PIEZO1 regulates leader cell formation and cellular coordination during collective keratinocyte migration.

The collective migration of keratinocytes during wound healing requires both the generation and transmission of mechanical forces for individual cellular locomotion and the coordination of movement across cells. Leader cells along the wound edge transmit mechanical and biochemical cues to ensuing follower cells, ensuring their coordinated direction of migration across multiple cells. Despite the observed importance of mechanical cues in leader cell formation and in controlling coordinated directionality of cell migration, the underlying biophysical mechanisms remain elusive. The mechanically-activated ion channel PIEZO1 was recently identified to play an inhibitory role during the reepithelialization of wounds. Here, through an integrative experimental and mathematical modeling approach, we elucidate PIEZO1's contributions to collective migration. Time-lapse microscopy reveals that PIEZO1 activity inhibits leader cell formation at the wound edge. To probe the relationship between PIEZO1 activity, leader cell formation and inhibition of reepithelialization, we developed an integrative 2D continuum model of wound closure that links observations at the single cell and collective cell migration scales. Through numerical simulations and subsequent experimental validation, we found that coordinated directionality plays a key role during wound closure and is inhibited by upregulated PIEZO1 activity. We propose that PIEZO1-mediated retraction suppresses leader cell formation which inhibits coordinated directionality between cells during collective migration.

Phase field modeling and computation of vesicle growth or shrinkage.

We present a phase field model for vesicle growth or shrinkage induced by an osmotic pressure due to a chemical potential gradient. The model consists of an Allen-Cahn equation describing the evolution of the phase field parameter that describes the shape of the vesicle and a Cahn-Hilliard-type equation describing the evolution of the ionic fluid. We establish conditions for vesicle growth or shrinkage via a common tangent construction using free energy curves. During the membrane deformation, the model ensures total mass conservation of the ionic fluid, and we weakly enforce a surface area constraint of the vesicle. We develop a stable numerical scheme and an efficient nonlinear multigrid solver to evolve the phase and concentration fields, and we use this to evolve the fields to near equilibrium for 2D vesicles. Convergence tests confirm an [Formula: see text] accuracy for our scheme and near-optimal convergence for our multigrid solver. Numerical results reveal that the diffuse interface model captures the main features of cell shape dynamics: for a growing vesicle, there exist circle-like equilibrium shapes if the concentration difference across the membrane and the initial osmotic pressure are large enough; while for a shrinking vesicle, there exists a rich collection of finger-like equilibrium morphologies.

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.

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.

Optimal experimental design for mathematical models of haematopoiesis.

The haematopoietic system has a highly regulated and complex structure in which cells are organized to successfully create and maintain new blood cells. It is known that feedback regulation is crucial to tightly control this system, but the specific mechanisms by which control is exerted are not completely understood. In this work, we aim to uncover the underlying mechanisms in haematopoiesis by conducting perturbation experiments, where animal subjects are exposed to an external agent in order to observe the system response and evolution. We have developed a novel Bayesian hierarchical framework for optimal design of perturbation experiments and proper analysis of the data collected. We use a deterministic model that accounts for feedback and feedforward regulation on cell division rates and self-renewal probabilities. A significant obstacle is that the experimental data are not longitudinal, rather each data point corresponds to a different animal. We overcome this difficulty by modelling the unobserved cellular levels as latent variables. We then use principles of Bayesian experimental design to optimally distribute time points at which the haematopoietic cells are quantified. We evaluate our approach using synthetic and real experimental data and show that an optimal design can lead to better estimates of model parameters.

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.

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.

Modeling solvent evaporation during thin film formation in phase separating polymer mixtures.

Preparation of thin films by dissolving polymers in a common solvent followed by evaporation of the solvent has become a routine processing procedure. However, modeling of thin film formation in an evaporating solvent has been challenging due to a need to simulate processes at multiple length and time scales. In this work, we present a methodology based on the principles of linear non-equilibrium thermodynamics, which allows systematic study of various effects such as the changes in the solvent properties due to phase transformation from liquid to vapor and polymer thermodynamics resulting from such solvent transformations. The methodology allows for the derivation of evaporative flux and boundary conditions near each surface for simulations of systems close to the equilibrium. We apply it to study thin film microstructural evolution in phase segregating polymer blends dissolved in a common volatile solvent and deposited on a planar substrate. Effects of the evaporation rates, interactions of the polymers with the underlying substrate and concentration dependent mobilities on the kinetics of thin film formation are studied.

Activation of the HGF/c-Met axis in the tumor microenvironment: A multispecies model.

