CS1: Cancer Dynamics

Monday, July 17, 10:30-12:30, City Creek Room

In multicellular organisms, when cells divide, random mutations occur in the genome. Most of these mutations may be silent or corrected by repairing mechanisms. In some rare cases, a mutation can confer a cell, neoplastic characteristics that accumulate over time until cancer develops.

At a cellular scale, cancers present a set of distinctive hallmarks, including mutations associated with proliferation, apoptosis, genome instability, angiogenesis and metastasis. In a previous paper (Spencer et al., 2004), we proposed a set of ordinary differential equations to model the multistep transformation to cancer; here I present an extension of those ideas but using a discrete model with stochastic events.

The proposed model starts with a relatively small numbers of cells in which three kinds of different mutations may occur, involving either: apoptosis, proliferation or genome instability. Cells then proliferate at discrete times, allowing for mutations to occur until cells cannot sustain growth unless they acquire further mutations associated with angiogenesis and metastasis. As discrete events are stochastic, simulations were replicated 100 times, and the mean values for the number of each mutant were estimated.

Although the dynamics for single mutants depends on the first mutation, the dynamics for the tumour cells is independent of the starting mutation. The proposed model is relatively simple and can be extended to simulate other events, besides cancer.

Colorectal cancer (CRC) is one of the major causes of death in the developed world. CRC arises when the cells lining the gut (which consist of millions of distinct test tube shape domains called crypts) undergo a string of mutations causing them to no longer respond to the usual control mechanisms. Possible treatments attempt to restore these control mechanisms by providing drugs, which change how cells respond to stimuli.

Much mathematical and computational modelling has been undertaken in order to try to under- stand how crypts develop and how things go wrong under disease. We have developed a framework, which combines mechanical models of cell interactions with detailed representations of the subcel- lular machinery. The framework allows the specification of arbitrary subcellular networks in the Systems Biology Markup Language (SBML) within a selection of multicellular representations of the crypt. Allowing the multiscale models to be simulated and the effects of mutations in individual cells, and their inflence on the crypt, to be investigated.

In this talk we present the coupled multicellular models underlying the framework; discuss its implementation; and present example applications of the framework, including the inclusion of mutations in the system and their effect on the Crypt.

Angiogenesis is the process by which new blood vessels develop from existing vasculature, and is important in pathological conditions, such as cancer. Tumor angiogenesis has been modeled phenomenologically using the well-known snail-trail approach. We use an alternative methodology, utillizing an agent-based approach to model the behavior of individual endothelial cells during angiogenesis. We incorporate crowding effects through volume exclusion, motility of cells through biased random walks, and include birth and death-like processes to represent sprout formation and anastomosis. We use the transition probabilities associated with the discrete model and a mean-field approximation to systematically derive a series of partial differential equation (PDE) descriptions of collective cell behavior that vary in complexity depending on the extent of volume exclusion incorporated on the microscale. This general framework generates non-linear PDEs that impose physically realistic density restrictions, and are structurally different from existing snail-trail models which implicitly view cells as point particles. We compare solutions to our PDEs and a well-known snail- trail model to discrete simulation results to determine conditions under which it is necessary to account for cell volume in the continuum models. We find that for a single-species exclusion process, once cell density exceeds a threshold, the implicit point particle assumption in snail-trail models leads to non- negligible errors. In the case of multi-species exclusion, both model types perform poorly once interaction between the cell species becomes pronounced. In general, snail-trail models perform well when exclusion effects are diminished at the macroscale; in other cases, non-linear models should be used. In future, we aim to distinguish model performance based on network morphology. This may impact drug development strategies based on PDE models.

Clonal hematopoiesis — where mature myeloid cells deriving from a single stem cell are over-represented — is detected in at least 10% of people aged 70 and over and represents a major risk factor for overt hematologic malignancies. To quantify how likely this phenomena is, we combine existing quantitative observations with a novel stochastic model of hematopoietic stem cell dynamics. We include migration between stem cell compartments — in mammals these cells reside mostly in the bone marrow, but also transitively passage in small numbers in the blood. We find in mice that cells require a selective advantage, i.e. it can not be a by-chance neutral expansion, if a clone is to be detectable within a lifetime. In humans this prediction is dependent on the number of stem cells, which is frequently debated. Combining our predictions with incidence data establishes a lower bound for the number of hematopoietic stem cells in man. The compartmental nature of our model further captures scenarios of stem cell transplantation in preconditioned and non- preconitioned hosts. Our analyses support existing findings that niche-space saturation decreases the engraftment efficiency of donor cells.

