Predicting the efficacy of radiotherapy in individual glioblastoma patients in vivo: a mathematical modeling approach

Mathematical deconvolution of CAR T-cell proliferation and exhaustion from real-time killing assay data

Chimeric antigen receptor (CAR) T-cell therapy has shown promise in the treatment of haematological cancers and is currently being investigated for solid tumours, including high-grade glioma brain tumours. There is a desperate need to quantitatively study the factors that contribute to the efficacy of CAR T-cell therapy in solid tumours. In this work, we use a mathematical model of predator-prey dynamics to explore the kinetics of CAR T-cell killing in glioma: the Chimeric Antigen Receptor T-cell treatment Response in GliOma (CARRGO) model. The model includes rates of cancer cell proliferation, CAR T-cell killing, proliferation, exhaustion, and persistence. We use patient-derived and engineered cancer cell lines with an in vitro real-time cell analyser to parametrize the CARRGO model. We observe that CAR T-cell dose correlates inversely with the killing rate and correlates directly with the net rate of proliferation and exhaustion. This suggests that at a lower dose of CAR T-cells, individual T-cells kill more cancer cells but become more exhausted when compared with higher doses. Furthermore, the exhaustion rate was observed to increase significantly with tumour growth rate and was dependent on level of antigen expression. The CARRGO model highlights nonlinear dynamics involved in CAR T-cell therapy and provides novel insights into the kinetics of CAR T-cell killing. The model suggests that CAR T-cell treatment may be tailored to individual tumour characteristics including tumour growth rate and antigen level to maximize therapeutic benefit.

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.

Distinct Phenotypic Clusters of Glioblastoma Growth and Response Kinetics Predict Survival

PURPOSE

Despite the intra- and intertumoral heterogeneity seen in glioblastoma multiforme (GBM), there is little definitive data on the underlying cause of the differences in patient survivals. Serial imaging assessment of tumor growth allows quantification of tumor growth kinetics (TGK) measured in terms of changes in the velocity of radial expansion seen on imaging. Because a systematic study of this entire TGK phenotype-growth before treatment and during each treatment to recurrence -has never been coordinately studied in GBMs, we sought to identify whether patients cluster into discrete groups on the basis of their TGK.

PATIENTS AND METHODS

From our multi-institutional database, we identified 48 patients who underwent maximally safe resection followed by radiotherapy with imaging follow-up through the time of recurrence. The patients were then clustered into two groups through a k-means algorithm taking as input only the TGK before and during treatment.

RESULTS

There was a significant survival difference between the clusters ( P = .003). Paradoxically, patients among the long-lived cluster had significantly larger tumors at diagnosis ( P = .027) and faster growth before treatment ( P = .003) but demonstrated a better response to adjuvant chemotherapy ( P = .048). A predictive model was built to identify which cluster patients would likely fall into on the basis of information that would be available to clinicians immediately after radiotherapy (accuracy, 90.3%).

CONCLUSION

Dichotomizing the heterogeneity of GBMs into two populations-one faster growing yet more responsive with increased survival and one slower growing yet less responsive with shorter survival-suggests that many patients who receive standard-of-care treatments may get better benefit from select alternative treatments.

Three-dimensional tumor growth in time-varying chemical fields: a modeling framework and theoretical study

Background

Contemporary biological observations have revealed a large variety of mechanisms acting during the expansion of a tumor. However, there are still many qualitative and quantitative aspects of the phenomenon that remain largely unknown. In this context, mathematical and computational modeling appears as an invaluable tool providing the means for conducting in silico experiments, which are cheaper and less tedious than real laboratory experiments.

Results

This paper aims at developing an extensible and computationally efficient framework for in silico modeling of tumor growth in a 3-dimensional, inhomogeneous and time-varying chemical environment. The resulting model consists of a set of mathematically derived and algorithmically defined operators, each one addressing the effects of a particular biological mechanism on the state of the system. These operators may be extended or re-adjusted, in case a different set of starting assumptions or a different simulation scenario needs to be considered.

Conclusion

In silico modeling provides an alternative means for testing hypotheses and simulating scenarios for which exact biological knowledge remains elusive. However, finer tuning of pertinent methods presupposes qualitative and quantitative enrichment of available biological evidence. Validation in a strict sense would further require comprehensive, case-specific simulations and detailed comparisons with biomedical observations.

Mechanistic modelling of prostate-specific antigen dynamics shows potential for personalized prediction of radiation therapy outcome

External beam radiation therapy is a widespread treatment for prostate cancer. The ensuing patient follow-up is based on the evolution of the prostate-specific antigen (PSA). Serum levels of PSA decay due to the radiation-induced death of tumour cells and cancer recurrence usually manifest as a rising PSA. The current definition of biochemical relapse requires that PSA reaches nadir and starts increasing, which delays the use of further treatments. Also, these methods do not account for the post-radiation tumour dynamics that may contain early information on cancer recurrence. Here, we develop three mechanistic models of post-radiation PSA evolution. Our models render superior fits of PSA data in a patient cohort and provide a biological justification for the most common empirical formulation of PSA dynamics. We also found three model-based prognostic variables: the proliferation rate of the survival fraction, the ratio of radiation-induced cell death rate to the survival proliferation rate, and the time to PSA nadir since treatment termination. We argue that these markers may enable the early identification of biochemical relapse, which would permit physicians to subsequently adapt patient monitoring to optimize the detection and treatment of cancer recurrence.