The tumor microenvironment is an integral component in promoting tumor development. Cancer-associated fibroblasts (CAFs), which reside in the tumor stroma, produce Hepatocyte Growth Factor (HGF), an important trigger for invasive and metastatic tumor behavior. HGF contributes to a pro-tumorigenic environment by activating its cognate receptor, c-Met, on tumor cells. Tumor cells, in turn, secrete growth factors that upregulate HGF production in CAFs, thereby establishing a dynamic tumor-host signaling program. Using a spatiotemporal multispecies model of tumor growth, we investigate how the development and spread of a tumor is impacted by the initiation of a dynamic interaction between tumor-derived growth factors and CAF-derived HGF. We show that establishment of such an interaction results in increased tumor growth and morphological instability, the latter due in part to increased cell species heterogeneity at the tumor-host boundary. Invasive behavior is further increased if the tumor lowers responsiveness to paracrine pro-differentiation signals, which is a hallmark of neoplastic development. By modeling anti-HGF and anti-c-Met therapy, we show how disruption of the HGF/c-Met axis can reduce tumor invasiveness and growth, thereby providing theoretical evidence that targeting tumor-microenvironment interactions is a promising avenue for therapeutic development.

Three-Dimensional Spatiotemporal Modeling of Colon Cancer Organoids Reveals that Multimodal Control of Stem Cell Self-Renewal is a Critical Determinant of Size and Shape in Early Stages of Tumor Growth.

We develop a three-dimensional multispecies mathematical model to simulate the growth of colon cancer organoids containing stem, progenitor and terminally differentiated cells, as a model of early (prevascular) tumor growth. Stem cells (SCs) secrete short-range self-renewal promoters (e.g., Wnt) and their long-range inhibitors (e.g., Dkk) and proliferate slowly. Committed progenitor (CP) cells proliferate more rapidly and differentiate to produce post-mitotic terminally differentiated cells that release differentiation promoters, forming negative feedback loops on SC and CP self-renewal. We demonstrate that SCs play a central role in normal and cancer colon organoids. Spatial patterning of the SC self-renewal promoter gives rise to SC clusters, which mimic stem cell niches, around the organoid surface, and drive the development of invasive fingers. We also study the effects of externally applied signaling factors. Applying bone morphogenic proteins, which inhibit SC and CP self-renewal, reduces invasiveness and organoid size. Applying hepatocyte growth factor, which enhances SC self-renewal, produces larger sizes and enhances finger development at low concentrations but suppresses fingers at high concentrations. These results are consistent with recent experiments on colon organoids. Because many cancers are hierarchically organized and are subject to feedback regulation similar to that in normal tissues, our results suggest that in cancer, control of cancer stem cell self-renewal should influence the size and shape in similar ways, thereby opening the door to novel therapies.

3D Mathematical Modeling of Glioblastoma Suggests That Transdifferentiated Vascular Endothelial Cells Mediate Resistance to Current Standard-of-Care Therapy.

Glioblastoma (GBM), the most aggressive brain tumor in human patients, is decidedly heterogeneous and highly vascularized. Glioma stem/initiating cells (GSC) are found to play a crucial role by increasing cancer aggressiveness and promoting resistance to therapy. Recently, cross-talk between GSC and vascular endothelial cells has been shown to significantly promote GSC self-renewal and tumor progression. Furthermore, GSC also transdifferentiate into bona fide vascular endothelial cells (GEC), which inherit mutations present in GSC and are resistant to traditional antiangiogenic therapies. Here we use three-dimensional mathematical modeling to investigate GBM progression and response to therapy. The model predicted that GSCs drive invasive fingering and that GEC spontaneously form a network within the hypoxic core, consistent with published experimental findings. Standard-of-care treatments using DNA-targeted therapy (radiation/chemo) together with antiangiogenic therapies reduced GBM tumor size but increased invasiveness. Anti-GEC treatments blocked the GEC support of GSCs and reduced tumor size but led to increased invasiveness. Anti-GSC therapies that promote differentiation or disturb the stem cell niche effectively reduced tumor invasiveness and size, but were ultimately limited in reducing tumor size because GECs maintain GSCs. Our study suggests that a combinatorial regimen targeting the vasculature, GSCs, and GECs, using drugs already approved by the FDA, can reduce both tumor size and invasiveness and could lead to tumor eradication. Cancer Res; 77(15); 4171-84. ©2017 AACR.

Recapitulating the human tumor microenvironment: Colon tumor-derived extracellular matrix promotes angiogenesis and tumor cell growth.

