Publications by authors named "John S Lowengrub"

31 Publications

Optimal experimental design for mathematical models of haematopoiesis.

J R Soc Interface 2021 01 27;18(174):20200729. Epub 2021 Jan 27.

Center for Complex Biological Systems, University of California Irvine, Irvine, CA, USA.

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.
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http://dx.doi.org/10.1098/rsif.2020.0729DOI Listing
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC7879761PMC
January 2021

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

J Theor Biol 2021 01 29;509:110499. Epub 2020 Oct 29.

Department of Mathematics, University of California Irvine, Irvine, CA 92697, United States. Electronic address:

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.
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http://dx.doi.org/10.1016/j.jtbi.2020.110499DOI Listing
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC8445233PMC
January 2021

Tumor growth and calcification in evolving microenvironmental geometries.

J Theor Biol 2019 02 5;463:138-154. Epub 2018 Dec 5.

Department of Mathematics, Department of Biomedical Engineering, Center for Complex Biological Systems, University of California, Irvine, USA. Electronic address:

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.
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http://dx.doi.org/10.1016/j.jtbi.2018.12.006DOI Listing
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC6476430PMC
February 2019

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

Clin Cancer Res 2018 12 6;24(23):5883-5894. Epub 2018 Aug 6.

Sheikh Ahmed Center for Pancreatic Cancer Research, The University of Texas MD Anderson Cancer Center, Houston, Texas.

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.
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http://dx.doi.org/10.1158/1078-0432.CCR-17-3668DOI Listing
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC6279613PMC
December 2018

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

Soft Matter 2018 Mar;14(10):1833-1846

Department of Mathematics, The University of Tennessee, Knoxville, TN-37996, USA.

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.
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http://dx.doi.org/10.1039/c7sm02560bDOI Listing
March 2018

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

J Theor Biol 2018 02 5;439:86-99. Epub 2017 Dec 5.

Department of Mathematics, University of California, Irvine, CA, USA; Center for Complex Biological Systems, University of California, Irvine, CA, USA; Department of Biomedical Engineering, University of California, Irvine, CA, USA; Chao Family Comprehensive Cancer Center, University of California, Irvine, USA. Electronic address:

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.
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http://dx.doi.org/10.1016/j.jtbi.2017.11.025DOI Listing
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC5799021PMC
February 2018

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.

Bull Math Biol 2018 05 5;80(5):1404-1433. Epub 2017 Jul 5.

Department of Mathematics, Department of Biomedical Engineering, Center for Complex Biological Systems, and Chao Comprehensive Cancer Center, University of California, Irvine, Irvine, CA, 92697, USA.

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.
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http://dx.doi.org/10.1007/s11538-017-0294-1DOI Listing
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC5756149PMC
May 2018

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

Cancer Res 2017 08 23;77(15):4171-4184. Epub 2017 May 23.

Department of Mathematics, University of California, Irvine, California.

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. .
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http://dx.doi.org/10.1158/0008-5472.CAN-16-3094DOI Listing
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC5540807PMC
August 2017

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

Biomaterials 2017 02 24;116:118-129. Epub 2016 Nov 24.

Department of Biomedical Engineering, The Henry Samueli School of Engineering, UC Irvine, USA; Department of Molecular Biology and Biochemistry, School of Biological Sciences, UC Irvine, USA; The Edwards Lifesciences Center for Advanced Cardiovascular Technology, UC Irvine, USA. Electronic address:

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.
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http://dx.doi.org/10.1016/j.biomaterials.2016.11.034DOI Listing
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC5226635PMC
February 2017

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

IEEE Trans Biomed Eng 2017 03 7;64(3):538-548. Epub 2016 Oct 7.

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.
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http://dx.doi.org/10.1109/TBME.2016.2615566DOI Listing
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC5501302PMC
March 2017

Feedback, Lineages and Self-Organizing Morphogenesis.

PLoS Comput Biol 2016 Mar 18;12(3):e1004814. Epub 2016 Mar 18.

Department of Biomedical Engineering, University of California, Irvine, Irvine, California, United States of America.

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.
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http://dx.doi.org/10.1371/journal.pcbi.1004814DOI Listing
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4798729PMC
March 2016

Collective Properties of a Transcription Initiation Model Under Varying Environment.