A Review of Mathematical Models for Tumor Dynamics and Treatment Resistance Evolution of Solid Tumors

Increasing knowledge of intertumor heterogeneity, intratumor heterogeneity, and cancer evolution has improved the understanding of anticancer treatment resistance. A better characterization of cancer evolution and subsequent use of this knowledge for personalized treatment would increase the chance to overcome cancer treatment resistance. Model‐based approaches may help achieve this goal. In this review, we comprehensively summarized mathematical models of tumor dynamics for solid tumors and of drug resistance evolution. Models displayed by ordinary differential equations, algebraic equations, and partial differential equations for characterizing tumor burden dynamics are introduced and discussed. As for tumor resistance evolution, stochastic and deterministic models are introduced and discussed. The results may facilitate a novel model‐based analysis on anticancer treatment response and the occurrence of resistance, which incorporates both tumor dynamics and resistance evolution. The opportunities of a model‐based approach as discussed in this review can be of great benefit for future optimizing and personalizing anticancer treatment.

ENvironmental Dynamics Underlying Responsive Extreme Survivors (ENDURES) of Glioblastoma: A Multidisciplinary Team-based, Multifactorial Analytical Approach

Although glioblastoma (GBM) is a fatal primary brain cancer with short median survival of 15 months, a small number of patients survive >5 years after diagnosis; they are known as extreme survivors (ES). Because of their rarity, very little is known about what differentiates these outliers from other patients with GBM. For the purpose of identifying unknown drivers of extreme survivorship in GBM, the ENDURES consortium (ENvironmental Dynamics Underlying Responsive Extreme Survivors of GBM) was developed. This consortium is a multicenter collaborative network of investigators focused on the integration of multiple types of clinical data and the creation of patient-specific models of tumor growth informed by radiographic and histologic parameters. Leveraging our combined resources, the goals of the ENDURES consortium are 2-fold: (1) to build a curated, searchable, multilayered repository housing clinical and outcome data on a large cohort of ES patients with GBM; and (2) to leverage the ENDURES repository for new insights into tumor behavior and novel targets for prolonging survival for all patients with GBM. In this article, the authors review the available literature and discuss what is already known about ES. The authors then describe the creation of their consortium and some preliminary results.

Integration of machine learning and mechanistic models accurately predicts variation in cell density of glioblastoma using multiparametric MRI

Glioblastoma (GBM) is a heterogeneous and lethal brain cancer. These tumors are followed using magnetic resonance imaging (MRI), which is unable to precisely identify tumor cell invasion, impairing effective surgery and radiation planning. We present a novel hybrid model, based on multiparametric intensities, which combines machine learning (ML) with a mechanistic model of tumor growth to provide spatially resolved tumor cell density predictions. The ML component is an imaging data-driven graph-based semi-supervised learning model and we use the Proliferation-Invasion (PI) mechanistic tumor growth model. We thus refer to the hybrid model as the ML-PI model. The hybrid model was trained using 82 image-localized biopsies from 18 primary GBM patients with pre-operative MRI using a leave-one-patient-out cross validation framework. A Relief algorithm was developed to quantify relative contributions from the data sources. The ML-PI model statistically significantly outperformed (p < 0.001) both individual models, ML and PI, achieving a mean absolute predicted error (MAPE) of 0.106 ± 0.125 versus 0.199 ± 0.186 (ML) and 0.227 ± 0.215 (PI), respectively. Associated Pearson correlation coefficients for ML-PI, ML, and PI were 0.838, 0.518, and 0.437, respectively. The Relief algorithm showed the PI model had the greatest contribution to the result, emphasizing the importance of the hybrid model in achieving the high accuracy.

Spatial optimization for radiation therapy of brain tumours

Glioblastomas are the most common primary brain tumours. They are known for their highly aggressive growth and invasion, leading to short survival times. Treatments for glioblastomas commonly involve a combination of surgical intervention, chemotherapy, and external beam radiation therapy (XRT). Previous works have not only successfully modelled the natural growth of glioblastomas in vivo, but also show potential for the prediction of response to radiation prior to treatment. This suggests that the efficacy of XRT can be optimized before treatment in order to yield longer survival times. However, while current efforts focus on optimal scheduling of radiotherapy treatment, they do not include a similarly sophisticated spatial optimization. In an effort to improve XRT, we present a method for the spatial optimization of radiation profiles. We expand upon previous results in the general problem and examine the more physically reasonable cases of 1-step and 2-step radiation profiles during the first and second XRT fractions. The results show that by including spatial optimization in XRT, while retaining a constant prescribed total dose amount, we are able to increase the total cell kill from the clinically-applied uniform case.

The 2019 mathematical oncology roadmap

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

Improved model prediction of glioma growth utilizing tissue-specific boundary effects

Kinetic parameter estimates for mathematical models of glioblastoma multiforme (GBM), derived from clinical scans, have been used to predict the occurrence of hypoxia, necrosis, response to radiation therapy, and overall survival. Modeling GBM growth in a cerebral model encounters anatomical boundaries that interfere with model calibration from clinical measurements.

METHODS

The effect of boundaries is examined on both spherically symmetric and anatomical models of tumor growth. This effect is incorporated into a method that updates kinetic parameters. The efficacy of this method in reproducing clinical image-derived subject data is evaluated.

RESULTS

Spherically symmetric simulations of tumor growth with simple boundaries behave predictably when in a linear phase of growth. Anatomic simulations of eleven out of twenty subjects demonstrated improved fit to subject data with the new method. When only subjects exhibiting linear growth are considered, eight out of nine subject demonstrate improved fit to the data.

CONCLUSION

Anatomical boundaries to tumor growth measurably deflect progression and affect estimates of kinetic parameters. The presented method reliably updates kinetic parameters to fit anatomic computational models to clinically derived subject data when those data are in a linear regime.

