Mathematical Analysis of Glioma Growth in a Murine Model

Role of modeled high-grade glioma cell invasion and survival on the prediction of tumor progression after radiotherapy

Objective.Glioblastoma (GBM) prognosis remains poor despite progress in radiotherapy and imaging techniques. Tumor recurrence has been attributed to the widespread tumor invasion of normal tissue. Since the complete extension of invasion is undetectable on imaging, it is not deliberately treated. To improve the treatment outcome, models have been developed to predict tumor invasion based standard imaging data. This study aimed to investigate whether a tumor invasion model, together with the predicted number of surviving cells after radiotherapy, could predict tumor progression post-treatment.Approach.A tumor invasion model was applied to 56 cases of GBMs treated with radiotherapy. The invasion was quantified as the volume encompassed by the 100 cells mm-3isocontour (V100). A new metric, cell-volume-product, was defined as the product of the volume with cell density greater than a threshold value (in cells mm-3), and the number of surviving cells within that volume, post-treatment. Tumor progression was assessed at 20 ± 10 d and 90 ± 20 d after treatment. Correlations between the disease progression and the gross tumor volume (GTV),V100, and cell-volume-product, were determined using receiver operating characteristic curves.Main results.For the early follow-up time, the correlation between GTV and tumor progression was not statistically significant (p= 0.684). However, statistically significant correlations with progression were found betweenV100and cell-volume-product with a cell threshold of 10-6cells mm-3with areas-under-the-curve of 0.69 (p= 0.023) and 0.66 (p= 0.045), respectively. No significant correlations were found for the late follow-up time.Significance.Modeling tumor spread otherwise undetectable on conventional imaging, as well as radiobiological model predictions of cell survival after treatment, may provide useful information regarding the likelihood of tumor progression at an early follow-up time point, which could potentially lead to improved treatment decisions for patients with GBMs.

Mathematical modeling of multicellular tumor spheroids quantifies inter-patient and intra-tumor heterogeneity

In the study of brain tumors, patient-derived three-dimensional sphere cultures provide an important tool for studying emerging treatments. The growth of such spheroids depends on the combined effects of proliferation and migration of cells, but it is challenging to make accurate distinctions between increase in cell number versus the radial movement of cells. To address this, we formulate a novel model in the form of a system of two partial differential equations (PDEs) incorporating both migration and growth terms, and show that it more accurately fits our data compared to simpler PDE models. We show that traveling-wave speeds are strongly associated with population heterogeneity. Having fitted the model to our dataset we show that a subset of the cell lines are best described by a "Go-or-Grow"-type model, which constitutes a special case of our model. Finally, we investigate whether our fitted model parameters are correlated with patient age and survival.

Heterogeneous biomechanical/mathematical modeling of spatial prediction of glioblastoma progression using magnetic resonance imaging-based finite element method

BACKGROUND AND OBJECTIVE

Brain tumors are one of the most common diseases and causes of death in humans. Since the growth of brain tumors has irreparable risks for the patient, predicting the growth of the tumor and knowing its effect on the brain tissue will increase the efficiency of treatment strategies.

METHODS

This study examines brain tumor growth using mathematical modeling based on the Reaction-Diffusion equation and the biomechanical model based on continuum mechanics principles. With the help of the image threshold technique of magnetic resonance images, a heterogeneous and close-to-reality environment of the brain has been modeled and experimental data validated the results to achieve maximum accuracy in predicting growth.

RESULTS

The obtained results have been compared with the reported conventional models to evaluate the presented model. In addition to incorporating the chemotherapy effects in governing equations, the real-time finite element analysis of the stress tensors of the surrounding tissue of tumor cells and considering its role in changing the shape and growth of the tumor has added to the importance and accuracy of the current model.

CONCLUSIONS

The comparison of the obtained results with conventional models shows that the heterogeneous model has higher reliability due to the consideration of the appropriate properties for the different regions of the brain. The presented model can contribute to personalized medicine, aid in understanding the dynamics of tumor growth, optimize treatment regimens, and develop adaptive therapy strategies.

