Andrea Hawkins-Daarud

The biology and mathematical modelling of glioma invasion: a review.

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

Simulating PDGF-Driven Glioma Growth and Invasion in an Anatomically Accurate Brain Domain

Gliomas are the most common of all primary brain tumors. They are characterized by their diffuse infiltration of the brain tissue and are uniformly fatal, with glioblastoma being the most aggressive form of the disease. In recent years, the over-expression of platelet-derived growth factor (PDGF) has been shown to produce tumors in experimental rodent models that closely resemble this human disease, specifically the proneural subtype of glioblastoma. We have previously modeled this system, focusing on the key attribute of these experimental tumors—the “recruitment” of oligodendroglial progenitor cells (OPCs) to participate in tumor formation by PDGF-expressing retrovirally transduced cells—in one dimension, with spherical symmetry. However, it has been observed that these recruitable progenitor cells are not uniformly distributed throughout the brainxa0and that tumor cells migrate at different rates depending on the material properties in different regions of the brain. Here we model the differential diffusion of PDGF-expressing and recruited cell populations via a system of partial differential equations with spatially variable diffusion coefficients and solve the equations in two spatial dimensions on a mouse brain atlas using a flux-differencing numerical approach. Simulations of our in silico model demonstrate qualitative agreement with the observed tumor distribution in the experimental animal system. Additionally, we show that while there are higher concentrations of OPCs in white matter, the level of recruitment of these plays little role in the appearance of “white matter disease,” where the tumor shows a preponderance for white matter. Instead, simulations show that this is largely driven by the ratio of the diffusion rate in white matter as compared to gray. However, this ratio has less effect on the speed of tumor growth than does the degree of OPC recruitment in the tumor. It was observed that tumor simulations with greater degrees of recruitment grow faster and develop more nodular tumors than if there is no recruitment at all, similar to our prior results from implementing our model in one dimension.xa0Combined, these results show that recruitment remains an important consideration in understanding and slowing glioma growth.

MultiCellDS: a community-developed standard for curating microenvironment-dependent multicellular data

Exchanging and understanding scientific data and their context represents a significant barrier to advancing research, especially with respect to information siloing. Maintaining information provenance and providing data curation and quality control help overcome common concerns and barriers to the effective sharing of scientific data. To address these problems in and the unique challenges of multicellular systems, we assembled a panel composed of investigators from several disciplines to create the MultiCellular Data Standard (MultiCellDS) with a use-case driven development process. The standard includes (1) digital cell lines, which are analogous to traditional biological cell lines, to record metadata, cellular microenvironment, and cellular phenotype variables of a biological cell line, (2) digital snapshots to consistently record simulation, experimental, and clinical data for multicellular systems, and (3) collections that can logically group digital cell lines and snapshots. We have created a MultiCellular DataBase (MultiCellDB) to store digital snapshots and the 200+ digital cell lines we have generated. MultiCellDS, by having a fixed standard, enables discoverability, extensibility, maintainability, searchability, and sustainability of data, creating biological applicability and clinical utility that permits us to identify upcoming challenges to uplift biology and strategies and therapies for improving human health.

Comparative dynamics of microglial and glioma cell motility at the infiltrative margin of brain tumours

Microglia are a major cellular component of gliomas, and abundant in the centre of the tumour and at the infiltrative margins. While glioma is a notoriously infiltrative disease, the dynamics of microglia and glioma migratory patterns have not been well characterized. To investigate the migratory behaviour of microglia and glioma cells at the infiltrative edge, we performed two-colour time-lapse fluorescence microscopy of brain slices generated from a platelet-derived growth factor-B (PDGFB)-driven rat model of glioma, in which glioma cells and microglia were each labelled with one of two different fluorescent markers. We used mathematical techniques to analyse glioma cells and microglia motility with both single cell tracking and particle image velocimetry (PIV). Our results show microglia motility is strongly correlated with the presence of glioma, while the correlation of the speeds of glioma cells and microglia was variable and weak. Additionally, we showed that microglia and glioma cells exhibit different types of diffusive migratory behaviour. Microglia movement fit a simple random walk, while glioma cell movement fits a super diffusion pattern. These results show that glioma cells stimulate microglia motility at the infiltrative margins, creating a correlation between the spatial distribution of glioma cells and the pattern of microglia motility.