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

Purpose: Radiotherapy prescription dose and dose fractionation protocols vary little between individual patients having the same tumor grade and stage. To personalize radiotherapy a predictive model is needed to simulate radiation response. Previous modeling attempts with multiple variables and parameters have been shown to yield excellent data fits at the cost of non-identifiability and clinically unrealistic results. Materials and methods: We develop a mathematical model based on a proliferation saturation index (PSI) that is a measurement of pre-treatment tumor volume-to-carrying capacity ratio that modulates intrinsic tumor growth and radiation response rates. In an adaptive Bayesian approach, we utilize an increasing number of data points for individual patients to predict patient-specific responses to subsequent radiation doses. Results: Model analysis shows that using PSI as the only patient-specific parameter, model simulations can fit longitudinal clinical data with high accuracy (R²=0.84). By analyzing tumor response to radiation using daily CT scans early in the treatment, response to the remaining treatment fractions can be predicted after two weeks with high accuracy (c-index = 0.89). Conclusion: The PSI model may be suited to forecast treatment response for individual patients and offers actionable decision points for mid-treatment protocol adaptation. The presented work provides an actionable image-derived biomarker prior to and during therapy to personalize and adapt radiotherapy.

Biophysical model-based parameters to classify tumor recurrence from radiation-induced necrosis for brain metastases

PURPOSE

Stereotactic radiosurgery (SRS) is used for local control treatment of patients with intracranial metastases. As a result of SRS, some patients develop radiation-induced necrosis. Radiographically, radiation-induced necrosis can appear similar to tumor recurrence in magnetic resonance (MR) T₁ -weighted contrast-enhanced imaging, T₂ -weighted MR imaging, and Fluid-Attenuated Inversion Recovery (FLAIR) MR imaging. Radiographic ambiguities often necessitate invasive brain biopsies to determine lesion etiology or cause delayed subsequent therapy initiation. We use a biomechanically coupled tumor growth model to estimate patient-specific model parameters and model-derived measures to noninvasively classify etiology of enhancing lesions in this patient population.

METHODS

In this initial, preliminary retrospective study, we evaluated five patients with tumor recurrence and five with radiation-induced necrosis. Longitudinal patient-specific MR imaging data were used in conjunction with the model to parameterize tumor cell proliferation rate and tumor cell diffusion coefficient, and Dice correlation coefficients were used to quantify degree of correlation between model-estimated mechanical stress fields and edema visualized from MR imaging.

RESULTS

Results found four statistically relevant parameters which can differentiate tumor recurrence and radiation-induced necrosis.

CONCLUSIONS

This preliminary investigation suggests potential of this framework to noninvasively determine the etiology of enhancing lesions in patients who previously underwent SRS for intracranial metastases.

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.

Kinetic modeling of tumor regression incorporating the concept of cancer stem-like cells for patients with locally advanced lung cancer

Background

Personalized medicine for patients receiving radiation therapy remains an elusive goal due, in part, to the limits in our understanding of the underlying mechanisms governing tumor response to radiation. The purpose of this study was to develop a kinetic model, in the context of locally advanced lung cancer, connecting cancer cell subpopulations with tumor volumes measured during the course of radiation treatment for understanding treatment outcome for individual patients.

Methods

The kinetic model consists of three cell compartments: cancer stem-like cells (CSCs), non-stem tumor cells (TCs) and dead cells (DCs). A set of ordinary differential equations were developed to describe the time evolution of each compartment, and the analytic solution of these equations was iterated to be aligned with the day-to-day tumor volume changes during the course of radiation treatment. A least squares fitting method was used to estimate the parameters of the model that include the proportion of CSCs and their radio-sensitivities. This model was applied to five patients with stage III lung cancer, and tumor volumes were measured from 33 cone-beam computed tomography (CBCT) images for each of these patients. The analytical solution of these differential equations was compared with numerically simulated results.

Results

For the five patients with late stage lung cancer, the derived proportions of CSCs are 0.3 on average, the average probability of the symmetry division is 0.057 and the average surviving fractions of CSCs is 0.967, respectively. The derived parameters are comparable to the results from literature and our experiments. The preliminary results suggest that the CSC self-renewal rate is relatively small, compared to the proportion of CSCs for locally advanced lung cancers.

Conclusions

A novel mathematical model has been developed to connect the population of cancer stem-like cells with tumor volumes measured from a sequence of CBCT images. This model may help improve our understanding of tumor response to radiation therapy, and is valuable for development of new treatment regimens for patients with locally advanced lung cancer.

Mathematical Modeling of Bacteria-Enabled Drug Delivery System Penetration into Multicellular Tumor Spheroids

Bacteria-based cancer treatment is a promising approach to address the need for targeted tumor therapies in an effort to avoid the systemic toxicity inherent in conventional chemotherapy. A number of bacterial strains have been shown to preferentially colonize tumors and impart therapeutic benefits. However, the physical underpinnings of bacteria intratumoral transport remain poorly studied. It is hypothesized that cell Iysis in hypoxic and necrotic regions of tumors creates a niche in which some bacteria thrive. To understand if preferential growth plausibly explains the experimentally observed bacterial colonization profiles, we have developed a mathematical model incorporating transport and growth dependent on tumor cell Iysate. We fit model parameters to experimental data, showing that our formulation captures experimentally observed trends. Moreover, we find that bacteria have a higher effective diffusivity than nanoparticles alone, demonstrating transport advantages to designing bacteria-based cancer therapy. This model serves as a first step towards building computational tools for designing optimized bacteria- based chemotherapeutic delivery systems.

A Coupled Modified Reaction Diffusion and Biomechanical Models for Cerebral Tumor Growth Modeling in Presence of Treatment

Tumor growth modeling at macroscopic level from multimodal images can help in predicting the future evolution of tumor and the treatment planning. This can be achieved using mathematical models where multi time-point images are available. In this paper, we propose a coupled modified reaction diffusion model that measures tumor invasion and infiltration with biomechanical model to consider tumor mass effect. In addition, our model considers treatment effects from radiotherapy and/or chemotherapy if any. The chemotherapy effect is included via a modified log-kill method to consider tissue heterogeneity while radiotherapy effect is considered using the linear quadratic model. We test the proposed model on both synthetic and 6 real datasets of low grade glioma cases with and without treatments. Experimental results of the proposed model on the clinical magnetic resonance images show that our model can simulate the tumor growth with good accuracy and effectively include the treatment effects.

