The Physics of Cancer

The following sections are included:

• CONTENTS
• Preface
• About the Editor

Multiscale, agent-based mathematical models of biological systems are often associated with model uncertainty and sensitivity to parameter perturbations. Here, three uncertainty and sensitivity analyses methods, that are suitable to use when working with agent-based models, are discussed. These methods are namely Consistency Analysis, Robustness Analysis and Latin Hypercube Analysis. This introductory review discusses origins, conventions, implementation and result interpretation of the aforementioned methods. Information on how to implement the discussed methods in MATLAB is included.

We describe an evolutionary game theory model that has been used to predict the population dynamics of interacting cancer and stromal cells. We first consider the mean field case assuming homogeneous and nondiscrete populations. Interacting Particle Systems (IPS) are then presented as a discrete and spatial alternative to the mean field approach. Finally, we discuss cases where IPS gives results different from the mean field approach.

Favorable outcomes have been associated with high densities of tumor infiltrating lymphocytes (TILs) such as cytotoxic (CD8⁺) T cells. However, the clinical significance of the spatial distribution of TILs is less well understood. We have developed novel statistical techniques to characterize the spatial distribution of TILs at various length scales. These include a box counting method that we call “occupancy” and novel applications of fractal dimensions. We apply these techniques to the spatial distribution of CD8⁺ T cells in the tumor microenvironment of tissue resected from 35 triple negative breast cancer patients. We find that there is a distinct difference in the spatial distribution of CD8⁺ T cells between good clinical outcome (no recurrence within at least 5 years of diagnosis) and poor clinical outcome (recurrence within 3 years of diagnosis). The statistical significance of the difference between good and poor outcome in the occupancy, fractal dimension (FD), and FD difference of CD8⁺ T cells is comparable to that of the CD8⁺ T cell density. Even when we randomly exclude some of the cells so that the images have the same cell density, we still find that the fractal dimension at short length scales is correlated with cancer recurrence, implying that the actual spatial distribution of CD8⁺ cells, and not just the CD8⁺ cell density, is associated with clinical outcome. The occupancy and FD difference indicate that the CD8⁺ T cells are more spatially dispersed in good outcome and more aggregated in poor outcome. We discuss possible interpretations.

Heterogeneity is a hallmark of all cancers. Tumor heterogeneity is found at different levels — interpatient, intrapatient, and intratumor heterogeneity. All of them pose challenges for clinical treatments. The latter two scenarios can also increase the risk of developing drug resistance. Although the existence of tumor heterogeneity has been known for two centuries, a clear understanding of its origin is still elusive, especially at the level of intratumor heterogeneity (ITH). The coexistence of different subpopulations within a single tumor has been shown to play crucial roles during all stages of carcinogenesis. Here, using concepts from evolutionary game theory and public goods game, often invoked in the context of the tragedy of commons, we explore how the interactions among subclone populations influence the establishment of ITH. By using an evolutionary model, which unifies several experimental results in distinct cancer types, we develop quantitative theoretical models for explaining data from in vitro experiments involving pancreatic cancer as well as in vivo data in glioblastoma multiforme. Such physical and mathematical models complement experimental studies, and could optimistically provide new ideas for the design of efficacious therapies for cancer patients.

The classical migration modes, such as mesenchymal or amoeboid migration modes, are essentially determined by molecular, morphological or biochemical properties of the cells. These specific properties facilitate the cell migration and invasion through artificial extracellular matrices mimicking the environmental conditions of connective tissues. However, during the migration of cells through narrow extracellular matrix constrictions, the specific extracellular matrix environments can either support or impair the invasion of cells. Beyond the classical molecular or biochemical properties, the migration and invasion of cells depends on intracellular cell mechanical characteristics and extracellular matrix mechanical features. The switch between cell states, such as epithelial, mesenchymal or amoeboid states, seems to be mainly based on epigenetic changes and environmental cues that induce the reversible transition of cells toward another state and thereby promote a specific migration mode. However, the exact number of migration modes is not yet clear. Moreover, it is also unclear whether every individual cell, independent of the type, can undergo a transition between all different migration modes in general. A newer theory states that the transition from the jamming to unjamming phase of clustered cells enables cells to migrate as single cells through extracellular matrix confinements. This review will highlight the mechanical features of cells and their matrix environment that regulate and subsequently determine individual migration modes. It is discussed whether each migration mode in each cell type is detectable or whether some migration modes are limited to artificially engineered matrices in vitro and can therefore not or only rarely be detected in vivo. It is specifically pointed out how the intracellular architecture and its contribution to cellular stiffness or contractility favors the employment of a distinct migration mode. Finally, this review envisions a connection between mechanical properties of cells and matrices and the choice of a distinct migration mode in confined 3D microenvironments.

The physical properties of the extracellular matrix strongly influence tumor progression and malignancy. This impact can be direct by physically impeding or promoting tumor progression. Furthermore, the ECM properties can also indirectly modulate tumors via interaction with cellular signaling pathways, such as integrin or YAP/TAZ signaling. Conversely, tumors remodel and rebuild the extracellular matrix. This leads to a strong mutual influence of tumor and extracellular matrix including feedback loops and cascades of mutual interaction. Combinations of therapies that treat tumor signaling directly and indirectly via modulation of the tumor’s interaction with the extracellular matrix may thus leverage the success of cancer therapy.

