements. Our mathematical model successfully calibrated the spatiotemporal cell growth dynamics, achieving median [range] concordance correlation coefficients of > .99 [.88, >.99] and .73 [.58, .85] across the whole well and individual pixels, respectively. Our proposed data assimilation-prediction approach achieved values of .97 [.44, >.99] and .69 [.35, .79] for the whole well and individual pixels, respectively. Thus, our model can capture and predict the spatiotemporal dynamics of MCF7 cells treated with doxorubicin in an in-vitro setting.

Patient-Specific, Mechanistic Models of Tumor Growth Incorporating Artificial Intelligence and Big Data

Despite the remarkable advances in cancer diagnosis, treatment, and management over the past decade, malignant tumors remain a major public health problem. Further progress in combating cancer may be enabled by personalizing the delivery of therapies according to the predicted response for each individual patient. The design of personalized therapies requires the integration of patient-specific information with an appropriate mathematical model of tumor response. A fundamental barrier to realizing this paradigm is the current lack of a rigorous yet practical mathematical theory of tumor initiation, development, invasion, and response to therapy. We begin this review with an overview of different approaches to modeling tumor growth and treatment, including mechanistic as well as data-driven models based on big data and artificial intelligence. We then present illustrative examples of mathematical models manifesting their utility and discuss the limitations of stand-alone mechanistic and data-driven models. We then discuss the potential of mechanistic models for not only predicting but also optimizing response to therapy on a patient-specific basis. We describe current efforts and future possibilities to integrate mechanistic and data-driven models. We conclude by proposing five fundamental challenges that must be addressed to fully realize personalized care for cancer patients driven by computational models.

The Temozolomide-Doxorubicin paradox in Glioblastoma in vitro-in silico preclinical drug-screening

Adjuvant Temozolomide is considered the front-line Glioblastoma chemotherapeutic treatment; yet not all patients respond. Latest trends in clinical trials usually refer to Doxorubicin; yet it can lead to severe side-effects if administered in high doses. While Glioblastoma prognosis remains poor, little is known about the combination of the two chemotherapeutics. Patient-derived spheroids were generated and treated with a range of Temozolomide/Doxorubicin concentrations either as monotherapy or in combination. Optical microscopy was used to monitor the growth pattern and cell death. Based on the monotherapy experiments, we developed a probabilistic mathematical framework in order to describe the drug-induced effect at the single-cell level and simulate drug doses in combination assuming probabilistic independence. Doxorubicin was found to be effective in doses even four orders of magnitude less than Temozolomide in monotherapy. The combination therapy doses tested in vitro were able to lead to irreversible growth inhibition at doses where monotherapy resulted in relapse. In our simulations, we assumed both drugs are anti-mitotic; Temozolomide has a growth-arrest effect, while Doxorubicin is able to cumulatively cause necrosis. Interestingly, under no mechanistic synergy assumption, the in silico predictions underestimate the in vitro results. In silico models allow the exploration of a variety of potential underlying hypotheses. The simulated-biological discrepancy at certain doses indicates a supra-additive response when both drugs are combined. Our results suggest a Temozolomide-Doxorubicin dual chemotherapeutic scheme to both disable proliferation and increase cytotoxicity against Glioblastoma.

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.

Predicting the spatio-temporal response of recurrent glioblastoma treated with rhenium-186 labelled nanoliposomes

Rhenium-186 (186Re) labeled nanoliposome (RNL) therapy for recurrent glioblastoma patients has shown promise to improve outcomes by locally delivering radiation to affected areas. To optimize the delivery of RNL, we have developed a framework to predict patient-specific response to RNL using image-guided mathematical models.

Methods

We calibrated a family of reaction-diffusion type models with multi-modality imaging data from ten patients (NCR01906385) to predict the spatio-temporal dynamics of each patient's tumor. The data consisted of longitudinal magnetic resonance imaging (MRI) and single photon emission computed tomography (SPECT) to estimate tumor burden and local RNL activity, respectively. The optimal model from the family was selected and used to predict future growth. A simplified version of the model was used in a leave-one-out analysis to predict the development of an individual patient's tumor, based on cohort parameters.

Results

Across the cohort, predictions using patient-specific parameters with the selected model were able to achieve Spearman correlation coefficients (SCC) of 0.98 and 0.93 for tumor volume and total cell number, respectively, when compared to the measured data. Predictions utilizing the leave-one-out method achieved SCCs of 0.89 and 0.88 for volume and total cell number across the population, respectively.

Conclusion

We have shown that patient-specific calibrations of a biology-based mathematical model can be used to make early predictions of response to RNL therapy. Furthermore, the leave-one-out framework indicates that radiation doses determined by SPECT can be used to assign model parameters to make predictions directly following the conclusion of RNL treatment.

Statement of Significance

This manuscript explores the application of computational models to predict response to radionuclide therapy in glioblastoma. There are few, to our knowledge, examples of mathematical models used in clinical radionuclide therapy. We have tested a family of models to determine the applicability of different radiation coupling terms for response to the localized radiation delivery. We show that with patient-specific parameter estimation, we can make accurate predictions of future glioblastoma response to the treatment. As a comparison, we have shown that population trends in response can be used to forecast growth from the moment the treatment has been delivered.In addition to the high simulation and prediction accuracy our modeling methods have achieved, the evaluation of a family of models has given insight into the response dynamics of radionuclide therapy. These dynamics, while different than we had initially hypothesized, should encourage future imaging studies involving high dosage radiation treatments, with specific emphasis on the local immune and vascular response.