Patterns of re-irradiation for recurrent gliomas and validation of a prognostic score

PURPOSE OR OBJECTIVE

Re-irradiation is a generally accepted method for salvage treatment in patients with recurrent glioma. However, no standard radiation regimen has been defined. This study aims to compare the efficacy and safety of different treatment regimens and to independently externally validate a recently published reirradiation risk score.

MATERIAL AND METHODS

We retrospectively analyzed a cohort of patients with recurrent malignant glioma treated with salvage conventionally fractionated (CFRT), hypofractionated (HFRT) or stereotactic radiotherapy (SRT) between 2007 and 2017 at the University Medical Centers in Utrecht and Groningen.

RESULTS

Of the 121 patients included, 60 patients (50%) underwent CFRT, 22 (18%) HFRT and 39 (32%) SRT. The primary tumor was grade II-III in 52 patients and grade IV in 69 patients with median Overall Survival (mOS) since first surgery of 113 [Interquartile range: 53.2-137] and 39.7 [24.6-64.9] months respectively (p < 0.01). Overall, mOS from the first day of re-irradiation was 9.7 months [6.5-14.6]. No significant difference in mOS was found between the treatment groups. In multivariate analysis, the Karnofsky performance scale ≥70% (p < 0.01), re-irradiation for first recurrence (p = 0.02), longer time interval between RT start dates (p < 0.01) and smaller planning target volume (p < 0.05) were significant favorable prognostic factors. The reirradiation risk score was validated.

CONCLUSION

In our series, mOS after reirradiation was sufficient to justify use of this modality. Until a reliable treatment decision tool is developed based on larger retrospective research, the decision for re-irradiation schedule should remain personalized and based on a multidisciplinary evaluation of each patient.

The biology and mathematical modelling of glioma invasion: a review

Adult gliomas are aggressive brain tumours associated with low patient survival rates and limited life expectancy. The most important hallmark of this type of tumour is its invasive behaviour, characterized by a markedly phenotypic plasticity, infiltrative tumour morphologies and the ability of malignant progression from low- to high-grade tumour types. Indeed, the widespread infiltration of healthy brain tissue by glioma cells is largely responsible for poor prognosis and the difficulty of finding curative therapies. Meanwhile, mathematical models have been established to analyse potential mechanisms of glioma invasion. In this review, we start with a brief introduction to current biological knowledge about glioma invasion, and then critically review and highlight future challenges for mathematical models of glioma invasion.

Cognitive impairment and morphological changes after radiation therapy in brain tumors: A review

Life expectancy of patients treated for brain tumors has lengthened due to the therapeutic improvements. Cognitive impairment has been described following brain radiotherapy, but the mechanisms leading to this adverse event remain mostly unknown. Technical evolutions aim at enhancing the therapeutic ratio. Sparing of the healthy tissues has been improved using various approaches; however, few dose constraints have been established regarding brain structures associated with cognitive functions. The aims of this literature review are to report the main brain areas involved in cognitive adverse effects induced by radiotherapy as described in literature, to better understand brain radiosensitivity and to describe potential future improvements.

Data Analysis in Radiotherapy Treatments

Radiotherapy is one of the main cancer treatments available today, together with chemotherapy and surgery. Radiotherapy treatments have to be planned for each patient in an individualized manner. The knowledge acquired from one single treatment can be used to improve the treatment planning and outcome of several other patients. In the last years, attention has been drawn to the added value of using data analysis for radiotherapy treatment planning, prediction of treatment outcomes, survival analysis and quality assurance. In this article, existing literature is reviewed.

Temporally feathered intensity-modulated radiation therapy: A planning technique to reduce normal tissue toxicity

PURPOSE

Intensity-modulated radiation therapy (IMRT) has allowed optimization of three-dimensional spatial radiation dose distributions permitting target coverage while reducing normal tissue toxicity. However, radiation-induced normal tissue toxicity is a major contributor to patients' quality of life and often a dose-limiting factor in the definitive treatment of cancer with radiation therapy. We propose the next logical step in the evolution of IMRT using canonical radiobiological principles, optimizing the temporal dimension through which radiation therapy is delivered to further reduce radiation-induced toxicity by increased time for normal tissue recovery. We term this novel treatment planning strategy "temporally feathered radiation therapy" (TFRT).

METHODS

Temporally feathered radiotherapy plans were generated as a composite of five simulated treatment plans each with altered constraints on particular hypothetical organs at risk (OARs) to be delivered sequentially. For each of these TFRT plans, OARs chosen for feathering receive higher doses while the remaining OARs receive lower doses than the standard fractional dose delivered in a conventional fractionated IMRT plan. Each TFRT plan is delivered a specific weekday, which in effect leads to a higher dose once weekly followed by four lower fractional doses to each temporally feathered OAR. We compared normal tissue toxicity between TFRT and conventional fractionated IMRT plans by using a dynamical mathematical model to describe radiation-induced tissue damage and repair over time.

RESULTS

Model-based simulations of TFRT demonstrated potential for reduced normal tissue toxicity compared to conventionally planned IMRT. The sequencing of high and low fractional doses delivered to OARs by TFRT plans suggested increased normal tissue recovery, and hence less overall radiation-induced toxicity, despite higher total doses delivered to OARs compared to conventional fractionated IMRT plans. The magnitude of toxicity reduction by TFRT planning was found to depend on the corresponding standard fractional dose of IMRT and organ-specific recovery rate of sublethal radiation-induced damage.

