Biocomputing: Numerical simulation of glioblastoma growth and comparison with conventional irradiation margins

Toward image-based personalization of glioblastoma therapy: A clinical and biological validation study of a novel, deep learning-driven tumor growth model

Background

The diffuse growth pattern of glioblastoma is one of the main challenges for accurate treatment. Computational tumor growth modeling has emerged as a promising tool to guide personalized therapy. Here, we performed clinical and biological validation of a novel growth model, aiming to close the gap between the experimental state and clinical implementation.

Methods

One hundred and twenty-four patients from The Cancer Genome Archive (TCGA) and 397 patients from the UCSF Glioma Dataset were assessed for significant correlations between clinical data, genetic pathway activation maps (generated with PARADIGM; TCGA only), and infiltration (Dw) as well as proliferation (ρ) parameters stemming from a Fisher-Kolmogorov growth model. To further evaluate clinical potential, we performed the same growth modeling on preoperative magnetic resonance imaging data from 30 patients of our institution and compared model-derived tumor volume and recurrence coverage with standard radiotherapy plans.

Results

The parameter ratio Dw/ρ (P < .05 in TCGA) as well as the simulated tumor volume (P < .05 in TCGA/UCSF) were significantly inversely correlated with overall survival. Interestingly, we found a significant correlation between 11 proliferation pathways and the estimated proliferation parameter. Depending on the cutoff value for tumor cell density, we observed a significant improvement in recurrence coverage without significantly increased radiation volume utilizing model-derived target volumes instead of standard radiation plans.

Conclusions

Identifying a significant correlation between computed growth parameters and clinical and biological data, we highlight the potential of tumor growth modeling for individualized therapy of glioblastoma. This might improve the accuracy of radiation planning in the near future.

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

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

On PTV definition for glioblastoma based on fiber tracking of diffusion tensor imaging data

Radiotherapy (RT) is commonly applied for the treatment of glioblastoma multiforme (GBM). Following the planning target volume (PTV) definition procedure standardized in guidelines, a 20% risk of missing non-local recurrences is present. Purpose of this study was to evaluate whether diffusion tensor imaging (DTI)-based fiber tracking may be beneficial for PTV definition taking into account the prediction of distant recurrences. 56 GBM patients were examined with magnetic resonance imaging (MRI) including DTI performed before RT after resection of the primary tumor. Follow-up MRIs were acquired in three month intervals. For the seven patients with a distant recurrence, fiber tracking was performed with three algorithms and it was evaluated whether connections existed from the primary tumor region to the distant recurrence. It depended strongly on the used tracking algorithm and the used tracking parameters whether a connection was observed. Most of the connections were weak and thus not usable for PTV definition. Only in one of the seven patients with a recurring tumor, a clear connection was present. It seems unlikely that DTI-based fiber tracking can be beneficial for predicting distant recurrences in the planning of PTVs for glioblastoma multiforme.

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.

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.

[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.

Diffusion tensor magnetic resonance imaging driven growth modeling for radiotherapy target definition in glioblastoma

BACKGROUND

The clinical target volume (CTV) in radiotherapy is routinely based on gadolinium contrast enhanced T1 weighted (T1w + Gd) and T2 weighted fluid attenuated inversion recovery (T2w FLAIR) magnetic resonance imaging (MRI) sequences which have been shown to over- or underestimate the microscopic tumor cell spread. Gliomas favor spread along the white matter fiber tracts. Tumor growth models incorporating the MRI diffusion tensors (DTI) allow to account more consistently for the glioma growth. The aim of the study was to investigate the potential of a DTI driven growth model to improve target definition in glioblastoma (GBM).

MATERIAL AND METHODS

Eleven GBM patients were scanned using T1w, T2w FLAIR, T1w + Gd and DTI. The brain was segmented into white matter, gray matter and cerebrospinal fluid. The Fisher-Kolmogorov growth model was used assuming uniform proliferation and a difference in white and gray matter diffusion of a ratio of 10. The tensor directionality was tested using an anisotropy weighting parameter set to zero (γ0) and twenty (γ20). The volumetric comparison was performed using Hausdorff distance, Dice similarity coefficient (DSC) and surface area.

