Personalized Predictions of Glioblastoma Infiltration: Mathematical Models, Physics-Informed Neural Networks and Multimodal Scans.

Predicting the infiltration of Glioblastoma (GBM) from medical MRI scans is crucial for understanding tumor growth dynamics and designing personalized radiotherapy treatment plans.Mathematical models of GBM growth can complement the data in the prediction of spatial distributions of tumor cells. However, this requires estimating patient-specific parameters of the model from clinical data, which is a challenging inverse problem due to limited temporal data and the limited time between imaging and diagnosis. This work proposes a method that uses Physics-Informed Neural Networks (PINNs) to estimate patient-specific parameters of a reaction-diffusion PDE model of GBM growth from a single 3D structural MRI snapshot. PINNs embed both the data and the PDE into a loss function, thus integrating theory and data. Key innovations include the identification and estimation of characteristic non-dimensional parameters, a pre-training step that utilizes the non-dimensional parameters and a fine-tuning step to determine the patient specific parameters. Additionally, the diffuse domain method is employed to handle the complex brain geometry within the PINN framework. Our method is validated both on synthetic and patient datasets, and shows promise for real-time parametric inference in the clinical setting for personalized GBM treatment.

Predictive nonlinear modeling of malignant myelopoiesis and tyrosine kinase inhibitor therapy.

Chronic myeloid leukemia (CML) is a blood cancer characterized by dysregulated production of maturing myeloid cells driven by the product of the Philadelphia chromosome, the BCR-ABL1 tyrosine kinase. Tyrosine kinase inhibitors (TKIs) have proved effective in treating CML, but there is still a cohort of patients who do not respond to TKI therapy even in the absence of mutations in the BCR-ABL1 kinase domain that mediate drug resistance. To discover novel strategies to improve TKI therapy in CML, we developed a nonlinear mathematical model of CML hematopoiesis that incorporates feedback control and lineage branching. Cell-cell interactions were constrained using an automated model selection method together with previous observations and new in vivo data from a chimeric BCR-ABL1 transgenic mouse model of CML. The resulting quantitative model captures the dynamics of normal and CML cells at various stages of the disease and exhibits variable responses to TKI treatment, consistent with those of CML patients. The model predicts that an increase in the proportion of CML stem cells in the bone marrow would decrease the tendency of the disease to respond to TKI therapy, in concordance with clinical data and confirmed experimentally in mice. The model further suggests that, under our assumed similarities between normal and leukemic cells, a key predictor of refractory response to TKI treatment is an increased maximum probability of self-renewal of normal hematopoietic stem cells. We use these insights to develop a clinical prognostic criterion to predict the efficacy of TKI treatment and design strategies to improve treatment response. The model predicts that stimulating the differentiation of leukemic stem cells while applying TKI therapy can significantly improve treatment outcomes.

A human vascularized microtumor model of patient-derived colorectal cancer recapitulates clinical disease.

Accurately modeling tumor biology and testing novel therapies on patient-derived cells is critically important to developing therapeutic regimens personalized to a patient's specific disease. The vascularized microtumor (VMT), or "tumor-on-a-chip," is a physiologic preclinical cancer model that incorporates key features of the native human tumor microenvironment within a transparent microfluidic platform, allowing rapid drug screening in vitro. Herein we optimize methods for generating patient-derived VMT (pVMT) using fresh colorectal cancer (CRC) biopsies and surgical resections to test drug sensitivities at the individual patient level. In response to standard chemotherapy and TGF-ßR1 inhibition, we observe heterogeneous responses between pVMT derived from 6 patient biopsies, with the pVMT recapitulating tumor growth, histological features, metabolic heterogeneity, and drug responses of actual CRC tumors. Our results suggest that a translational infrastructure providing rapid information from patient-derived tumor cells in the pVMT, as established in this study, will support efforts to improve patient outcomes.

Tumor growth towards lower extracellular matrix conductivity regions under Darcy's Law and steady morphology.

We study a classic Darcy's law model for tumor cell motion with inhomogeneous and isotropic conductivity. The tumor cells are assumed to be a constant density fluid flowing through porous extracellular matrix (ECM). The ECM is assumed to be rigid and motionless with constant porosity. One and two dimensional simulations show that the tumor mass grows from high to low conductivity regions when the tumor morphology is steady. In the one-dimensional case, we proved that when the tumor size is steady, the tumor grows towards lower conductivity regions. We conclude that this phenomenon is produced by the coupling of a special inward flow pattern in the steady tumor and Darcy's law which gives faster flow speed in higher conductivity regions.

