Mechanically Coupled Reaction-Diffusion Model to Predict Glioma Growth: Methodological Details

Hormuth, David A.; Eldridge, Stephanie L.; Weis, Jared A.; Miga, Michael I.; Yankeelov, Thomas E.

doi:10.1007/978-1-4939-7493-1_11

Cited by 35 publications

(35 citation statements)

References 37 publications

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“…For example, the diffusion coefficient of cancer cells (i.e., the parameter “ D ” in Eq. (8)) can be correlated with the local von Mises stress by solving an elasticity boundary problem between the tumor and host tissue [13, 53, 54]. Hormuth et al assumed that solid stress would primarily affect the motility of tumor cells rather than the proliferation of cells [13].…”

Section: Mathematical Modeling Of Proliferation and Therapy Across Scmentioning

confidence: 99%

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

Jarrett

Lima

Hormuth

et al. 2018

Expert Review of Anticancer Therapy

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107

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

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Section: Mathematical Modeling Of Proliferation and Therapy Across Scmentioning

confidence: 99%

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

Jarrett

Lima

Hormuth

et al. 2018

Expert Review of Anticancer Therapy

Self Cite

107

View full text Add to dashboard Cite

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“…where N T (x -, t) is the tumor cell fraction at three-dimensional position xand time t, θ T is the tumor cell carrying capacity (i.e., the maximum packing fraction that a voxel can functionally support), and k p,T is the tumor cell proliferation rate. D T (x -, t) changes spatially and temporally as a function of the local tissue stress 8,14,15 In vitro experiments have demonstrated that tumor expansion is restricted as local mechanical stresses increase 11 . Thus, it is natural to relate D T (x -, t) to stress as shown in Eq.…”

Section: Mechanically Coupled Reaction Diffusion Model Of Tumor Growth-mentioning

confidence: 99%

“…where G is the shear modulus, ν is the Poisson's Ratio, and λ is second coupling constant. Literature values are used to assign G and v for different tissue regions within the brain (described in detail in 15 ) as we assume that these tissue properties will not change dramatically from animal to animal (we return to this important point in the Discussions section). Eqs.…”

Section: Mechanically Coupled Reaction Diffusion Model Of Tumor Growth-mentioning

confidence: 99%

“…Fourth, if the stopping criteria are not met, the model's parameters are updated based on the Levenberg-Marquardt algorithm, and the process returns to the first step. We used a hybrid, simulatedannealing Levenberg-Marquardt algorithm, which works identically to a standard Levenberg-Marquardt approach 15 , with the exception of a stochastic component to determine if the change in model parameters (and thus change in objective function) is acceptable. All of the model parameters were calibrated for a global value (spatially uniform) unless otherwise indicated in Table 2, in which case a locally varying parameter was estimated.…”

Section: Model Parameter Calibration-mentioning

confidence: 99%

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Calibrating a Predictive Model of Tumor Growth and Angiogenesis with Quantitative MRI

et al. 2019

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The spatiotemporal variations in tumor vasculature inevitably alters cell proliferation and treatment efficacy. Thus, rigorous characterization of tumor dynamics must include a description of this phenomenon. We have developed a family of biophysical models of tumor growth and angiogenesis that are calibrated with diffusion-weighted magnetic resonance imaging (DW-MRI) and dynamic contrast-enhanced (DCE-) MRI data to provide individualized tumor growth forecasts. Tumor and blood volume fractions were evolved using two, coupled partial differential equations consisting of proliferation, diffusion, and death terms. To evaluate these models, rats (n=8) with C6 gliomas were imaged seven times. The tumor volume fraction was estimated using DW-MRI, while DCE-MRI provided estimates of the blood volume fraction. The first three time points were used to calibrate model parameters, which were then used to predict growth at the remaining four time points and compared directly to the measurements. The best performing model predicted tumor growth with less than 10.3% error in tumor volume and with less than 9.4% error at the voxel-level at all prediction time points. The best performing model resulted in less than 9.3% error in blood volume at the voxel-level. This pre-clinical study demonstrates the potential for image-based, mechanistic modeling of tumor growth and angiogenesis.

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“…And the third one is the solution algorithm.Regarding the underlying model, like us, most researchers focus on parameter calibration of a handful of model parameters using single-species reaction-diffusion equations [7,22,24,28,33,39,51,58,59]. While more complex models describing processes like mass effect, angiogenesis and chemotaxis [23,26,48,54,60] exist, they have not been considered for calibration due to theoretical and computational challenges. However, several groups, including ours, are working to address these challenges.Regarding the inverse problem setup, in most studies the authors use scans from two or more time points and assume that the tumor concentration is fully observed.…”

mentioning

confidence: 99%

Where did the tumor start? An inverse solver with sparse localization for tumor growth models

et al. 2020

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We present a numerical scheme for solving an inverse problem for parameter estimation in tumor growth models for glioblastomas, a form of aggressive primary brain tumor. The growth model is a reaction-diffusion partial differential equation (PDE) for the tumor concentration. We use a PDE-constrained optimization formulation for the inverse problem. The unknown parameters are the reaction coefficient (proliferation), the diffusion coefficient (infiltration), and the initial condition field for the tumor PDE. Segmentation of Magnetic Resonance Imaging (MRI) scans drive the inverse problem where segmented tumor regions serve as partial observations of the tumor concentration. Like most cases in clinical practice, we use data from a single time snapshot. Moreover, the precise time relative to the initiation of the tumor is unknown, which poses an additional difficulty for inversion. We perform a frozen-coefficient spectral analysis and show that the inverse problem is severely ill-posed. We introduce a biophysically motivated regularization on the structure and magnitude of the tumor initial condition. In particular, we assume that the tumor starts at a few locations (enforced with a sparsity constraint on the initial condition of the tumor) and that the initial condition magnitude in the maximum norm is equal to one. We solve the resulting optimization problem using an inexact quasi-Newton method combined with a compressive sampling algorithm for the sparsity constraint. Our implementation uses PETSc and AccFFT libraries. We conduct numerical experiments on synthetic and clinical images to highlight the improved performance of our solver over a previously existing solver that uses standard two-norm regularization for the calibration parameters. The existing solver is unable to localize the initial condition. Our new solver can localize the initial condition and recover infiltration and proliferation. In clinical datasets (for which the ground truth is unknown), our solver results in qualitatively different solutions compared to the two-norm regularized solver. Related work.Although there has been a lot of work on forward problems for tumor growth, there has been less work on inverse problems. The latter has different aspects. The first is the underlying biophysical model. The second is the inverse problem setup, observation operators and the existence of scans at multiple points, the noise models, inversion parameters and constraints. And the third one is the solution algorithm.Regarding the underlying model, like us, most researchers focus on parameter calibration of a handful of model parameters using single-species reaction-diffusion equations [7,22,24,28,33,39,51,58,59]. While more complex models describing processes like mass effect, angiogenesis and chemotaxis [23,26,48,54,60] exist, they have not been considered for calibration due to theoretical and computational challenges. However, several groups, including ours, are working to address these challenges.Regarding the inverse problem setup, in most studies t...

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Mechanically Coupled Reaction-Diffusion Model to Predict Glioma Growth: Methodological Details

Cited by 35 publications

References 37 publications

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

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

Calibrating a Predictive Model of Tumor Growth and Angiogenesis with Quantitative MRI

Where did the tumor start? An inverse solver with sparse localization for tumor growth models

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