Extracellular matrix (ECM) is an essential and dynamic component of all tissues and directly affects cellular behavior by providing both mechanical and biochemical signaling cues. Changes in ECM can alter tissue homeostasis, potentially leading to promotion of cellular transformation and the generation of tumors. Therefore, understanding ECM compositional changes during cancer progression is vital to the development of targeted treatments. Previous efforts to reproduce the native 3D cellular microenvironment have utilized protein gels and scaffolds that incompletely recapitulate the complexity of native tissues. Here, we address this problem by extracting and comparing ECM from normal human colon and colon tumor that had metastasized to liver. We found differences in protein composition and stiffness, and observed significant differences in vascular network formation and tumor growth in each of the reconstituted matrices, both in vitro and in vivo. We studied free/bound ratios of NADH in the tumor and endothelial cells using Fluorescence Lifetime Imaging Microscopy as a surrogate for the metabolic state of the cells. We observed that cells seeded in tumor ECM had higher relative levels of free NADH, consistent with a higher glycolytic rate, than those seeded in normal ECM. These results demonstrate that the ECM plays an important role in the growth of cancer cells and their associated vasculature.

Multiscale Modeling of Glioblastoma Suggests that the Partial Disruption of Vessel/Cancer Stem Cell Crosstalk Can Promote Tumor Regression Without Increasing Invasiveness.

OBJECTIVE: In glioblastoma, the crosstalk between vascular endothelial cells (VECs) and glioma stem cells (GSCs) has been shown to enhance tumor growth. We propose a multiscale mathematical model to study this mechanism, explore tumor growth under various initial and microenvironmental conditions, and investigate the effects of blocking this crosstalk. METHODS: We develop a hybrid continuum-discrete model of highly organized vascularized tumors. VEC-GSC crosstalk is modeled via vascular endothelial growth factor (VEGF) production by tumor cells and by secretion of soluble factors by VECs that promote GSC self-renewal and proliferation. RESULTS: VEC-GSC crosstalk increases both tumor size and GSC fraction by enhancing GSC activity and neovascular development. VEGF promotes vessel formation, and larger VEGF sources typically increase vessel numbers, which enhances tumor growth and stabilizes the tumor shape. Increasing the initial GSC fraction has a similar effect. Partially disrupting the crosstalk by blocking VEC secretion of GSC promoters reduces tumor size but does not increase invasiveness, which is in contrast to antiangiogenic therapies, which reduce tumor size but may significantly increase tumor invasiveness. SIGNIFICANCE: Multiscale modeling supports the targeting of VEC-GSC crosstalk as a promising approach for cancer therapy.

Feedback, Lineages and Self-Organizing Morphogenesis.

Feedback regulation of cell lineage progression plays an important role in tissue size homeostasis, but whether such feedback also plays an important role in tissue morphogenesis has yet to be explored. Here we use mathematical modeling to show that a particular feedback architecture in which both positive and negative diffusible signals act on stem and/or progenitor cells leads to the appearance of bistable or bi-modal growth behaviors, ultrasensitivity to external growth cues, local growth-driven budding, self-sustaining elongation, and the triggering of self-organization in the form of lamellar fingers. Such behaviors arise not through regulation of cell cycle speeds, but through the control of stem or progenitor self-renewal. Even though the spatial patterns that arise in this setting are the result of interactions between diffusible factors with antagonistic effects, morphogenesis is not the consequence of Turing-type instabilities.

Collective Properties of a Transcription Initiation Model Under Varying Environment.

The dynamics of gene transcription is tightly regulated in eukaryotes. Recent experiments have revealed various kinds of transcriptional dynamics, such as RNA polymerase II pausing, that involves regulation at the transcription initiation stage, and the choice of different regulation pattern is closely related to the physiological functions of the target gene. Here we consider a simplified model of transcription initiation, a process including the assembly of transcription complex and the pausing and releasing of the RNA polymerase II. Focusing on the collective behaviors of a population level, we explore the potential regulatory functions this model can offer. These functions include fast and synchronized response to environmental change, or long-term memory about the transcriptional status. As a proof of concept we also show that, by selecting different control mechanisms cells can adapt to different environments. These findings may help us better understand the design principles of transcriptional regulation.

Tumor growth in complex, evolving microenvironmental geometries: a diffuse domain approach.