The self-organized structure of glioma oncostreams and the disruptive role of passive cells

Oncostreams are self-organized structures formed by spindle-like, elongated, self-propelled cells recently described in glioblastomas and especially in gliosarcomas. Cells within these structures either move as large clusters in one main direction, flocks, or as linear, intermingling collections of cells advancing in opposite directions, streams. Round, passive cells are also observed, either inside or segregated from the oncostreams. Here we generalize a recently formulated particle-field approach to investigate the genesis and evolution of these structures, first showing that, in systems consisting only of identical self-propelled cells, both flocks and streams emerge as self-organized dynamic configurations. Flocks are the more stable configurations, while streams are transient and usually originate in collisions between flocks. Stream degradation is easier at low self-propulsion speeds. In systems consisting of both motile and passive cells, the latter block stream formation and accelerate their degradation and flock stabilization. Since the flock appears to be the most effective invasive structure, we thus argue that a phenotype mixture (motile and passive cells) may favor glioblastoma invasion. hlBy relating cellular properties to the observed outcome, our model shows that oncostreams are self-organized structures that result from the interplay between speed, shape, and steric repulsion.

Calibrating tumor growth and invasion parameters with spectral spatial analysis of cancer biopsy tissues

The reaction-diffusion equation is widely used in mathematical models of cancer. The calibration of model parameters based on limited clinical data is critical to using reaction-diffusion equation simulations for reliable predictions on a per-patient basis. Here, we focus on cell-level data as routinely available from tissue biopsies used for clinical cancer diagnosis. We analyze the spatial architecture in biopsy tissues stained with multiplex immunofluorescence. We derive a two-point correlation function and the corresponding spatial power spectral distribution. We show that this data-deduced power spectral distribution can fit the power spectrum of the solution of reaction-diffusion equations that can then identify patient-specific tumor growth and invasion rates. This approach allows the measurement of patient-specific critical tumor dynamical properties from routinely available biopsy material at a single snapshot in time.

Global stability and parameter analysis reinforce therapeutic targets of PD-L1-PD-1 and MDSCs for glioblastoma

Glioblastoma (GBM) is an aggressive primary brain cancer that currently has minimally effective treatments. Like other cancers, immunosuppression by the PD-L1-PD-1 immune checkpoint complex is a prominent axis by which glioma cells evade the immune system. Myeloid-derived suppressor cells (MDSCs), which are recruited to the glioma microenviroment, also contribute to the immunosuppressed GBM microenvironment by suppressing T cell functions. In this paper, we propose a GBM-specific tumor-immune ordinary differential equations model of glioma cells, T cells, and MDSCs to provide theoretical insights into the interactions between these cells. Equilibrium and stability analysis indicates that there are unique tumorous and tumor-free equilibria which are locally stable under certain conditions. Further, the tumor-free equilibrium is globally stable when T cell activation and the tumor kill rate by T cells overcome tumor growth, T cell inhibition by PD-L1-PD-1 and MDSCs, and the T cell death rate. Bifurcation analysis suggests that a treatment plan that includes surgical resection and therapeutics targeting immune suppression caused by the PD-L1-PD1 complex and MDSCs results in the system tending to the tumor-free equilibrium. Using a set of preclinical experimental data, we implement the approximate Bayesian computation (ABC) rejection method to construct probability density distributions that estimate model parameters. These distributions inform an appropriate search curve for global sensitivity analysis using the extended fourier amplitude sensitivity test. Sensitivity results combined with the ABC method suggest that parameter interaction is occurring between the drivers of tumor burden, which are the tumor growth rate and carrying capacity as well as the tumor kill rate by T cells, and the two modeled forms of immunosuppression, PD-L1-PD-1 immune checkpoint and MDSC suppression of T cells. Thus, treatment with an immune checkpoint inhibitor in combination with a therapeutic targeting the inhibitory mechanisms of MDSCs should be explored.

Estimation of Parameter Distributions for Reaction-Diffusion Equations with Competition using Aggregate Spatiotemporal Data

Reaction-diffusion equations have been used to model a wide range of biological phenomenon related to population spread and proliferation from ecology to cancer. It is commonly assumed that individuals in a population have homogeneous diffusion and growth rates; however, this assumption can be inaccurate when the population is intrinsically divided into many distinct subpopulations that compete with each other. In previous work, the task of inferring the degree of phenotypic heterogeneity between subpopulations from total population density has been performed within a framework that combines parameter distribution estimation with reaction-diffusion models. Here, we extend this approach so that it is compatible with reaction-diffusion models that include competition between subpopulations. We use a reaction-diffusion model of glioblastoma multiforme, an aggressive type of brain cancer, to test our approach on simulated data that are similar to measurements that could be collected in practice. We use Prokhorov metric framework and convert the reaction-diffusion model to a random differential equation model to estimate joint distributions of diffusion and growth rates among heterogeneous subpopulations. We then compare the new random differential equation model performance against other partial differential equation models' performance. We find that the random differential equation is more capable at predicting the cell density compared to other models while being more time efficient. Finally, we use k-means clustering to predict the number of subpopulations based on the recovered distributions.

