Ardith W. El-Kareh

Effect of cell arrangement and interstitial volume fraction on the diffusivity of monoclonal antibodies in tissue.

We present theoretical calculations relating the effective diffusivity of monoclonal antibodies in tissue (Deff) to the actual diffusivity in the interstitium (Dint) and the interstitial volume fraction phi. Measured diffusivity values are effective values, deduced from concentration profiles with the tissue treated as a continuum. By using homogenization theory, the ratio Deff/Dint is calculated for a range of interstitial volume fractions from 10 to 65%. It is assumed that only diffusion in the interstitial spaces between cells contributes to the effective diffusivity. The geometries considered have cuboidal cells arranged periodically, with uniform gaps between cells. Deff/Dint is found to generally be between (2/3) phi and phi for these geometries. In general, the pathways for diffusion between cells are not straight. The effect of winding pathways on Deff/Dint is examined by varying the arrangement of the cells, and found to be slight. Also, the estimates of Deff/Dint are shown to be insensitive to typical nonuniformities in the widths of gaps between cells. From our calculations and from published experimental measurements of the effective diffusivity of an IgG polyclonal antibody both in water and in tumor tissue, we deduce that the diffusivity of this molecule in the interstitium is one-tenth to one-twentieth its diffusivity in water. We also conclude that exclusion of molecules from cells (an effect independent of molecular weight) contributes as much as interstitial hindrance to the reduction of effective diffusivity, for small interstitial volume fractions (around 20%). This suggests that the increase in the rate of delivery to tissues resulting from the use of smaller molecular-weight molecules (such as antibody fragments or bifunctional antibodies) may be less than expected.

A mathematical model for cisplatin cellular pharmacodynamics.

A simple theoretical model for the cellular pharmacodynamics of cisplatin is presented. The model, which takes into account the kinetics of cisplatin uptake by cells and the intracellular binding of the drug, can be used to predict the dependence of survival (relative to controls) on the time course of extracellular exposure. Cellular pharmacokinetic parameters are derived from uptake data for human ovarian and head and neck cancer cell lines. Survival relative to controls is assumed to depend on the peak concentration of DNA-bound intracellular platinum. Model predictions agree well with published data on cisplatin cytotoxicity for three different cancer cell lines, over a wide range of exposure times. In comparison with previously published mathematical models for anticancer drug pharmacodynamics, the present model provides a better fit to experimental data sets including long exposure times (approximately 100 hours). The model provides a possible explanation for the fact that cell kill correlates well with area under the extracellular concentration-time curve in some data sets, but not in others. The model may be useful for optimizing delivery schedules and for the dosing of cisplatin for cancer therapy.

A mathematical model of tumor–immune interactions

A mathematical model of the interactions between a growing tumor and the immune system is presented. The equations and parameters of the model are based on experimental and clinical results from published studies. The model includes the primary cell populations involved in effector T-cell mediated tumor killing: regulatory T cells, helper T cells, and dendritic cells. A key feature is the inclusion of multiple mechanisms of immunosuppression through the main cytokines and growth factors mediating the interactions between the cell populations. Decreased access of effector cells to the tumor interior with increasing tumor size is accounted for. The model is applied to tumors with different growth rates and antigenicities to gauge the relative importance of various immunosuppressive mechanisms. The most important factors leading to tumor escape are TGF-β-induced immunosuppression, conversion of helper T cells into regulatory T cells, and the limitation of immune cell access to the full tumor at large tumor sizes. The results suggest that for a given tumor growth rate, there is an optimal antigenicity maximizing the response of the immune system. Further increases in antigenicity result in increased immunosuppression, and therefore a decrease in tumor killing rate. This result may have implications for immunotherapies which modulate the effective antigenicity. Simulation of dendritic cell therapy with the model suggests that for some tumors, there is an optimal dose of transfused dendritic cells.