RESULTS

The median of the standard CTV (CTVstandard) was 180 cm³. The median surface area of CTVstandard was 211 cm². The median surface area of respective CTV_γ0 and CTV_γ20 significantly increased to 338 and 376 cm², respectively. The Hausdorff distance was greater than zero and significantly increased for both CTV_γ0 and CTV_γ20 with respective median of 18.7 and 25.2 mm. The DSC for both CTV_γ0 and CTV_γ20 were significantly below one with respective median of 0.74 and 0.72, which means that 74 and 72% of CTVstandard were included in CTV_γ0 and CTV_γ20, respectively.

CONCLUSIONS

DTI driven growth models result in CTVs with a significantly increased surface area, a significantly increased Hausdorff distance and decreased overlap between the standard and model derived volume.

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.

Interhemispheric Difference Images from Postoperative Diffusion Tensor Imaging of Gliomas

Introduction

Determining the full extent of gliomas during radiotherapy planning can be challenging with conventional T1 and T2 magnetic resonance imaging (MRI). The purpose of this study was to develop a method to automatically calculate differences in the fractional anisotropy (FA) and mean diffusivity (MD) values in target volumes obtained with diffusion tensor imaging (DTI) by comparing with values from anatomically homologous voxels on the contralateral side of the brain.

Methods

Seven patients with a histologically confirmed glioma underwent postoperative radiotherapy planning with 1.5 T MRI and computed tomography. DTI was acquired using echo planar imaging for 20 noncolinear directions with b = 1000 s/mm² and one additional image with b = 0, repeated four times for signal averaging. The distribution of FA and MD was calculated in the gross tumor volume (GTV), shells 0-5 mm, 5-10 mm, 10-15 mm, 15-20 mm, and 20-25 mm outside the GTV, and the GTV mirrored in the left-right direction (mirGTV). All images were aligned to a template image, and FA and MD interhemispheric difference images were calculated. The difference in mean FA and MD between the regions of interest was statistically tested using two-sided paired t-tests with α = 0.05.

Results

The mean FA in mirGTV was 0.20 ± 0.04, which was larger than the FA in the GTV (0.12 ± 0.03) and shells 0-5 mm (0.15 ± 0.03) and 5-10 mm (0.17 ± 0.03) outside the GTV. The mean MD (×10^-3 mm²/s) in mirGTV was 0.93 ± 0.09, which was smaller than the MD in the GTV (1.48 ± 0.19) and the peritumoral shells. The distribution of FA and MD interhemispheric differences followed the same trends as FA and MD values.

Conclusions

This study successfully implemented a method for calculation of FA and MD differences by comparison of voxel values with anatomically homologous voxels on the contralateral side of the brain. Further research is warranted to determine if radiotherapy planning using these images can be used to improve target delineation.

Diffusion and perfusion weighted magnetic resonance imaging for tumor volume definition in radiotherapy of brain tumors

Accurate target volume delineation is crucial for the radiotherapy of tumors. Diffusion and perfusion magnetic resonance imaging (MRI) can provide functional information about brain tumors, and they are able to detect tumor volume and physiological changes beyond the lesions shown on conventional MRI. This review examines recent studies that utilized diffusion and perfusion MRI for tumor volume definition in radiotherapy of brain tumors, and it presents the opportunities and challenges in the integration of multimodal functional MRI into clinical practice. The results indicate that specialized and robust post-processing algorithms and tools are needed for the precise alignment of targets on the images, and comprehensive validations with more clinical data are important for the improvement of the correlation between histopathologic results and MRI parameter images.

Modelling the radiotherapy effect in the reaction-diffusion equation

PURPOSE

In recent years, the reaction-diffusion (Fisher-Kolmogorov) equation has received much attention from the oncology research community due to its ability to describe the infiltrating nature of glioblastoma multiforme and its extraordinary resistance to any type of therapy. However, in a number of previous papers in the literature on applications of this equation, the term (R) expressing the 'External Radiotherapy effect' was incorrectly derived. In this note we derive an analytical expression for this term in the correct form to be included in the reaction-diffusion equation.

METHODS

The R term has been derived starting from the Linear-Quadratic theory of cell killing by ionizing radiation. The correct definition of R was adopted and the basic principles of differential calculus applied.

RESULTS

The compatibility of the R term derived here with the reaction-diffusion equation was demonstrated. Referring to a typical glioblastoma tumour, we have compared the results obtained using our expression for the R term with the 'incorrect' expression proposed by other authors.