Correction: Stress generation, relaxation and size control in confined tumor growth.

[This corrects the article DOI: 10.1371/journal.pcbi.1009701.].

Modelling glioma progression, mass effect and intracranial pressure in patient anatomy.

Increased intracranial pressure is the source of most critical symptoms in patients with glioma, and often the main cause of death. Clinical interventions could benefit from non-invasive estimates of the pressure distribution in the patient's parenchyma provided by computational models. However, existing glioma models do not simulate the pressure distribution and they rely on a large number of model parameters, which complicates their calibration from available patient data. Here we present a novel model for glioma growth, pressure distribution and corresponding brain deformation. The distinct feature of our approach is that the pressure is directly derived from tumour dynamics and patient-specific anatomy, providing non-invasive insights into the patient's state. The model predictions allow estimation of critical conditions such as intracranial hypertension, brain midline shift or neurological and cognitive impairments. A diffuse-domain formalism is employed to allow for efficient numerical implementation of the model in the patient-specific brain anatomy. The model is tested on synthetic and clinical cases. To facilitate clinical deployment, a high-performance computing implementation of the model has been publicly released.

Modeling of Tumor Growth with Input from Patient-Specific Metabolomic Data.

Advances in omic technologies have provided insight into cancer progression and treatment response. However, the nonlinear characteristics of cancer growth present a challenge to bridge from the molecular- to the tissue-scale, as tumor behavior cannot be encapsulated by the sum of the individual molecular details gleaned experimentally. Mathematical modeling and computational simulation have been traditionally employed to facilitate analysis of nonlinear systems. In this study, for the first time tumor metabolomic data are linked via mathematical modeling to the tumor tissue-scale behavior, showing the capability to mechanistically simulate cancer progression personalized to omic information obtainable from patient tumor core biopsy analysis. Generally, a higher degree of metabolic dysregulation has been correlated with more aggressive tumor behavior. Accordingly, key parameters influenced by metabolomic data in this model include tumor proliferation, vascularization, aggressiveness, lactic acid production, monocyte infiltration and macrophage polarization, and drug effect. The model enables evaluating interactions of interest between these parameters which drive tumor growth based on the metabolomic data. The results show that the model can group patients consistently with the clinically observed outcomes of response/non-response to chemotherapy. This modeling approach provides a first step towards evaluation of tumor growth based on tumor-specific metabolomic data.

Mathematical modeling of cancer immunotherapy for personalized clinical translation.

Encouraging advances are being made in cancer immunotherapy modeling, especially in the key areas of developing personalized treatment strategies based on individual patient parameters, predicting treatment outcomes and optimizing immunotherapy synergy when used in combination with other treatment approaches. Here we present a focused review of the most recent mathematical modeling work on cancer immunotherapy with a focus on clinical translatability. It can be seen that this field is transitioning from pure basic science to applications that can make impactful differences in patients' lives. We discuss how researchers are integrating experimental and clinical data to fully inform models so that they can be applied for clinical predictions, and present the challenges that remain to be overcome if widespread clinical adaptation is to be realized. Lastly, we discuss the most promising future applications and areas that are expected to be the focus of extensive upcoming modeling studies.

Stress generation, relaxation and size control in confined tumor growth.

Experiments on tumor spheroids have shown that compressive stress from their environment can reversibly decrease tumor expansion rates and final sizes. Stress release experiments show that nonuniform anisotropic elastic stresses can be distributed throughout. The elastic stresses are maintained by structural proteins and adhesive molecules, and can be actively relaxed by a variety of biophysical processes. In this paper, we present a new continuum model to investigate how the growth-induced elastic stresses and active stress relaxation, in conjunction with cell size control feedback machinery, regulate the cell density and stress distributions within growing tumors as well as the tumor sizes in the presence of external physical confinement and gradients of growth-promoting chemical fields. We introduce an adaptive reference map that relates the current position with the reference position but adapts to the current position in the Eulerian frame (lab coordinates) via relaxation. This type of stress relaxation is similar to but simpler than the classical Maxwell model of viscoelasticity in its formulation. By fitting the model to experimental data from two independent studies of tumor spheroid growth and their cell density distributions, treating the tumors as incompressible, neo-Hookean elastic materials, we find that the rates of stress relaxation of tumor tissues can be comparable to volumetric growth rates. Our study provides insight on how the biophysical properties of the tumor and host microenvironment, mechanical feedback control and diffusion-limited differential growth act in concert to regulate spatial patterns of stress and growth. When the tumor is stiffer than the host, our model predicts tumors are more able to change their size and mechanical state autonomously, which may help to explain why increased tumor stiffness is an established hallmark of malignant tumors.

