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

Response classification based on a minimal model of glioblastoma growth is prognostic for clinical outcomes and distinguishes progression from pseudoprogression

Glioblastoma multiforme is the most aggressive type of primary brain tumor. Glioblastoma growth dynamics vary widely across patients, making it difficult to accurately gauge their response to treatment. We developed a model-based metric of therapy response called Days Gained that accounts for this heterogeneity. Here, we show in 63 newly diagnosed patients with glioblastoma that Days Gained scores from a simple glioblastoma growth model computed at the time of the first postradiotherapy MRI scan are prognostic for time to tumor recurrence and overall patient survival. After radiation treatment, Days Gained also distinguished patients with pseudoprogression from those with true progression. Because Days Gained scores can be easily computed with routinely available clinical imaging devices, this model offers immediate potential to be used in ongoing prospective studies.

Discriminating survival outcomes in patients with glioblastoma using a simulation-based, patient-specific response metric

Accurate clinical assessment of a patient's response to treatment for glioblastoma multiforme (GBM), the most malignant type of primary brain tumor, is undermined by the wide patient-to-patient variability in GBM dynamics and responsiveness to therapy. Using computational models that account for the unique geometry and kinetics of individual patients' tumors, we developed a method for assessing treatment response that discriminates progression-free and overall survival following therapy for GBM. Applying these models as untreated virtual controls, we generate a patient-specific “Days Gained” response metric that estimates the number of days a therapy delayed imageable tumor progression. We assessed treatment response in terms of Days Gained scores for 33 patients at the time of their first MRI scan following first-line radiation therapy. Based on Kaplan-Meier analyses, patients with Days Gained scores of 100 or more had improved progression-free survival, and patients with scores of 117 or more had improved overall survival. Our results demonstrate that the Days Gained response metric calculated at the routinely acquired first post-radiation treatment time point provides prognostic information regarding progression and survival outcomes. Applied prospectively, our model-based approach has the potential to improve GBM treatment by accounting for patient-to-patient heterogeneity in GBM dynamics and responses to therapy.

A Monte Carlo based formalism to identify potential locations at high risk of tumor recurrence with a numerical model for glioblastoma multiforme

PURPOSE

The strategy currently used to treat glioblastoma multiforme (GBM) patients, which mostly relies on population-based failure patterns, does not consider the important variability in such patterns reported in the literature. As part of the multidisciplinary efforts being made to develop personalized therapeutic approaches, numerical models of tumor growth and treatment are increasingly being used by different groups around the world. In this study, a new formalism relying on the proliferation-invasion model is developed to identify potential locations of GBM recurrences. The authors assess the sensitivity of the location of potential tumor recurrences to the input parameter values predicted for a given patient by varying those values using a Monte-Carlo based approach. Our approach is designed to be prospective in the sense that it relies on patient-specific imaging data that can be gathered in one single preradiotherapy imaging session.

METHODS

The authors modeled the infiltration paths of glial cells using patient-specific diffusion tensor imaging (DTI) data. Nine GBM patients with preradiotherapy DTI data are considered in this study. The possible locations of tumor recurrences are determined by randomly selecting many ensembles of values for each of the growth and radiobiological parameters in the GBM growth model. A novel concept, the occurrence probability (OP), is introduced to assess the sensitivity of potential tumor recurrence locations to the input parameter values. For a given patient, the OP map is derived from a superposition of all potential tumor recurrence locations obtained with all sets of parameter values.

RESULTS

For eight out of nine of patients, the authors have identified a statistically significant region where the OP is above 50%. For two patients, these high risk regions are found to be located at a distance greater than 3.9 cm from the border of the gross tumor volume highlighting the inaccuracy of current margins for some patients. The exact location and size of these volumes with OP > 50 % are, however, sensitive to the number N of ensembles of parameter values for N ≲ 400. On the other hand, the authors have identified for each patient a threshold OP, the OP(T), which defines a volume that converges more rapidly with increasing N. The OP(T) for each patient varies between 20% and 40%. The volume defined by OP > OP(T) may be an adequate candidate to define a personalized margin for radiotherapy treatment planning of GBM patients.