J Comput Biol 2016 Jan 8;23(1):56-66. Epub 2015 Dec 8.

2 Department of Mathematics, University of California-Irvine , Irvine, California.

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.
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http://dx.doi.org/10.1089/cmb.2015.0144DOI Listing
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4700395PMC
January 2016

Predicting mechanism of biphasic growth factor action on tumor growth using a multi-species model with feedback control.

J Coupled Syst Multiscale Dyn 2013 Dec;1(4):459-467

Department of Mathematics, University of California, Irvine, CA 92697-3875, USA ; Center for Complex Biological Systems, University of California, 2620 Biological Sciences III, Irvine, CA 92697-2280, USA ; Department of Biomedical Engineering, University of California, 3120 Natural Sciences II, Irvine, CA 92697-2715, USA.

A large number of growth factors and drugs are known to act in a biphasic manner: at lower concentrations they cause increased division of target cells, whereas at higher concentrations the mitogenic effect is inhibited. Often, the molecular details of the mitogenic effect of the growth factor are known, whereas the inhibitory effect is not. Hepatoctyte Growth Factor, HGF, has recently been recognized as a strong mitogen that is present in the microenvironment of solid tumors. Recent evidence suggests that HGF acts in a biphasic manner on tumor growth. We build a multi-species model of HGF action on tumor cells using different hypotheses for high dose-HGF activation of a growth inhibitor and show that the shape of the dose-response curve is directly related to the mechanism of inhibitor activation. We thus hypothesize that the shape of a dose-response curve is informative of the molecular action of the growth factor on the growth inhibitor.
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http://dx.doi.org/10.1166/jcsmd.2013.1028DOI Listing
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4112130PMC
December 2013

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

J Theor Biol 2014 Nov 9;361:14-30. Epub 2014 Jul 9.

Department of Mathematics, Department of Biomedical Engineering, Center for Complex Biological Systems, University of California, Irvine, USA. Electronic address:

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.
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http://dx.doi.org/10.1016/j.jtbi.2014.06.024DOI Listing
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4252595PMC
November 2014

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

J Theor Biol 2014 Aug 19;355:194-207. Epub 2014 Apr 19.

Department of Mathematics, University of California, Irvine, United States; Department of Biomedical Engineering, University of California, Irvine, United States. Electronic address:

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.
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http://dx.doi.org/10.1016/j.jtbi.2014.04.012DOI Listing
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4098870PMC
August 2014

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

Int J Numer Method Biomed Eng 2014 Jul 17;30(7):726-54. Epub 2014 Jan 17.

Department of Mathematics, University of California, Irvine, CA, USA.

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.
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http://dx.doi.org/10.1002/cnm.2624DOI Listing
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4149601PMC
July 2014

Dynamic density functional theory of solid tumor growth: Preliminary models.

AIP Adv 2012 Mar 22;2(1):11210. Epub 2012 Mar 22.

Cancer is a disease that can be seen as a complex system whose dynamics and growth result from nonlinear processes coupled across wide ranges of spatio-temporal scales. The current mathematical modeling literature addresses issues at various scales but the development of theoretical methodologies capable of bridging gaps across scales needs further study. We present a new theoretical framework based on Dynamic Density Functional Theory (DDFT) extended, for the first time, to the dynamics of living tissues by accounting for cell density correlations, different cell types, phenotypes and cell birth/death processes, in order to provide a biophysically consistent description of processes across the scales. We present an application of this approach to tumor growth.
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http://dx.doi.org/10.1063/1.3699065DOI Listing
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3321520PMC
March 2012

Axisymmetric multicomponent vesicles: A comparison of hydrodynamic and geometric models.

Int J Numer Method Biomed Eng 2012 Mar;28(3):346-68

Department of Mathematics, University of California, Los Angeles, Los Angeles, CA, USA.

