Oncologic Mathematics: Title and subTitle BreakEvolution of a New Specialty FREE

DURING THE past quarter century, tremendous strides have been made in the diagnosis and treatment of cancer. Technology now permits the diagnosis and treatment of tumors of ever-diminishing size, as with breast cancers; ductal carcinoma in situ now comprises 25% to 30% of all newly diagnosed breast cancers at most medical centers.¹ With earlier detection, an understanding of growth patterns reflective of the natural biological characteristics of these tumors must also evolve. Surgeons have always led the fields of technological and basic scientific medical advances. Current concepts, be they either the physiological characteristics of shock, organ transplantation, antisepsis, wound healing, or gene therapy, have been forged by surgical investigators.

The field of mathematics has undergone a similar evolution. Topology, fractals, chaos theory, and development of nonlinear descriptive methods have provided mathematicians new creative tools that permit the development of models of tumor growth and behavior at the microenvironmental level.^{2 - 3} Specific formulas have been described for growth, angiogenesis,⁴ cell-to-cell adhesion,⁵ and even pH regulation and drug delivery.⁶ From a clinical viewpoint many of these formulas may seem oversimplified, but they collectively form an important foundation for descriptive insight.

What has been lacking is the linkage of these two naturally and mutually beneficial research endeavors. For oncologic surgeons, the ability to mathematically describe (or, even better, predict) patterns of tumor behavior provides an exciting, new, and precise method that may benefit both current and future therapies. For the mathematician, an understanding of the clinical factors essential for tumor development and metastasis provides realistic insight into these complex biological processes, in turn permitting the development of accurate, clinically relevant mathematical formulas.

In most medical centers, surgeons lead the team that provides comprehensive cancer care. Oncologic mathematics provides surgeons another opportunity to expand their leadership role and to better understand tumor behavior and optimize cancer treatment.

Rather than simply reviewing various mathematical formulas, we have chosen to organize this report around pertinent clinical issues of cancer. Using mathematical models, we describe certain important aspects of tumor behavior. Although these examples are theoretical, their hypotheses are experimentally testable.

To understand the nature of tumor behavior, we borrow the following terms from mathematics.

Not to be confused with the colloquial use of the term, chaos alludes mathematically to the complex nature of an object that is the end result of small subtle alterations in simple nonlinear systems.^{7 - 8} Classically, chaos results when 2 or more factors affect a given system. As described by Rew,⁹ the mathematics of chaos represents a series of events that alter a system exponentially, making prediction challenging.

Based on the theory of chaos, catastrophe represents a dramatic severe change in response to small changes either in inputs, variables, or parameters. The transforming event that precedes the catastrophic change is called "bifurcation." In clinical terms, bifurcation could be used to describe a genomic mutation in a cell that drastically alters its behavior from normal to malignant or even a transformation from in situ lesion to invasive cancer.

Hysteresis is a delayed response to a changing stimulus. Occasionally, tumor growth may follow one path at one end of the stimulus and another in response to a different level of stimulation. A prime clinical example is the wide variation of tumor behavior in breast cancer and its metastasis, as proposed recently by Baum et al,¹⁰ or the response of breast cancer to hormonal manipulation.

Mathematically, every process can be described as an equation, which, broadly speaking, is the relationship of cause to effect. For example, differential equations describe a function over time. An ordinary differential equation is one in which an unknown variable is a function of a single independent variable. For example, the description of tumor growth (T) dependent on a single growth factor (A) is dependent on only one stimulus. Such an equation would be written as T = K × Diffusion of A where K is a constant. A differential equation, for example the rate of tumor growth (G), would be equal to the change in size (Δs) over a time period (Δt). In an equation it would be expressed as G = Δs/Δt.

In contrast, a partial differential equation allows for the inclusion of multiple unknowns as functions over time and space. Intuitively, these equations are more applicable to in vivo situations, wherein several interactive variables are at work. The variables may either be single or grouped together for collective effect: Tⁿ = K × Diffusion of A^N where K is a constant; n, groups of growth parameters; and N, numbers of growth promoters.