CONCLUSIONS

A new Monte-Carlo based formalism was described and used to assess the variability of sites of potential recurrence predicted by the proliferation-invasion model to input parameter values. The authors have shown that high risk areas could be consistently identified with a limited number of sets (N ≲ 400) of randomly chosen parameter values. A major strength of this formalism is its potential prospective nature. Although a validation of the accuracy of the model-predicted tumor recurrence location still remains to be done, our method is potentially applicable to orient patient-specific definition of margins.

A multiscale road map of cancer spheroids – incorporating experimental and mathematical modelling to understand cancer progression

Computational models represent a highly suitable framework, not only for testing biological hypotheses and generating new ones but also for optimising experimental strategies. As one surveys the literature devoted to cancer modelling, it is obvious that immense progress has been made in applying simulation techniques to the study of cancer biology, although the full impact has yet to be realised. For example, there are excellent models to describe cancer incidence rates or factors for early disease detection, but these predictions are unable to explain the functional and molecular changes that are associated with tumour progression. In addition, it is crucial that interactions between mechanical effects, and intracellular and intercellular signalling are incorporated in order to understand cancer growth, its interaction with the extracellular microenvironment and invasion of secondary sites. There is a compelling need to tailor new, physiologically relevant in silico models that are specialised for particular types of cancer, such as ovarian cancer owing to its unique route of metastasis, which are capable of investigating anti-cancer therapies, and generating both qualitative and quantitative predictions. This Commentary will focus on how computational simulation approaches can advance our understanding of ovarian cancer progression and treatment, in particular, with the help of multicellular cancer spheroids, and thus, can inform biological hypothesis and experimental design.

Acute and fractionated irradiation differentially modulate glioma stem cell division kinetics

Glioblastoma multiforme (GBM) is one of the most aggressive human malignancies with a poor patient prognosis. Ionizing radiation either alone or adjuvant after surgery is part of standard treatment for GBM but remains primarily noncurative. The mechanisms underlying tumor radioresistance are manifold and, in part, accredited to a special subpopulation of tumorigenic cells. The so-called glioma stem cells (GSC) are bestowed with the exclusive ability to self-renew and repopulate the tumor and have been reported to be less sensitive to radiation-induced damage through preferential activation of DNA damage checkpoint responses and increased capacity for DNA damage repair. During each fraction of radiation, non-stem cancer cells (CC) die and GSCs become enriched and potentially increase in number, which may lead to accelerated repopulation. We propose a cellular Potts model that simulates the kinetics of GSCs and CCs in glioblastoma growth and radiation response. We parameterize and validate this model with experimental data of the U87-MG human glioblastoma cell line. Simulations are conducted to estimate GSC symmetric and asymmetric division rates and explore potential mechanisms for increased GSC fractions after irradiation. Simulations reveal that in addition to their higher radioresistance, a shift from asymmetric to symmetric division or a fast cycle of GSCs following fractionated radiation treatment is required to yield results that match experimental observations. We hypothesize a constitutive activation of stem cell division kinetics signaling pathways during fractionated treatment, which contributes to the frequently observed accelerated repopulation after therapeutic irradiation.

Hypoxic cell waves around necrotic cores in glioblastoma: a biomathematical model and its therapeutic implications

Glioblastoma is a rapidly evolving high-grade astrocytoma that is distinguished pathologically from lower grade gliomas by the presence of necrosis and microvascular hyperplasia. Necrotic areas are typically surrounded by hypercellular regions known as "pseudopalisades" originated by local tumor vessel occlusions that induce collective cellular migration events. This leads to the formation of waves of tumor cells actively migrating away from central hypoxia. We present a mathematical model that incorporates the interplay among two tumor cell phenotypes, a necrotic core and the oxygen distribution. Our simulations reveal the formation of a traveling wave of tumor cells that reproduces the observed histologic patterns of pseudopalisades. Additional simulations of the model equations show that preventing the collapse of tumor microvessels leads to slower glioma invasion, a fact that might be exploited for therapeutic purposes.