MultiCellDS: a standard and a community for sharing multicellular data

Cell biology is increasingly focused on cellular heterogeneity and multicellular systems. To make the fullest use of experimental, clinical, and computational efforts, we need standardized data formats, community-curated “public data libraries”, and tools to combine and analyze shared data. To address these needs, our multidisciplinary community created MultiCellDS (MultiCellular Data Standard): an extensible standard, a library of digital cell lines and tissue snapshots, and support software. With the help of experimentalists, clinicians, modelers, and data and library scientists, we can grow this seed into a community-owned ecosystem of shared data and tools, to the benefit of basic science, engineering, and human health.

Glioblastoma recurrence and the role of MGMT promoter methylation

Tumor recurrence in glioblastoma multiforme (GBM) is often attributed to acquired resistance to the standard chemotherapeutic agent temozolomide (TMZ). Promoter methylation of the DNA repair gene MGMT has been associated with sensitivity to TMZ, while increased expression of MGMT has been associated with TMZ resistance. Clinical studies have observed a downward shift in MGMT methylation percentage from primary to recurrent stage tumors. However, the evolutionary processes driving this shift, and more generally the emergence and growth of TMZ-resistant tumor subpopulations, are still poorly understood. Here we develop a mathematical model, parameterized using clinical and experimental data, to investigate the role of MGMT methylation in TMZ resistance during the standard treatment regimen for GBM (surgery, chemotherapy and radiation). We first find that the observed downward shift in MGMT promoter methylation status between detection and recurrence cannot be explained solely by evolutionary selection. Next, our model suggests that TMZ has an inhibitory effect on maintenance methylation of MGMT after cell division. Finally, incorporating this inhibitory effect, we study the optimal number of TMZ doses per adjuvant cycle for GBM patients with high and low levels of MGMT methylation at diagnosis.

Connecting Patterns of Tumor Growth with Sex Differences in Extreme Survivorship for Primary Glioblastoma Patients

Purpose: Patient sex is recognized as a significant determinant of outcome but the relative prognostic importance of molecular, imaging, and other clinical features of GBM has not yet been thoroughly explored for male versus female patients. Combining multi-modal MR images and patient clinical information, this investigation assesses which pretreatment MRI-based and clinical variables impact sex-specific survivorship in glioblastoma patients. Methods: We considered the multi-modal MRI and clinical data of 494 patients newly diagnosed with primary glioblastoma (299 males and 195 females). Patient MR images (T1Gd, T2, and T2-FLAIR) were segmented to quantify imageable tumor volumes for each MR sequence. Cox proportional hazard (CPH) models and Student’s t-tests were used to assess which variables were significantly associated with survival outcomes. We used machine learning algorithms to develop pruned decision trees to integrate the impact of these variables on patient survival. Results: Among males, tumor (T1Gd) radius was a significant predictor of overall survival (HR=1.027, p=0.044). Among females, higher tumor cell net invasion rate was a significant detriment to overall survival (HR=1.011, p<0.001). Female extreme survivors had significantly smaller tumors (T1Gd) (p=0.010 t-test), but tumor size was not significantly correlated with female overall survival (p=0.955 CPH). Both male and female extreme survivors had significantly lower tumor cell net proliferation rates than patients in other survival groups (M p=0.004, F p=0.001, t-test). Age at diagnosis was a significant predictive factor for overall survival length for both males and females (M HR= 1.030, F HR=1.022). Additional variables like extent of resection, tumor laterality, and IDH1 mutation status were also found to have sex-specific effects on overall survival. Conclusion: The results indicated that some variables, like the tumor cell diffuse invasion rate and tumor size, had sex-specific implications for survival, while other variables, such as age at diagnosis and tumor cell proliferation rate, impacted both sexes in the same way. Despite similar distributions of the MR imaging parameters between males and females, there was a sex-specific difference in how these parameters related to outcomes. The sex differences in the predictive value of these and other variables emphasizes the importance of considering sex as a biological factor when determining patient prognosis and treatment approach.Background Sex is recognized as a significant determinant of outcome among glioblastoma patients, but the relative prognostic importance of glioblastoma features has not been thoroughly explored for sex differences. Methods Combining multi-modal MR images, biomathematical models, and patient clinical information, this investigation assesses which pretreatment variables have a sex-specific impact on the survival of glioblastoma patients. Pretreatment MR images of 494 glioblastoma patients (299 males and 195 females) were segmented to quantify tumor volumes. Cox proportional hazard (CPH) models and Student’s t-tests were used to assess which variables were associated with survival outcomes. Results Among males, tumor (T1Gd) radius was a predictor of overall survival (HR=1.027, p=0.044). Among females, higher tumor cell net invasion rate was a significant detriment to overall survival (HR=1.011, p Conclusion Despite similar distributions of the MR imaging parameters between males and females, there was a sex-specific difference in how these parameters related to outcomes, which emphasizes the importance of considering sex as a biological factor when determining patient prognosis and treatment approach.