A phase-field model for non-small cell lung cancer under the effects of immunotherapy

Formulating mathematical models that estimate tumor growth under therapy is vital for improving patient-specific treatment plans. In this context, we present our recent work on simulating non-small-scale cell lung cancer (NSCLC) in a simple, deterministic setting for two different patients receiving an immunotherapeutic treatment. At its core, our model consists of a Cahn-Hilliard-based phase-field model describing the evolution of proliferative and necrotic tumor cells. These are coupled to a simplified nutrient model that drives the growth of the proliferative cells and their decay into necrotic cells. The applied immunotherapy decreases the proliferative cell concentration. Here, we model the immunotherapeutic agent concentration in the entire lung over time by an ordinary differential equation (ODE). Finally, reaction terms provide a coupling between all these equations. By assuming spherical, symmetric tumor growth and constant nutrient inflow, we simplify this full 3D cancer simulation model to a reduced 1D model. We can then resort to patient data gathered from computed tomography (CT) scans over several years to calibrate our model. Our model covers the case in which the immunotherapy is successful and limits the tumor size, as well as the case predicting a sudden relapse, leading to exponential tumor growth. Finally, we move from the reduced model back to the full 3D cancer simulation in the lung tissue. Thereby, we demonstrate the predictive benefits that a more detailed patient-specific simulation including spatial information as a possible generalization within our framework could yield in the future.

Bayesian Inference of Tissue Heterogeneity for Individualized Prediction of Glioma Growth

Reliably predicting the future spread of brain tumors using imaging data and on a subject-specific basis requires quantifying uncertainties in data, biophysical models of tumor growth, and spatial heterogeneity of tumor and host tissue. This work introduces a Bayesian framework to calibrate the two-/three-dimensional spatial distribution of the parameters within a tumor growth model to quantitative magnetic resonance imaging (MRI) data and demonstrates its implementation in a pre-clinical model of glioma. The framework leverages an atlas-based brain segmentation of grey and white matter to establish subject-specific priors and tunable spatial dependencies of the model parameters in each region. Using this framework, the tumor-specific parameters are calibrated from quantitative MRI measurements early in the course of tumor development in four rats and used to predict the spatial development of the tumor at later times. The results suggest that the tumor model, calibrated by animal-specific imaging data at one time point, can accurately predict tumor shapes with a Dice coefficient 0.89. However, the reliability of the predicted volume and shape of tumors strongly relies on the number of earlier imaging time points used for calibrating the model. This study demonstrates, for the first time, the ability to determine the uncertainty in the inferred tissue heterogeneity and the model-predicted tumor shape.

A data assimilation framework to predict the response of glioma cells to radiation

We incorporate a practical data assimilation methodology into our previously established experimental-computational framework to predict the heterogeneous response of glioma cells receiving fractionated radiation treatment. Replicates of 9L and C6 glioma cells grown in 96-well plates were irradiated with six different fractionation schemes and imaged via time-resolved microscopy to yield 360- and 286-time courses for the 9L and C6 lines, respectively. These data were used to calibrate a biology-based mathematical model and then make predictions within two different scenarios. For Scenario 1, 70% of the time courses are fit to the model and the resulting parameter values are averaged. These average values, along with the initial cell number, initialize the model to predict the temporal evolution for each test time course (10% of the data). In Scenario 2, the predictions for the test cases are made with model parameters initially assigned from the training data, but then updated with new measurements every 24 hours via four versions of a data assimilation framework. We then compare the predictions made from Scenario 1 and the best version of Scenario 2 to the experimentally measured microscopy measurements using the concordance correlation coefficient (CCC). Across all fractionation schemes, Scenario 1 achieved a CCC value (mean ± standard deviation) of 0.845 ± 0.185 and 0.726 ± 0.195 for the 9L and C6 cell lines, respectively. For the best data assimilation version from Scenario 2 (validated with the last 20% of the data), the CCC values significantly increased to 0.954 ± 0.056 (p = 0.002) and 0.901 ± 0.061 (p = 8.9e-5) for the 9L and C6 cell lines, respectively. Thus, we have developed a data assimilation approach that incorporates an experimental-computational system to accurately predict the in vitro response of glioma cells to fractionated radiation therapy.

Opportunities for improving brain cancer treatment outcomes through imaging-based mathematical modeling of the delivery of radiotherapy and immunotherapy

Immunotherapy has become a fourth pillar in the treatment of brain tumors and, when combined with radiation therapy, may improve patient outcomes and reduce the neurotoxicity. As with other combination therapies, the identification of a treatment schedule that maximizes the synergistic effect of radiation- and immune-therapy is a fundamental challenge. Mechanism-based mathematical modeling is one promising approach to systematically investigate therapeutic combinations to maximize positive outcomes within a rigorous framework. However, successful clinical translation of model-generated combinations of treatment requires patient-specific data to allow the models to be meaningfully initialized and parameterized. Quantitative imaging techniques have emerged as a promising source of high quality, spatially and temporally resolved data for the development and validation of mathematical models. In this review, we will present approaches to personalize mechanism-based modeling frameworks with patient data, and then discuss how these techniques could be leveraged to improve brain cancer outcomes through patient-specific modeling and optimization of treatment strategies.

Quantitative system pharmacology as a legitimate approach to examine extrapolation strategies used to support pediatric drug development

Extrapolation strategies from adult data for designing pediatric drug development programs are explored using the quantitative systems pharmacology (QSP) modeling approach, a mechanistic drug and disease modeling framework that can predict clinical response and guide pediatric drug development in general. This innovative model‐informed drug discovery and development approach can leverage adult‐pediatric pharmacology and disease similarity metrics to validate extrapolation assumptions. We describe the QSP model strategy and framework for extrapolation to design pediatric drug development programs by leveraging adult data across a wide range of therapeutic areas and illustrating stage‐gate decisions informed by such an approach.

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.