Adaptive IMRT using a multiobjective evolutionary algorithm integrated with a diffusion-invasion model of glioblastoma

We demonstrate a patient-specific method of adaptive IMRT treatment for glioblastoma using a multiobjective evolutionary algorithm (MOEA). The MOEA generates spatially optimized dose distributions using an iterative dialogue between the MOEA and a mathematical model of tumor cell proliferation, diffusion and response. Dose distributions optimized on a weekly basis using biological metrics have the potential to substantially improve and individualize treatment outcomes. Optimized dose distributions were generated using three different decision criteria for the tumor and compared with plans utilizing standard dose of 1.8 Gy/fraction to the CTV (T2-visible MRI region plus a 2.5 cm margin). The sets of optimal dose distributions generated using the MOEA approach the Pareto Front (the set of IMRT plans that delineate optimal tradeoffs amongst the clinical goals of tumor control and normal tissue sparing). MOEA optimized doses demonstrated superior performance as judged by three biological metrics according to simulated results. The predicted number of reproductively viable cells 12 weeks after treatment was found to be the best target objective for use in the MOEA.

Simulating radiotherapy effect in high-grade glioma by using diffusive modeling and brain atlases

Applying diffusive models for simulating the spatiotemporal change of concentration of tumour cells is a modern application of predictive oncology. Diffusive models are used for modelling glioblastoma, the most aggressive type of glioma. This paper presents the results of applying a linear quadratic model for simulating the effects of radiotherapy on an advanced diffusive glioma model. This diffusive model takes into consideration the heterogeneous velocity of glioma in gray and white matter and the anisotropic migration of tumor cells, which is facilitated along white fibers. This work uses normal brain atlases for extracting the proportions of white and gray matter and the diffusion tensors used for anisotropy. The paper also presents the results of applying this glioma model on real clinical datasets.

Biophysical modeling of brain tumor progression: from unconditionally stable explicit time integration to an inverse problem with parabolic PDE constraints for model calibration

PURPOSE

A novel unconditionally stable, explicit numerical method is introduced to the field of modeling brain cancer progression on a tissue level together with an inverse problem (IP) based on optimal control theory that allows for automated model calibration with respect to observations in clinical imaging data.

METHODS

Biophysical models of cancer progression on a tissue level are in general based on the assumption that the spatiotemporal spread of cancerous cells is determined by cell division and net migration. These processes are typically described in terms of a parabolic partial differential equation (PDE). In the present work a parallelized implementation of an unconditionally stable, explicit Euler (EE(⋆)) time integration method for the solution of this PDE is detailed. The key idea of the discussed EE(⋆) method is to relax the strong stability requirement on the spectral radius of the coefficient matrix by introducing a subdivision regime for a given outer time step. The performance is related to common implicit numerical methods. To quantify the numerical error, a simplified model that has a closed form solution is considered. To allow for a systematic, phenomenological validation a novel approach for automated model calibration on the basis of observations in medical imaging data is developed. The resulting IP is based on optimal control theory and manifests as a large scale, PDE constrained optimization problem.

RESULTS

The numerical error of the EE(⋆) method is at the order of standard implicit numerical methods. The computing times are well below those obtained for implicit methods and by that demonstrate efficiency. Qualitative and quantitative analysis in 12 patients demonstrates that the obtained results are in strong agreement with observations in medical imaging data. Rating simulation success in terms of the mean overlap between model predictions and manual expert segmentations yields a success rate of 75% (9 out of 12 patients).

CONCLUSIONS

The discussed EE(⋆) method provides desirable features for image-based model calibration or hybrid image registration algorithms in which the model serves as a biophysical prior. This is due to (i) ease of implementation, (ii) low memory requirements, (iii) efficiency, (iv) a straightforward interface for parameter updates, and (v) the fact that the method is inherently matrix-free. The explicit time integration method is confirmed via experiments for automated model calibration. Qualitative and quantitative analysis demonstrates that the proposed framework allows for recovering observations in medical imaging data and by that phenomenological model validity.

In silico modelling of tumour margin diffusion and infiltration: review of current status

As a result of advanced treatment techniques, requiring precise target definitions, a need for more accurate delineation of the Clinical Target Volume (CTV) has arisen. Mathematical modelling is found to be a powerful tool to provide fairly accurate predictions for the Microscopic Extension (ME) of a tumour to be incorporated in a CTV. In general terms, biomathematical models based on a sequence of observations or development of a hypothesis assume some links between biological mechanisms involved in cancer development and progression to provide quantitative or qualitative measures of tumour behaviour as well as tumour response to treatment. Generally, two approaches are taken: deterministic and stochastic modelling. In this paper, recent mathematical models, including deterministic and stochastic methods, are reviewed and critically compared. It is concluded that stochastic models are more promising to provide a realistic description of cancer tumour behaviour due to being intrinsically probabilistic as well as discrete, which enables incorporation of patient-specific biomedical data such as tumour heterogeneity and anatomical boundaries.

