The mathematics of cancer: integrating quantitative models

Key Points

  • Mathematical models have become an integral part of cancer biology. They are useful tools for deriving a mechanistic understanding of dynamic processes in cancer.

  • The somatic evolutionary process, which maintains tissues and can initiate cancer, has served as a hallmark of mathematical descriptions of tumours. Mathematical models have helped in the understanding of interactions among homeostatic mechanisms, environmental factors and mutation accumulation that drive tumorigenesis.

  • Using cell-based hierarchical models of tissue structure, theoretical insights have influenced the prediction of the cell of origin of human cancers, which may drive an understanding of metastasis and treatment response.

  • The temporal order of events during tumour development can be recapitulated using mathematical modelling and genomics data sets.

  • Mathematical models have also been used to explore the role of the tumour microenvironment in cancer progression. Such models help to elucidate important microenvironmental barriers to effective cancer treatment and how to overcome them.

  • Metastasis evolution and immunotherapy have attracted increasing interest but still offer a wide range of opportunities for mathematical modelling.

  • In combination with pharmacological considerations, quantitative models have a decisive role in the exploration of novel treatment modalities of cancer. This includes drug scheduling and the effect of combination therapy to avoid the evolution of resistance.

  • The key role of mathematical modelling in the future will not only be to describe what is known, but also to point to gaps in our understanding of which complex interactions drive tumour growth, treatment dynamics and resistance evolution.

Abstract

Mathematical modelling approaches have become increasingly abundant in cancer research. The complexity of cancer is well suited to quantitative approaches as it provides challenges and opportunities for new developments. In turn, mathematical modelling contributes to cancer research by helping to elucidate mechanisms and by providing quantitative predictions that can be validated. The recent expansion of quantitative models addresses many questions regarding tumour initiation, progression and metastases as well as intra-tumour heterogeneity, treatment responses and resistance. Mathematical models can complement experimental and clinical studies, but also challenge current paradigms, redefine our understanding of mechanisms driving tumorigenesis and shape future research in cancer biology.

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Figure 1: Tumour initiation and progression.
Figure 2: Microenvironmental interactions and the emergence of metastases.
Figure 3: Treatment response and dosing strategies.

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Glossary

Mathematical models

Models can describe a system by means of abstraction and mathematical formalism. They enable extrapolation beyond situations originally analysed, quantitative predictions, inferrence of mechanisms, falsification of underlying biological hypotheses and quantitative description of relationships between different components of a system.

Hybrid models

A modelling approach that combines several modelling techniques in one. For example, a hybrid model that describes the tumour microenvironment in which stromal cells follow a continuous nonlinear description, whereas tumour cells obey a discrete stochastic process.

Branching process

A stochastic process model of cell division, mutation events and cell death that leads on average to an exponential increase or decrease in the total population size. The branching process is based on the assumption that each individual event occurs at the same rate, independently of, for example, the population size or composition, or the point in time. The branching process is a Markov process; that is, the probability of the next event happening depends only on the current state of the population, and not on its earlier history.

Passenger mutations

Genetic changes that have no obvious or direct effect on cell fitness or cancer development, and may occur and potentially vanish again during any stage of tissue development and homeostasis. According to some definitions, passengers might also be (slightly) deleterious.

Driver mutations

Genetic changes that are causally involved in cancer development, typically conferring a functional change and a somatic evolutionary advantage.

Epistatic interactions

Interactions that occur when the functional effect of one genetic alteration depends on the genetic background of the cell; that is, the state of one or more other genes.

Homeostasis

A property of a system in which variables are regulated so that internal conditions remain stable and relatively constant. An example is the constant tissue size of most organs in the absence of neoplasms.

Deterministic model

Given a specific initial condition, a deterministic process always yields the same output, and no randomness is involved. Deterministic processes can be chaotic in that a small deviation in the initial condition may yield a large deviation after some time. However, this effect is different from the effect of a stochastic process in which the same initial condition can lead to different results.

Hierarchical tissue structures

Structures according to which most tissues are organized, ranging from slowly proliferating stem and progenitor cells to more quickly proliferating precursors and terminally differentiated cells.

Phylogenetic tree

A branching, tree-structured graph that represents the evolutionary relationships among different (mutational) stages of a tumour cell population, quantified by some measure of distance between individual cells or patient samples.

Graphical models

Mathematical structures that describes pairwise relations (called edges) between objects (called nodes), possibly on several layers. An acyclic graph does not have any cycles. Undirected graphs imply that there is no direction in the relationships along any edge. A tree-like graph has the property that every node can be traced back to a central node, called the root node, while final nodes of a tree are called leaves.

Longitudinal data

Repeated observations of the same system or set of systems over time.

Agent-based simulation

A computational approach that models complex systems consisting of interacting discretized items or 'agents'. In cancer modelling, these agents often represent cells, which can mutate into other types, divide into two cells, die or move in space. These simulations can be implemented according to either probabilistic or deterministic laws.

Stochastic process

This describes how a random variable (or set of random variables) changes over time and/or space. A stochastic process ascribes a probability to each event and allows for the prediction of the probability of a certain outcome. In contrast to a deterministic process, the initial condition yields an entire probability distribution over possible events at any later point in time.

Markov process

A memoryless stochastic process in which the conditional probability distribution over all future events depends only on the present state. A Markov chain explicitly addresses stochastic dynamics between discrete states in discrete time, thus allowing for a full characterization using a transition matrix in which the entries describe the probability of transitioning from one state to another.

Biased random walk

The movements of an object or changes in a variable that on average follow a specific pattern or trend.

Linear-quadratic model

A prominent heuristic to describe cell survival under radiation. The number of surviving cells after a certain dose of radiation has been administered takes the form of an exponential function with a linear and a quadratic term in its argument.

Predator–prey models

These models, also known as Lotka–Volterra dynamics, are used to describe the dynamics of ecological species, or types, as a nonlinear deterministic process. They were originally used to describe population dynamics of predators and prey, taking into account abundance, interactions, and population growth and diminution. They can also be used to describe mutualistic and competitive evolutionary dynamics; for example, between cellular types.

Sigmoidal growth curve

An S-shaped growth pattern in which the population size starts from a low density with positive acceleration, then transitions to negative acceleration at high density. An equilibrium population size can be characterized, for instance, by a proliferation–self renewal–death balance, or by a carrying capacity. Examples are Gompertzian growth and logistic growth.

Power-law growth model

A functional relationship between two quantities (for example, time and tumour size), where one quantity varies with the power (that is, exponent) of the other. The exponent can typically be inferred from linear regression analysis of a doubly logarithmic transformation of the data.

Luria–Delbrück model

The Luria-Delbrück experiment investigated whether mutations occur independently from, or owing to, selection. Data from growth experiments in which Escherichia coli were challenged with a virus were compared to a stochastic process model used to calculate the probability of having a certain number of resistant mutants. The findings suggested that mutations occurred randomly over time and were not a response to selection.

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Altrock, P., Liu, L. & Michor, F. The mathematics of cancer: integrating quantitative models. Nat Rev Cancer 15, 730–745 (2015). https://doi.org/10.1038/nrc4029

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