Determinism is the ability to predict. In a deterministic equation, depending on the input variables, the outcomes are predicted and can be determined. Three variables play a role in deterministic equations: the substrate, the causative factors, and the conditions under which the interaction occurs. As shown in Figure 1, the deterministic potential of cancer diminishes with advancing stage. Simultaneously, as chaos increases, the ability to determine diminishes. This has been elegantly demonstrated in cellular studies as well.¹¹ Clinically, the diminishing determinism is similar to the lack of reproducibility in terms of results, described by Menoret and Chandawarkar,¹² who showed that clinical pertinence is inversely proportional to reproducibility or determinism. Experimental models are typically highly reproducible because of the controlled nature of experimental conditions. In clinical settings this reproducibility lessens considerably owing to the heterogeneity of patients' conditions, which makes the results less reproducible.

It must be remembered that mathematical modeling is not a single discipline. There are different ways of representing biological systems theoretically, depending on the available inputs and observed results.

As clinicians, we seek out relationships in all biological processes we study, either between genetics and development, etiology and disease, pharmacokinetics and efficacy, or simply in the broad spectrum of cause and effect. These simple proportional relationships are mathematically described by linear equations. Integrating the work of several mathematicians, including Rew, Chaplain, Anderson, Byrne, and Baum, we have generated the following ideas.

In a linear system, the measured response is proportional to the stimulus. In contrast, a nonlinear system does not exhibit this property. Cancer, by virtue of its complex genesis and unconventional patterns of growth and metastasis, eludes prediction. It seems to be a totally random and unpredictable sequence of events. We reexamined these issues and came up with a lasting hypothesis to the contrary.

First, neoplastic disease classically follows a nonlinear growth pattern.¹³ Its complexity in terms of predication cannot be explained using linear mathematics. However, this seemingly complex end point is a result of several serial linear predictable events, which, on a large scale, add up to create incomprehensible complexity. Therefore, it follows that, once initiated, cancer is not a random event but the sum of predictable linear and nonlinear events that remain deterministic.

Second is the inherent predictability of most natural biological processes. For example, in vitro cell cultures exhibit an inordinate amount of genetic and morphologic stability over several passages, in comparison to the number of points at which this linearity could have been broken.¹⁴ Physiologic processes at cellular levels, for example the gastrointestinal tract, skin, or the reproductive system, display inherent controlled growth patterns despite the massive turnover of cells.

Macroscopically, natural processes such as puberty, pregnancy, aging, and even death represent larger versions of the same predictability. Several extrinsic deterents, such as environment, or intrinsic disruptions, such as mutations, could potentially derail this deterministic process. However, inherent control mechanisms prevent digression from the predetermined biological goal.

That vital processes within nature are largely deterministic despite chaotic extrinsic or intrinsic stressors suggests that tumors may be deterministic. Although a clinically detectable tumor, by virtue of its age, is already complex in form and behavior, it seems that the generation of its complexity is the sum of small pathogenic steps. Each of these events could have been determined or predicted individually. To do so clinically is impractical, but as molecular tools develop, an integration of these events, their prediction, and their summative value will become feasible.

Being clinicians, we recognize that it is not the physical presence of the tumor alone, but what it ultimately does, in terms of alterations to the host, that makes tumors deadly. With this underlying thought, we concentrate on 4 fundamental biological processes that ultimately govern tumor behavior: loss of genetic stability (LoGS), tumor growth, immunological identity, and genesis of diversity. Mathematical descriptions of these processes are hypothetical by definition. With the growing understanding of biological processes, their relevance and applicability to a given cancer will only increase.

Surgeons realize that a minor genetic alteration can lead to profound morphologic consequences at a much later point. The well-established concept of LoGS forms a working definition of catastrophe theory as it applies to the biological characteristics of tumors, seen in the genetic evolution of non–small-cell lung cancer, wherein a p53 to HER2/NEU to Ras sequence is described.¹⁵ Genetic components that lead to carcinogenesis include either addition or deletion of genetic material in a milieu that either promotes genetic repair or leads to programmed cell death (Figure 2A); LoGS manifests as chromosomal mutations, DNA insertions, deletions, translocations, gene amplifications, and allelic loss.¹⁷ Albeit random in their individual occurrences, collectively these events follow stochastic principles. Two pathways augment understanding of LoGS events.

