# Cellular interactions constrain tumor growth

^{a}Integrated Mathematical Oncology, Moffitt Cancer Center, Tampa, FL 33612;^{b}Department of Aerospace & Mechanical Engineering and Mathematics, University of Southern California, Los Angeles, CA 90089-1191;^{c}Norris Comprehensive Cancer Center, Keck School of Medicine, University of Southern California, Los Angeles, CA 90089-1191

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Edited by Katepalli R. Sreenivasan, New York University, New York, NY, and approved December 24, 2018 (received for review March 9, 2018)

## Significance

A mathematical model relating tumor heterogeneity at the cellular level to tumor growth at the macroscopic level is described based on a statistical mechanics framework. The model takes into account the number of accessible states available to each cell as well as their long-range coupling (population cooperation) to other cells. We show that the degree to which cell populations cooperate determines the number of independent cell states, which in turn dictates the macroscopic (volumetric) growth law. It follows that targeting cell-to-cell interactions or functional coupling among cell populations could be a way of mitigating and controlling tumor growth.

## Abstract

A tumor is made up of a heterogeneous collection of cell types, all competing on a fitness landscape mediated by microenvironmental conditions that dictate their interactions. Despite the fact that much is known about cell signaling, cellular cooperation, and the functional constraints that affect cellular behavior, the specifics of how these constraints (and the range over which they act) affect the macroscopic tumor growth laws that govern total volume, mass, and carrying capacity remain poorly understood. We develop a statistical mechanics approach that focuses on the total number of possible states each cell can occupy and show how different assumptions on correlations of these states give rise to the many different macroscopic tumor growth laws used in the literature. Although it is widely understood that molecular and cellular heterogeneity within a tumor is a driver of growth, here we emphasize that focusing on the functional coupling of states at the cellular level is what determines macroscopic growth characteristics.

A typical tumor comprises a remarkably heterogeneous agglomeration of cell types, both at the molecular level (1) and at the phenotypic/morphological level (2, 3). Why this is often the case is still somewhat open to debate (1, 4), but clearly mutational instability (5, 6), ecological niches (7, 8), and tissue microenvironmental factors (9⇓–11) all contribute to this diversity of cell types, which, in turn, enables natural selection to act to shape the fitness landscape of the tumor (3, 12), drive tumor growth (13, 14), and select for resistant subpopulations during treatment (15, 16). A question we address in this paper is: What are the consequences of tumor heterogeneity at the cellular scale with respect to key aspects of tumor growth at the macroscale? It has long been known that cellular communication is essential for embryonic development, for example, but it has been less widely appreciated in cancer. In particular, how do cellular diversity, on the one hand, and cell–cell interactions and intercellular communication (via hormones, growth factors, neurotransmitters, and cytokines mediated by gap junction channels), on the other hand, both work in concert to determine the macroscopic growth laws of a tumor? It has been observed that the cancer problem is not merely a cell problem, it is a problem of cell interaction, not only within tissues, but with distant cells in other tissues (17). We know that communication processes among cells ultimately control many aspects of cell growth, including proliferation, differentiation, apoptosis, and a cell’s ability to adapt and respond to microenvironmental cues. We also know that disruption of cell–cell communication can lead to increased or decreased proliferation, abnormal differentiation, apoptotic alteration, and abnormal adaptive responses (18⇓–20). In fact, it has been hypothesized that cancer essentially is a consequence of dysfunctional gap-junctional intercellular communication (21, 22).

In this paper, we present a mathematical framework that links heterogeneity/diversity at the cellular level to the many different deterministic growth laws at the volumetric level. At the cellular level, heterogeneity is modeled by allowing each of the cells to freely “occupy” one of m possible independent states in a statistical mechanics formulation. A cell state could be defined either genotypically or phenotypically, keeping in mind that a cell’s phenotype can be heavily influenced by neighboring cells, microenviromental factors, or even longer-range coupling among cells. The combination of all of these various local cellular influences leads to an emergent volumetric growth law. In the early stages of progression (from a single mutated cell to a small collection of rapidly proliferating cells), the cell population is relatively unconstrained, gradually becoming more and more constrained over time due to extrinsic factors (e.g., fibroblasts, immune cells, blood vessels, and nutritional landscapes) as well as intrinsic factors (e.g., necrosis, DNA mismatch repair, chromosomal instability). The collection of all of these extrinsic and intrinsic factors is represented by a constraining function