Individualized radiotherapy by combining high-end irradiation and magnetic resonance imaging

Image-guided radiotherapy (IGRT) has been integrated into daily clinical routine and can today be considered the standard especially with high-dose radiotherapy. Currently imaging is based on MV- or kV-CT, which has clear limitations especially in soft-tissue contrast. Thus, combination of magnetic resonance (MR) imaging and high-end radiotherapy opens a new horizon. The intricate technical properties of MR imagers pose a challenge to technology when combined with radiation technology. Several solutions that are almost ready for routine clinical application have been developed. The clinical questions include dose-escalation strategies, monitoring of changes during treatment as well as imaging without additional radiation exposure during treatment.

Evaluation of current clinical target volume definitions for glioblastoma using cell-based dosimetry stochastic methods

OBJECTIVE

Determination of an optimal clinical target volume (CTV) is complex and remains uncertain. The aim of this study was to develop a glioblastoma multiforme (GBM) model to be used for evaluation of current CTV practices for external radiotherapy.

METHODS

The GBM model was structured as follows: (1) a Geant4 cellular model was developed to calculate the absorbed dose in individual cells represented by cubic voxels of 20 μm sides. The system was irradiated with opposing 6 MV X-ray beams. The beams encompassed planning target volumes corresponding to 2.0- and 2.5-cm CTV margins; (2) microscopic extension probability (MEP) models were developed using MATLAB(®) 2012a (MathWorks(®), Natick, MA), based on clinical studies reporting on GBM clonogenic spread; (3) the cellular dose distribution was convolved with the MEP models to evaluate cellular survival fractions (SFs) for both CTV margins.

RESULTS

A CTV margin of 2.5 cm, compared to a 2.0-cm CTV margin, resulted in a reduced total SF from 12.9% ± 0.9% to 3.6% ± 0.2%, 5.5% ± 0.4% to 1.2% ± 0.1% and 11.1% ± 0.7% to 3.0% ± 0.2% for circular, elliptical and irregular MEP distributions, respectively.

CONCLUSION

A Monte Carlo model was developed to quantitatively evaluate the impact of GBM CTV margins on total and penumbral SF. The results suggest that the reduction in total SF ranges from 3.5 to 5, when the CTV is extended by 0.5 cm.

ADVANCES IN KNOWLEDGE

The model provides a quantitative tool for evaluation of different CTV margins in terms of cell kill efficacy. Cellular platform of the tool allows future incorporation of cellular properties of GBM.

Optimal radiation fractionation for low-grade gliomas: Insights from a mathematical model

We study optimal radiotherapy fractionations for low-grade glioma using mathematical models. Both space-independent and space-dependent models are studied. Two different optimization criteria have been developed, the first one accounting for the global effect of the tumor mass on the disease symptoms and the second one related to the delay of the malignant transformation of the tumor. The models are studied theoretically and numerically using the method of feasible directions. We have searched for optimal distributions of the daily doses dj in the standard protocol of 30 fractions using both models and the two different optimization criteria. The optimal results found in all cases are minor deviations from the standard protocol and provide only marginal potential gains. Thus, our results support the optimality of current radiation fractionations over the standard 6 week treatment period. This is also in agreement with the observation that minor variations of the fractionation have failed to provide measurable gains in survival or progression free survival, pointing out to a certain optimality of the current approach.

Extreme protraction for low-grade gliomas: theoretical proof of concept of a novel therapeutical strategy

Grade II gliomas are slowly growing primary brain tumours that affect mostly young patients and become fatal after a variable time period. Current clinical handling includes surgery as first-line treatment. Cytotoxic therapies (radiotherapy RT or chemotherapy QT) are used initially only for patients having a bad prognosis. Therapies are administered following the 'maximum dose in minimum time' principle, which is the same schedule used for high-grade brain tumours. Using mathematical models describing the growth of these tumours in response to radiotherapy, we find that an extreme protraction therapeutical strategy, i.e. enlarging substantially the time interval between RT fractions, may lead to better tumour control. Explicit formulas are found providing the optimal spacing between doses in a very good agreement with the simulations of the full 3D mathematical model approximating the tumour spatiotemporal dynamics. This idea, although breaking the well-established paradigm, has biological meaning since, in these slowly growing tumours, it may be more favourable to treat the tumour as the tumour cells leave the quiescent compartment and move into the cell cycle.