Multispecies model of cell lineages and feedback control in solid tumors

We develop a multispecies continuum model to simulate the spatiotemporal dynamics of cell lineages in solid tumors. The model accounts for protein signaling factors produced by cells in lineages, and nutrients supplied by the microenvironment. Together, these regulate the rates of proliferation, self-renewal and differentiation of cells within the lineages, and control cell population sizes and distributions. Terminally differentiated cells release proteins (e.g., from the TGFβ superfamily) that feedback upon less differentiated cells in the lineage both to promote differentiation and decrease rates of proliferation (and self-renewal). Stem cells release a short-range factor that promotes self-renewal (e.g., representative of Wnt signaling factors), as well as a long-range inhibitor of this factor (e.g., representative of Wnt inhibitors such as Dkk and SFRPs). We find that the progression of the tumors and their response to treatment is controlled by the spatiotemporal dynamics of the signaling processes. The model predicts the development of spatiotemporal heterogeneous distributions of the feedback factors (Wnt, Dkk and TGFβ) and tumor cell populations with clusters of stem cells appearing at the tumor boundary, consistent with recent experiments. The nonlinear coupling between the heterogeneous expressions of growth factors and the heterogeneous distributions of cell populations at different lineage stages tends to create asymmetry in tumor shape that may sufficiently alter otherwise homeostatic feedback so as to favor escape from growth control. This occurs in a setting of invasive fingering, and enhanced aggressiveness after standard therapeutic interventions. We find, however, that combination therapy involving differentiation promoters and radiotherapy is very effective in eradicating such a tumor.

Mathematical modeling of human glioma growth based on brain topological structures: study of two clinical cases

Gliomas are the most common primary brain tumors and yet almost incurable due mainly to their great invasion capability. This represents a challenge to present clinical oncology. Here, we introduce a mathematical model aiming to improve tumor spreading capability definition. The model consists in a time dependent reaction-diffusion equation in a three-dimensional spatial domain that distinguishes between different brain topological structures. The model uses a series of digitized images from brain slices covering the whole human brain. The Talairach atlas included in the model describes brain structures at different levels. Also, the inclusion of the Brodmann areas allows prediction of the brain functions affected during tumor evolution and the estimation of correlated symptoms. The model is solved numerically using patient-specific parametrization and finite differences. Simulations consider an initial state with cellular proliferation alone (benign tumor), and an advanced state when infiltration starts (malign tumor). Survival time is estimated on the basis of tumor size and location. The model is used to predict tumor evolution in two clinical cases. In the first case, predictions show that real infiltrative areas are underestimated by current diagnostic imaging. In the second case, tumor spreading predictions were shown to be more accurate than those derived from previous models in the literature. Our results suggest that the inclusion of differential migration in glioma growth models constitutes another step towards a better prediction of tumor infiltration at the moment of surgical or radiosurgical target definition. Also, the addition of physiological/psychological considerations to classical anatomical models will provide a better and integral understanding of the patient disease at the moment of deciding therapeutic options, taking into account not only survival but also life quality.

Glioblastoma brain tumours: estimating the time from brain tumour initiation and resolution of a patient survival anomaly after similar treatment protocols

A practical mathematical model for glioblastomas (brain tumours), which incorporates the two key parameters of tumour growth, namely the cancer cell diffusion and the cell proliferation rate, has been shown to be clinically useful and predictive. Previous studies explain why multifocal recurrence is inevitable and show how various treatment scenarios have been incorporated in the model. In most tumours, it is not known when the cancer started. Based on patient in vivo parameters, obtained from two brain scans, it is shown how to estimate the time, after initial detection, when the tumour started. This is an input of potential importance in any future controlled clinical study of any connection between cell phone radiation and brain tumour incidence. It is also used to estimate more accurately survival times from detection. Finally, based on patient parameters, the solution of the model equation of the tumour growth helps to explain why certain patients live longer than others after similar treatment protocols specifically surgical resection (removal) and irradiation.