## 1. Heterogeneity, Diversity, and Growth

Genetic instability, a hallmark of cancer, is generally believed to be acquired early in tumorigenesis and thought to lead in a multistep fashion to other cancer hallmarks (24⇓–26). This instability can be thought of as increasing the potential of diversity within a tumor, leading to a large number of potential genetic or morphological “states” each cell can occupy, which in turn gives rise to a combinatorial mushrooming of overall molecular/cellular configuration of the tumor. This makes a statistical mechanics approach to cancer modeling an attractive option. Kendal (27) introduced such a model based on these types of considerations, which we take as a point of departure for our work, so it is useful to first review the main features of his simple argument. Consider a population of n cells where the jth cell has the potential to assume one of **4**] allows for the possibility that functional coupling reduces the number of effective states (

To compare with Kendal’s original formulation (27), if we begin with the simplest completely uncoupled case, where we let **1**] as:**6**] and [**8**], we see that:

By contrast, suppose that the number of states each cell can achieve is reduced as the total cell population increases, due to functional coupling of the cell population. Instead of each cell acting independently as previously assumed, we now let

## 2. Functional Cellular Constraints and Volumetric Growth

As the tumor grows, cell signaling and microenvironmental factors act to couple many functional aspects of the individual cells (much the way the uncoupled degrees of freedom in a mechanical system can sync, thereby reducing the effective degrees of freedom of the system as a whole). The two previous examples, **1**], the growth Eq. **2** then becomes:

Broadly, the macroscopic growth models described below can be binned into three categories: (*i*) exponential, (*ii*) sigmoidal, and (*iii*) power law models. Exponential models are characterized by a long period of constant proliferation cell cycle time, while sigmoidal models have eventual slowed growth until an eventual plateau. A power law model also gives rise to a similarly slowed growth, but without a specified plateau. Each of these categories is associated with a certain functional coupling: approximately constant (exponential), increasing (sigmoidal), or decreasing (power). Explicit solutions for each of the growth models are detailed below, along with a discussion on the implications of the functional form of coupling (i.e., the denominator of Eq. **15**).

## 3. Exponential Models

A simple model of tumor growth assumes a constant cell cycle time for all proliferating tumor cells, **17** and **18**).**19**), first introduced here (34):**20**, τ is uniquely determined as:

The exponential models are shown in Fig. 2*A* in blue (*m*, a single cell can occupy is restricted by the exponential of a function form of **15**). The functional coupling,

## 4. Sigmoidal Growth Models

The logistic (Eq. **21**), generalized logistic (Eq. **23**), and Gompertz (Eq. **26**) equations are all in a general class of equations that quantify tumor growth in a sigmoidal shape, where growth is slowed with increasing tumor size (29⇓–31, 35).

Logistic growth (Eq. **21**) is characterized by a linear decrease of relative growth rate and is often interpreted as a competition between proliferating tumor cells for space or nutrients. Logistic growth models have been used by many to describe tumor dynamics (e.g., refs. 31 and 36).**25**) gives rise to the logistic model (Eq. **21**) when **26**, below) when *A* in red (**15**) as the tumor size increases. This is shown in red in Fig. 2*B* and given by

## 5. Power Law Models

Another class of growth models, the power law model, has been used to derive general laws of tumor dynamics from a relationship between growth and metabolism (38, 39). Metabolic rates within the tumor often scale with a power of the total tumor size (29, 38). The Von Bertalanffy equation is written in Eq. **29**, where the power law model is a special case, derived by neglecting the loss term (*A* in green (*B* and given by