Post-Radiotherapy PET Image Outcome Prediction by Deep Learning Under Biological Model Guidance: A Feasibility Study of Oropharyngeal Cancer Application

Purpose

To develop a method of biologically guided deep learning for post-radiation 18FDG-PET image outcome prediction based on pre-radiation images and radiotherapy dose information.

Methods

Based on the classic reaction–diffusion mechanism, a novel biological model was proposed using a partial differential equation that incorporates spatial radiation dose distribution as a patient-specific treatment information variable. A 7-layer encoder–decoder-based convolutional neural network (CNN) was designed and trained to learn the proposed biological model. As such, the model could generate post-radiation 18FDG-PET image outcome predictions with breakdown biological components for enhanced explainability. The proposed method was developed using 64 oropharyngeal patients with paired 18FDG-PET studies before and after 20-Gy delivery (2 Gy/day fraction) by intensity-modulated radiotherapy (IMRT). In a two-branch deep learning execution, the proposed CNN learns specific terms in the biological model from paired 18FDG-PET images and spatial dose distribution in one branch, and the biological model generates post-20-Gy 18FDG-PET image prediction in the other branch. As in 2D execution, 718/233/230 axial slices from 38/13/13 patients were used for training/validation/independent test. The prediction image results in test cases were compared with the ground-truth results quantitatively.

Results

The proposed method successfully generated post-20-Gy 18FDG-PET image outcome prediction with breakdown illustrations of biological model components. Standardized uptake value (SUV) mean values in 18FDG high-uptake regions of predicted images (2.45 ± 0.25) were similar to ground-truth results (2.51 ± 0.33). In 2D-based Gamma analysis, the median/mean Gamma Index (<1) passing rate of test images was 96.5%/92.8% using the 5%/5 mm criterion; such result was improved to 99.9%/99.6% when 10%/10 mm was adopted.

Conclusion

The developed biologically guided deep learning method achieved post-20-Gy 18FDG-PET image outcome predictions in good agreement with ground-truth results. With the breakdown biological modeling components, the outcome image predictions could be used in adaptive radiotherapy decision-making to optimize personalized plans for the best outcome in the future.

Geometry-Aware Neural Solver for Fast Bayesian Calibration of Brain Tumor Models

Modeling of brain tumor dynamics has the potential to advance therapeutic planning. Current modeling approaches resort to numerical solvers that simulate the tumor progression according to a given differential equation. Using highly-efficient numerical solvers, a single forward simulation takes up to a few minutes of compute. At the same time, clinical applications of tumor modeling often imply solving an inverse problem, requiring up to tens of thousands of forward model evaluations when used for a Bayesian model personalization via sampling. This results in a total inference time prohibitively expensive for clinical translation. While recent data-driven approaches become capable of emulating physics simulation, they tend to fail in generalizing over the variability of the boundary conditions imposed by the patient-specific anatomy. In this paper, we propose a learnable surrogate for simulating tumor growth which maps the biophysical model parameters directly to simulation outputs, i.e. the local tumor cell densities, whilst respecting patient geometry. We test the neural solver in a Bayesian model personalization task for a cohort of glioma patients. Bayesian inference using the proposed surrogate yields estimates analogous to those obtained by solving the forward model with a regular numerical solver. The near real-time computation cost renders the proposed method suitable for clinical settings. The code is available at https://github.com/IvanEz/tumor-surrogate.

Effect of energy requirements in the growth of brain tumor: a theoretical approach

A malignant tumor is an uncontrolled growth of tissues receiving energy in form of the nutrients provided by the microvascular networks. It is proposed that the supplied energy to a tumor is used for three purposes: the creation of new cells, maintenance of tumor cells, and tumor volume expansion by overcoming external pressure. A mathematical model studying the effects of energy required for maintenance and overcoming external pressure, the energy required creating a single cell, death rate, and tumor cell density on tumor development has been formulated. Including a term, residual energy for tumor growth in the tumor growth equation, the well-known logistic equation has been re-derived for tumors. Analytical solutions have been developed, and numerical analysis for the growth in brain tumors with the variation of parameters related to energy supply, the energy required for maintenance, and expansion of tumor has been performed. Expressions for the tumor growth rate(r) and carrying capacity(C) of the tumor are formulated in terms of the parameters used in the model. The range of 'r', estimated using our model is found within the ranges of tumor growth rates in gliomas reported by the other researchers. Selecting the model parameters precisely for a particular individual, the tumor growth rate and carrying capacity could be estimated accurately. Our study indicates that the actual growth rate and carrying capacity of a tumor reduce and tumor saturation time increases with the increase of death rate, the energy required for a single cell division, and energy requirement for the tumor cell maintenance.

A Deep Convolutional Neural Network for Annotation of Magnetic Resonance Imaging Sequence Type

The explosion of medical imaging data along with the advent of big data analytics has launched an exciting era for clinical research. One factor affecting the ability to aggregate large medical image collections for research is the lack of infrastructure for automated data annotation. Among all imaging modalities, annotation of magnetic resonance (MR) images is particularly challenging due to the non-standard labeling of MR image types. In this work, we aimed to train a deep neural network to annotate MR image sequence type for scans of brain tumor patients. We focused on the four most common MR sequence types within neuroimaging: T1-weighted (T1W), T1-weighted post-gadolinium contrast (T1Gd), T2-weighted (T2W), and T2-weighted fluid-attenuated inversion recovery (FLAIR). Our repository contains images acquired using a variety of pulse sequences, sequence parameters, field strengths, and scanner manufacturers. Image selection was agnostic to patient demographics, diagnosis, and the presence of tumor in the imaging field of view. We used a total of 14,400 two-dimensional images, each visualizing a different part of the brain. Data was split into train, validation, and test sets (9600, 2400, and 2400 images, respectively) and sets consisted of equal-sized groups of image types. Overall, the model reached an accuracy of 99% on the test set. Our results showed excellent performance of deep learning techniques in predicting sequence types for brain tumor MR images. We conclude deep learning models can serve as tools to support clinical research and facilitate efficient database management.

