A multiscale model for avascular tumor growth

The desire to understand tumor complexity has given rise to mathematical models to describe the tumor microenvironment. We present a new mathematical model for avascular tumor growth and development that spans three distinct scales. At the cellular level, a lattice Monte Carlo model describes cellular dynamics (proliferation, adhesion, and viability). At the subcellular level, a Boolean network regulates the expression of proteins that control the cell cycle. At the extracellular level, reaction-diffusion equations describe the chemical dynamics (nutrient, waste, growth promoter, and inhibitor concentrations). Data from experiments with multicellular spheroids were used to determine the parameters of the simulations. Starting with a single tumor cell, this model produces an avascular tumor that quantitatively mimics experimental measurements in multicellular spheroids. Based on the simulations, we predict: 1), the microenvironmental conditions required for tumor cell survival; and 2), growth promoters and inhibitors have diffusion coefficients in the range between 10(-6) and 10(-7) cm2/h, corresponding to molecules of size 80-90 kDa. Using the same parameters, the model also accurately predicts spheroid growth curves under different external nutrient supply conditions.     

A Mathematical Model of the Enhanced Permeability and Retention Effect for Liposome Transport in Solid Tumors

The discovery of the enhanced permeability and retention (EPR) effect has resulted in the development of nanomedicines, including liposome-based formulations of drugs, as cancer therapies. The use of liposomes has resulted in substantial increases in accumulation of drugs in solid tumors; yet, significant improvements in therapeutic efficacy have yet to be achieved. Imaging of the tumor accumulation of liposomes has revealed that this poor or variable performance is in part due to heterogeneous inter-subject and intra-tumoral liposome accumulation, which occurs as a result of an abnormal transport microenvironment. A mathematical model that relates liposome accumulation to the underlying transport properties in solid tumors could provide insight into inter and intra-tumoral variations in the EPR effect. In this paper, we present a theoretical framework to describe liposome transport in solid tumors. The mathematical model is based on biophysical transport equations that describe pressure driven fluid flow across blood vessels and through the tumor interstitium. The model was validated by direct comparison with computed tomography measurements of tumor accumulation of liposomes in three preclinical tumor models. The mathematical model was fit to liposome accumulation curves producing predictions of transport parameters that reflect the tumor microenvironment. Notably, all fits had a high coefficient of determination and predictions of interstitial fluid pressure agreed with previously published independent measurements made in the same tumor type. Furthermore, it was demonstrated that the model attributed inter-subject heterogeneity in liposome accumulation to variations in peak interstitial fluid pressure. These findings highlight the relationship between transvascular and interstitial flow dynamics and variations in the EPR effect. In conclusion, we have presented a theoretical framework that predicts inter-subject and intra-tumoral variations in the EPR effect based on fundamental properties of the tumor microenvironment and forms the basis for transport modeling of liposome drug delivery.     

Theory and Experimental Validation of a Spatio-temporal Model of Chemotherapy Transport to Enhance Tumor Cell Kill

It has been hypothesized that continuously releasing drug molecules into the tumor over an extended period of time may significantly improve the chemotherapeutic efficacy by overcoming physical transport limitations of conventional bolus drug treatment. In this paper, we present a generalized space- and time-dependent mathematical model of drug transport and drug-cell interactions to quantitatively formulate this hypothesis. Model parameters describe: perfusion and tissue architecture (blood volume fraction and blood vessel radius); diffusion penetration distance of drug (i.e., a function of tissue compactness and drug uptake rates by tumor cells); and cell death rates (as function of history of drug uptake). We performed preliminary testing and validation of the mathematical model using in vivo experiments with different drug delivery methods on a breast cancer mouse model. Experimental data demonstrated a 3-fold increase in response using nano-vectored drug vs. free drug delivery, in excellent quantitative agreement with the model predictions. Our model results implicate that therapeutically targeting blood volume fraction, e.g., through vascular normalization, would achieve a better outcome due to enhanced drug delivery.