Data-driven spatio-temporal modelling of glioblastoma

Mathematical oncology provides unique and invaluable insights into tumour growth on both the microscopic and macroscopic levels. This review presents state-of-the-art modelling techniques and focuses on their role in understanding glioblastoma, a malignant form of brain cancer. For each approach, we summarize the scope, drawbacks and assets. We highlight the potential clinical applications of each modelling technique and discuss the connections between the mathematical models and the molecular and imaging data used to inform them. By doing so, we aim to prime cancer researchers with current and emerging computational tools for understanding tumour progression. By providing an in-depth picture of the different modelling techniques, we also aim to assist researchers who seek to build and develop their own models and the associated inference frameworks. Our article thus strikes a unique balance. On the one hand, we provide a comprehensive overview of the available modelling techniques and their applications, including key mathematical expressions. On the other hand, the content is accessible to mathematicians and biomedical scientists alike to accommodate the interdisciplinary nature of cancer research.

Practical Understanding of Cancer Model Identifiability in Clinical Applications

Mathematical models are a core component in the foundation of cancer theory and have been developed as clinical tools in precision medicine. Modeling studies for clinical applications often assume an individual's characteristics can be represented as parameters in a model and are used to explain, predict, and optimize treatment outcomes. However, this approach relies on the identifiability of the underlying mathematical models. In this study, we build on the framework of an observing-system simulation experiment to study the identifiability of several models of cancer growth, focusing on the prognostic parameters of each model. Our results demonstrate that the frequency of data collection, the types of data, such as cancer proxy, and the accuracy of measurements all play crucial roles in determining the identifiability of the model. We also found that highly accurate data can allow for reasonably accurate estimates of some parameters, which may be the key to achieving model identifiability in practice. As more complex models required more data for identification, our results support the idea of using models with a clear mechanism that tracks disease progression in clinical settings. For such a model, the subset of model parameters associated with disease progression naturally minimizes the required data for model identifiability.

Spatiotemporal dynamics of a glioma immune interaction model

We report a mathematical model which depicts the spatiotemporal dynamics of glioma cells, macrophages, cytotoxic-T-lymphocytes, immuno-suppressive cytokine TGF-β and immuno-stimulatory cytokine IFN-γ through a system of five coupled reaction-diffusion equations. We performed local stability analysis of the biologically based mathematical model for the growth of glioma cell population and their environment. The presented stability analysis of the model system demonstrates that the temporally stable positive interior steady state remains stable under the small inhomogeneous spatiotemporal perturbations. The irregular spatiotemporal dynamics of gliomas, macrophages and cytotoxic T-lymphocytes are discussed extensively and some numerical simulations are presented. Performed some numerical simulations in both one and two dimensional spaces. The occurrence of heterogeneous pattern formation of the system has both biological and mathematical implications and the concepts of glioma cell progression and invasion are considered. Simulation of the model shows that by increasing the value of time, the glioma cell population, macrophages and cytotoxic-T-lymphocytes spread throughout the domain.

Anti-PD-1 checkpoint blockade monotherapy in the orthotopic GL261 glioma model: the devil is in the detail

The GL261 cell line, syngeneic on the C57BL/6 background, has, since its establishment half a century ago in 1970, become the most commonly used immunocompetent murine model of glioblastoma. As immunotherapy has entered the mainstream of clinical discourse in the past decade, this model has proved its worth as a formidable opponent against various immunotherapeutic combinations. Although advances in surgical, radiological, and chemotherapeutic interventions have extended mean glioblastoma patient survival by several months, 5-year survival postdiagnosis remains below 5%. Immunotherapeutic interventions, such as the ones explored in the murine GL261 model, may prove beneficial for patients with glioblastoma. However, even common immunotherapeutic interventions in the GL261 model still have unclear efficacy, with wildly discrepant conclusions being made in the literature regarding this topic. Here, we focus on anti-PD-1 checkpoint blockade monotherapy as an example of this pattern. We contend that a fine-grained analysis of how biological variables (age, sex, tumor location, etc.) predict treatment responsiveness in this preclinical model will better enable researchers to identify glioblastoma patients most likely to benefit from checkpoint blockade immunotherapy moving forward.