Enhancing CHK1 inhibitor lethality in glioblastoma

The present studies were initiated to determine whether inhibitors of MEK1/2 or SRC signaling, respectively, enhance CHK1 inhibitor lethality in primary human glioblastoma cells. Multiple MEK1/2 inhibitors (CI-1040 (PD184352); AZD6244 (ARRY-142886)) interacted with multiple CHK1 inhibitors (UCN-01, AZD7762) to kill multiple primary human glioma cell isolates that have a diverse set of genetic alterations typically found in the disease. Inhibition of SRC family proteins also enhanced CHK1 inhibitor lethality. Combined treatment of glioma cells with (MEK1/2 + CHK1) inhibitors enhanced radiosensitivity. Combined (MEK1/2 + CHK1) inhibitor treatment led to dephosphorylation of ERK1/2 and S6 ribosomal protein, whereas the phosphorylation of JNK and p38 was increased. MEK1/2 + CHK1 inhibitor-stimulated cell death was associated with the cleavage of pro-caspases 3 and 7 as well as the caspase substrate (PARP). We also observed activation of pro-apoptotic BCL-2 effector proteins BAK and BAX and reduced levels of pro-survival BCL-2 family protein BCL-XL. Overexpression of BCL-XL alleviated but did not completely abolish MEK1/2 + CHK1 inhibitor cytotoxicity in GBM cells. These findings argue that multiple inhibitors of the SRC-MEK pathway have the potential to interact with multiple CHK1 inhibitors to kill glioma cells.

Increased re-entry into cell cycle mitigates age-related neurogenic decline in the murine subventricular zone

Although new neurons are produced in the subventricular zone (SVZ) of the adult mammalian brain, fewer functional neurons are produced with increasing age. The age-related decline in neurogenesis has been attributed to a decreased pool of neural progenitor cells (NPCs), an increased rate of cell death, and an inability to undergo neuronal differentiation and develop functional synapses. The time between mitotic events has also been hypothesized to increase with age, but this has not been directly investigated. Studying primary-cultured NPCs from the young adult and aged mouse forebrain, we observe that fewer aged cells are dividing at a given time; however, the mitotic cells in aged cultures divide more frequently than mitotic cells in young cultures during a 48-hour period of live-cell time-lapse imaging. Double-thymidine-analog labeling also demonstrates that fewer aged cells are dividing at a given time, but those that do divide are significantly more likely to re-enter the cell cycle within a day, both in vitro and in vivo. Meanwhile, we observed that cellular survival is impaired in aged cultures. Using our live-cell imaging data, we developed a mathematical model describing cell cycle kinetics to predict the growth curves of cells over time in vitro and the labeling index over time in vivo. Together, these data surprisingly suggest that progenitor cells remaining in the aged SVZ are highly proliferative.

Integrating Imaging Data into Predictive Biomathematical and Biophysical Models of Cancer

While there is a mature literature on biomathematical and biophysical modeling in cancer, many of the existing approaches are not of clinical utility, as they require input data that are extremely difficult to obtain in an intact organism, and/or require a large number of assumptions on the free parameters included in the models. Thus, there has only been very limited application of such models to solve problems of clinical import. More recently, however, there has been increased activity at the interface of quantitative, noninvasive imaging data, and tumor mathematical modeling. In addition to reporting on bulk tumor morphology and volume, emerging imaging techniques can quantitatively report on for example tumor vascularity, glucose metabolism, cell density and proliferation, and hypoxia. In this paper, we first motivate the problem of predicting therapy response by highlighting some (acknowledged) shortcomings in existing methods. We then provide introductions to a number of representative quantitative imaging methods and describe how they are currently (and potentially can be) used to initialize and constrain patient specific mathematical and biophysical models of tumor growth and treatment response, thereby increasing the clinical utility of such approaches. We conclude by highlighting some of the exciting research directions when one integrates quantitative imaging and tumor modeling.

Improving the time-machine: estimating date of birth of grade II gliomas

OBJECTIVES

Here we present a model aiming to provide an estimate of time from tumour genesis, for grade II gliomas. The model is based on a differential equation describing the diffusion-proliferation process. We have applied our model to situations where tumour diameter was shown to increase linearly with time, with characteristic diametric velocity.

