In this chapter, we present some of the key biological concepts necessary to motivate, develop, and understand the tumor models introduced in this book. We introduce the molecular and cellular biology of noncancerous tissue (Section 2.1) and then discuss how this biology is altered during cancer progression (Section 2.2). The discussion may be more detailed in some areas than is necessary for the models that we present; the intention is to offer a sample of the rich world of molecular and cellular biology, helping the reader to consider how these and other details may need to be incorporated in the work of cancer modeling. For greater depth on any of the topics, refer to such excellent texts as for molecular and cellular biology and for cancer cell biology.

Key molecular and cellular biology

We focus upon the molecular and cellular biology of epithelial cells, the stroma, and the mesenchymal cells that create and maintain the stroma (Section 2.1.1). Specific and often anisotropic adhesive forces help to maintain tissue architecture (Section 2.1.2). Epithelial and stromal cells have the same basic subcellular structure (Section 2.1.3) and share much in common. They progress through a cell cycle when preparing to divide, can control their entry into and exit from the cycle, and can self-terminate (apoptose) when they detect irreparable DNA errors or other damage (Section 2.1.4). Their behavior is governed by a signaling network that integrates genetic and proteomic information with extracellular signals received through membrane-bound receptors (Section 2.1.5).

Tumors are complex systems dominated by large numbers of processes with highly nonlinear dynamics spanning a wide range of dimensions. Typically, such complex systems can be understood only through complementary experimental investigation and mathematical modeling. New methodologies are needed to integrate and quantify the myriad variables and enable the prediction of outcomes, the selection of existing therapies, and the development of new treatments, possibly on a personalized, individual, basis. Mathematical modeling can provide a rigorous, precise, approach for quantifying correlations between tumor parameters, prognosis, and treatment outcomes. Integration of these elements into a multidisciplinary model of tumor progression would be an important tool to advance clinical decision-making. Thus, there is a critical need for biologically realistic and predictive multiscale and multivariate models of tumor growth and invasion and, as we have seen in this book, there has been much recent effort directed towards this goal.

The tumor models and simulation results that we have reviewed demonstrate that, while mathematical analyses still lag behind experimentation, significant progress has been made in providing important insights into the root causes of solid-tumor invasion and metastasis and in providing and assessing effective treatment strategies. For example, the results provide a means to associate quantitatively tumor growth and the effects of a limited supply of cell substrates. Inhomogeneities in the substrate availability and/or host tissue biomechanical properties can result in tumor morphological instability. These instabilities may allow a tumor to overcome diffusional limitations on growth and to grow to sizes larger than would be possible if it had a compact shape.

In this chapter, we introduce discrete cancer-cell modeling, assess the strengths and weaknesses of the available discrete cell modeling approaches, sample the major discrete cell modeling approaches employed in current computational cancer modeling, and introduce a discrete agent-based cell modeling framework. This framework currently being developed by the present authors and collaborators will be used to implement the next-generation multiscale cancer-modeling framework detailed in Chapter 7.

A brief review of discrete modeling in cancer biology

Thus far we have discussed continuum modeling, in which cancer is modeled at the tissue scale and the effects of individual cells are averaged out. We now turn our attention to discrete models, in which the behavior of one or more individual cells as they interact with one another and the microenvironment is addressed.

Discrete modeling has enjoyed a long history in applied mathematics and biology, dating as far back as the 1940s when John von Neumann applied lattice crystal models to study the necessary rule sets for self-replicating robots. Perhaps the most famous early example of discrete biological modeling is John Conway's 1970 “game of life,” a two-dimensional rectangular lattice of “cells” that changed color according to rules based upon the colors of the neighboring cells. Even simple rules can lead to complex emergent behavior, and Conway's model was later shown to be Turing complete. Today, discrete cell modeling has advanced to study a broad swath of cancer biology, spanning carcinogenesis, tumor growth, invasion, and angiogenesis.

In this book we describe recent efforts to model tumor growth and invasion using an interdisciplinary approach that integrates mathematical and computational models of cancer with laboratory experiments and clinical data. The aim of these efforts has been to provide insight into the root causes of solid tumor invasion and metastasis, to aid in the understanding of experimental and clinical observations, and to help design new, targeted, experiments and treatment strategies. The ultimate goal is for modeling and simulation to aid in the development of individualized therapy protocols that minimize patient suffering while maximizing treatment effectiveness. In order to achieve this objective, mathematical and computational models are needed that quantify the links of three dimensional tumor-tissue architecture with the growth, invasion, and underlying microscale cellular and environmental characteristics. This approach requires a multiscale modeling framework that is capable of linking the molecular and cell scales directly to the patient data.