Immunotherapies with anti-programmed death-1 antibodies (anti-PD-1) have revolutionized cancer treatment. However, the majority of patients still fail to respond and the reasons remain largely unknown. In order to identify tumor characteristics associated with treatment failure, we developed a computational model capable of simulating tumor response to anti-PD-1 immunotherapy, validated with on-site in vitro and in vivo experiments.

The experiments were conducted using non-responsive and responsive murine cell lines, 4T1 mammary carcinoma and CT26 colon carcinoma, respectively. Tumors were induced in athymic nude mice, wild-type mice, and wild-type mice receiving anti-PD-1 immunotherapy to: (1) assess Gompertzian parameters describing intrinsic tumor growth in the absence of T cells, (2) assess the immune-related parameters, and (3) validate model’s predictive ability, respectively. Flow-cytometric analyses of major histocompatibility complex (MHC) class I and PD-1 ligand (PD-L1) expression were performed to assess initial heterogeneity in tumor cell subpopulations. Model parameter sensitivity studies were performed to establish importance of different tumor characteristics on the observed tumor response to anti-PD-1 immunotherapy.

The model correctly predicted the observed trend of response to anti-PD-1 in 4T1, while the predictions in CT26, where dichotomous responses were observed experimentally, were less accurate. Sensitivity studies of measured model parameters and initial conditions suggested that complete responses to anti-PD-1 immunotherapy might be possible only in the case of homogeneous MHC class I positive tumors. On the other hand, heterogeneous tumors containing MHC class I negative tumor cell subpopulations were associated with resistance to anti-PD-1 therapy.

The tumor microenvironment plays a critical role in tumorigenesis and metastasis. As tightly controlled extracellular matrix homeostasis is lost during tumor progression, a dysregulated extracellular matrix can significantly alter cellular phenotype and drive malignancy. Altered physical properties of the tumor microenvironment alter cancer cell behavior, limit delivery and efficacy of therapies, and correlate with tumorigenesis and patient prognosis. The physical features of the extracellular matrix during tumor progression have been characterized; however, a wide range of methods have been used between studies and cancer types resulting in a large range of reported values. Here, we discuss the significant mechanical and structural properties of the tumor microenvironment, summarizing their reported values and clinical impact across cancer type and grade. We attempt to integrate the values in the literature to identify sources of reported differences and commonalities to better understand how aberrant extracellular matrix dynamics contribute to cancer progression. An intimate understanding of altered matrix properties during malignant transformation will be crucial in effectively detecting, monitoring, and treating cancer.

Tumor invasion, the process by which tumor cells break away from their primary tumor and gain access to vascular systems, is an important step in cancer metastasis. Most current 3D tumor invasion assays consisted of a single tumor cell embedded within an extracellular matrix (ECM). These assays taught us much of what we know today on how key biophysical (e.g., ECM stiffiness) and biochemical (e.g., cytokine gradients) parameters within the tumor microenvironment guided and regulated tumor invasion. One limitation of the single tumor cell invasion assays was that it did not account for cell–cell adhesion within the tumor. In this chapter, we developed a micrometer scale 3D co-culture spheroid invasion assay that recapitulated physiologically realistic tumor microenvironment and was compatible with microscopic imaging. Micrometer scale co-culture spheroids (1:1 ratio of metastatic breast cancer MDA-MB-231 and non-tumorigenic epithelial MCF-10A cells) were made using an array of microwells, and then were embedded within a collagen matrix in a microfluidic platform. Real time imaging of tumor spheroid invasion revealed that the spatial distribution of the two cell types within the tumor spheroid critically regulated tumor invasion. This work linked tumor architecture with tumor invasion and highlighted the importance of the biophysical cues within the bulk of the tumor in tumor invasion.

Altretamine (ALT) is a Food and Drug Administration (FDA) approved antineoplastic drug particularly used for ovarian cancer. This study examined, at a molecular level, the interactions of the drug with model membranes composed of phospholipids with different acyl chain lengths and head group charges at varied ALT concentrations based on temperature. For this purpose, spectroscopic studies of the liposomes in multilamellar vesicles form were conducted by Fourier transform-infrared spectroscopy (FTIR) and their calorimetric studies were carried out by differential scanning calorimetry (DSC) techniques. The results of the study showed that ALT clearly interacted with lipids and that these interactions were more significant in multilamellar vesicles made up of short chain phospholipids. Moreover, the results suggested that ALT settled into the tail group region, in particular the region that formed the hydrophobic part of lipids, and this effects the whole section of the membranes including glycerol backbones and head groups. This study is expected to contribute, on molecular level, to the studies on the knowledge of the mechanism in cancer, which is still very much a dangerous disease, and its related treatment.

The understanding of the interplay between cancer and the immune system is still limited. Herein, I focus on two aspects of this interplay. First, I propose a kinetic model describing the likely role of the immune system in the lifetime risk of cancer at the level of the whole human population. For each tissue, the risk is predicted to be influenced by the heterogeneity of the population and to depend exponentially on time. The expression for the risk does not, however, depend explicitly on the total number of divisions of the corresponding stem cells. For this reason, the correlation with the latter number can only be indirect. Second, using another kinetic framework, I describe how the growth of a few tumors can depend on their interaction via the immune system. The analysis shows that depending on specific details, the tumors of different sizes tend either to reach the same size or remain to be of different sizes.

We review the atavistic model of cancer and compare it with the leading model: Somatic Mutation Theory (SMT). We identify their differences and describe specific predictions of the atavistic model that make it more easily tested than SMT. The increasingly dense phylogenetic tree of all life on Earth has permitted a range of phylostratigraphic analyses which support the atavistic model.

The following section is included:

• Index