CONCLUSIONS

TFRT is a novel technique for treatment planning and optimization of therapeutic radiotherapy that considers the nonlinear aspects of normal tissue repair to optimize toxicity profiles. Model-based simulations of TFRT to carefully conceptualized clinical cases have demonstrated potential for radiation-induced toxicity reduction in a previously described dynamical model of normal tissue complication probability (NTCP).

Precision Medicine with Imprecise Therapy: Computational Modeling for Chemotherapy in Breast Cancer

Medical oncology is in need of a mathematical modeling toolkit that can leverage clinically-available measurements to optimize treatment selection and schedules for patients. Just as the therapeutic choice has been optimized to match tumor genetics, the delivery of those therapeutics should be optimized based on patient-specific pharmacokinetic/pharmacodynamic properties. Under the current approach to treatment response planning and assessment, there does not exist an efficient method to consolidate biomarker changes into a holistic understanding of treatment response. While the majority of research on chemotherapies focus on cellular and genetic mechanisms of resistance, there are numerous patient-specific and tumor-specific measures that contribute to treatment response. New approaches that consolidate multimodal information into actionable data are needed. Mathematical modeling offers a solution to this problem. In this perspective, we first focus on the particular case of breast cancer to highlight how mathematical models have shaped the current approaches to treatment. Then we compare chemotherapy to radiation therapy. Finally, we identify opportunities to improve chemotherapy treatments using the model of radiation therapy. We posit that mathematical models can improve the application of anticancer therapeutics in the era of precision medicine. By highlighting a number of historical examples of the contributions of mathematical models to cancer therapy, we hope that this contribution serves to engage investigators who may not have previously considered how mathematical modeling can provide real insights into breast cancer therapy.

Biophysical Modeling of In Vivo Glioma Response After Whole-Brain Radiation Therapy in a Murine Model of Brain Cancer

PURPOSE

To develop and investigate a set of biophysical models based on a mechanically coupled reaction-diffusion model of the spatiotemporal evolution of tumor growth after radiation therapy.

METHODS AND MATERIALS

Post-radiation therapy response is modeled using a cell death model (Md), a reduced proliferation rate model (Mp), and cell death and reduced proliferation model (Mdp). To evaluate each model, rats (n = 12) with C6 gliomas were imaged with diffusion-weighted magnetic resonance imaging (MRI) and contrast-enhanced MRI at 7 time points over 2 weeks. Rats received either 20 or 40 Gy between the third and fourth imaging time point. Diffusion-weighted MRI was used to estimate tumor cell number within enhancing regions in contrast-enhanced MRI data. Each model was fit to the spatiotemporal evolution of tumor cell number from time point 1 to time point 5 to estimate model parameters. The estimated model parameters were then used to predict tumor growth at the final 2 imaging time points. The model prediction was evaluated by calculating the error in tumor volume estimates, average surface distance, and voxel-based cell number.

RESULTS

For both the rats treated with either 20 or 40 Gy, significantly lower error in tumor volume, average surface distance, and voxel-based cell number was observed for the Mdp and Mp models compared with the Md model. The Mdp model fit, however, had significantly lower sum squared error compared with the Mp and Md models.

CONCLUSIONS

The results of this study indicate that for both doses, the Mp and Mdp models result in accurate predictions of tumor growth, whereas the Md model poorly describes response to radiation therapy.

A Brownian dynamics tumor progression simulator with application to glioblastoma

Tumor progression modeling offers the potential to predict tumor-spreading behavior to improve prognostic accuracy and guide therapy development. Common simulation methods include continuous reaction-diffusion (RD) approaches that capture mean spatio-temporal tumor spreading behavior and discrete agent-based (AB) approaches which capture individual cell events such as proliferation or migration. The brain cancer glioblastoma (GBM) is especially appropriate for such proliferation-migration modeling approaches because tumor cells seldom metastasize outside of the central nervous system and cells are both highly proliferative and migratory. In glioblastoma research, current RD estimates of proliferation and migration parameters are derived from computed tomography or magnetic resonance images. However, these estimates of glioblastoma cell migration rates, modeled as a diffusion coefficient, are approximately 1-2 orders of magnitude larger than single-cell measurements in animal models of this disease. To identify possible sources for this discrepancy, we evaluated the fundamental RD simulation assumptions that cells are point-like structures that can overlap. To give cells physical size (~10 μm), we used a Brownian dynamics approach that simulates individual single-cell diffusive migration, growth, and proliferation activity via a gridless, off-lattice, AB method where cells can be prohibited from overlapping each other. We found that for realistic single-cell parameter growth and migration rates, a non-overlapping model gives rise to a jammed configuration in the center of the tumor and a biased outward diffusion of cells in the tumor periphery, creating a quasi-ballistic advancing tumor front. The simulations demonstrate that a fast-progressing tumor can result from minimally diffusive cells, but at a rate that is still dependent on single-cell diffusive migration rates. Thus, modeling with the assumption of physically-grounded volume conservation can account for the apparent discrepancy between estimated and measured diffusion of GBM cells and provide a new theoretical framework that naturally links single-cell growth and migration dynamics to tumor-level progression.