A model for red blood cell motion in bifurcating microvessels

Abstract A theoretical model is developed for red blood cell motion in a diverging microvessel bifurcation, where the downstream branches are equal in size but receive different flows. The model is used to study migration of red cells across streamlines of the underlying flow, due to particle shape and flow asymmetry. Effects of cell–cell interactions are neglected. Shapes of flowing red cells are approximated by rigid spherical caps. In uniform shear flows, such particles rotate periodically and oscillate about fluid streamlines with no net migration. However, net migration can occur in non-uniform flows due to the particles’ lack of fore-aft symmetry. A nonuniform flow field representative of a bifurcation is developed: flow bounded by two parallel plates, and divided by a cylindrical post. Significant migration is found to occur only with a nonuniform and asymmetric distribution of upstream orientations. The model suggests that the assumption made in previous models of bifurcations, that red cells follow fluid streamlines, is justified if cells approach the bifurcations with random orientations.

Cell Cycle Checkpoint Models for Cellular Pharmacology of Paclitaxel and Platinum Drugs

A pharmacokinetic–pharmacodynamic mathematical model is developed for cellular pharmacology of chemotherapeutic drugs for which the decisive step towards cell death occurs at a point in the cell cycle, presumably corresponding to a cell cycle checkpoint. For each cell, the model assumes a threshold level of some intracellular species at that checkpoint, beyond which the cell dies. The threshold level is assumed to have a log-normal distribution in the cell population. The kinetics of formation of the lethal intracellular species depends on the drug, and on the cellular pharmacokinetics and binding kinetics of the cell. Specific models are developed for paclitaxel and for platinum drugs (cisplatin, oxaliplatin and carboplatin). In the case of paclitaxel, two separate mechanisms of cell death necessitate a model that accounts for two checkpoints, with different intracellular species. The model was tested on a number of in vitro cytotoxicity data sets for these drugs, and found overall to give significantly better fits than previously proposed cellular pharmacodynamic models. It provides an explanation for the asymptotic convergence of dose-response curves as exposure time becomes long.

A model for effects of adaptive immunity on tumor response to chemotherapy and chemoimmunotherapy

Complete clinical regressions of solid tumors in response to chemotherapy are difficult to explain by direct cytotoxicity alone, because of low growth fractions and obstacles to drug delivery. A plausible indirect mechanism that might reconcile this is the action of the immune system. A model for interaction between tumors and the adaptive immune system is presented here, and used to examine controllability of tumors through the interplay of cytotoxic, cytostatic and immunogenic effects of chemotherapy and the adaptive immune response. The model includes cytotoxic and helper T cells, T regulatory cells (Tregs), dendritic cells, memory cells, and several key cytokines. Nearly all parameter estimates are derived from experimental and clinical data. Individual tumors are characterized by two parameters: growth rate and antigenicity, and regions of tumor control are identified in this parameter space. The model predicts that inclusion of the immune response significantly expands the region of tumor control for both cytostatic and cytotoxic chemotherapies. Moreover, outside the control zone, tumor growth is delayed significantly. An optimal fractionation schedule is predicted, for a fixed cumulative dose. The model further predicts expanded regions of tumor control when several forms of immunotherapy (adoptive T cell transfer, Treg depletion, and dendritic cell vaccination) are combined with chemotherapy. Outcomes depend greatly on tumor characteristics, the schedule of administration, and the type of immunotherapy chosen, suggesting promising opportunities for personalized medicine. Overall, the model provides insight into the role of the adaptive immune system in chemotherapy, and how scheduling and immunotherapeutic interventions might improve efficacy.

The additive damage model: a mathematical model for cellular responses to drug combinations.

Mathematical models to describe dose-dependent cellular responses to drug combinations are an essential component of computational simulations for predicting therapeutic responses. Here, a new model, the additive damage model, is introduced and tested in cases where varying concentrations of two drugs are applied with a fixed exposure schedule. In the model, cell survival is determined by whether cellular damage, which depends on the concentrations of the drugs, exceeds a lethal threshold, which varies randomly in the cell population with a prescribed statistical distribution. Cellular damage is assumed to be additive, and is expressed as a sum of separate terms for each drug. Each term has a saturable dependence on drug concentration. The model has appropriate behavior over the entire range of drug concentrations, and is predictive, given single-agent dose-response data for each drug. The proposed model is compared with several other models, by testing their ability to fit 24 data sets for platinum-taxane combinations and 21 data sets for various other combinations. The Akaike Information Criterion is used to assess goodness of fit, taking into account the number of unknown parameters in each model. Overall, the additive damage model provides a better fit to the data sets than any previous model. The proposed model provides a basis for computational simulations of therapeutic responses. It predicts responses to drug combinations based on data for each drug acting as a single agent, and can be used as an improved null reference model for assessing synergy in the action of drug combinations.