Author Summary

Cancer treatment efficacy can be significantly enhanced through the elution of drug from nano-carriers that can temporarily stay in the tumor vasculature. Here we present a relatively simple yet powerful mathematical model that accounts for both spatial and temporal heterogeneities of drug dosing to help explain, examine, and prove this concept. We find that the delivery of systemic chemotherapy through a certain form of nano-carriers would have enhanced tumor kill by a factor of 2 to 4 over the standard therapy that the patients actually received. We also find that targeting blood volume fraction (a parameter of the model) through vascular normalization can achieve more effective drug delivery and tumor kill. More importantly, this model only requires a limited number of parameters which can all be readily assessed from standard clinical diagnostic measurements (e.g., histopathology and CT). This addresses an important challenge in current translational research and justifies further development of the model towards clinical translation.     

An application of Lacker's mathematical model for the prediction of ovarian response to superstimulation

Introduction. A mathematical model of ovarian follicular growth is applied to the problem of predicting ovarian response in a superstimulation protocol. Methods. Fifty-four women enrolled in an ovarian superstimulation program of therapy for the amelioration of idiopathic infertility had their ovarian cycles synchronized by taking Demulen 30 for two weeks prior to the study. Daily ultrasonographic imaging, measurements of serum estradiol and doses of hMG began on day 5 after the patients stopped taking Demulen. The diameters of individual follicles were measured and followed daily. When the largest follicle attained a diameter of 19 mm, hCG was given to induce ovulation. Individual follicle growth data were fit to a mathematical model of ovarian follicle maturation and the resulting parameters were used to classify patients into low and high ovarian response groups. Results. The parameters computed from the mathematical model fit were found to be predictive of ovarian response with a sensitivity of 71% and a specificity of 70%. The parameters were also meaningful within the context of the original mathematical model and have value for determining how doses of hMG may be adjusted during the course therapy to increase the ovarian response in individuals. Conclusion. Mathematical modeling of ultrasonographically derived follicular growth data has significant potential for clinical application in ovarian superstimulation protocols. The method of fitting follicular growth data to a mathematical follicle maturation surface furthermore provides a straightforward approach for the characterization of ovarian follicular dynamics in general.     

Classical mathematical models for prediction of response to chemotherapy and immunotherapy

Classical mathematical models of tumor growth have shaped our understanding of cancer and have broad practical implications for treatment scheduling and dosage. However, even the simplest textbook models have been barely validated in real world-data of human patients. In this study, we fitted a range of differential equation models to tumor volume measurements of patients undergoing chemotherapy or cancer immunotherapy for solid tumors. We used a large dataset of 1472 patients with three or more measurements per target lesion, of which 652 patients had six or more data points. We show that the early treatment response shows only moderate correlation with the final treatment response, demonstrating the need for nuanced models. We then perform a head-to-head comparison of six classical models which are widely used in the field: the Exponential, Logistic, Classic Bertalanffy, General Bertalanffy, Classic Gompertz and General Gompertz model. Several models provide a good fit to tumor volume measurements, with the Gompertz model providing the best balance between goodness of fit and number of parameters. Similarly, when fitting to early treatment data, the general Bertalanffy and Gompertz models yield the lowest mean absolute error to forecasted data, indicating that these models could potentially be effective at predicting treatment outcome. In summary, we provide a quantitative benchmark for classical textbook models and state-of-the art models of human tumor growth. We publicly release an anonymized version of our original data, providing the first benchmark set of human tumor growth data for evaluation of mathematical models.     

Quantitative analysis of gradient sensing: towards building predictive models of chemotaxis in cancer

Chemotaxis of tumor cells in response to a gradient of extracellular ligand is an important step in cancer metastasis. The heterogeneity of chemotactic responses in cancer has not been widely addressed by experimental or mathematical modeling techniques. However, recent advancements in chemoattractant presentation, fluorescent-based signaling probes, and phenotypic analysis paradigms provide rich sources for building data-driven relational models that describe tumor cell chemotaxis in response to a wide variety of stimuli. Here we present gradient sensing, and the resulting chemotactic behavior, in a 'cue-signal-response' framework and suggest methods for utilizing recently reported experimental methods in data-driven modeling ventures.     