In linear pathways, altering events (noted in the diagram as "change") follows a sequential pattern. For example, Genetic Change 1 → Change 2 → Change 3.

Clinical examples of this type of change can be described by tracking genetic alterations in a colon cancer model, as shown in Figure 2B. Linear pathways also allow for the development of therapeutic strategies that interrupt the process of transformation. Treatment with aspirin, folates, or estrogens, as shown in Figure 2B, emphasizes this clinical advantage. Importantly, LoGS may also induce nonlinear behavior. It is conceivable that in triggering carcinogenesis, linear and nonlinear pathways interrelate. Laterally expanding pathways provide an example of this interrelation.

Change 1 can potentially follow multiple directions and elicit several seemingly unrelated change patterns. Laterally expanding pathways may be self-repeating and lead to a fractal pattern of events. Typically, environmental factors of carcinogenesis follow laterally expanding patterns of behavior. As shown in Figure 3, tumor cells, which are inherently unstable, can transform into a variety of cell types. This inherent heterogeneity translates into a variable pattern of growth, metastasis, and clinical response. As shown in Figure 3, the carcinogen provides a stimulus for a genetic change that either leads to apoptosis or allows the altered genomic cells to continue growing. As the carcinogenic stimulus continues, LoGS transforms into an overt invasive cancer. Using the basics of chaos theory, these seemingly chaotic signals can be traced back to either the structure or its biochemical components. Conversely, identifying methods that mitigate destabilizing effects could be of therapeutic benefit. Genetically, chaos manifests as a gain or loss of complexity, ie, allelic loss, tumor suppressor genes, p53 deletion,¹⁸ and overexpressed oncogenic markers.^{19 - 20} Occasionally, accumulation of multiple genetic abnormalities in individual tumor cells can also be charted to predict tumor behavior.²¹ Similarly, if tumor growth is complex, chaotic, and dynamic, then radiation therapy or chemotherapy, although effective in linear laboratory models, would have an unpredictable response in nonlinear tumor models in humans.

Chaos theory may be insufficient in explaining the fundamentally indeterministic nature of LoGS. However, awareness that insignificant linear and nonlinear stimuli can summatively cause apparently unpredictable random effects represents a quantum change in the clinical thought processes. As we learn more about the genome, our ability to trace genetic patterns will increase. Because these events can ultimately be traced, they are classic examples of deterministic chaos, which allows mathematical description.^{22 - 23}

The ongoing or intermittent presence of genetically driven changes in the nuclear DNA is required for neoplastic growth. These oncogenes are thought to control or initiate malignant processes, which once begun, proceed inevitably.^{24 - 26} Triggered by genetic change, cellular growth relies heavily on extrinsic factors. We have broadly categorized these factors as intratumoral and extratumoral.

Morphologically, a tumor is a composite of viable cells, an extracellular matrix, and necrotic debris. Estimates suggest that a 1-cm tumor may contain only 10⁶ viable cells; the remaining bulk comprises an extracellular matrix and perhaps a necrotic core. Early mathematical equations proposed that tumor growth was a differential between cell multiplication and cell death: A = B + C − D − E where A is the total number of living cells; B, the initial number of cells at time t = 0; C, the number of cells produced in time >0; D, the necrotic cell count at time t; and E, cells lost in the necrotic core at time t. Tumor growth was thought to be exponential and dependent on diffusion of nutrients and local biological processes, but the theory did not account for several complex milieus that exist in vivo, such as neovascularization, growth of the extracellular matrix, and apoptosis. Figure 4 shows possible relationships between tumor size and availability of nutrients or angiogenesis. Although these relationships are hypothetical, these equations provide a framework for future experimental designs. Charting growth curves in this manner is not only mathematically useful but also clinically accurate. It accounts for the biological process that promotes and limits growth.