We develop a mathematical model of tumor growth in complex, dynamic microenvironments with active, deformable membranes. Using a diffuse domain approach, the complex domain is captured implicitly using an auxiliary function and the governing equations are appropriately modified, extended and solved in a larger, regular domain. The diffuse domain method enables us to develop an efficient numerical implementation that does not depend on the space dimension or the microenvironmental geometry. We model homotypic cell-cell adhesion and heterotypic cell-basement membrane (BM) adhesion with the latter being implemented via a membrane energy that models cell-BM interactions. We incorporate simple models of elastic forces and the degradation of the BM and ECM by tumor-secreted matrix degrading enzymes. We investigate tumor progression and BM response as a function of cell-BM adhesion and the stiffness of the BM. We find tumor sizes tend to be positively correlated with cell-BM adhesion since increasing cell-BM adhesion results in thinner, more elongated tumors. Prior to invasion of the tumor into the stroma, we find a negative correlation between tumor size and BM stiffness as the elastic restoring forces tend to inhibit tumor growth. In order to model tumor invasion of the stroma, we find it necessary to downregulate cell-BM adhesiveness, which is consistent with experimental observations. A stiff BM promotes invasiveness because at early stages the opening in the BM created by MDE degradation from tumor cells tends to be narrower when the BM is stiffer. This requires invading cells to squeeze through the narrow opening and thus promotes fragmentation that then leads to enhanced growth and invasion. In three dimensions, the opening in the BM was found to increase in size even when the BM is stiff because of pressure induced by growing tumor clusters. A larger opening in the BM can increase the potential for further invasiveness by increasing the possibility that additional tumor cells could invade the stroma.

The effect of interstitial pressure on therapeutic agent transport: coupling with the tumor blood and lymphatic vascular systems.

Vascularized tumor growth is characterized by both abnormal interstitial fluid flow and the associated interstitial fluid pressure (IFP). Here, we study the effect that these conditions have on the transport of therapeutic agents during chemotherapy. We apply our recently developed vascular tumor growth model which couples a continuous growth component with a discrete angiogenesis model to show that hypertensive IFP is a physical barrier that may hinder vascular extravasation of agents through transvascular fluid flux convection, which drives the agents away from the tumor. This result is consistent with previous work using simpler models without blood flow or lymphatic drainage. We consider the vascular/interstitial/lymphatic fluid dynamics to show that tumors with larger lymphatic resistance increase the agent concentration more rapidly while also experiencing faster washout. In contrast, tumors with smaller lymphatic resistance accumulate less agents but are able to retain them for a longer time. The agent availability (area-under-the curve, or AUC) increases for less permeable agents as lymphatic resistance increases, and correspondingly decreases for more permeable agents. We also investigate the effect of vascular pathologies on agent transport. We show that elevated vascular hydraulic conductivity contributes to the highest AUC when the agent is less permeable, but to lower AUC when the agent is more permeable. We find that elevated interstitial hydraulic conductivity contributes to low AUC in general regardless of the transvascular agent transport capability. We also couple the agent transport with the tumor dynamics to simulate chemotherapy with the same vascularized tumor under different vascular pathologies. We show that tumors with an elevated interstitial hydraulic conductivity alone require the strongest dosage to shrink. We further show that tumors with elevated vascular hydraulic conductivity are more hypoxic during therapy and that the response slows down as the tumor shrinks due to the heterogeneity and low concentration of agents in the tumor interior compared with the cases where other pathological effects may combine to flatten the IFP and thus reduce the heterogeneity. We conclude that dual normalizations of the micronevironment - both the vasculature and the interstitium - are needed to maximize the effects of chemotherapy, while normalization of only one of these may be insufficient to overcome the physical resistance and may thus lead to sub-optimal outcomes.

A stable scheme for a nonlinear, multiphase tumor growth model with an elastic membrane.

In this paper, we extend the 3D multispecies diffuse-interface model of the tumor growth, which was derived in Wise et al. (Three-dimensional multispecies nonlinear tumor growth-I: model and numerical method, J. Theor. Biol. 253 (2008) 524-543), and incorporate the effect of a stiff membrane to model tumor growth in a confined microenvironment. We then develop accurate and efficient numerical methods to solve the model. When the membrane is endowed with a surface energy, the model is variational, and the numerical scheme, which involves adaptive mesh refinement and a nonlinear multigrid finite difference method, is demonstrably shown to be energy stable. Namely, in the absence of cell proliferation and death, the discrete energy is a nonincreasing function of time for any time and space steps. When a simplified model of membrane elastic energy is used, the resulting model is derived analogously to the surface energy case. However, the elastic energy model is actually nonvariational because certain coupling terms are neglected. Nevertheless, a very stable numerical scheme is developed following the strategy used in the surface energy case. 2D and 3D simulations are performed that demonstrate the accuracy of the algorithm and illustrate the shape instabilities and nonlinear effects of membrane elastic forces that may resist or enhance growth of the tumor. Compared with the standard Crank-Nicholson method, the time step can be up to 25 times larger using the new approach.