A Mathematical-Biological Joint Effort to Investigate the Tumor-Initiating Ability of Cancer Stem Cells

The involvement of Cancer Stem Cells (CSCs) in tumor progression and tumor recurrence is one of the most studied subjects in current cancer research. The CSC hypothesis states that cancer cell populations are characterized by a hierarchical structure that affects cancer progression. Due to the complex dynamics involving CSCs and the other cancer cell subpopulations, a robust theory explaining their action has not been established yet. Some indications can be obtained by combining mathematical modeling and experimental data to understand tumor dynamics and to generate new experimental hypotheses. Here, we present a model describing the initial phase of ErbB2⁺ mammary cancer progression, which arises from a joint effort combing mathematical modeling and cancer biology. The proposed model represents a new approach to investigate the CSC-driven tumorigenesis and to analyze the relations among crucial events involving cancer cell subpopulations. Using in vivo and in vitro data we tuned the model to reproduce the initial dynamics of cancer growth, and we used its solution to characterize observed cancer progression with respect to mutual CSC and progenitor cell variation. The model was also used to investigate which association occurs among cell phenotypes when specific cell markers are considered. Finally, we found various correlations among model parameters which cannot be directly inferred from the available biological data and these dependencies were used to characterize the dynamics of cancer subpopulations during the initial phase of ErbB2⁺ mammary cancer progression.     

Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis

We present a mathematical model of the cytotoxic T lymphocyte response to the growth of an immunogenic tumor. The model exhibits a number of phenomena that are seenin vivo, including immunostimulation of tumor growth, “sneaking through” of the tumor, and formation of a tumor “dormant state”. The model is used to describe the kinetics of growth and regression of the B-lymphoma BCL₁ in the spleen of mice. By comparing the model with experimental data, numerical estimates of parameters describing processes that cannot be measuredin vivo are derived. Local and global bifurcations are calculated for realistic values of the parameters. For a large set of parameters we predict that the course of tumor growth and its clinical manifestation have a recurrent profile with a 3- to 4-month cycle, similar to patterns seen in certain leukemias.     

BCG-Mediated Bladder Cancer Immunotherapy: Identifying Determinants of Treatment Response Using a Calibrated Mathematical Model

Intravesical Bacillus Calmette Guérin (BCG) immunotherapy is considered the standard of care for treatment of non-muscle invasive bladder cancer; however the treatment parameters were established empirically. In order to evaluate potential optimization of clinical parameters of BCG induction therapy, we constructed and queried a new mathematical model. Specifically, we assessed the impact of (1) duration between resection and the first instillation; (2) BCG dose; (3) indwelling time; and (4) treatment interval of induction therapy – using cure rate as the primary endpoint. Based on available clinical and in vitro experimental data, we constructed and parameterized a stochastic mathematical model describing the interactions between BCG, the immune system, the bladder mucosa and tumor cells. The primary endpoint of the model was the probability of tumor extinction following BCG induction therapy in patients with high risk for tumor recurrence. We theoretically demonstrate that extending the duration between the resection and the first BCG instillation negatively influences treatment outcome. Simulations of higher BCG doses and longer indwelling times both improved the probability of tumor extinction. A remarkable finding was that an inter-instillation interval two times longer than the seven-day interval used in the current standard of care would substantially improve treatment outcome. We provide insight into relevant clinical questions using a novel mathematical model of BCG immunotherapy. Our model predicts an altered regimen that may decrease side effects of treatment while improving response to therapy.     