Bridging cell-scale simulations and radiologic images to explain short-time intratumoral oxygen fluctuations

Radiologic images provide a way to monitor tumor development and its response to therapies in a longitudinal and minimally invasive fashion. However, they operate on a macroscopic scale (average value per voxel) and are not able to capture microscopic scale (cell-level) phenomena. Nevertheless, to examine the causes of frequent fast fluctuations in tissue oxygenation, models simulating individual cells’ behavior are needed. Here, we provide a link between the average data values recorded for radiologic images and the cellular and vascular architecture of the corresponding tissues. Using hybrid agent-based modeling, we generate a set of tissue morphologies capable of reproducing oxygenation levels observed in radiologic images. We then use these in silico tissues to investigate whether oxygen fluctuations can be explained by changes in vascular oxygen supply or by modulations in cellular oxygen absorption. Our studies show that intravascular changes in oxygen supply reproduce the observed fluctuations in tissue oxygenation in all considered regions of interest. However, larger-magnitude fluctuations cannot be recreated by modifications in cellular absorption of oxygen in a biologically feasible manner. Additionally, we develop a procedure to identify plausible tissue morphologies for a given temporal series of average data from radiology images. In future applications, this approach can be used to generate a set of tissues comparable with radiology images and to simulate tumor responses to various anti-cancer treatments at the tissue-scale level.

Low levels of oxygen, called hypoxia, are observable in many solid tumors. They are associated with more aggressive malignant cells that are resistant to chemo-, radio-, and immunotherapies. Recently developed imaging techniques provide a way to measure the magnitude of frequent short-term oxygen fluctuations, but they operate on a macro-scale voxel level. To examine the possible causes of rapid oxygen fluctuations at the cell level, we developed a hybrid agent-based mathematical model. We tested two different mechanisms that may be responsible for these cyclic effects on tissue oxygenation: temporal variations in vascular influx of oxygen and modulations in cellular oxygen absorption. Additionally, we developed a procedure to identify plausible tissue morphologies from data collected from radiological images. This can provide a bridge between the micro-scale simulations with individual cells and the longitudinal medical images containing average values. In future applications, this approach can be used to generate a set of tissues compatible with radiology images and to simulate tumor responses to various anticancer treatments at the cell-scale level.

A mathematical framework for modelling 3D cell motility: applications to glioblastoma cell migration

The collection of 3D cell tracking data from live images of micro-tissues is a recent innovation made possible due to advances in imaging techniques. As such there is increased interest in studying cell motility in 3D in vitro model systems but a lack of rigorous methodology for analysing the resulting data sets. One such instance of the use of these in vitro models is in the study of cancerous tumours. Growing multicellular tumour spheroids in vitro allows for modelling of the tumour microenvironment and the study of tumour cell behaviours, such as migration, which improves understanding of these cells and in turn could potentially improve cancer treatments. In this paper, we present a workflow for the rigorous analysis of 3D cell tracking data, based on the persistent random walk model, but adaptable to other biologically informed mathematical models. We use statistical measures to assess the fit of the model to the motility data and to estimate model parameters and provide confidence intervals for those parameters, to allow for parametrization of the model taking correlation in the data into account. We use in silico simulations to validate the workflow in 3D before testing our method on cell tracking data taken from in vitro experiments on glioblastoma tumour cells, a brain cancer with a very poor prognosis. The presented approach is intended to be accessible to both modellers and experimentalists alike in that it provides tools for uncovering features of the data set that may suggest amendments to future experiments or modelling attempts.

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.

A neuro evolutionary algorithm for patient calibrated prediction of survival in Glioblastoma patients

BACKGROUND AND OBJECTIVES

Glioblastoma multiforme (GBM) is the most common and malignant type of primary brain tumors. Radiation therapy (RT) plus concomitant and adjuvant Temozolomide (TMZ) constitute standard treatment of GBM. Existing models for GBM growth do not consider the effect of different schedules on tumor growth and patient survival. However, clinical trials show that treatment schedule and drug dosage significantly affect patient survival. The goal is to provide a patient calibrated model for predicting survival according to the treatment schedule.

METHODS

We propose a top-down method based on artificial neural networks (ANN) and genetic algorithm (GA) to predict survival of GBM patients. A feed forward undercomplete Autoencoder network is integrated with the neuro-evolutionary (NE) algorithm in order to extract a compressed representation of input clinical data. The proposed NE algorithm uses GA to obtain optimal architecture of a multi-layer perceptron (MLP). Taguchi L₁₆ orthogonal design of experiments is used to tune parameters of the proposed NE algorithm. Finally, the optimal MLP is used to predict survival of GBM patients.

RESULTS

Data from 8 related clinical trials have been collected and integrated to train the model. From 847 evaluable cases, 719 were used for train and validation and the remaining 128 cases were used to test the model. Mean absolute error of the predictions on the test data is 0.087 months which shows excellent performance of the proposed model in predicting survival of the patients. Also, the results show that the proposed NE algorithm is superior to other existing models in both the mean and variability of the prediction error.

Integrating Quantitative Assays with Biologically Based Mathematical Modeling for Predictive Oncology

We provide an overview on the use of biological assays to calibrate and initialize mechanism-based models of cancer phenomena. Although artificial intelligence methods currently dominate the landscape in computational oncology, mathematical models that seek to explicitly incorporate biological mechanisms into their formalism are of increasing interest. These models can guide experimental design and provide insights into the underlying mechanisms of cancer progression. Historically, these models have included a myriad of parameters that have been difficult to quantify in biologically relevant systems, limiting their practical insights. Recently, however, there has been much interest calibrating biologically based models with the quantitative measurements available from (for example) RNA sequencing, time-resolved microscopy, and in vivo imaging. In this contribution, we summarize how a variety of experimental methods quantify tumor characteristics from the molecular to tissue scales and describe how such data can be directly integrated with mechanism-based models to improve predictions of tumor growth and treatment response.