There are many ways to evaluate the progression of these efforts. Here we follow the major stages that progressively incorporate the complexity of the tumor environment: (i) modeling of avascular tumors in vitro and in silico to assess stages of tumor growth; (ii) interactions between a tumor and its in vivo microenvironment; (iii) modeling of vascularized tumors in silico to assess angiogenesis and vascular growth; (iv) modeling of vascularized tumors in vivo and in silico to assess tumor progression in the body.

In the past several decades there has been significant progress in understanding and identifying the causes of cancer and in developing effective treatment strategies. Nevertheless, a cure remains frustratingly elusive. At its most essential level, cancer involves the abnormal growth and spread of tissues within a body. Yet each cancer is unique, based on the tissue in the body where it originates and the particular person who has it. While molecular mechanisms and cell-scale dynamics governing tumor cell migration and proliferation are well studied from a biological perspective, cancer progression actually involves events that occur at multiple time and spatial scales. What occurs at the nano-scale of molecules and micro-scale of cells affects the behavior of tissue at the centimeter-scale – and vice versa. In order to better understand these multiscale linkages, mathematical modeling, analysis, and simulation have been employed to study tumor behavior. The complex shapes and invasive behavior of tumors requires a nonlinear approach, meaning that effects at various physical scales within the tissue do not necessarily influence each other additively. Hence, the combination of events may yield a response greater or less than of each component, depending whether there is synchrony. The application of such computational models in the clinical setting, however, is still in its infancy.

In this book we outline recent advances in the field of mathematical modeling and the simulation of cancer, particularly with respect to multiscale, nonlinear, computational models that integrate theory and experiment.

Introduction

In this chapter we describe recent multidisciplinary efforts that combine the models of cancer described in this book with laboratory experiments to model tumor growth and invasion. We focus on morphology, invasion, and anti-angiogenic therapy. The work has been accomplished in stages that progressively incorporate the complexity of the tumor environment: (i) the modeling of avascular tumors in vitro and in silico to assess the stages of tumor growth; (ii) the modeling of vascularized tumors in silico to assess the angiogenic switch and vascular growth; (iii) the modeling of vascularized tumors in vivo and in silico to assess tumor progression in the body.

Avascular growth

Even though in vitro conditions offer tumor cells unlimited space for expansion and no interaction with the host environment, the study of tumor growth and invasion using a three-dimensional multicellular tumor spheroid in vitro model has provided novel insights. A spheroid can be viewed as a network of individual cells, each with its own proliferation potential. Growth of the spheroid is the outcome of the balance between individual cell proliferation and internal cohesive forces. Variations in spheroid growth rates and in the extent of central necrosis have been attributed to fluctuations in the oxygen and nutrient concentrations, and tumor growth characteristics in vitro and in vivo have been extensively modeled. Spheroids may have the complexity of self-organized dynamical systems, which are regulated by both environmental and internal noise.

In Chapter 6 we discussed an agent-based cell model that can be applied to a variety of biological systems, with a particular emphasis on epithelial cancers. We now illustrate the model by applying it to breast cancer and demonstrating its use in obtaining theoretical biologic and clinical insights, including quantitative predictions that can be assessed using patient immunohistochemistry and histopathology data. The theoretical biologic and clinical significance is discussed.

Introduction

Ductal carcinoma in situ (DCIS) is the most prevalent precursor to invasive breast cancer (IC), the second-leading cause of death in women in the United States. The American Cancer Society predicted that 50 000 new cases of DCIS alone (excluding lobular carcinoma in situ) and 180 000 new cases of IC would be diagnosed in 2007. Co-existing DCIS is expected in 80% of IC, or 144 000 cases. Because DCIS is a known precursor to IC, this leads us to hypothesize that up to 75% of DCIS cases progress to invasion prior to detection by screening mammography. While DCIS itself is not life-threatening, it is a very important precursor to IC because (i) it can be treated and (ii) if left untreated it is likely to progress to IC, which is a deadly disease.

Women tend to prefer breast-conserving surgery (BCS), also known as lumpectomy, to complete mastectomy for the treatment of DCIS: in the United States today, approximately two-thirds of women diagnosed with DCIS will opt for BCS over mastectomy. Women who undergo BCS face two problems.

Background

In this chapter we develop and analyze multiphase mixture models of solid tumor growth and include their coupling via angiogenesis to the tumor-induced vasculature. Mixture models incorporate more detailed biophysical processes than can be accounted for in single-phase models, and they provide a more general framework for modeling solid tumor growth. Phenotypic and genetic heterogeneity (e.g., mutations) in the cell population can be represented, and this is a critical improvement towards realistically simulating mutation-driven heterogeneity.