[Hypofractionated radiotherapy for glioblastoma: changing the radiation treatment paradigm]

Hypofractionation has the dual advantage of increased cell death with a higher dose per fraction and a reduced effect of accelerated tumor cell repopulation due to a shorter overall treatment time. However, the potential advantage may be offset by increased toxicity in the late-responding neural tissues. Recently, investigators have attempted delivering radical doses of HFRT by escalating the dose in the immediate vicinity of the enhancing tumor and postoperative surgical cavity and reported reasonable outcomes with acceptable toxicity levels. Three different studies of high-dose HFRT have reported on the paradoxical phenomenon of improved survival in patients developing radiation necrosis at the primary tumor site. The toxicity criteria of RTOG and EORTC have defined clinically or radiographically suspected radionecrosis as Grade 4 toxicity. However, most patients diagnosed with radiation necrosis in the above studies remained asymptomatic. Furthermore, the probable association with improved survival would strongly argue against adopting a blind approach for classifying radiation necrosis as Grade 4 toxicity. The data emerging from the above studies is encouraging and strongly argues for further research. However, the majority of these studies are predominantly retrospective or relatively small single-arm prospective series that add little to the overall quality of evidence. Notwithstanding the above limitations, HFRT appears to be a safe and feasible strategy for glioblastoma patients.

A stochastic model for the normal tissue complication probability (NTCP) and applicationss

The normal tissue complication probability (NTCP) is a measure for the estimated side effects of a given radiation treatment schedule. Here we use a stochastic logistic birth-death process to define an organ-specific and patient-specific NTCP. We emphasize an asymptotic simplification which relates the NTCP to the solution of a logistic differential equation. This framework is based on simple modelling assumptions and it prepares a framework for the use of the NTCP model in clinical practice. As example, we consider side effects of prostate cancer brachytherapy such as increase in urinal frequency, urinal retention and acute rectal dysfunction.

Integrating evolutionary dynamics into treatment of metastatic castrate-resistant prostate cancer

Abiraterone treats metastatic castrate-resistant prostate cancer by inhibiting CYP17A, an enzyme for testosterone auto-production. With standard dosing, evolution of resistance with treatment failure (radiographic progression) occurs at a median of ~16.5 months. We hypothesize time to progression (TTP) could be increased by integrating evolutionary dynamics into therapy. We developed an evolutionary game theory model using Lotka–Volterra equations with three competing cancer “species”: androgen dependent, androgen producing, and androgen independent. Simulations with standard abiraterone dosing demonstrate strong selection for androgen-independent cells and rapid treatment failure. Adaptive therapy, using patient-specific tumor dynamics to inform on/off treatment cycles, suppresses proliferation of androgen-independent cells and lowers cumulative drug dose. In a pilot clinical trial, 10 of 11 patients maintained stable oscillations of tumor burdens; median TTP is at least 27 months with reduced cumulative drug use of 47% of standard dosing. The outcomes show significant improvement over published studies and a contemporaneous population.

Evolution of resistance is a common cause of cancer treatment failure and tumor progression. Here, the authors present a method for integrating evolutionary principles based on adaptive therapy into abiraterone therapy for metastatic castrate-resistant prostate cancer and show the positive results of an interim analysis of a trial cohort.

The role of Allee effect in modelling post resection recurrence of glioblastoma

Resection of the bulk of a tumour often cannot eliminate all cancer cells, due to their infiltration into the surrounding healthy tissue. This may lead to recurrence of the tumour at a later time. We use a reaction-diffusion equation based model of tumour growth to investigate how the invasion front is delayed by resection, and how this depends on the density and behaviour of the remaining cancer cells. We show that the delay time is highly sensitive to qualitative details of the proliferation dynamics of the cancer cell population. The typically assumed logistic type proliferation leads to unrealistic results, predicting immediate recurrence. We find that in glioblastoma cell cultures the cell proliferation rate is an increasing function of the density at small cell densities. Our analysis suggests that cooperative behaviour of cancer cells, analogous to the Allee effect in ecology, can play a critical role in determining the time until tumour recurrence.

Mathematical models of propagating fronts have been used to represent a wide variety of biological phenomena from action potentials in neural cells to invasive species in ecology and epidemic spreading. Here we show that when such models are used to predict the effects of external perturbations the results can be very sensitive to certain details of the local dynamics. For example, the post resection recurrence of tumour growth depends strongly on the density dependence of the proliferation of cancer cells. This suggests that targeting the cooperative behaviour of cancer cells could be an efficient strategy for delaying the recurrence of diffuse aggressive brain tumours.

Computational Modeling in Liver Surgery

The need for extended liver resection is increasing due to the growing incidence of liver tumors in aging societies. Individualized surgical planning is the key for identifying the optimal resection strategy and to minimize the risk of postoperative liver failure and tumor recurrence. Current computational tools provide virtual planning of liver resection by taking into account the spatial relationship between the tumor and the hepatic vascular trees, as well as the size of the future liver remnant. However, size and function of the liver are not necessarily equivalent. Hence, determining the future liver volume might misestimate the future liver function, especially in cases of hepatic comorbidities such as hepatic steatosis. A systems medicine approach could be applied, including biological, medical, and surgical aspects, by integrating all available anatomical and functional information of the individual patient. Such an approach holds promise for better prediction of postoperative liver function and hence improved risk assessment. This review provides an overview of mathematical models related to the liver and its function and explores their potential relevance for computational liver surgery. We first summarize key facts of hepatic anatomy, physiology, and pathology relevant for hepatic surgery, followed by a description of the computational tools currently used in liver surgical planning. Then we present selected state-of-the-art computational liver models potentially useful to support liver surgery. Finally, we discuss the main challenges that will need to be addressed when developing advanced computational planning tools in the context of liver surgery.

Clinical implications of in silico mathematical modeling for glioblastoma: a critical review

Glioblastoma remains a clinical challenge in spite of years of extensive research. Novel approaches are needed in order to integrate the existing knowledge. This is the potential role of mathematical oncology. This paper reviews mathematical models on glioblastoma from the clinical doctor's point of view, with focus on 3D modeling approaches of radiation response of in vivo glioblastomas based on contemporary imaging techniques. As these models aim to provide a clinically useful tool in the era of personalized medicine, the integration of the latest advances in molecular and imaging science and in clinical practice by the in silico models is crucial for their clinical relevance. Our aim is to indicate areas of GBM research that have not yet been addressed by in silico models and to point out evidence that has come up from in silico experiments, which may be worth considering in the clinic. This review examines how close these models have come in predicting the outcome of treatment protocols and in shaping the future of radiotherapy treatments.