Towards integration of 64Cu-DOTA-trastuzumab PET-CT and MRI with mathematical modeling to predict response to neoadjuvant therapy in HER2 + breast cancer

While targeted therapies exist for human epidermal growth factor receptor 2 positive (HER2 +) breast cancer, HER2 + patients do not always respond to therapy. We present the results of utilizing a biophysical mathematical model to predict tumor response for two HER2 + breast cancer patients treated with the same therapeutic regimen but who achieved different treatment outcomes. Quantitative data from magnetic resonance imaging (MRI) and 64Cu-DOTA-trastuzumab positron emission tomography (PET) are used to estimate tumor density, perfusion, and distribution of HER2-targeted antibodies for each individual patient. MRI and PET data are collected prior to therapy, and follow-up MRI scans are acquired at a midpoint in therapy. Given these data types, we align the data sets to a common image space to enable model calibration. Once the model is parameterized with these data, we forecast treatment response with and without HER2-targeted therapy. By incorporating targeted therapy into the model, the resulting predictions are able to distinguish between the two different patient responses, increasing the difference in tumor volume change between the two patients by > 40%. This work provides a proof-of-concept strategy for processing and integrating PET and MRI modalities into a predictive, clinical-mathematical framework to provide patient-specific predictions of HER2 + treatment response.

A Mechanistic Investigation into Ischemia-Driven Distal Recurrence of Glioblastoma

Glioblastoma (GBM) is the most aggressive primary brain tumor with a short median survival. Tumor recurrence is a clinical expectation of this disease and usually occurs along the resection cavity wall. However, previous clinical observations have suggested that in cases of ischemia following surgery, tumors are more likely to recur distally. Through the use of a previously established mechanistic model of GBM, the Proliferation Invasion Hypoxia Necrosis Angiogenesis (PIHNA) model, we explore the phenotypic drivers of this observed behavior. We have extended the PIHNA model to include a new nutrient-based vascular efficiency term that encodes the ability of local vasculature to provide nutrients to the simulated tumor. The extended model suggests sensitivity to a hypoxic microenvironment and the inherent migration and proliferation rates of the tumor cells are key factors that drive distal recurrence.

Learning Equations from Biological Data with Limited Time Samples

Equation learning methods present a promising tool to aid scientists in the modeling process for biological data. Previous equation learning studies have demonstrated that these methods can infer models from rich datasets; however, the performance of these methods in the presence of common challenges from biological data has not been thoroughly explored. We present an equation learning methodology comprised of data denoising, equation learning, model selection and post-processing steps that infers a dynamical systems model from noisy spatiotemporal data. The performance of this methodology is thoroughly investigated in the face of several common challenges presented by biological data, namely, sparse data sampling, large noise levels, and heterogeneity between datasets. We find that this methodology can accurately infer the correct underlying equation and predict unobserved system dynamics from a small number of time samples when the data are sampled over a time interval exhibiting both linear and nonlinear dynamics. Our findings suggest that equation learning methods can be used for model discovery and selection in many areas of biology when an informative dataset is used. We focus on glioblastoma multiforme modeling as a case study in this work to highlight how these results are informative for data-driven modeling-based tumor invasion predictions.

QSP-IO: A Quantitative Systems Pharmacology Toolbox for Mechanistic Multiscale Modeling for Immuno-Oncology Applications

Immunotherapy has shown great potential in the treatment of cancer; however, only a fraction of patients respond to treatment, and many experience autoimmune‐related side effects. The pharmaceutical industry has relied on mathematical models to study the behavior of candidate drugs and more recently, complex, whole‐body, quantitative systems pharmacology (QSP) models have become increasingly popular for discovery and development. QSP modeling has the potential to discover novel predictive biomarkers as well as test the efficacy of treatment plans and combination therapies through virtual clinical trials. In this work, we present a QSP modeling platform for immuno‐oncology (IO) that incorporates detailed mechanisms for important immune interactions. This modular platform allows for the construction of QSP models of IO with varying degrees of complexity based on the research questions. Finally, we demonstrate the use of the platform through two example applications of immune checkpoint therapy.

Mechanism-Based Modeling of Tumor Growth and Treatment Response Constrained by Multiparametric Imaging Data

Multiparametric imaging is a critical tool in the noninvasive study and assessment of cancer. Imaging methods have evolved over the past several decades to provide quantitative measures of tumor and healthy tissue characteristics related to, for example, cell number, blood volume fraction, blood flow, hypoxia, and metabolism. Mechanistic models of tumor growth also have matured to a point where the incorporation of patient-specific measures could provide clinically relevant predictions of tumor growth and response. In this review, we identify and discuss approaches that use multiparametric imaging data, including diffusion-weighted magnetic resonance imaging, dynamic contrast-enhanced magnetic resonance imaging, diffusion tensor imaging, contrast-enhanced computed tomography, [¹⁸F]fluorodeoxyglucose positron emission tomography, and [¹⁸F]fluoromisonidazole positron emission tomography to initialize and calibrate mechanistic models of tumor growth and response. We focus the discussion on brain and breast cancers; however, we also identify three emerging areas of application in kidney, pancreatic, and lung cancers. We conclude with a discussion of the future directions for incorporating multiparametric imaging data and mechanistic modeling into clinical decision making for patients with cancer.