Using a mathematical model, we investigate the role of hydrodynamic forces on three-dimensional axisymmetric multicomponent vesicles. The equations are derived using an energy variation approach that accounts for different surface phases, the excess energy associated with surface domain boundaries, bending energy and inextensibility. The equations are high-order (fourth order) nonlinear and nonlocal. To solve the equations numerically, we use a nonstiff, pseudo-spectral boundary integral method that relies on an analysis of the equations at small scales. We also derive equations governing the dynamics of inextensible vesicles evolving in the absence of hydrodynamic forces and simulate numerically the evolution of this geometric model. We find that compared with the geometric model, hydrodynamic forces provide additional paths for relaxing inextensible vesicles. The presence of hydrodynamic forces may enable the dynamics to overcome local energy barriers and reach lower energy states than those accessible by geometric motion or energy minimization algorithms. Because of the intimate connection between morphology, surface phase distribution and biological function, these results have important consequences in understanding the role vesicles play in biological processes.
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http://dx.doi.org/10.1002/cnm.2475DOI Listing
March 2012

Physical oncology: a bench-to-bedside quantitative and predictive approach.

Cancer Res 2011 Jan 11;71(2):298-302. Epub 2011 Jan 11.

Department of Bioengineering and James Graham Brown Cancer Center, University of Louisville, Louisville, Kentucky 40208, USA.

Cancer models relating basic science to clinical care in oncology may fail to address the nuances of tumor behavior and therapy, as in the case, discussed herein, of the complex multiscale dynamics leading to the often-observed enhanced invasiveness, paradoxically induced by the very antiangiogenic therapy designed to destroy the tumor. Studies would benefit from approaches that quantitatively link the multiple physical and temporal scales from molecule to tissue in order to offer outcome predictions for individual patients. Physical oncology is an approach that applies fundamental principles from the physical and biological sciences to explain certain cancer behaviors as observable characteristics arising from the underlying physical and biochemical events. For example, the transport of oxygen molecules through tissue affects phenotypic characteristics such as cell proliferation, apoptosis, and adhesion, which in turn underlie the patient-scale tumor growth and invasiveness. Our review of physical oncology illustrates how tumor behavior and treatment response may be a quantifiable function of marginally stable molecular and/or cellular conditions modulated by inhomogeneity. By incorporating patient-specific genomic, proteomic, metabolomic, and cellular data into multiscale physical models, physical oncology could complement current clinical practice through enhanced understanding of cancer behavior, thus potentially improving patient survival.
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http://dx.doi.org/10.1158/0008-5472.CAN-10-2676DOI Listing
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3073485PMC
January 2011

Mathematical Oncology: How Are the Mathematical and Physical Sciences Contributing to the War on Breast Cancer?

Curr Breast Cancer Rep 2010 Sep 22;2(3):121-129. Epub 2010 Jul 22.

Mathematical modeling has recently been added as a tool in the fight against cancer. The field of mathematical oncology has received great attention and increased enormously, but over-optimistic estimations about its ability have created unrealistic expectations. We present a critical appraisal of the current state of mathematical models of cancer. Although the field is still expanding and useful clinical applications may occur in the future, managing over-expectation requires the proposal of alternative directions for mathematical modeling. Here, we propose two main avenues for this modeling: 1) the identification of the elementary biophysical laws of cancer development, and 2) the development of a multiscale mathematical theory as the framework for models predictive of tumor growth. Finally, we suggest how these new directions could contribute to addressing the current challenges of understanding breast cancer growth and metastasis.
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http://dx.doi.org/10.1007/s12609-010-0020-6DOI Listing
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2987530PMC
September 2010

Dynamics of multicomponent vesicles in a viscous fluid.

J Comput Phys 2010 ;229(1):119-144

Department of Mathematics, University of California, Irvine, USA.

We develop and investigate numerically a thermodynamically consistent model of two-dimensional multicomponent vesicles in an incompressible viscous fluid. The model is derived using an energy variation approach that accounts for different lipid surface phases, the excess energy (line energy) associated with surface phase domain boundaries, bending energy, spontaneous curvature, local inextensibility and fluid flow via the Stokes equations. The equations are high-order (fourth order) nonlinear and nonlocal due to incompressibil-ity of the fluid and the local inextensibility of the vesicle membrane. To solve the equations numerically, we develop a nonstiff, pseudo-spectral boundary integral method that relies on an analysis of the equations at small scales. The algorithm is closely related to that developed very recently by Veerapaneni et al. [81] for homogeneous vesicles although we use a different and more efficient time stepping algorithm and a reformulation of the inextensibility equation. We present simulations of multicomponent vesicles in an initially quiescent fluid and investigate the effect of varying the average surface concentration of an initially unstable mixture of lipid phases. The phases then redistribute and alter the morphology of the vesicle and its dynamics. When an applied shear is introduced, an initially elliptical vesicle tank-treads and attains a steady shape and surface phase distribution. A sufficiently elongated vesicle tumbles and the presence of different surface phases with different bending stiffnesses and spontaneous curvatures yields a complex evolution of the vesicle morphology as the vesicle bends in regions where the bending stiffness and spontaneous curvature are small.
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http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2929801PMC
http://dx.doi.org/10.1016/j.jcp.2009.09.017DOI Listing
January 2010