Growth-dependent factors need angiogenesis for delivery of nutrients. As the metabolic needs of cells outstrip the available nutrient supply, necrosis sets in, which in itself stimulates development of neovascularity. Diffusion of locally available factors may act in a paracrine manner, contributing to differing rates of cell multiplication. Mathematicians describe the process of diffusion largely with respect to cellular nutrients.²⁷ Early hypotheses were based on visualization of the tumor as a symmetrical sphere where diffusion occurred in a state of equilibrium.²⁸ Although this model accounted for a necrotic center, there were many practical problems associated with this assumption. Our criticism of this assumption is that it is oversimplified. Aside from the physical description of tumors, both the extent and rate of diffusion of nutrients remain unidentified. Moreover, these nutrients may be essential for cell growth in a sequential manner, and that sequence remains poorly understood.

Although the morphologic characteristics of tumors in vivo appear to be guided by chaotic signals, mathematical equations that include tumor composition can be used to determine rates of tumor growth and the availability of nutrients, thereby tracing these signals to either the structural or biochemical components of tumors. As shown in Figure 5, tumor size is affected differently by the various extratumoral processes that govern it. As shown by Byrne and Chaplain,²⁹ tumor growth varies significantly depending on external factors. Although immune escape seems to cause the most rapid tumor growth rate (as seen in immunocompromised individuals), age-related tumors are the slowest to grow.

For example, Baum et al¹⁰ studied 3 variables of tumor angiogenesis, namely endothelial cells, tumor angiogenic factors, such as vascular endothelial growth factor, α–fibroblast growth factor, or β–fibroblast growth factor, and matrix angiogenic factors.

The change in endothelial cell density denoted by Δn/Δt (where n is the number of cells and t is time) is the net gain in cells derived from random motility, chemotaxis, and haplotaxis. In an equation this is expressed as Δn/Δt = Random Motility − Chemotaxis − Haplotaxis

The rate of change of tumor angiogenic factor concentration (Δc) can be expressed as Δc/Δt = Production of Tumor Angiogenic Factors − Uptake by the Endothelial Cells

Equations for each component are also derivable and add to the mathematic complexity. Details of these equations are presented by Baum et al¹⁰ and provide an insight into the vast possibilities mathematical methods create.

Although under optimal conditions tumor cell–host interactions fundamentally favor the host, tumor cells prevail in circumstances wherein they evade detection and annihilation by immune surveillance mechanisms. Several adaptive strategies developed by tumor cells allow for changes in their immunologic identity, facilitating immune escape. Tomlinson and Bodmer³⁰ provide a contrast to the cell-autonomous effects of mutations that are assumed in classical models of tumorigenesis. They also illustrate how tumor cells employ different strategies that affect other cells, thereby favoring their own replication, perhaps at the expense of other cells. These models identify which strategies are likely to be most successful in the tumor cell population and provide a theoretical basis for the data. Notably, all mutations and thereby their phenotypic antigenic variations do not necessarily occur at an oncogenic level but at loci involved in interactions with other cells within the tumor substance. The possibility that understanding these immunologic functions may lead to the development of better vaccination strategies is currently being studied.³¹

The genomic structure of tumor cells is an excellent example; tumor cells are phenotypically and behaviorally vastly different from normal cells, yet they share a common genetic structure.³² This shared code allows for the production of shared proteomic material that is identical to that in normal cells. The shared identity makes immune detection difficult because the immune system is unable to distinguish self from mutated self. As the uniqueness of an antigen increases, so does the ability of the host to mount an antigen-specific immune response.³¹ Conversely, antigens that are highly shared, as in self-proteins, do not elicit any immune reaction, thereby preventing autoimmunity. Mathematically, if one analyzes the antigenic components of normal cells and tumor cells, one notices that shared antigens continue to decrease in number because the unique antigens are the only distinguishing factors that immune mechanisms can reliably activate themselves against.³³

To describe tumor spread and the genesis of metastatic disease we use the principle of fractal geometry. Typically, fractals are self-replicating images, mathematically denoted by simple quadratic equations. In human tissues, fractals are routinely seen in neural branching, biliary ductal systems, vascular branching, and bronchiolar architecture. Subunits of these branching patterns resemble the larger structure. Extrinsic and intrinsic factors that drive this process remain continuous and hence similar.