How heterogeneity drives tumour growth: a computational study

Although cancerous tumours usually originate from a single cell, they normally evolve into a remarkably heterogeneous agglomeration of cells. Heterogeneity is a pervasive and almost universal feature of tumours, but its origin and consequences remain poorly understood. Tumour heterogeneity has been usually associated with poor prognosis, but a better understanding of it may lead to more personalized diagnosis and therapy. Here, we study tumour heterogeneity developing a computational model in which different cell subpopulations compete for space. The model suggests that aggressive tumour subpopulations may become even more aggressive when they grow with a non-aggressive subpopulation. The model also provides a mechanistic explanation of how heterogeneity drives growth. In particular, we observed that even a mild heterogeneity in the proliferation rates of different cell subpopulations leads to a much faster overall tumour growth when compared to a homogeneous tumour. The proposed model may be a starting point to study tumour heterogeneity computationally and to suggest new hypotheses to be tested experimentally. This article is part of the theme issue 'Patterns in soft and biological matters'.

Optimal Control Theory for Personalized Therapeutic Regimens in Oncology: Background, History, Challenges, and Opportunities

Optimal control theory is branch of mathematics that aims to optimize a solution to a dynamical system. While the concept of using optimal control theory to improve treatment regimens in oncology is not novel, many of the early applications of this mathematical technique were not designed to work with routinely available data or produce results that can eventually be translated to the clinical setting. The purpose of this review is to discuss clinically relevant considerations for formulating and solving optimal control problems for treating cancer patients. Our review focuses on two of the most widely used cancer treatments, radiation therapy and systemic therapy, as they naturally lend themselves to optimal control theory as a means to personalize therapeutic plans in a rigorous fashion. To provide context for optimal control theory to address either of these two modalities, we first discuss the major limitations and difficulties oncologists face when considering alternate regimens for their patients. We then provide a brief introduction to optimal control theory before formulating the optimal control problem in the context of radiation and systemic therapy. We also summarize examples from the literature that illustrate these concepts. Finally, we present both challenges and opportunities for dramatically improving patient outcomes via the integration of clinically relevant, patient-specific, mathematical models and optimal control theory.

Identifying density-dependent interactions in collective cell behaviour

Scratch assays are routinely used to study collective cell behaviour in vitro. Typical experimental protocols do not vary the initial density of cells, and typical mathematical modelling approaches describe cell motility and proliferation based on assumptions of linear diffusion and logistic growth. Jin et al. (Jin et al. 2016 J. Theor. Biol. 390, 136-145 (doi:10.1016/j.jtbi.2015.10.040)) find that the behaviour of cells in scratch assays is density-dependent, and show that standard modelling approaches cannot simultaneously describe data initiated across a range of initial densities. To address this limitation, we calibrate an individual-based model to scratch assay data across a large range of initial densities. Our model allows proliferation, motility, and a direction bias to depend on interactions between neighbouring cells. By considering a hierarchy of models where we systematically and sequentially remove interactions, we perform model selection analysis to identify the minimum interactions required for the model to simultaneously describe data across all initial densities. The calibrated model is able to match the experimental data across all densities using a single parameter distribution, and captures details about the spatial structure of cells. Our results provide strong evidence to suggest that motility is density-dependent in these experiments. On the other hand, we do not see the effect of crowding on proliferation in these experiments. These results are significant as they are precisely the opposite of the assumptions in standard continuum models, such as the Fisher-Kolmogorov equation and its generalizations.

Learning partial differential equations for biological transport models from noisy spatio-temporal data

We investigate methods for learning partial differential equation (PDE) models from spatio-temporal data under biologically realistic levels and forms of noise. Recent progress in learning PDEs from data have used sparse regression to select candidate terms from a denoised set of data, including approximated partial derivatives. We analyse the performance in using previous methods to denoise data for the task of discovering the governing system of PDEs. We also develop a novel methodology that uses artificial neural networks (ANNs) to denoise data and approximate partial derivatives. We test the methodology on three PDE models for biological transport, i.e. the advection-diffusion, classical Fisher-Kolmogorov-Petrovsky-Piskunov (Fisher-KPP) and nonlinear Fisher-KPP equations. We show that the ANN methodology outperforms previous denoising methods, including finite differences and both local and global polynomial regression splines, in the ability to accurately approximate partial derivatives and learn the correct PDE model.

Personalized image-based tumor growth prediction in a convection-diffusion-reaction model

Inter-individual heterogeneity of tumors leads to non-effectiveness of unique therapy plans. This issue has caused a growing interest in the field of personalized medicine and its application in tumor growth evaluation. Accordingly, in this paper, a framework of personalized medicine is presented for growth prediction of brain glioma tumors. A convection-diffusion-reaction model is used as the patient-specific tumor growth model which is associated with multimodal magnetic resonance images (MRIs). Two parameters of intracellular area fraction (ICAF) and metabolic rate have been used to incorporate the physiological data obtained from medical images into the model. The framework is tested on the data of two cases of glioma tumors to document the approach; parameter estimation is made using particle swarm optimization (PSO) and genetic algorithm (GA) and the model is evaluated by comparing the predicted tumors with the observed tumors in terms of root mean square error of the ICAF maps (IRMSE), relative area difference (RAD) and Dice's coefficient (DC). Results show the differences of IRMSE, RAD and DC in 4.1 ∓ 1.15%, 0.099 ∓ 0.041 and 85.5 ∓ 7.5%, respectively. Survival times are estimated by assuming the tumor radius of 35 mm as the fatal burden. Results confirm that less-diffusive tumors lead to higher survival times. The represented framework makes it possible to personally predict the growth behavior of glioma tumors only based on patients' routine MRIs and provides a basis for modeling the personalized therapy and walking in the path of personalized medicine.

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.