Image-based personalization of computational models for predicting response of high-grade glioma to chemoradiation

High-grade gliomas are an aggressive and invasive malignancy which are susceptible to treatment resistance due to heterogeneity in intratumoral properties such as cell proliferation and density and perfusion. Non-invasive imaging approaches can measure these properties, which can then be used to calibrate patient-specific mathematical models of tumor growth and response. We employed multiparametric magnetic resonance imaging (MRI) to identify tumor extent (via contrast-enhanced T1-weighted, and T2-FLAIR) and capture intratumoral heterogeneity in cell density (via diffusion-weighted imaging) to calibrate a family of mathematical models of chemoradiation response in nine patients with unresected or partially resected disease. The calibrated model parameters were used to forecast spatially-mapped individual tumor response at future imaging visits. We then employed the Akaike information criteria to select the most parsimonious member from the family, a novel two-species model describing the enhancing and non-enhancing components of the tumor. Using this model, we achieved low error in predictions of the enhancing volume (median: - 2.5%, interquartile range: 10.0%) and a strong correlation in total cell count (Kendall correlation coefficient 0.79) at 3-months post-treatment. These preliminary results demonstrate the plausibility of using multiparametric MRI data to inform spatially-informative, biologically-based predictive models of tumor response in the setting of clinical high-grade gliomas.

Towards an Image-Informed Mathematical Model of In Vivo Response to Fractionated Radiation Therapy

Simple Summary

Using medical imaging data and computational models, we develop a modeling framework to provide personalized treatment response forecasts to fractionated radiation therapy for individual tumors. We evaluate this approach in an animal model of brain cancer and forecast changes in tumor cellularity and vasculature.

Abstract

Fractionated radiation therapy is central to the treatment of numerous malignancies, including high-grade gliomas where complete surgical resection is often impractical due to its highly invasive nature. Development of approaches to forecast response to fractionated radiation therapy may provide the ability to optimize or adapt treatment plans for radiotherapy. Towards this end, we have developed a family of 18 biologically-based mathematical models describing the response of both tumor and vasculature to fractionated radiation therapy. Importantly, these models can be personalized for individual tumors via quantitative imaging measurements. To evaluate this family of models, rats (n = 7) with U-87 glioblastomas were imaged with magnetic resonance imaging (MRI) before, during, and after treatment with fractionated radiotherapy (with doses of either 2 Gy/day or 4 Gy/day for up to 10 days). Estimates of tumor and blood volume fractions, provided by diffusion-weighted MRI and dynamic contrast-enhanced MRI, respectively, were used to calibrate tumor-specific model parameters. The Akaike Information Criterion was employed to select the most parsimonious model and determine an ensemble averaged model, and the resulting forecasts were evaluated at the global and local level. At the global level, the selected model’s forecast resulted in less than 16.2% error in tumor volume estimates. At the local (voxel) level, the median Pearson correlation coefficient across all prediction time points ranged from 0.57 to 0.87 for all animals. While the ensemble average forecast resulted in increased error (ranging from 4.0% to 1063%) in tumor volume predictions over the selected model, it increased the voxel wise correlation (by greater than 12.3%) for three of the animals. This study demonstrates the feasibility of calibrating a model of response by serial quantitative MRI data collected during fractionated radiotherapy to predict response at the conclusion of treatment.

Estimating the extent of glioblastoma invasion : Approximate stationalization of anisotropic advection-diffusion-reaction equations in the context of glioblastoma invasion

Glioblastoma Multiforme is a malignant brain tumor with poor prognosis. There have been numerous attempts to model the invasion of tumorous glioma cells via partial differential equations in the form of advection–diffusion–reaction equations. The patient-wise parametrization of these models, and their validation via experimental data has been found to be difficult, as time sequence measurements are mostly missing. Also the clinical interest lies in the actual (invisible) tumor extent for a particular MRI/DTI scan and not in a predictive estimate. Therefore we propose a stationalized approach to estimate the extent of glioblastoma (GBM) invasion at the time of a given MRI/DTI scan. The underlying dynamics can be derived from an instationary GBM model, falling into the wide class of advection-diffusion-reaction equations. The stationalization is introduced via an analytic solution of the Fisher-KPP equation, the simplest model in the considered model class. We investigate the applicability in 1D and 2D, in the presence of inhomogeneous diffusion coefficients and on a real 3D DTI-dataset.