Selection in spatial stochastic models of cancer: migration as a key modulator of fitness.

Biol Direct 2010 Apr 20;5:21. Epub 2010 Apr 20.

Department of Mathematics, University of California Irvine, Irvine, California, USA.

Background: We study the selection dynamics in a heterogeneous spatial colony of cells. We use two spatial generalizations of the Moran process, which include cell divisions, death and migration. In the first model, migration is included explicitly as movement to a proximal location. In the second, migration is implicit, through the varied ability of cell types to place their offspring a distance away, in response to another cell's death.

Results: In both models, we find that migration has a direct positive impact on the ability of a single mutant cell to invade a pre-existing colony. Thus, a decrease in the growth potential can be compensated by an increase in cell migration. We further find that the neutral ridges (the set of all types with the invasion probability equal to that of the host cells) remain invariant under the increase of system size (for large system sizes), thus making the invasion probability a universal characteristic of the cells selection status. We find that repeated instances of large scale cell-death, such as might arise during therapeutic intervention or host response, strongly select for the migratory phenotype.

Conclusions: These models can help explain the many examples in the biological literature, where genes involved in cell's migratory and invasive machinery are also associated with increased cellular fitness, even though there is no known direct effect of these genes on the cellular reproduction. The models can also help to explain how chemotherapy may provide a selection mechanism for highly invasive phenotypes.
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http://dx.doi.org/10.1186/1745-6150-5-21DOI Listing
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2873940PMC
April 2010

Three-dimensional multispecies nonlinear tumor growth-II: Tumor invasion and angiogenesis.

J Theor Biol 2010 Jun 18;264(4):1254-78. Epub 2010 Mar 18.

School of Health Information Sciences, The University of Texas Health Science Center, Houston, TX 77054, USA.

We extend the diffuse interface model developed in Wise et al. (2008) to study nonlinear tumor growth in 3-D. Extensions include the tracking of multiple viable cell species populations through a continuum diffuse-interface method, onset and aging of discrete tumor vessels through angiogenesis, and incorporation of individual cell movement using a hybrid continuum-discrete approach. We investigate disease progression as a function of cellular-scale parameters such as proliferation and oxygen/nutrient uptake rates. We find that heterogeneity in the physiologically complex tumor microenvironment, caused by non-uniform distribution of oxygen, cell nutrients, and metabolites, as well as phenotypic changes affecting cellular-scale parameters, can be quantitatively linked to the tumor macro-scale as a mechanism that promotes morphological instability. This instability leads to invasion through tumor infiltration of surrounding healthy tissue. Models that employ a biologically founded, multiscale approach, as illustrated in this work, could help to quantitatively link the critical effect of heterogeneity in the tumor microenvironment with clinically observed tumor growth and invasion. Using patient tumor-specific parameter values, this may provide a predictive tool to characterize the complex in vivo tumor physiological characteristics and clinical response, and thus lead to improved treatment modalities and prognosis.
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http://dx.doi.org/10.1016/j.jtbi.2010.02.036DOI Listing
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2912164PMC
June 2010

Control of viscous fingering patterns in a radial Hele-Shaw cell.

Phys Rev Lett 2009 May 30;102(17):174501. Epub 2009 Apr 30.

Department of Applied Mathematics, Illinois Institute of Technology, Chicago, Illinois 60616, USA.