An in vitro vascularized micro-tumor model of human colorectal cancer recapitulates in vivo responses to standard-of-care therapy.

Around 95% of anti-cancer drugs that show promise during preclinical study fail to gain FDA-approval for clinical use. This failure of the preclinical pipeline highlights the need for improved, physiologically-relevant in vitro models that can better serve as reliable drug-screening and disease modeling tools. The vascularized micro-tumor (VMT) is a novel three-dimensional model system (tumor-on-a-chip) that recapitulates the complex human tumor microenvironment, including perfused vasculature, within a transparent microfluidic device, allowing real-time study of drug responses and tumor-stromal interactions. Here we have validated this microphysiological system (MPS) platform for the study of colorectal cancer (CRC), the second leading cause of cancer-related deaths, by showing that gene expression, tumor heterogeneity, and treatment responses in the VMT more closely model CRC tumor clinicopathology than current standard drug screening modalities, including 2-dimensional monolayer culture and 3-dimensional spheroids.

Effect of feedback regulation on stem cell fractions in tissues and tumors: Understanding chemoresistance in cancer.

While resistance mutations are often implicated in the failure of cancer therapy, lack of response also occurs without such mutants. In bladder cancer mouse xenografts, repeated chemotherapy cycles have resulted in cancer stem cell (CSC) enrichment, and consequent loss of therapy response due to the reduced susceptibility of CSCs to drugs. A particular feedback loop present in the xenografts has been shown to promote CSC enrichment in this system. Yet, many other regulatory loops might also be operational and might promote CSC enrichment. Their identification is central to improving therapy response. Here, we perform a comprehensive mathematical analysis to define what types of regulatory feedback loops can and cannot contribute to CSC enrichment, providing guidance to the experimental identification of feedback molecules. We derive a formula that reveals whether or not the cell population experiences CSC enrichment over time, based on the properties of the feedback. We find that negative feedback on the CSC division rate or positive feedback on differentiated cell death rate can lead to CSC enrichment. Further, the feedback mediators that achieve CSC enrichment can be secreted by either CSCs or by more differentiated cells. The extent of enrichment is determined by the CSC death rate, the CSC self-renewal probability, and by feedback strength. Defining these general characteristics of feedback loops can guide the experimental screening for and identification of feedback mediators that can promote CSC enrichment in bladder cancer and potentially other tumors. This can help understand and overcome the phenomenon of CSC-based therapy resistance.

Dynamics of nevus development implicate cell cooperation in the growth arrest of transformed melanocytes.

Mutational activation of the BRAF proto-oncogene in melanocytes reliably produces benign nevi (pigmented 'moles'), yet the same change is the most common driver mutation in melanoma. The reason nevi stop growing, and do not progress to melanoma, is widely attributed to a cell-autonomous process of 'oncogene-induced senescence'. Using a mouse model of Braf-driven nevus formation, analyzing both proliferative dynamics and single-cell gene expression, we found no evidence that nevus cells are senescent, either compared with other skin cells, or other melanocytes. We also found that nevus size distributions could not be fit by any simple cell-autonomous model of growth arrest, yet were easily fit by models based on collective cell behavior, for example in which arresting cells release an arrest-promoting factor. We suggest that nevus growth arrest is more likely related to the cell interactions that mediate size control in normal tissues, than to any cell-autonomous, 'oncogene-induced' program of senescence.

Melanocytes are pigment-producing cells found throughout the skin. Mutations that activate a gene called BRAF cause these cells to divide and produce melanocytic nevi, also known as "moles". These mutations are oncogenic, meaning they can cause cancer. Indeed, BRAF is the most commonly mutated gene in melanoma, a deadly skin cancer that arises from melanocytes. Yet, moles hardly ever progress to melanoma. A proposed explanation for this behavior is that, once activated, BRAF initiates a process called "oncogene-induced senescence" in each melanocyte. This process, likened to premature aging, is thought to be what causes cells in a mole to quit dividing. Although this hypothesis is widely accepted, it has proved difficult to test directly. To investigate this notion, Ruiz-Vega et al. studied mice with hundreds of moles created by the same BRAF mutation found in human moles. Analyzing the activity of genes in individual cells revealed that nevus melanocytes that have stopped growing are no more senescent than other skin cells, including non-mole melanocytes. Ruiz-Vega et al. then analyzed the sizes at which moles stopped growing, estimating the number of cells in each mole. The data were then compared with the results of a simulation and mathematical modeling. This revealed that any model based on the idea of cells independently shutting down after a number of random events could not reproduce the distribution of mole sizes that had been experimentally observed. On the other hand, models based on melanocytes acting collectively to shut down each other's growth fit the observed data much better. These findings suggest that moles do not stop growing as a direct result of the activation of BRAF, but because they sense and respond to their own overgrowth. The same kind of collective sensing is observed in normal tissues that maintain a constant size. Discovering that melanocytes do this not only sheds light on why moles stop growing, it could also help researchers devise new ways to prevent melanomas from forming.