## 6. Underlying Biological Mechanisms

It is useful to focus on the changes in cell–cell coupling throughout the entire life history of the tumor and to discuss the biological mechanisms that give rise to these changes. To do so, we reexamine classic data reproduced from ref. 30 showing tumor growth data from 19 samples of 12 different tumors of the rat, mouse, and rabbit superimposed and normalized appropriately so that the full range of growth can be tracked from a single cell to the full carrying capacity of a tumor. We replot the data in Fig. 3*A* with best fits from two of the most common classical models: exponential and Gompertz. The cellular constraints (Eq. **14**) implied by model fits of data from the full range of tumor initiation (a single cell) to tumor saturation (carrying capacity) give insight into how the constraints shape the tumor growth through its time history. To view how changes in the constraints affect the growth law of the tumor over time, the derivative of the function (*B*. As the tumor grows further, the Gompertz function predicts an increase in the effect of the constraints (*A*, corresponding to the maximum point in Fig. 3*B*. Nonetheless, the cell–cell coupling is continually increasing over time until saturation.

There are two distinct biological mechanisms that are at play during this history that are most likely responsible for these changes in coupling behavior among the tumor cells. In the early stages, the short-range communication between cells is mediated by cell gap junctions (43), which are known to be quite permeable to a wide size range of molecules and can provide a direct path for the flow of these molecules between cell interiors. A review of how this exchange of molecular information between nearby cells can control cell division is discussed in ref. 43. A failure in junctional communication is thus implicated as a key biological mechanism responsible for a transition in volumetric growth of the tumor near the time marked

## 7. Therapeutic Implications

Traditional therapeutic treatments target the cancer cells directly by surgical removal or maximal eradication (chemotherapy and radiation). The linkage between cellular coupling and tumor growth leads naturally to the idea of therapeutic methods to try to disrupt or enhance the functional coupling of the cellular states to control tumor growth characteristics. This idea has been touched upon with the suggestion of multitargeted therapeutic approaches disrupting or coopting ecological interactions of tumor–host or tumor–tumor cell interactions. These approaches have been termed “ecological therapies” (20). The model developed here provides a framework for determining how those interactions guide volumetric growth, which may lead to models that optimize timing of new therapies targeting cell–cell interactions, by targeting the mediators of those interactions.

In general terms, the growth law that a tumor follows has definitive therapeutic implications. Cancers that follow exponential growth laws can be assumed to have constant cell–cell interactions (or lack thereof, in the case of blood cancers), and targeted therapies may have more promise in these scenarios, while the therapies aimed at limiting cell–cell interactions described previously will likely have little effect. Likewise, growth models with decreased coupling over time (green curves in Fig. 2*B*) will benefit from ecological therapies early in tumor development but not as tumor sizes increase to clinical relevant sizes. Otherwise, growth laws associated with increased coupling (red curves in Fig. 2*B*) show promise with respect to therapies targeting interactions.

Some have described the therapeutic implications of tumors viewed as ecosystems (47), characterized by competing subpopulations, their spatial and temporal assortment, and their interactions with their physical and chemical microenvironments (20). Much research has been done in the area of collateral sensitivity, to determine if there is pharmacological interaction (additivity or synergy) of multiple drugs in combination or sequence (see refs. 48⇓–50), showing that drugs may have complex downstream effects on cell–cell interactions. Communication or feedback between tumor cells may provide negative (competition, predation, amensalism, parasitism) or positive (commensalism, synergism, mutualism) (19, 20) growth. Organisms compete for limited resources and cooperate for mutual advantage with interactions fluctuating with resource consumption or cell turnover. Additionally, tumor cell interactions may also be dependent on benefits derived from noncancer cells (endothelial cells, cancer-associated fibroblasts, and tumor-associated macrophages) (19, 20). Targeting these noncancer cells, from which the cancer cells are receiving benefit, should also provide therapeutic benefits during the process by which a tumor transitions from a closed system (primary tumor) to an open system of circulating cells to distant colonies (metastatic cancers). But a clear understanding of the relationship between functional coupling among cells and volumetric tumor growth is a necessary step in the direction of exploiting these connections.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: jeffrey.west{at}moffitt.org.

Author contributions: P.K.N. designed research; J.W. and P.K.N. performed research; J.W. analyzed data; and J.W. and P.K.N. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

Published under the PNAS license.

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