We study numerically and experimentally the dynamics and control of viscous fingering patterns in a circular Hele-Shaw cell. The nonlocality and nonlinearity of the system, especially interactions among developing fingers, make the emergent pattern difficult to predict and control. By controlling the injection rate of the less viscous fluid, we can precisely suppress the evolving interfacial instabilities. There exist denumerable attractive, self-similarly evolving symmetric, universal shapes. Experiments confirm the feasibility of the control strategy, which is summarized in a morphology diagram.
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http://dx.doi.org/10.1103/PhysRevLett.102.174501DOI Listing
May 2009

Phase-field modeling of the dynamics of multicomponent vesicles: Spinodal decomposition, coarsening, budding, and fission.

Phys Rev E Stat Nonlin Soft Matter Phys 2009 Mar 31;79(3 Pt 1):031926. Epub 2009 Mar 31.

Department of Mathematics, University of California, Irvine, California 92697-3875, USA.

We develop a thermodynamically consistent phase-field model to simulate the dynamics of multicomponent vesicles. The model accounts for bending stiffness, spontaneous curvature, excess (surface) energy, and a line tension between the coexisting surface phases. Our approach is similar to that recently used by Wang and Du [J. Math. Biol. 56, 347 (2008)] with a key difference. Here, we concentrate on the dynamic evolution and solve the surface mass conservation equation explicitly; this equation was not considered by Wang and Du. The resulting fourth-order strongly coupled system of nonlinear nonlocal equations are solved numerically using an adaptive finite element numerical method. Although the system is valid for three dimensions, we limit our studies here to two dimensions where the vesicle is a curve. Differences between the spontaneous curvatures and the bending rigidities of the surface phases are found numerically to lead to the formation of buds, asymmetric vesicle shapes and vesicle fission even in two dimensions. In addition, simulations of configurations far from equilibrium indicate that phase separation via spinodal decomposition and coarsening not only affect the vesicle shape but also that the vesicle shape affects the phase separation dynamics, especially the coarsening and may lead to lower energy states than might be achieved by evolving initially phase-separated configurations.
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http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3037283PMC
http://dx.doi.org/10.1103/PhysRevE.79.031926DOI Listing
March 2009

Multiparameter computational modeling of tumor invasion.

Cancer Res 2009 May 14;69(10):4493-501. Epub 2009 Apr 14.

Department of Pathology and Laboratory Medicine, and Division of Engineering, Brown University, Providence, RI, USA.

Clinical outcome prognostication in oncology is a guiding principle in therapeutic choice. A wealth of qualitative empirical evidence links disease progression with tumor morphology, histopathology, invasion, and associated molecular phenomena. However, the quantitative contribution of each of the known parameters in this progression remains elusive. Mathematical modeling can provide the capability to quantify the connection between variables governing growth, prognosis, and treatment outcome. By quantifying the link between the tumor boundary morphology and the invasive phenotype, this work provides a quantitative tool for the study of tumor progression and diagnostic/prognostic applications. This establishes a framework for monitoring system perturbation towards development of therapeutic strategies and correlation to clinical outcome for prognosis.
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http://dx.doi.org/10.1158/0008-5472.CAN-08-3834DOI Listing
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2835777PMC
May 2009

Nonlinear simulations of solid tumor growth using a mixture model: invasion and branching.

J Math Biol 2009 Apr 12;58(4-5):723-63. Epub 2008 Sep 12.

The University of Texas MD Anderson Cancer Center, School of Health Information Sciences, University of Texas Health Science Center, Houston, TX 77030, USA.

We develop a thermodynamically consistent mixture model for avascular solid tumor growth which takes into account the effects of cell-to-cell adhesion, and taxis inducing chemical and molecular species. The mixture model is well-posed and the governing equations are of Cahn-Hilliard type. When there are only two phases, our asymptotic analysis shows that earlier single-phase models may be recovered as limiting cases of a two-phase model. To solve the governing equations, we develop a numerical algorithm based on an adaptive Cartesian block-structured mesh refinement scheme. A centered-difference approximation is used for the space discretization so that the scheme is second order accurate in space. An implicit discretization in time is used which results in nonlinear equations at implicit time levels. We further employ a gradient stable discretization scheme so that the nonlinear equations are solvable for very large time steps. To solve those equations we use a nonlinear multilevel/multigrid method which is of an optimal order O(N) where N is the number of grid points. Spherically symmetric and fully two dimensional nonlinear numerical simulations are performed. We investigate tumor evolution in nutrient-rich and nutrient-poor tissues. A number of important results have been uncovered. For example, we demonstrate that the tumor may suffer from taxis-driven fingering instabilities which are most dramatic when cell proliferation is low, as predicted by linear stability theory. This is also observed in experiments. This work shows that taxis may play a role in tumor invasion and that when nutrient plays the role of a chemoattractant, the diffusional instability is exacerbated by nutrient gradients. Accordingly, we believe this model is capable of describing complex invasive patterns observed in experiments.
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http://dx.doi.org/10.1007/s00285-008-0215-xDOI Listing
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3037274PMC
April 2009