Complex Far-Field Geometries Determine the Stability of Solid Tumor Growth with Chemotaxis.

In this paper, we develop a sharp interface tumor growth model to study the effect of the tumor microenvironment using a complex far-field geometry that mimics a heterogeneous distribution of vasculature. Together with different nutrient uptake rates inside and outside the tumor, this introduces variability in spatial diffusion gradients. Linear stability analysis suggests that the uptake rate in the tumor microenvironment, together with chemotaxis, may induce unstable growth, especially when the nutrient gradients are large. We investigate the fully nonlinear dynamics using a spectrally accurate boundary integral method. Our nonlinear simulations reveal that vascular heterogeneity plays an important role in the development of morphological instabilities that range from fingering and chain-like morphologies to compact, plate-like shapes in two dimensions.

A Multiscale Agent-Based Model of Ductal Carcinoma In Situ.

OBJECTIVE: we present a multiscale agent-based model of Ductal Carcinoma in Situ (DCIS) in order to gain a detailed understanding of the cell-scale population dynamics, phenotypic distributions, and the associated interplay of important molecular signaling pathways that are involved in DCIS ductal invasion into the duct cavity (a process we refer to as duct advance rate here). METHODS: DCIS is modeled mathematically through a hybridized discrete cell-scale model and a continuum molecular scale model, which are explicitly linked through a bidirectional feedback mechanism. RESULTS: we find that duct advance rates occur in two distinct phases, characterized by an early exponential population expansion, followed by a long-term steady linear phase of population expansion, a result that is consistent with other modeling work. We further found that the rates were influenced most strongly by endocrine and paracrine signaling intensity, as well as by the effects of cell density induced quiescence within the DCIS population. CONCLUSION: our model analysis identified a complex interplay between phenotypic diversity that may provide a tumor adaptation mechanism to overcome proliferation limiting conditions, allowing for dynamic shifts in phenotypic populations in response to variation in molecular signaling intensity. Further, sensitivity analysis determined DCIS axial advance rates and calcification rates were most sensitive to cell cycle time variation. SIGNIFICANCE: this model may serve as a useful tool to study the cell-scale dynamics involved in DCIS initiation and intraductal invasion, and may provide insights into promising areas of future experimental research.

Personalized Radiotherapy Design for Glioblastoma: Integrating Mathematical Tumor Models, Multimodal Scans, and Bayesian Inference.

Glioblastoma (GBM) is a highly invasive brain tumor, whose cells infiltrate surrounding normal brain tissue beyond the lesion outlines visible in the current medical scans. These infiltrative cells are treated mainly by radiotherapy. Existing radiotherapy plans for brain tumors derive from population studies and scarcely account for patient-specific conditions. Here, we provide a Bayesian machine learning framework for the rational design of improved, personalized radiotherapy plans using mathematical modeling and patient multimodal medical scans. Our method, for the first time, integrates complementary information from high-resolution MRI scans and highly specific FET-PET metabolic maps to infer tumor cell density in GBM patients. The Bayesian framework quantifies imaging and modeling uncertainties and predicts patient-specific tumor cell density with credible intervals. The proposed methodology relies only on data acquired at a single time point and, thus, is applicable to standard clinical settings. An initial clinical population study shows that the radiotherapy plans generated from the inferred tumor cell infiltration maps spare more healthy tissue thereby reducing radiation toxicity while yielding comparable accuracy with standard radiotherapy protocols. Moreover, the inferred regions of high tumor cell densities coincide with the tumor radioresistant areas, providing guidance for personalized dose-escalation. The proposed integration of multimodal scans and mathematical modeling provides a robust, non-invasive tool to assist personalized radiotherapy design.

Tumor growth and calcification in evolving microenvironmental geometries.