Radiation protraction schedules for low-grade gliomas: a comparison between different mathematical models

We optimize radiotherapy (RT) administration strategies for treating low-grade gliomas. Specifically, we consider different tumour growth laws, both with and without spatial effects. In each scenario, we find the optimal treatment in the sense of maximizing the overall survival time of a virtual low-grade glioma patient, whose tumour progresses according to the examined growth laws. We discover that an extreme protraction therapeutic strategy, which amounts to substantially extending the time interval between RT sessions, may lead to better tumour control. The clinical implications of our results are also presented.

Personalized Radiotherapy Design for Glioblastoma: Integrating Mathematical Tumor Models, Multimodal Scans, and Bayesian Inference

Glioblastoma (GBM) is a highly invasive brain tumor, whose cells infiltrate surrounding normal brain tissue beyond the lesion outlines visible in the current medical scans. These infiltrative cells are treated mainly by radiotherapy. Existing radiotherapy plans for brain tumors derive from population studies and scarcely account for patient-specific conditions. Here, we provide a Bayesian machine learning framework for the rational design of improved, personalized radiotherapy plans using mathematical modeling and patient multimodal medical scans. Our method, for the first time, integrates complementary information from high-resolution MRI scans and highly specific FET-PET metabolic maps to infer tumor cell density in GBM patients. The Bayesian framework quantifies imaging and modeling uncertainties and predicts patient-specific tumor cell density with credible intervals. The proposed methodology relies only on data acquired at a single time point and, thus, is applicable to standard clinical settings. An initial clinical population study shows that the radiotherapy plans generated from the inferred tumor cell infiltration maps spare more healthy tissue thereby reducing radiation toxicity while yielding comparable accuracy with standard radiotherapy protocols. Moreover, the inferred regions of high tumor cell densities coincide with the tumor radioresistant areas, providing guidance for personalized dose-escalation. The proposed integration of multimodal scans and mathematical modeling provides a robust, non-invasive tool to assist personalized radiotherapy design.

An adjoint-based method for a linear mechanically-coupled tumor model: Application to estimate the spatial variation of murine glioma growth based on diffusion weighted magnetic resonance imaging

We present an efficient numerical method to quantify the spatial variation of glioma growth based on subject-specific medical images using a mechanically-coupled tumor model. The method is illustrated in a murine model of glioma in which we consider the tumor as a growing elastic mass that continuously deforms the surrounding healthy-appearing brain tissue. As an inverse parameter identification problem, we quantify the volumetric growth of glioma and the growth component of deformation by fitting the model predicted cell density to the cell density estimated using the diffusion-weighted magnetic resonance imaging (DW-MRI) data. Numerically, we developed an adjoint-based approach to solve the optimization problem. Results on a set of experimentally measured, in vivo rat glioma data indicate good agreement between the fitted and measured tumor area and suggest a wide variation of in-plane glioma growth with the growth-induced Jacobian ranging from 1.0 to 6.0.

The many faces of mathematical modelling in oncology

The application of modelling to solve problems in biology and medicine, and specifically in oncology and radiation therapy, is increasingly established and holds big promise. We provide an overview of the basic concepts of the field and its current state, along with new tools available and future directions for research. We will outline radiobiology models, examples of other anticancer therapy models, multiscale modelling, and we will discuss mechanistic and phenomenological approaches to modelling.

Mathematical models of tumor cell proliferation: A review of the literature

INTRODUCTION

A defining hallmark of cancer is aberrant cell proliferation. Efforts to understand the generative properties of cancer cells span all biological scales: from genetic deviations and alterations of metabolic pathways to physical stresses due to overcrowding, as well as the effects of therapeutics and the immune system. While these factors have long been studied in the laboratory, mathematical and computational techniques are being increasingly applied to help understand and forecast tumor growth and treatment response. Advantages of mathematical modeling of proliferation include the ability to simulate and predict the spatiotemporal development of tumors across multiple experimental scales. Central to proliferation modeling is the incorporation of available biological data and validation with experimental data. Areas covered: We present an overview of past and current mathematical strategies directed at understanding tumor cell proliferation. We identify areas for mathematical development as motivated by available experimental and clinical evidence, with a particular emphasis on emerging, non-invasive imaging technologies. Expert commentary: The data required to legitimize mathematical models are often difficult or (currently) impossible to obtain. We suggest areas for further investigation to establish mathematical models that more effectively utilize available data to make informed predictions on tumor cell proliferation.

Advances in Glioblastoma Multiforme Treatment: New Models for Nanoparticle Therapy

The most lethal form of brain cancer, glioblastoma multiforme, is characterized by rapid growth and invasion facilitated by cell migration and degradation of the extracellular matrix. Despite technological advances in surgery and radio-chemotherapy, glioblastoma remains largely resistant to treatment. New approaches to study glioblastoma and to design optimized therapies are greatly needed. One such approach harnesses computational modeling to support the design and delivery of glioblastoma treatment. In this paper, we critically summarize current glioblastoma therapy, with a focus on emerging nanomedicine and therapies that capitalize on cell-specific signaling in glioblastoma. We follow this summary by discussing computational modeling approaches focused on optimizing these emerging nanotherapeutics for brain cancer. We conclude by illustrating how mathematical analysis can be used to compare the delivery of a high potential anticancer molecule, delphinidin, in both free and nanoparticle loaded forms across the blood-brain barrier for glioblastoma.

Combining radiation with hyperthermia: a multiscale model informed by in vitro experiments

Combined radiotherapy and hyperthermia offer great potential for the successful treatment of radio-resistant tumours through thermo-radiosensitization. Tumour response heterogeneity, due to intrinsic, or micro-environmentally induced factors, may greatly influence treatment outcome, but is difficult to account for using traditional treatment planning approaches. Systems oncology simulation, using mathematical models designed to predict tumour growth and treatment response, provides a powerful tool for analysis and optimization of combined treatments. We present a framework that simulates such combination treatments on a cellular level. This multiscale hybrid cellular automaton simulates large cell populations (up to 107 cells) in vitro, while allowing individual cell-cycle progression, and treatment response by modelling radiation-induced mitotic cell death, and immediate cell kill in response to heating. Based on a calibration using a number of experimental growth, cell cycle and survival datasets for HCT116 cells, model predictions agreed well (R2 > 0.95) with experimental data within the range of (thermal and radiation) doses tested (0-40 CEM43, 0-5 Gy). The proposed framework offers flexibility for modelling multimodality treatment combinations in different scenarios. It may therefore provide an important step towards the modelling of personalized therapies using a virtual patient tumour.