Estimating the Total Rate of DNA Replication Using Branching Processes

Increasing the knowledge of various cell cycle kinetic parameters, such as the length of the cell cycle and its different phases, is of considerable importance for several purposes including tumor diagnostics and treatment in clinical health care and a deepened understanding of tumor growth mechanisms. Of particular interest as a prognostic factor in different cancer forms is the S phase, during which DNA is replicated. In the present paper, we estimate the DNA replication rate and the S phase length from bromodeoxyuridine-DNA flow cytometry data. The mathematical analysis is based on a branching process model, paired with an assumed gamma distribution for the S phase duration, with which the DNA distribution of S phase cells can be expressed in terms of the DNA replication rate. Flow cytometry data typically contains rather large measurement variations, however, and we employ nonparametric deconvolution to estimate the underlying DNA distribution of S phase cells; an estimate of the DNA replication rate is then provided by this distribution and the mathematical model.     

Secreted Gaussia Luciferase as a Biomarker for Monitoring Tumor Progression and Treatment Response of Systemic Metastases

Background

Currently, only few techniques are available for quantifying systemic metastases in preclinical model. Thus techniques that can sensitively detect metastatic colonization and assess treatment response in real-time are urgently needed. To this end, we engineered tumor cells to express a naturally secreted Gaussia luciferase (Gluc), and investigated its use as a circulating biomarker for monitoring viable metastatic or primary tumor growth and their treatment responses.

Methodology/Principal Findings

We first developed orthotopic primary and metastatic breast tumors with derivative of MDA-MB-231 cells expressing Gluc. We then correlated tumor burden with Gluc activity in the blood and urine along with bioluminescent imaging (BLI). Second, we utilized blood Gluc assay to monitor treatment response to lapatinib in an experimental model of systemic metastasis. We observed good correlation between the primary tumor volume and Gluc concentration in blood (R² = 0.84) and urine (R² = 0.55) in the breast tumor model. The correlation deviated as a primary tumor grew due to a reduction in viable tumor fraction. This was also supported by our mathematical models for tumor growth to compare the total and viable tumor burden in our model. In the experimental metastasis model, we found numerous brain metastases as well as systemic metastases including bone and lungs. Importantly, blood Gluc assay revealed early growth of metastatic tumors before BLI could visualize their presence. Using secreted Gluc, we localized systemic metastases by BLI and quantitatively monitored the total viable metastatic tumor burden by blood Gluc assay during the course of treatment with lapatinib, a dual tyrosine kinase inhibitor of EGFR and HER2.

Conclusion/Significance

We demonstrated secreted Gluc assay accurately reflects the amount of viable cancer cells in primary and metastatic tumors. Blood Gluc activity not only tracks metastatic tumor progression but also serves as a longitudinal biomarker for tumor response to treatments.     

A Delayed Nonlinear PBPK Model for Genistein Dosimetry in Rats

Genistein is an endocrine-active compound (EAC) found in soy products. It has been linked to beneficial effects such as mammary tumor growth suppression and adverse endocrine-related effects such as reduced birth weight in rats and humans. In its conjugated form, genistein is excreted in the bile, which is a significant factor in its pharmacokinetics. Experimental data suggest that genistein induces a concentration-dependent suppression of biliary excretion. In this article, we describe a physiologically based pharmacokinetic (PBPK) model that focuses on biliary excretion with the goal of accurately simulating the observed suppression. The mathematical model is a system of nonlinear differential equations with state-dependent delay to describe biliary excretion. The model was analyzed to examine local existence and uniqueness of a solution to the equations. Furthermore, unknown parameters were estimated, and the mathematical model was compared against published experimental data. This research was supported by the American Chemistry Council (formerly the Chemical Manufacturers Association, CMA Agreement Reference Number 9121).     

Model parameter uncertainties in a dual-species biofilm competition model affect ecological output parameters much stronger than morphological ones

Bacterial biofilms are complex microbial depositions on immersed interfaces that form wherever the environmental conditions sustain microbial growth. Despite their name, biofilms can develop in highly irregular structures. Recently several mathematical concepts have been introduced to model these spatially structured microbial populations. Regardless of the type of model, they all have, even for microbially relatively simple systems, many parameters which generally are known at most approximately. We investigate the effect of uncertainties in model parameters on four morphological and four ecological output parameters using a nonlinear diffusion model for a biofilm in which two species compete for a shared nutrient. To this end we conduct an extensive computer simulation experiment for two different levels of data uncertainty, three different hydrodynamic conditions, and two different scenarios of bulk substrate availability. Our results indicate that input model parameter uncertainties have a much larger effect on ecological than on morphological output parameters.     