In this paper, we apply the diffuse domain framework developed in Chen and Lowengrub (Tumor growth in complex, evolving microenvironmental geometries: A diffuse domain approach, J. Theor. Biol. 361 (2014) 14-30) to study the effects of a deformable basement membrane (BM) on the growth of a tumor in a confined, ductal geometry, such as ductal carcinoma in situ (DCIS). We use a continuum model of tumor microcalcification and investigate the tumor extent beyond the microcalcification. In order to solve the governing equations efficiently, we develop a stable nonlinear multigrid finite difference method. Two dimensional simulations are performed where the adhesion between tumor cells and the basement membrane is varied. Additional simulations considering the variation of duct radius and membrane stiffness are also conducted. The results demonstrate that enhanced membrane deformability promotes tumor growth and tumor calcification. When the duct radius is small, the cell-BM adhesion is weak or when the membrane is slightly deformed, the mammographic and pathologic tumor extents are linearly correlated, as predicted by Macklin et al. (J. Theor. Biol. 301 (2012) 122-140) using an agent-based model that does not account for the deformability of the basement membrane and the active forces that the membrane imparts on the tumor cells. Interestingly, we predict that when the duct radius is large, there is strong cell-BM adhesion or the membrane is highly deformed, the extents of the mammographic and pathologic tumors are instead quadratically correlated. The simulations can help surgeons to measure DCIS surgical margins while removing less non-cancerous tissue, and can improve targeting of intra- and post-operative radiotherapy.

A Visually Apparent and Quantifiable CT Imaging Feature Identifies Biophysical Subtypes of Pancreatic Ductal Adenocarcinoma.

PURPOSE: Pancreatic ductal adenocarcinoma (PDAC) is a heterogeneous disease with variable presentations and natural histories of disease. We hypothesized that different morphologic characteristics of PDAC tumors on diagnostic computed tomography (CT) scans would reflect their underlying biology. EXPERIMENTAL DESIGN: We developed a quantitative method to categorize the PDAC morphology on pretherapy CT scans from multiple datasets of patients with resectable and metastatic disease and correlated these patterns with clinical/pathologic measurements. We modeled macroscopic lesion growth computationally to test the effects of stroma on morphologic patterns, hypothesizing that the balance of proliferation and local migration rates of the cancer cells would determine tumor morphology. RESULTS: In localized and metastatic PDAC, quantifying the change in enhancement on CT scans at the interface between tumor and parenchyma (delta) demonstrated that patients with conspicuous (high-delta) tumors had significantly less stroma, higher likelihood of multiple common pathway mutations, more mesenchymal features, higher likelihood of early distant metastasis, and shorter survival times compared with those with inconspicuous (low-delta) tumors. Pathologic measurements of stromal and mesenchymal features of the tumors supported the mathematical model's underlying theory for PDAC growth. CONCLUSIONS: At baseline diagnosis, a visually striking and quantifiable CT imaging feature reflects the molecular and pathological heterogeneity of PDAC, and may be used to stratify patients into distinct subtypes. Moreover, growth patterns of PDAC may be described using physical principles, enabling new insights into diagnosis and treatment of this deadly disease.

Nonlinear studies of tumor morphological stability using a two-fluid flow model.

We consider the nonlinear dynamics of an avascular tumor at the tissue scale using a two-fluid flow Stokes model, where the viscosity of the tumor and host microenvironment may be different. The viscosities reflect the combined properties of cell and extracellular matrix mixtures. We perform a linear morphological stability analysis of the tumors, and we investigate the role of nonlinearity using boundary-integral simulations in two dimensions. The tumor is non-necrotic, although cell death may occur through apoptosis. We demonstrate that tumor evolution is regulated by a reduced set of nondimensional parameters that characterize apoptosis, cell-cell/cell-extracellular matrix adhesion, vascularization and the ratio of tumor and host viscosities. A novel reformulation of the equations enables the use of standard boundary integral techniques to solve the equations numerically. Nonlinear simulation results are consistent with linear predictions for nearly circular tumors. As perturbations develop and grow, the linear and nonlinear results deviate and linear theory tends to underpredict the growth of perturbations. Simulations reveal two basic types of tumor shapes, depending on the viscosities of the tumor and microenvironment. When the tumor is more viscous than its environment, the tumors tend to develop invasive fingers and a branched-like structure. As the relative ratio of the tumor and host viscosities decreases, the tumors tend to grow with a more compact shape and develop complex invaginations of healthy regions that may become encapsulated in the tumor interior. Although our model utilizes a simplified description of the tumor and host biomechanics, our results are consistent with experiments in a variety of tumor types that suggest that there is a positive correlation between tumor stiffness and tumor aggressiveness.