MATERIALS AND METHODS

We have performed numerical simulations to analyse data, on patients with grade II gliomas and to extract information concerning time of tumour biological onset, as well as radiology and distribution of model parameters.

RESULTS AND CONCLUSIONS

We show that the estimate of tumour onset obtained from extrapolation using a constant velocity assumption, always underestimates biological tumour age, and that the correction one should add to this estimate is given roughly by 20/v (year), where v is the diametric velocity of expansion of the tumour (expressed in mm/year). Within the assumptions of the model, we have identified two types of tumour: the first corresponds to very slowly growing tumours that appear during adolescence, and the second type corresponds to slowly growing tumours that appear later, during early adulthood. That all these tumours become detectable around a mean patient age of 30 years could be interesting for formulation of strategies for early detection of tumours.

High-grade glioma diffusive modeling using statistical tissue information and diffusion tensors extracted from atlases

Glioma, especially glioblastoma, is a leading cause of brain cancer fatality involving highly invasive and neoplastic growth. Diffusive models of glioma growth use variations of the diffusion-reaction equation in order to simulate the invasive patterns of glioma cells by approximating the spatiotemporal change of glioma cell concentration. The most advanced diffusive models take into consideration the heterogeneous velocity of glioma in gray and white matter, by using two different discrete diffusion coefficients in these areas. Moreover, by using diffusion tensor imaging (DTI), they simulate the anisotropic migration of glioma cells, which is facilitated along white fibers, assuming diffusion tensors with different diffusion coefficients along each candidate direction of growth. Our study extends this concept by fully exploiting the proportions of white and gray matter extracted by normal brain atlases, rather than discretizing diffusion coefficients. Moreover, the proportions of white and gray matter, as well as the diffusion tensors, are extracted by the respective atlases; thus, no DTI processing is needed. Finally, we applied this novel glioma growth model on real data and the results indicate that prognostication rates can be improved.

Quantifying the role of angiogenesis in malignant progression of gliomas: in silico modeling integrates imaging and histology

Gliomas are uniformly fatal forms of primary brain neoplasms that vary from low- to high-grade (glioblastoma). Whereas low-grade gliomas are weakly angiogenic, glioblastomas are among the most angiogenic tumors. Thus, interactions between glioma cells and their tissue microenvironment may play an important role in aggressive tumor formation and progression. To quantitatively explore how tumor cells interact with their tissue microenvironment, we incorporated the interactions of normoxic glioma cells, hypoxic glioma cells, vascular endothelial cells, diffusible angiogenic factors, and necrosis formation into a first-generation, biologically based mathematical model for glioma growth and invasion. Model simulations quantitatively described the spectrum of in vivo dynamics of gliomas visualized with medical imaging. Furthermore, we investigated how proliferation and dispersal of glioma cells combine to induce increasing degrees of cellularity, mitoses, hypoxia-induced neoangiogenesis and necrosis, features that characterize increasing degrees of "malignancy," and we found that changes in the net rates of proliferation (ρ) and invasion (D) are not always necessary for malignant progression. Thus, although other factors, including the accumulation of genetic mutations, can change cellular phenotype (e.g., proliferation and invasion rates), this study suggests that these are not required for malignant progression. Simulated results are placed in the context of the current clinical World Health Organization grading scheme for studying specific patient examples. This study suggests that through the application of the proposed model for tumor-microenvironment interactions, predictable patterns of dynamic changes in glioma histology distinct from changes in cellular phenotype (e.g., proliferation and invasion rates) may be identified, thus providing a powerful clinical tool.

Bright solitary waves in malignant gliomas

We put forward a nonlinear wave model describing the fundamental dynamical features of an aggressive type of brain tumors. Our model accounts for the invasion of normal tissue by a proliferating and propagating rim of active glioma cancer cells in the tumor boundary and the subsequent formation of a necrotic core. By resorting to numerical simulations, phase space analysis, and exact solutions we prove that bright solitary tumor waves develop in such systems. Possible implications of our model as a tool to extract relevant patient specific tumor parameters in combination with standard preoperative clinical imaging are also discussed.