Theoretical Analyses and Simulations of Anticancer Drug Delivery

Large amounts of data on tumor cell survival as a function of exposure to anticancer drugs, drug pharmacokinetics, drug distribution in the body, and other aspects of drug delivery and effectiveness are continually being generated. Cancer therapies are becoming increasingly complex, and it is now possible to choose the time schedule of drug delivery, the site of delivery, the size, lipophilicity, release kinetics and other properties of a carrier, and numerous other options. However, it is clearly impossible to perform sufficient animal experiments or clinical trials to determine the optimal choices of all these variables. Even for drugs that have been used for decades, doses and schedules are often based on past experience and medical tradition rather than on rational analysis. These circumstances suggest an increasing need for theoretical models of anticancer drug delivery. Such models can provide a framework for synthesizing and interpreting available experimental data, and a rational basis for optimizing therapies using existing drugs and for guiding development of new drugs.

Additive Damage Models for Cellular Pharmacodynamics of Radiation–Chemotherapy Combinations

Many cancer patients receive combination treatments with radiation and chemotherapy. Available mathematical models for cellular pharmacodynamics have limited ability to represent observed in vitro responses to radiochemotherapy. Here, a family of additive damage models is proposed to describe cell kill resulting from radiochemotherapy with fixed schedule and variable doses. The pathways by which the agents produce cellular damage are assumed to converge in a single cell death process, so that survival depends on total damage, which can be represented as a sum of contributions from the various damage pathways. Heterogeneity in response across the cell population is ascribed to variations in the damage threshold for cell kill. The family of proposed models includes effects of one or two pathways of damage for each agent, saturation in drug responses, and cooperative or antagonistic interactions between agents. Models from this family with 4–7 unknown parameters are tested for their ability to fit 218 in vitro literature data sets for a range of drugs and cell lines. Overall, the additive damage models are found to outperform models based on the existing concept of independent cell kill, according to the corrected Akaike Information Criterion. The results are used to assess the importance of the various effects included in the models. These additive damage models have potential applications to the optimization of treatment and to the analysis and interpretation of in vitro screening data for new drug–radiation combinations.

Abstract 5336: Mathematical models for cellular response to radiochemotherapy

Proceedings: AACR Annual Meeting 2014; April 5-9, 2014; San Diego, CA To optimize radiochemotherapy, several clinically adjustable factors must be considered: sequence order (drug administered before, during, or after radiation exposure); time interval between the agents; dose of each agent; and fractionation. Because the combined effects of all these factors are complex, mathematical modeling can provide a useful framework for discerning the role of each individual factor and can be applied to predict response to future experimental protocols. To date, there has been scant effort devoted to mathematical models for cellular response to radiochemotherapy. One widely-used model is based on the assumption that the two modalities act completely independently; the total cell survival fraction is the product of survival fractions for each agent acting alone. However, an examination of a number of dose-response data sets for paclitaxel followed by radiation exposure shows that the independent cell kill model consistently underestimates cell kill. The additive damage model, which is based on the assumption that cell kill depends on a quantity called cellular damage, is constructed as the sum of terms from each agent and is shown to better describe some data sets for ascorbate and paclitaxel. Cellular dose-response data sets for radiochemotherapy have shown complex interactions between therapies, with strong dose-dependent antagonism in some cases, and significant, non-monotonic dependence on the time interval between the modalities in others. We present generalizations of the additive damage model, which include either dose-dependent cell-cycle blocking or transport kinetics, and show that they can describe some of these experimentally-observed complex behaviors. In conclusion, the generalized additive damage model is a promising tool to describe cellular pharmacodynamics of radiochemotherapy and is expected to have application to the optimization of dosage and schedule. Citation Format: Katherine S. Williams, Ardith W. El-Kareh, Timothy W. Secomb. Mathematical models for cellular response to radiochemotherapy. [abstract]. In: Proceedings of the 105th Annual Meeting of the American Association for Cancer Research; 2014 Apr 5-9; San Diego, CA. Philadelphia (PA): AACR; Cancer Res 2014;74(19 Suppl):Abstract nr 5336. doi:10.1158/1538-7445.AM2014-5336