Expert-validated CSF segmentation of MNI atlas enhances accuracy of virtual glioma growth patterns

Biomathematical modeling of glioma growth has been developed to optimize treatments delivery and to evaluate their efficacy. Simulations currently make use of anatomical knowledge from standard MRI atlases. For example, cerebrospinal fluid (CSF) spaces are obtained by automatic thresholding of the MNI atlas, leading to an approximate representation of real anatomy. To correct such inaccuracies, an expert-revised CSF segmentation map of the MNI atlas was built. Several virtual glioma growth patterns of different locations were generated, with and without using the expert-revised version of the MNI atlas. The adequacy between virtual and radiologically observed growth patterns was clearly higher when simulations were based on the expert-revised atlas. This work emphasizes the need for close collaboration between clinicians and researchers in the field of brain tumor modeling.

Radiotherapy planning for glioblastoma based on a tumor growth model: improving target volume delineation

Glioblastoma differ from many other tumors in the sense that they grow infiltratively into the brain tissue instead of forming a solid tumor mass with a defined boundary. Only the part of the tumor with high tumor cell density can be localized through imaging directly. In contrast, brain tissue infiltrated by tumor cells at low density appears normal on current imaging modalities. In current clinical practice, a uniform margin, typically two centimeters, is applied to account for microscopic spread of disease that is not directly assessable through imaging. The current treatment planning procedure can potentially be improved by accounting for the anisotropy of tumor growth, which arises from different factors: anatomical barriers such as the falx cerebri represent boundaries for migrating tumor cells. In addition, tumor cells primarily spread in white matter and infiltrate gray matter at lower rate. We investigate the use of a phenomenological tumor growth model for treatment planning. The model is based on the Fisher-Kolmogorov equation, which formalizes these growth characteristics and estimates the spatial distribution of tumor cells in normal appearing regions of the brain. The target volume for radiotherapy planning can be defined as an isoline of the simulated tumor cell density. This paper analyzes the model with respect to implications for target volume definition and identifies its most critical components. A retrospective study involving ten glioblastoma patients treated at our institution has been performed. To illustrate the main findings of the study, a detailed case study is presented for a glioblastoma located close to the falx. In this situation, the falx represents a boundary for migrating tumor cells, whereas the corpus callosum provides a route for the tumor to spread to the contralateral hemisphere. We further discuss the sensitivity of the model with respect to the input parameters. Correct segmentation of the brain appears to be the most crucial model input. We conclude that the tumor growth model provides a method to account for anisotropic growth patterns of glioma, and may therefore provide a tool to make target delineation more objective and automated.

High-grade glioma radiation therapy target volumes and patterns of failure obtained from magnetic resonance imaging and 18F-FDOPA positron emission tomography delineations from multiple observers

PURPOSE

The objective of this study was to compare recurrent tumor locations after radiation therapy with pretreatment delineations of high-grade gliomas from magnetic resonance imaging (MRI) and 3,4-dihydroxy-6-[(18)F]fluoro-L-phenylalanine ((18)F-FDOPA) positron emission tomography (PET) using contours delineated by multiple observers.

METHODS AND MATERIALS

Nineteen patients with newly diagnosed high-grade gliomas underwent computed tomography (CT), gadolinium contrast-enhanced MRI, and (18)F-FDOPA PET/CT. The image sets (CT, MRI, and PET/CT) were registered, and 5 observers contoured gross tumor volumes (GTVs) using MRI and PET. Consensus contours were obtained by simultaneous truth and performance level estimation (STAPLE). Interobserver variability was quantified by the percentage of volume overlap. Recurrent tumor locations after radiation therapy were contoured by each observer using CT or MRI. Consensus recurrence contours were obtained with STAPLE.

RESULTS

The mean interobserver volume overlap for PET GTVs (42% ± 22%) and MRI GTVs (41% ± 22%) was not significantly different (P=.67). The mean consensus volume was significantly larger for PET GTVs (58.6 ± 52.4 cm(3)) than for MRI GTVs (30.8 ± 26.0 cm(3), P=.003). More than 95% of the consensus recurrence volume was within the 95% isodose surface for 11 of 12 (92%) cases with recurrent tumor imaging. Ten (91%) of these cases extended beyond the PET GTV, and 9 (82%) were contained within a 2-cm margin on the MRI GTV. One recurrence (8%) was located outside the 95% isodose surface.

CONCLUSIONS

High-grade glioma contours obtained with (18)F-FDOPA PET had similar interobserver agreement to volumes obtained with MRI. Although PET-based consensus target volumes were larger than MRI-based volumes, treatment planning using PET-based volumes may not have yielded better treatment outcomes, given that all but 1 recurrence extended beyond the PET GTV and most were contained by a 2-cm margin on the MRI GTV.

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.