Low grade glioma growth modeling considering chemotherapy and radiotherapy effects from magnetic resonance images

Studying tumor growth using mathematical models from magnetic resonance (MR) images is an important application that is believed to play a major role in cancer treatment by predicting tumor evolution, quantifying the response to therapy, and treatment planning. Reaction diffusion is the most popular model because of its simplicity and consistency with the biological growth process. However, most of the current growth models focus on presurgical images and likely without treatment. In this paper, we propose a new reaction diffusion model to consider the chemotherapy and radiotherapy effects on the tumor growth modelling for patients with low grade glioma. The proposed model does not consider the tensor information from diffusion tensor imaging. Instead it uses a weighted parameter to promote higher diffusivity in white matter. The radiotherapy and chemotherapy effects are considered as a loss terms in the proposed model. The preliminary results of the proposed model on synthetic and 2 real MR images show that, our model can effectively simulate tumor growth with high accuracies when treatments are administrated to low grade glioma patients.

Homeostasis Back and Forth: An Ecoevolutionary Perspective of Cancer

The role of genetic mutations in cancer is indisputable: They are a key source of tumor heterogeneity and drive its evolution to malignancy. But, the success of these new mutant cells relies on their ability to disrupt the homeostasis that characterizes healthy tissues. Mutated clones unable to break free from intrinsic and extrinsic homeostatic controls will fail to establish a tumor. Here, we will discuss, through the lens of mathematical and computational modeling, why an evolutionary view of cancer needs to be complemented by an ecological perspective to understand why cancer cells invade and subsequently transform their environment during progression. Importantly, this ecological perspective needs to account for tissue homeostasis in the organs that tumors invade, because they perturb the normal regulatory dynamics of these tissues, often coopting them for its own gain. Furthermore, given our current lack of success in treating advanced metastatic cancers through tumor-centric therapeutic strategies, we propose that treatments that aim to restore homeostasis could become a promising venue of clinical research. This ecoevolutionary view of cancer requires mechanistic mathematical models to both integrate clinical with biological data from different scales but also to detangle the dynamic feedback between the tumor and its environment. Importantly, for these models to be useful, they need to embrace a higher degree of complexity than many mathematical modelers are traditionally comfortable with.

A mathematical model describes the malignant transformation of low grade gliomas: Prognostic implications

Gliomas are the most frequent type of primary brain tumours. Low grade gliomas (LGGs, WHO grade II gliomas) may grow very slowly for the long periods of time, however they inevitably cause death due to the phenomenon known as the malignant transformation. This refers to the transition of LGGs to more aggressive forms of high grade gliomas (HGGs, WHO grade III and IV gliomas). In this paper we propose a mathematical model describing the spatio-temporal transition of LGGs into HGGs. Our modelling approach is based on two cellular populations with transitions between them being driven by the tumour microenvironment transformation occurring when the tumour cell density grows beyond a critical level. We show that the proposed model describes real patient data well. We discuss the relationship between patient prognosis and model parameters. We approximate tumour radius and velocity before malignant transformation as well as estimate the onset of this process.

A Predictive Mathematical Modeling Approach for the Study of Doxorubicin Treatment in Triple Negative Breast Cancer

Doxorubicin forms the basis of chemotherapy regimens for several malignancies, including triple negative breast cancer (TNBC). Here, we present a coupled experimental/modeling approach to establish an in vitro pharmacokinetic/pharmacodynamic model to describe how the concentration and duration of doxorubicin therapy shape subsequent cell population dynamics. This work features a series of longitudinal fluorescence microscopy experiments that characterize (1) doxorubicin uptake dynamics in a panel of TNBC cell lines, and (2) cell population response to doxorubicin over 30 days. We propose a treatment response model, fully parameterized with experimental imaging data, to describe doxorubicin uptake and predict subsequent population dynamics. We found that a three compartment model can describe doxorubicin pharmacokinetics, and pharmacokinetic parameters vary significantly among the cell lines investigated. The proposed model effectively captures population dynamics and translates well to a predictive framework. In a representative cell line (SUM-149PT) treated for 12 hours with doxorubicin, the mean percent errors of the best-fit and predicted models were 14% (±10%) and 16% (±12%), which are notable considering these statistics represent errors over 30 days following treatment. More generally, this work provides both a template for studies quantitatively investigating treatment response and a scalable approach toward predictions of tumor response in vivo.

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

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

Evaluating the potential for maximized T cell redistribution entropy to improve abscopal responses to radiotherapy

The potential for local radiation therapy to elicit systemic (abscopal) anti-tumor immune responses has been receiving a significant amount of attention over the last decade. We recently developed a mathematical framework designed to simulate the systemic dissemination of activated T cells among multiple metastatic sites. This framework allowed the identification of non-intuitive patterns of T cell redistribution after localized therapy, and offered suggestions as to the optimal site to irradiate in order to increase the magnitude of an immune-mediated abscopal response. Here, we evaluate the potential for such a framework to provide clinical decision making support to radiation oncologists. Several challenges such as efficient segmentation and delineation of multiple tumor sites on PET/CT scans, validation of model prediction performance, and effective clinical trial design remain to be addressed prior to the incorporation of such a tool in the clinical setting.