A multiphase approach describes a tumor as a saturated medium, comprising at least one solid phase and one liquid phase. The approach can be generalized to incorporate any number of additional phases describing multiple cell species and extracellular matrix components. At a given physical location, the mass or volume fractions describe the relative amounts of different cell clones, necrotic tissue and host tissue. The governing equations consist of mass and momentum balance for each phase, mass and momentum exchange between phases, and appropriate constitutive laws to close these equations. The mixture-model approach eliminates the need to enforce complicated boundary conditions across the tumor–host interface (and other species–species interfaces) that would have to be satisfied if the interfaces were assumed to be sharp. Also, this methodology does not require the interface to be tracked as is the case for a sharp interface.

Recently, multiphase mixture models have been developed to account for heterogeneities in cell type and in the mechanical response of the cellular and liquid tumor phases.

Recently, hybrid modeling techniques have been proposed that combine the advantages of the continuum and the discrete approaches in order to simulate multiscale multibody problems; these provide more realistic descriptions of microscopic mechanisms while efficiently evolving the entire system to obtain macroscopic observations. Solid-tumor growth is one example of such multiscale problems, where the cellular and subcellular scale pathways have been intensively studied and are fairly well understood while the tissue-scale tumor morphology is of interest in clinical applications. We presented a sharp-interface model in Chapter 3 and a Cahn–Hilliard mixture model in Chapter 5, using the continuum approach to simulate the tissue-scale macroscopic morphology of tumors. The dynamics of how tumor cells interact with the microenvironment were studied in Chapter 6 by formulating agent-based models at the cellular and subcellular scales using the discrete approach. By combining the continuum and the discrete approaches, we now present a hybrid modeling framework that utilizes knowledge collected at the cellular and subcellular scales in the study of tissue-scale tumor evolution. Figure 7.1 illustrates the basic components constituting a hybrid model; each component will be detailed in this chapter.

We first review, in Section 7.1, some earlier related work that led to the development of hybrid models. Next, in Section 7.2, we discuss how the fundamental conservation laws are used to establish the general concept of hybrid tumor models. Then the hybrid coupling mechanisms through mass and momentum exchange will be rigorously formulated in Section 7.3.

In this chapter, we provide readers with a numerical framework for solving the multiphase mixture model described in Chapter 5. It is highly challenging to develop an accurate and efficient numerical method to solve such a model, which consists of high-order (fourth-order) nonlinear equations and is characterized by the interaction of small-scale features (e.g. diffuse tumor–host boundaries) with larger-scale features (e.g. the tumor morphology). Here we describe a finite-difference nonlinear multigrid method that can be used to solve the system accurately and efficiently.

The primary component of the model is a Cahn–Hilliard-type equation for the total tumor volume fraction; this is a fourth-order nonlinear advection–diffusion equation. The solutions are characterized by nearly constant states, with complex morphologies, separated by evolving narrow transition layers that describe the diffuse interfaces between the tumor and host tissues. “Slaved” to the Cahn–Hilliard (CH) equation are the equations for the remaining tumor-cell volume fractions, the cell-substrate concentrations (e.g., the oxygen and nutrient levels), and the mechanical pressure that determines cell velocity. The coupling of phenomena across widely varying length scales is a hallmark of multiphase problems. The use of a uniform mesh in such cases leads to either infeasibly large computational problems or significant limitations on the scales of problems that can be simulated. We overcome this difficulty by adopting multigrid methods with adaptive spatial refinement that uses block-structured Cartesian meshes. This balances the needs of localized fine spatial resolution and computational efficiency.

Introduction

We present a sharp-interface continuum model of tumor growth, in which the tumor is treated as a single-phase, or piecewise single-phase, biomaterial with sharp boundaries delineating the various tumor components (viable, quiescent, and necrotic cells). We present the model in order of increasing complexity, starting with avascular and vascular versions that incorporate varying degrees of microenvironmental heterogeneity and ending with a model that accounts for the nonlinear coupling between tumor growth and tumor-induced angiogenesis. In this chapter we focus on the presentation of the model. Analysis and numerical simulations are presented in Chapter 4.

Background

In larger-scale systems, continuum methods provide a good modeling strategy. The reader should refer to the Introduction (Chapter 1) for references and reviews on the continuum modeling of cancer. The approach draws upon principles from continuum mechanics to describe variables as continuous fields described by means of partial differential and integro-differential equations, and it allows for the development of fast numerical solvers. These models treat a tumor as a collection of tissue, for which densities or volume fractions of cells are described. Individual cells and other discrete elements are not tracked. The model parameters may describe volume fractions of various cell species as well as concentrations of cell substrates such as glucose, oxygen, and growth factors. On the one hand, the parameters at this macro-scale are somewhat easier to obtain, analyze, and control compared with those required for discrete models, which typically involve cellular and subcellular measurements (see Chapter 6).