Melanoma is the third most common cancer in Australia, and associated with high rates of mortality. Melanoma spreading involves the migration of cancer cells amongst other native skin cells. We explore the interactions between melanoma cells and fibroblast cells by performing a suite of cell barrier assays with one or two types of cells at different ratios, and compare the experimental results to mathematical models. This allows us to investigate the interactions between the two types of cell, and reveals no evidence of interactions other than cell-to-cell contact. 15

Plasma cells (PCs) are white blood cells, which represent an essential part of the immune system producing the main fraction of antibodies. After an antigen encounter, a large population of non- proliferative PCs is generated, which enters the bone marrow to reside therein. Interactions with the bone marrow microenvironment, termed the niche, permit PCs to survive for decades. Yet, competition for the niche leads to PC displacement and their concomitant death. In contrast to healthy PCs, malignant PCs proliferate. Their accumulation in the bone marrow characterizes malignant PC diseases. Asymptomatic multiple myeloma (AMM) evolves from monoclonal gammopathy of unknown significance (MGUS). These two disease stages are asymptomatic and delineated solely by surrogates of tumor mass. Disease progression leads to multiple myeloma (MM) involving clinical signs and symptoms such as end organ damages.

The goals are to understand the evolution of malignant PC diseases within an individual patient and to reveal the underlying mechanisms of cancer development. Three main questions are addressed by mathematical modeling. Firstly, how is growth of malignant PCs characterized? Secondly, how fast does progression from asymptomatic stages (AMM, MGUS) to symptomatic myeloma (MM) happen? And thirdly, can a single malignant PC explain the development of myeloma? Or is such initial event rather characterized by a large population of malignant PCs arriving at the bone marrow, thus comparable to healthy PC development induced by an antigen encounter?

To answer these questions, new mathematical models consisting of piecewise-smooth ordinary differential equations are derived, which describe the dynamics of healthy and malignant PCs in the bone marrow and its niche. The models are validated using clinical data of patients with AMM (n=322) and MGUS (n=196), eventually allowing for novel biologically and clinically relevant conclusions.

Breast cancer metastases are the leading cause of breast cancer related death. Hypoxia is common in breast cancer, is a negative prognostic factor, and is associated with breast cancer metastases. In this talk, we present data from new immunofluorescent and immunohistochemical stains for hypoxic exposure, which are retained after cells return to normal physiological oxygen conditions. The data are paired with traditional hypoxic markers, blood vessel labels, cell birth and death markers, spatial oxygenation meas- urements, and tumor size measurements. Based on these data obtained in an orthotopic mouse model, we develop and calibrate an agent-based model of vascularized primary tumor growth. We match this primary site model to a PDE-based metastatic site model, which includes the effects of tissue mechanics, oxygen- driven birth and death, and vascular remodeling. We encapsulate this PDE model into a “metastasis agent”, and network those agents to simulate multi-site metastatic progression. We present preliminary results on metastatic dissemination, compare our results with early in vivo data on hypoxic cell trafficking, and discuss how this system could be used to explore the systemic impact of anti-angiogenic treatments.

Antibody drug conjugates, ADCs, are one of the latest developed chemotherapeutics that treat some types of tumor cells. It consists of monoclonal antibodies, linkers, and potent cytotoxic drugs. Unlike common chemotherapies, ADCs combine selectively with the target at tumor cell surface and potent cytotoxic drug (payload) effectively prevents microtubule polymerization. This magical drug has theoretically perfect but linkers are not stable and payloads release in the circulation before binding the target. It may cause side effect in the body. In this work, we construct an ADCs model that considers both the target of antibodies and the receptor (tubulin) of the cytotoxic payloads and then presents the comparision with tumor size prohibition effects according to linker stabilities after estimating parameters and sustaining half-life. This research will propose the effect and limitation of upcoming new ADCs drugs about linker improving.