A mathematical model to study the effects of drugs administration on tumor growth dynamics

A mathematical model for describing the cancer growth dynamics in response to anticancer agents administration in xenograft models is discussed. The model consists of a system of ordinary differential equations involving five parameters (three for describing the untreated growth and two for describing the drug action). Tumor growth in untreated animals is modelled by an exponential growth followed by a linear growth. In treated animals, tumor growth rate is decreased by an additional factor proportional to both drug concentration and proliferating cells. The mathematical analysis conducted in this paper highlights several interesting properties of this tumor growth model. It suggests also effective strategies to design in vivo experiments in animals with potential saving of time and resources. For example, the drug concentration threshold for the tumor eradication, the delay between drug administration and tumor regression, and a time index that measures the efficacy of a treatment are derived and discussed. The model has already been employed in several drug discovery projects. Its application on a data set coming from one of these projects is discussed in this paper.     

Mathematical modeling of ovarian cancer treatments: sequencing of surgery and chemotherapy

Ovarian cancer has long been one of the most common forms of cancer in women. The main treatment for ovarian cancer comprises a combination of surgery and chemotherapy. In an effort to improve treatment strategies, a variety of mathematical models have been developed in the literature. In this paper, we consider a simple mathematical model that incorporates tumor growth as well as the effects of chemotherapeutic and surgical treatments in ovarian cancer. We consider several growth models and combine them with different cell-kill hypotheses. Surgery is assumed to eliminate a fixed fraction of tumor cells instantaneously. We discuss how different models predict the optimal sequencing of chemotherapeutic and surgical treatments. This work has been carried out in the context of ovarian cancer; however, the results may also be useful for other kind of cancers.     

The Tumor Growth Paradox and Immune System-Mediated Selection for Cancer Stem Cells

Cancer stem cells (CSCs) drive tumor progression, metastases, treatment resistance, and recurrence. Understanding CSC kinetics and interaction with their nonstem counterparts (called tumor cells, TCs) is still sparse, and theoretical models may help elucidate their role in cancer progression. Here, we develop a mathematical model of a heterogeneous population of CSCs and TCs to investigate the proposed “tumor growth paradox”—accelerated tumor growth with increased cell death as, for example, can result from the immune response or from cytotoxic treatments. We show that if TCs compete with CSCs for space and resources they can prevent CSC division and drive tumors into dormancy. Conversely, if this competition is reduced by death of TCs, the result is a liberation of CSCs and their renewed proliferation, which ultimately results in larger tumor growth. Here, we present an analytical proof for this tumor growth paradox. We show how numerical results from the model also further our understanding of how the fraction of cancer stem cells in a solid tumor evolves. Using the immune system as an example, we show that induction of cell death can lead to selection of cancer stem cells from a minor subpopulation to become the dominant and asymptotically the entire cell type in tumors.     

Mechanistic characterization of oscillatory patterns in unperturbed tumor growth dynamics: The interplay between cancer cells and components of tumor microenvironment