A New Ghost Cell/Level Set Method for Moving Boundary Problems: Application to Tumor Growth.

J Sci Comput 2008 Jun;35(2-3):266-299

SHIS, U. of Texas Health Science Center, 7000 Fannin, Suite 600, Houston, TX 77030, USA.

In this paper, we present a ghost cell/level set method for the evolution of interfaces whose normal velocity depend upon the solutions of linear and nonlinear quasi-steady reaction-diffusion equations with curvature-dependent boundary conditions. Our technique includes a ghost cell method that accurately discretizes normal derivative jump boundary conditions without smearing jumps in the tangential derivative; a new iterative method for solving linear and nonlinear quasi-steady reaction-diffusion equations; an adaptive discretization to compute the curvature and normal vectors; and a new discrete approximation to the Heaviside function. We present numerical examples that demonstrate better than 1.5-order convergence for problems where traditional ghost cell methods either fail to converge or attain at best sub-linear accuracy. We apply our techniques to a model of tumor growth in complex, heterogeneous tissues that consists of a nonlinear nutrient equation and a pressure equation with geometry-dependent jump boundary conditions. We simulate the growth of glioblastoma (an aggressive brain tumor) into a large, 1 cm square of brain tissue that includes heterogeneous nutrient delivery and varied biomechanical characteristics (white matter, gray matter, cerebrospinal fluid, and bone), and we observe growth morphologies that are highly dependent upon the variations of the tissue characteristics-an effect observed in real tumor growth.
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http://dx.doi.org/10.1007/s10915-008-9190-zDOI Listing
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3039490PMC
June 2008

Linear stability analysis for step meandering instabilities with elastic interactions and Ehrlich-Schwoebel barriers.

Phys Rev E Stat Nonlin Soft Matter Phys 2007 Jul 11;76(1 Pt 1):011601. Epub 2007 Jul 11.

Department of Materials Science and Engineering, University of Michigan, Ann Arbor, Michigan 48109, USA.

Vicinal surfaces are known to exhibit morphological instabilities during step-flow growth. Through a linear stability analysis of step meandering instabilities, we investigate two effects that are important in many heteroepitaxial systems: elastic monopole-monopole interactions arising from bulk stress and the Ehrlich-Schwoebel (ES) barriers due to the asymmetric adatom incorporation rates. The analysis shows that the effects of the ES barriers increase as the average terrace width increases, whereas the effects of elastic monopole-monopole interactions decrease. The ES barriers favor an in-phase step pattern with a zero phase shift between consecutive steps, while elastic stress favors an out-of-phase pattern with a phase shift of pi. However, our analysis shows that the instability growth rate becomes nearly independent of the phase shift when either the ES-barrier effect or the stress effect is large. In particular, for ES-barrier-driven instability, the in-phase step pattern develops only within an intermediate range of terrace widths when bulk stress exists. Similarly, for the elastic-interaction-driven instability, an out-of-phase pattern only forms within a certain range of monopole strength; if the strength is too small, the ES barrier effect dominates, and if it is too large, the peak in the instability growth rate becomes delocalized in the phase shift and no patterns form. This transition between patterned and random step morphologies depends on the monopole strength, but is independent of the terrace width. A phase diagram that describes the regions of the ES-barrier-dominant instability and the elastic-interaction-dominant instability is established, along with the morphological phase diagrams that predict the step configurations as a function of the controlling parameters for the two types of instabilities.
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http://dx.doi.org/10.1103/PhysRevE.76.011601DOI Listing
July 2007
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