Significance of Inactivated Genes in Leukemia: Pathogenesis and Prognosis

Epigenetic and genetic alterations are two mechanisms participating in leukemia, which can inactivate genes involved in leukemia pathogenesis or progression. The purpose of this review was to introduce various inactivated genes and evaluate their possible role in leukemia pathogenesis and prognosis. By searching the mesh words "Gene, Silencing AND Leukemia" in PubMed website, relevant English articles dealt with human subjects as of 2000 were included in this study. Gene inactivation in leukemia is largely mediated by promoter's hypermethylation of gene involving in cellular functions such as cell cycle, apoptosis, and gene transcription. Inactivated genes, such as ASPP1, TP53, IKZF1 and P15, may correlate with poor prognosis in acute lymphoid leukemia (ALL), chronic lymphoid leukemia (CLL), chronic myelogenous leukemia (CML) and acute myeloid leukemia (AML), respectively. Gene inactivation may play a considerable role in leukemia pathogenesis and prognosis, which can be considered as complementary diagnostic tests to differentiate different leukemia types, determine leukemia prognosis, and also detect response to therapy. In general, this review showed some genes inactivated only in leukemia (with differences between B-ALL, T-ALL, CLL, AML and CML). These differences could be of interest as an additional tool to better categorize leukemia types. Furthermore; based on inactivated genes, a diverse classification of Leukemias could represent a powerful method to address a targeted therapy of the patients, in order to minimize side effects of conventional therapies and to enhance new drug strategies.

Non-standard radiotherapy fractionations delay the time to malignant transformation of low-grade gliomas

Grade II gliomas are slowly growing primary brain tumors that affect mostly young patients. Cytotoxic therapies (radiotherapy and/or chemotherapy) are used initially only for patients having a bad prognosis. These therapies are planned following the “maximum dose in minimum time” principle, i. e. the same schedule used for high-grade brain tumors in spite of their very different behavior. These tumors transform after a variable time into high-grade gliomas, which significantly decreases the patient’s life expectancy. In this paper we study mathematical models describing the growth of grade II gliomas in response to radiotherapy. We find that protracted metronomic fractionations, i.e. therapeutical schedules enlarging the time interval between low-dose radiotherapy fractions, may lead to a better tumor control without an increase in toxicity. Other non-standard fractionations such as protracted or hypoprotracted schemes may also be beneficial. The potential survival improvement depends on the tumor’s proliferation rate and can be even of the order of years. A conservative metronomic scheme, still being a suboptimal treatment, delays the time to malignant progression by at least one year when compared to the standard scheme.

Image-driven modeling of the proliferation and necrosis of glioblastoma multiforme

Background

The heterogeneity of response to treatment in patients with glioblastoma multiforme suggests that the optimal therapeutic approach incorporates an individualized assessment of expected lesion progression. In this work, we develop a novel computational model for the proliferation and necrosis of glioblastoma multiforme.

Methods

The model parameters are selected based on the magnetic resonance imaging features of each tumor, and the proposed technique accounts for intrinsic cell division, tumor cell migration along white matter tracts, as well as central tumor necrosis. As a validation of this approach, tumor growth is simulated in the brain of a healthy adult volunteer using parameters derived from the imaging of a patient with glioblastoma multiforme. A mutual information metric is calculated between the simulated tumor profile and observed tumor.

Results

The tumor progression profile generated by the proposed model is compared with those produced by existing models and with the actual observed tumor progression. Both qualitative and quantitative analyses show that the model introduced in this work replicates the observed progression of glioblastoma more accurately relative to prior techniques.

Conclusions

This image-driven model generates improved tumor progression profiles and may contribute to the development of more reliable prognostic estimates in patients with glioblastoma multiforme.

Post-Surgery Glioma Growth Modeling from Magnetic Resonance Images for Patients with Treatment

Reaction diffusion is the most common growth modelling methodology due to its simplicity and consistency with the biological tumor growth process. However, current extensions of the reaction diffusion model lack one or more of the following: efficient inclusion of treatments' effects, taking into account the viscoelasticity of brain tissues, and guaranteed stability of the numerical solution. We propose a new model to overcome the aforementioned drawbacks. Guided by directional information derived from diffusion tensor imaging, our model relates tissue heterogeneity with the absorption of the chemotherapy, adopts the linear-quadratic term to simulate the radiotherapy effect, employs Maxwell-Weichert model to incorporate brain viscoelasticity, and ensures the stability of the numerical solution. The performance is verified through experiments on synthetic and real MR images. Experiments on 9 MR datasets of patients with low grade gliomas undergoing surgery with different treatment regimens are carried out and validated using Jaccard score and Dice coefficient. The growth simulation accuracies of the proposed model are in ranges of [0.673 0.822] and [0.805 0.902] for Jaccard scores and Dice coefficients, respectively. The accuracies decrease up to 4% and 2.4% when ignoring treatment effects and the tensor information, while brain viscoelasticity has no significant impact on the accuracies.

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.

Mathematical modeling links Wnt signaling to emergent patterns of metabolism in colon cancer

Cell-intrinsic metabolic reprogramming is a hallmark of cancer that provides anabolic support to cell proliferation. How reprogramming influences tumor heterogeneity or drug sensitivities is not well understood. Here, we report a self-organizing spatial pattern of glycolysis in xenograft colon tumors where pyruvate dehydrogenase kinase (PDK1), a negative regulator of oxidative phosphorylation, is highly active in clusters of cells arranged in a spotted array. To understand this pattern, we developed a reaction-diffusion model that incorporates Wnt signaling, a pathway known to upregulate PDK1 and Warburg metabolism. Partial interference with Wnt alters the size and intensity of the spotted pattern in tumors and in the model. The model predicts that Wnt inhibition should trigger an increase in proteins that enhance the range of Wnt ligand diffusion. Not only was this prediction validated in xenograft tumors but similar patterns also emerge in radiochemotherapy-treated colorectal cancer. The model also predicts that inhibitors that target glycolysis or Wnt signaling in combination should synergize and be more effective than each treatment individually. We validated this prediction in 3D colon tumor spheroids.

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.