Fractals are based on the principle that complexity in natural systems arises from relatively simple rules and mathematical formulas that are repetitive and continuously applied. The situation is similar to that of tumorigenesis or spread, wherein the genetic changes, immunologic mechanisms, and environmental factors when constantly applied can generate a seemingly complex larger entity with repetitive patterns of the smaller initial version. Using the fractal principle may enable development of a working model for several important processes that govern tumor mechanisms. It simplifies the process by which a seemingly complex diversity is generated from simple linear events. It remains true to our original argument that complex forces governing the biological characteristics of tumors are linear, predictable, and inherently a controlled chaos, as is the case with other natural biological processes.

Fractals have been used to understand several biological processes including embryology, developmental biology, hematogenous metastasis,³⁴ genetic function, radiologic imaging,^{35 - 36} mutation, and electrophysiology.^{37 - 38} The recognition that the stimuli for its generation be constant and continuous and that the innate components of each subunit be similar to the larger structure is important to fractal generation. Fractals have also been applied to diagnosis and histopathological analysis of tumors.^{38 - 40}

In Figure 6A, we generated a simple progression of cellular diversity based on the antigenic change and genetic mutation shown in Figure 3. It must be emphasized that the changes in cells are caused under the same specific conditions as the first set of changes. If these conditions persist, this pattern will recur at the end of each cycle, the only difference being that the starting cell will be the endpoint of the preceding fractal. If each of the cells at the branches of each fractal were subjected to the same rules, one would generate a larger pattern (Figure 6B and C) that is complex and will have generated diversity of extraordinary proportions. It is important to remember that the larger pattern continues to retain all the characteristics of each of its constituents. When applied to antigenic diversity generated within tumor progression, for example, it is fundamental to identify the subunit pattern first. This may enable one to understand the larger genesis of diversity simply by applying the pattern in a repeated fashion.

Two mathematical principles are fundamental to prognostication of tumor patterns: (1) the understanding that tumor behavior is deterministic and (2) that chaos is a mathematical example of determinism.

Complex interactions between several diverse clinicopathological variables make prognostication unreliable. Moreover, the ever-expanding numbers of prognostic factors implicated in breast cancer have failed to provide practical benefit in cancer management. Partly, the inability to integrate multiple prognostic variables into existing methods of evaluation, such as the TNM system, is the limiting factor. In addition, current statistical methods of evaluating complex patient data at best generate a population-specific response, which is not patient-specific and hence irrelevant to an individual patient.

Mathematical methods using artificial intelligence systems will be necessary to evaluate all factors meaningfully and generate a patient-specific prognostic assessment. Studies are currently underway to ascertain the feasibility of these applications.

Albeit experimental, the field of oncologic mathematics is rapidly growing. So are the clinical and laboratory data that will lead to better assessment of the biological processes of tumors. A formal interactive collaboration between the fields of mathematics and oncology is inevitable. Oncologic mathematics defines methods that could provide the basis for experimental observations. It may not immediately address all issues or lead to cancer cures in the near future. Nor will it replace clinicians' understanding of tumor behavior and patient care—the need for more research is certain. That this collaboration will help create a better understanding of cancer is indisputable. It may even change the way we treat patients.

We thank Daniel Levey, PhD, James A. Lehman, Jr, MD, and James M. Lewis, MD, for their useful comments and Judy Knight, Akron General Medical Center Library, Akron, Ohio, for help in preparing the manuscript.

Corresponding author and reprints: Rajiv Y. Chandawarkar, MD, Department of Plastic and Reconstructive Surgery, Unit 443, M. D. Anderson Cancer Center, 1515 Holcombe Rd, Houston, TX 77030 (e-mail: rchandaw@mdanderson.net).