Personalized Radiotherapy Planning Based on a Computational Tumor Growth Model

In this article, we propose a proof of concept for the automatic planning of personalized radiotherapy for brain tumors. A computational model of glioblastoma growth is combined with an exponential cell survival model to describe the effect of radiotherapy. The model is personalized to the magnetic resonance images (MRIs) of a given patient. It takes into account the uncertainty in the model parameters, together with the uncertainty in the MRI segmentations. The computed probability distribution over tumor cell densities, together with the cell survival model, is used to define the prescription dose distribution, which is the basis for subsequent Intensity Modulated Radiation Therapy (IMRT) planning. Depending on the clinical data available, we compare three different scenarios to personalize the model. First, we consider a single MRI acquisition before therapy, as it would usually be the case in clinical routine. Second, we use two MRI acquisitions at two distinct time points in order to personalize the model and plan radiotherapy. Third, we include the uncertainty in the segmentation process. We present the application of our approach on two patients diagnosed with high grade glioma. We introduce two methods to derive the radiotherapy prescription dose distribution, which are based on minimizing integral tumor cell survival using the maximum a posteriori or the expected tumor cell density. We show how our method allows the user to compute a patient specific radiotherapy planning conformal to the tumor infiltration. We further present extensions of the method in order to spare adjacent organs at risk by re-distributing the dose. The presented approach and its proof of concept may help in the future to better target the tumor and spare organs at risk.

Three-dimensional Image-based Mechanical Modeling for Predicting the Response of Breast Cancer to Neoadjuvant Therapy

The use of quantitative medical imaging data to initialize and constrain mechanistic mathematical models of tumor growth has demonstrated a compelling strategy for predicting therapeutic response. More specifically, we have demonstrated a data-driven framework for prediction of residual tumor burden following neoadjuvant therapy in breast cancer that uses a biophysical mathematical model combining reaction-diffusion growth/therapy dynamics and biomechanical effects driven by early time point imaging data. Whereas early work had been based on a limited dimensionality reduction (two-dimensional planar modeling analysis) to simplify the numerical implementation, in this work, we extend our framework to a fully volumetric, three-dimensional biophysical mathematical modeling approach in which parameter estimates are generated by an inverse problem based on the adjoint state method for numerical efficiency. In an in silico performance study, we show accurate parameter estimation with error less than 3% as compared to ground truth. We apply the approach to patient data from a patient with pathological complete response and a patient with residual tumor burden and demonstrate technical feasibility and predictive potential with direct comparisons between imaging data observation and model predictions of tumor cellularity and volume. Comparisons to our previous two-dimensional modeling framework reflect enhanced model prediction of residual tumor burden through the inclusion of additional imaging slices of patient-specific data.

Quantitative analysis of the proliferative-to-invasive transition of hypoxic glioma cells

Impact of the proliferative-to-invasive transformation of glioma cells on the global growth kinetics of the tumor.

Fast and high temperature hyperthermia coupled with radiotherapy as a possible new treatment for glioblastoma

Background

A new transcranial focused ultrasound device has been developed that can induce hyperthermia in a large tissue volume. The purpose of this work is to investigate theoretically how glioblastoma multiforme (GBM) can be effectively treated by combining the fast hyperthermia generated by this focused ultrasound device with external beam radiotherapy.

Methods/Design

To investigate the effect of tumor growth, we have developed a mathematical description of GBM proliferation and diffusion in the context of reaction–diffusion theory. In addition, we have formulated equations describing the impact of radiotherapy and heat on GBM in the reaction–diffusion equation, including tumor regrowth by stem cells. This formulation has been used to predict the effectiveness of the combination treatment for a realistic focused ultrasound heating scenario.

Our results show that patient survival could be significantly improved by this combined treatment modality.

Discussion

High priority should be given to experiments to validate the therapeutic benefit predicted by our model.

Electronic supplementary material

The online version of this article (doi:10.1186/s40349-016-0078-3) contains supplementary material, which is available to authorized users.

Why one-size-fits-all vaso-modulatory interventions fail to control glioma invasion: in silico insights

Gliomas are highly invasive brain tumours characterised by poor prognosis and limited response to therapy. There is an ongoing debate on the therapeutic potential of vaso-modulatory interventions against glioma invasion. Prominent vasculature-targeting therapies involve tumour blood vessel deterioration and normalisation. The former aims at tumour infarction and nutrient deprivation induced by blood vessel occlusion/collapse. In contrast, the therapeutic intention of normalising the abnormal tumour vasculature is to improve the efficacy of conventional treatment modalities. Although these strategies have shown therapeutic potential, it remains unclear why they both often fail to control glioma growth. To shed some light on this issue, we propose a mathematical model based on the migration/proliferation dichotomy of glioma cells in order to investigate why vaso-modulatory interventions have shown limited success in terms of tumour clearance. We found the existence of a critical cell proliferation/diffusion ratio that separates glioma responses to vaso-modulatory interventions into two distinct regimes. While for tumours, belonging to one regime, vascular modulations reduce the front speed and increase the infiltration width, for those in the other regime, the invasion speed increases and infiltration width decreases. We discuss how these in silico findings can be used to guide individualised vaso-modulatory approaches to improve treatment success rates.