A Mechanistic Investigation into Ischemia-Driven Distal Recurrence of Glioblastoma

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

Generating patient-specific virtual tumor populations with reaction-diffusion models and molecular imaging data

The use of mathematical tumor growth models coupled to noisy imaging data has been suggested as a possible component in the push towards precision medicine. We discuss the generation of population and patient-specific virtual populations in this context, providing in silico experiments to demonstrate how intra- and inter-patient heterogeneity can be estimated by applying rigorous statistical procedures to noisy molecular imaging data, and how the noise properties of such data can be analyzed to estimate uncertainties in predicted patient outcomes.

Learning Equations from Biological Data with Limited Time Samples

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

Integrated Biophysical Modeling and Image Analysis: Application to Neuro-Oncology

Central nervous system (CNS) tumors come with vastly heterogeneous histologic, molecular, and radiographic landscapes, rendering their precise characterization challenging. The rapidly growing fields of biophysical modeling and radiomics have shown promise in better characterizing the molecular, spatial, and temporal heterogeneity of tumors. Integrative analysis of CNS tumors, including clinically acquired multi-parametric magnetic resonance imaging (mpMRI) and the inverse problem of calibrating biophysical models to mpMRI data, assists in identifying macroscopic quantifiable tumor patterns of invasion and proliferation, potentially leading to improved (a) detection/segmentation of tumor subregions and (b) computer-aided diagnostic/prognostic/predictive modeling. This article presents a summary of (a) biophysical growth modeling and simulation,(b) inverse problems for model calibration, (c) these models' integration with imaging workflows, and (d) their application to clinically relevant studies. We anticipate that such quantitative integrative analysis may even be beneficial in a future revision of the World Health Organization (WHO) classification for CNS tumors, ultimately improving patient survival prospects.

Efficacy of direct current generated by multiple-electrode arrays on F3II mammary carcinoma: experiment and mathematical modeling

Background

The modified Gompertz equation has been proposed to fit experimental data for direct current treated tumors when multiple-straight needle electrodes are individually inserted into the base perpendicular to the tumor long axis. The aim of this work is to evaluate the efficacy of direct current generated by multiple-electrode arrays on F3II mammary carcinoma that grow in the male and female BALB/c/Cenp mice, when multiple-straight needle electrodes and multiple-pairs of electrodes are inserted in the tumor.

Methods

A longitudinal and retrospective preclinical study was carried out. Male and female BALB/c/Cenp mice, the modified Gompertz equation, intensities (2, 6 and 10 mA) and exposure times (10 and 20 min) of direct current, and three geometries of multiple-electrodes (one formed by collinear electrodes and two by pair-electrodes) were used. Tumor volume and mice weight were measured. In addition, the mean tumor doubling time, tumor regression percentage, tumor growth delay, direct current overall effectiveness and mice survival were calculated.

Results

The greatest growth retardation, mean doubling time, regression percentage and growth delay of the primary F3II mammary carcinoma in male and female mice were observed when the geometry of multiple-pairs of electrodes was arranged in the tumor at 45, 135, 225 and 325^o and the longest exposure time. In addition, highest direct current overall effectiveness (above 66%) was observed for this EChT scheme.

Conclusions

It is concluded that electrochemical therapy may be potentially addressed to highly aggressive and metastic primary F3II murine mammary carcinoma and the modified Gompertz equation may be used to fit data of this direct current treated carcinoma. Additionally, electrochemical therapy effectiveness depends on the exposure time, geometry of multiple-electrodes and ratio between the direct current intensity applied and the polarization current induced in the tumor.

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

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

Growth pattern of experimental glioblastoma

Glioblastoma multiforme (GBM) is an aggressive primary brain malignancy with a very poor prognosis. Researchers employ animal models to develop potential therapies. It is important that these models have clinical relevance. This means that old models, propagated for decades in cultures, should be questioned. Parameters to be evaluated include whether animals are immune competent or not, the infiltrative growth pattern of the tumor, tumor volume resulting in symptoms and growth rate. We here describe the growth pattern of an experimental glioblastoma model in detail with GFP positive glioblastoma cells in fully immune competent animals and study tumor growth rate and tumor mass as a function of time from inoculation. We were able to correlate findings made with classical immunohistochemistry and MR findings. The tumor growth rate was fitted by a Gompertz function. The model predicted the time until onset of symptoms for 5000 inoculated cells to 18.7±0.4 days, and the tumor mass at days 10 and 14, which are commonly used as the start of treatment in therapeutic studies, were 5.97±0.62 mg and 29.1±3.0 mg, respectively. We want to raise the question regarding the clinical relevance of the outline of glioblastoma experiments, where treatment is often initiated at a very early stage. The approach presented here could potentially be modified to gain information also from other tumor models.