Mathematical modeling of unperturbed and perturbed tumor growth dynamics (TGD) in preclinical experiments provides an opportunity to establish translational frameworks. The most commonly used unperturbed tumor growth models (i.e. linear, exponential, Gompertz and Simeoni) describe a monotonic increase and although they capture the mean trend of the data reasonably well, systematic model misspecifications can be identified. This represents an opportunity to investigate possible underlying mechanisms controlling tumor growth dynamics through a mathematical framework. The overall goal of this work is to develop a data-driven semi-mechanistic model describing non-monotonic tumor growth in untreated mice. For this purpose, longitudinal tumor volume profiles from different tumor types and cell lines were pooled together and analyzed using the population approach. After characterizing the oscillatory patterns (oscillator half-periods between 8–11 days) and confirming that they were systematically observed across the different preclinical experiments available (p<10^?9), a tumor growth model was built including the interplay between resources (i.e. oxygen or nutrients), angiogenesis and cancer cells. The new structure, in addition to improving the model diagnostic compared to the previously used tumor growth models (i.e. AIC reduction of 71.48 and absence of autocorrelation in the residuals (p>0.05)), allows the evaluation of the different oncologic treatments in a mechanistic way. Drug effects can potentially, be included in relevant processes taking place during tumor growth. In brief, the new model, in addition to describing non-monotonic tumor growth and the interaction between biological factors of the tumor microenvironment, can be used to explore different drug scenarios in monotherapy or combination during preclinical drug development.     

Predictive Modeling of In Vivo Response to Gemcitabine in Pancreatic Cancer

A clear contradiction exists between cytotoxic in-vitro studies demonstrating effectiveness of Gemcitabine to curtail pancreatic cancer and in-vivo studies failing to show Gemcitabine as an effective treatment. The outcome of chemotherapy in metastatic stages, where surgery is no longer viable, shows a 5-year survival <5%. It is apparent that in-vitro experiments, no matter how well designed, may fail to adequately represent the complex in-vivo microenvironmental and phenotypic characteristics of the cancer, including cell proliferation and apoptosis. We evaluate in-vitro cytotoxic data as an indicator of in-vivo treatment success using a mathematical model of tumor growth based on a dimensionless formulation describing tumor biology. Inputs to the model are obtained under optimal drug exposure conditions in-vitro. The model incorporates heterogeneous cell proliferation and death caused by spatial diffusion gradients of oxygen/nutrients due to inefficient vascularization and abundant stroma, and thus is able to simulate the effect of the microenvironment as a barrier to effective nutrient and drug delivery. Analysis of the mathematical model indicates the pancreatic tumors to be mostly resistant to Gemcitabine treatment in-vivo. The model results are confirmed with experiments in live mice, which indicate uninhibited tumor proliferation and metastasis with Gemcitabine treatment. By extracting mathematical model parameter values for proliferation and death from monolayer in-vitro cytotoxicity experiments with pancreatic cancer cells, and simulating the effects of spatial diffusion, we use the model to predict the drug response in-vivo, beyond what would have been expected from sole consideration of the cancer intrinsic resistance. We conclude that this integrated experimental/computational approach may enhance understanding of pancreatic cancer behavior and its response to various chemotherapies, and, further, that such an approach could predict resistance based on pharmacokinetic measurements with the goal to maximize effective treatment strategies.     

New stochastic carcinogenesis model with covariates: an approach involving intracellular barrier mechanisms

In this paper we present a new multiple-pathway stochastic model of carcinogenesis with potential of predicting individual incidence risks on the basis of biomedical measurements. The model incorporates the concept of intracellular barrier mechanisms in which cell malignization occurs due to an inefficient operation of barrier cell mechanisms, such as antioxidant defense, repair systems, and apoptosis. Mathematical formalism combines methodological innovations of mechanistic carcinogenesis models and stochastic process models widely used in studying biodemography of aging and longevity. An advantage of the modeling approach is in the natural combining of two types of measures expressed in terms of model parameters: age-specific hazard rate and means of barrier states. Results of simulation studies allow us to conclude that the model parameters can be estimated in joint analyses of epidemiological data and newly collected data on individual biomolecular measurements of barrier states. Respective experimental designs for such measurements are suggested and discussed. An analytical solution is obtained for the simplest design when only age-specific incidence rates are observed. Detailed comparison with TSCE model reveals advantages of the approach such as the possibility to describe decline in risk at advanced ages, possibilities to describe heterogeneous system of intermediate cells, and perspectives for individual prognoses of cancer risks. Application of the results to fit the SEER data on cancer risks demonstrates a strong predictive power of the model. Further generalizations of the model, opportunities to measure barrier systems, biomedical and mathematical aspects of the new model are discussed.