Quantifying Uncertainty and Robustness in a Biomathematical Model Based Patient-Specific Response Metric for Glioblastoma

Glioblastomas, lethal primary brain tumors, are known for their heterogeneity and invasiveness. A growing literature has been developed demonstrating the clinical relevance of a biomathematical model, the Proliferation-Invasion (PI) model, of glioblastoma growth. Of interest here is the development of a treatment response metric, Days Gained (DG). This metric is based on individual tumor kinetics estimated through segmented volumes of hyperintense regions on T1-weighted gadolinium enhanced (T1Gd) and T2-weighted magnetic resonance images (MRIs). This metric was shown to be prognostic of time to progression. Further, it was shown to be more prognostic of outcome than standard response metrics. While promising, the original paper did not account for uncertainty in the calculation of the DG metric leaving the robustness of this cutoff in question. We harness the Bayesian framework to consider the impact of two sources of uncertainty: 1) image acquisition and 2) interobserver error in image segmentation. We first utilize synthetic data to characterize what non-error variants are influencing the final uncertainty in the DG metric. We then consider the original patient cohort to investigate clinical patterns of uncertainty and to determine how robust this metric is for predicting time to progression and overall survival. Our results indicate that the key clinical variants are the time between pre-treatment images and the underlying tumor growth kinetics, matching our observations in the clinical cohort. Finally, we demonstrated that for this cohort there was a continuous range of cutoffs between 94 and 105 for which the prediction of the time to progression and was over 80% reliable. While further validation must be done, this work represents a key step in ascertaining the clinical utility of this metric.

Simulating magnetic resonance images based on a model of tumor growth incorporating microenvironment.

Glioblastoma (GBM), the most aggressive primary brain tumor, is primarily diagnosed and monitored using gadoliniumenhanced T1-weighted and T2-weighted (T2W) magnetic resonance imaging (MRI). Hyperintensity on T2W images is understood to correspond with vasogenic edema and infiltrating tumor cells. GBM’s inherent heterogeneity and resulting non-specific MRI image features complicate assessing treatment response. To better understand treatment response, we propose creating a patient-specific untreated virtual imaging control (UVIC), which represents an individual tumor’s growth if it had not been treated, for comparison with actual post-treatment images. We generated a T2W MRI UVIC by combining a patient-specific mathematical model of tumor growth with a multi-compartmental MRI signal equation. GBM growth was mathematically modeled using the previously developed Proliferation-Invasion-Hypoxia-Necrosis- Angiogenesis-Edema (PIHNA-E) model, which simulated tumor as being comprised of three cellular phenotypes: normoxic, hypoxic and necrotic cells interacting with a vasculature species, angiogenic factors and extracellular fluid. Within the PIHNA-E model, both hypoxic and normoxic cells emitted angiogenic factors, which recruited additional vessels and caused the vessels to leak, allowing fluid, or edema, to escape into the extracellular space. The model’s output was spatial volume fraction maps for each glioma cell type and edema/extracellular space. Volume fraction maps and corresponding T2 values were then incorporated into a multi-compartmental Bloch signal equation to create simulated T2W images. T2 values for individual compartments were estimated from the literature and a normal volunteer. T2 maps calculated from simulated images had normal white matter, normal gray matter, and tumor tissue T2 values within range of literature values.

Hybrid approach for parameter estimation in agent-based models

Agent-based models are valuable in cancer research to show how different behaviors emerge from individual interactions between cells and their environment. However, calibrating such models can be difficult, especially if the parameters that govern the underlying interactions are hard to measure experimentally. Herein, we detail a new method to converge on parameter sets that fit an agent-based model to multiscale data using a model of glioblastoma as an example.