A novel in silico platform for a fully automatic personalized brain tumor growth

Glioblastoma Multiforme is the most common and most aggressive type of brain tumors. Although accurate prediction of Glioblastoma borders and shape is absolutely essential for neurosurgeons, there are not many in silico platforms that can make such predictions. In the current study, an automatic patient-specific simulation of Glioblastoma growth would be described. A finite element approach is used to analyze the magnetic resonance images from patients in the early stages of their tumors. For segmentation of the tumor, the Support Vector Machine (SVM) method, which is an automatic segmentation algorithm, is used. Using in situ and in vivo data, the main parameters of tumor prediction and growth are estimated with high precision in proliferation-invasion partial differential equation, using the genetic algorithm optimization method. The results show that for a C57BL mouse, the differences between the area and perimeter of in vivo test and simulation prediction data, as the objective function, are 3.7% and 17.4%, respectively.

Inhibition of TGF-β Signaling in Gliomas by the Flavonoid Diosmetin Isolated from Dracocephalum peregrinum L

Background: Dracocephalum peregrinum L., a traditional Kazakh medicine, has good expectorant, anti-cough, and to some degree, anti-asthmatic effects. Diosmetin (3',5,7-trihydroxy-4'-methoxyflavone), a natural flavonoid found in traditional Chinese herbs, is the main flavonoid in D. peregrinum L. and has been used in various medicinal products because of its anticancer, antimicrobial, antioxidant, estrogenic, and anti-inflammatory effects. The present study aimed to investigate the effects of diosmetin on the proliferation, invasion, and migration of glioma cells, as well as the possible underlying mechanisms. Methods: 3-(4,5-dimethylthiazol-2-yl)-2,5-diphenyltetrazolium bromide (MTT), scratch wound, and Transwell assays were used to demonstrate the effects of diosmetin in glioma. Protein levels of Bcl-2, Bax, cleaved caspase-3, transforming growth factor-β (TGF-β), E-cadherin, and phosphorylated and unphosphorylated smad2 and smad3 were determined by Western blots. U251 glioma cell development and progression were measured in vivo in a mouse model. Results: Diosmetin inhibited U251 cell proliferation, migration, and invasion in vitro, the TGF-β signaling pathway, and Bcl-2 expression. In contrast, there was a significant increase in E-cadherin, Bax, and cleaved caspase-3 expression. Furthermore, it effectively reduced the tumorigenicity of glioma cells and promoted apoptosis in vivo. Conclusion: The results of this study suggest that diosmetin suppresses the growth of glioma cells in vitro and in vivo, possibly by activating E-cadherin expression and inhibiting the TGF-β signaling pathway.

Forecasting tumor and vasculature response dynamics to radiation therapy via image based mathematical modeling

BACKGROUND

Intra-and inter-tumoral heterogeneity in growth dynamics and vascularity influence tumor response to radiation therapy. Quantitative imaging techniques capture these dynamics non-invasively, and these data can initialize and constrain predictive models of response on an individual basis.

METHODS

We have developed a family of 10 biologically-based mathematical models describing the spatiotemporal dynamics of tumor volume fraction, blood volume fraction, and response to radiation therapy. To evaluate this family of models, rats (n = 13) with C6 gliomas were imaged with magnetic resonance imaging (MRI) three times before, and four times following a single fraction of 20 Gy or 40 Gy whole brain irradiation. The first five 3D time series data of tumor volume fraction, estimated from diffusion-weighted (DW-) MRI, and blood volume fraction, estimated from dynamic contrast-enhanced (DCE-) MRI, were used to calibrate tumor-specific model parameters. The most parsimonious and well calibrated of the 10 models, selected using the Akaike information criterion, was then utilized to predict future growth and response at the final two imaging time points. Model predictions were compared at the global level (percent error in tumor volume, and Dice coefficient) as well as at the local or voxel level (concordance correlation coefficient).

RESULT

The selected model resulted in < 12% error in tumor volume predictions, strong spatial agreement between predicted and observed tumor volumes (Dice coefficient > 0.74), and high level of agreement at the voxel level between the predicted and observed tumor volume fraction and blood volume fraction (concordance correlation coefficient > 0.77 and > 0.65, respectively).

CONCLUSIONS

This study demonstrates that serial quantitative MRI data collected before and following radiation therapy can be used to accurately predict tumor and vasculature response with a biologically-based mathematical model that is calibrated on an individual basis. To the best of our knowledge, this is the first effort to characterize the tumor and vasculature response to radiation therapy temporally and spatially using imaging-driven mathematical models.

Using a continuum model to decipher the mechanics of embryonic tissue spreading from time-lapse image sequences: An approximate Bayesian computation approach

Advanced imaging techniques generate large datasets capable of describing the structure and kinematics of tissue spreading in embryonic development, wound healing, and the progression of many diseases. These datasets can be integrated with mathematical models to infer biomechanical properties of the system, typically identifying an optimal set of parameters for an individual experiment. However, these methods offer little information on the robustness of the fit and are generally ill-suited for statistical tests of multiple experiments. To overcome this limitation and enable efficient use of large datasets in a rigorous experimental design, we use the approximate Bayesian computation rejection algorithm to construct probability density distributions that estimate model parameters for a defined theoretical model and set of experimental data. Here, we demonstrate this method with a 2D Eulerian continuum mechanical model of spreading embryonic tissue. The model is tightly integrated with quantitative image analysis of different sized embryonic tissue explants spreading on extracellular matrix (ECM) and is regulated by a small set of parameters including forces on the free edge, tissue stiffness, strength of cell-ECM adhesions, and active cell shape changes. We find statistically significant trends in key parameters that vary with initial size of the explant, e.g., for larger explants cell-ECM adhesion forces are weaker and free edge forces are stronger. Furthermore, we demonstrate that estimated parameters for one explant can be used to predict the behavior of other similarly sized explants. These predictive methods can be used to guide further experiments to better understand how collective cell migration is regulated during development.

THE PROHOROV METRIC FRAMEWORK AND AGGREGATE DATA INVERSE PROBLEMS FOR RANDOM PDEs

We consider nonparametric estimation of probability measures for parameters in problems where only aggregate (population level) data are available. We summarize an existing computational method for the estimation problem which has been developed over the past several decades [24, 5, 12, 28, 16]. Theoretical results are presented which establish the existence and consistency of very general (ordinary, generalized and other) least squares estimates and estimators for the measure estimation problem with specific application to random PDEs.

Advances in Glioblastoma Multiforme Treatment: New Models for Nanoparticle Therapy

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

A Spatio-Temporal Model of Macrophage-Mediated Drug Resistance in Glioma Immunotherapy

The emergence of drug resistance is often an inevitable obstacle that limits the long-term effectiveness of clinical cancer chemotherapeutics. Although various forms of cancer cell-intrinsic mechanisms of drug resistance have been experimentally revealed, the role and the underlying mechanism of tumor microenvironment in driving the development of acquired drug resistance remain elusive, which significantly impedes effective clinical cancer treatment. Recent experimental studies have revealed a macrophage-mediated drug resistance mechanism in which the tumor microenvironment undergoes adaptation in response to macrophage-targeted colony-stimulating factor-1 receptor (CSF1R) inhibition therapy in gliomas. In this study, we developed a spatio-temporal model to quantitatively describe the interplay between glioma cells and CSF1R inhibitor-targeted macrophages through CSF1 and IGF1 pathways. Our model was used to investigate the evolutionary kinetics of the tumor regrowth and the associated dynamic adaptation of the tumor microenvironment in response to the CSF1R inhibitor treatment. The simulation result obtained using this model was in agreement with the experimental data. The sensitivity analysis revealed the key parameters involved in the model, and their potential impacts on the model behavior were examined. Moreover, we demonstrated that the drug resistance is dose-dependent. In addition, we quantitatively evaluated the effects of combined CSFR inhibition and IGF1 receptor (IGF1R) inhibition with the goal of designing more effective therapies for gliomas. Our study provides quantitative and mechanistic insights into the microenvironmental adaptation mechanisms that operate during macrophage-targeted immunotherapy and has implications for drug dose optimization and the design of more effective combination therapies. Mol Cancer Ther; 17(4); 814-24. ©2018 AACR.