# Local contact inhibition leads to
universal principles of cell population growth

Gregory J. Kimmel  Department of Integrated Mathematical Oncology, H. Lee
Moffitt Cancer Center and Research Institute, Tampa, FL 33612, USA  Jeffrey
West  Department of Integrated Mathematical Oncology, H. Lee Moffitt Cancer
Center and Research Institute, Tampa, FL 33612, USA  Mehdi Damaghi  Department
of Cancer Physiology, H. Lee Moffitt Cancer Center and Research Institute,
Tampa, FL 33612, USA  Alexander R. A. Anderson  Department of Integrated
Mathematical Oncology, H. Lee Moffitt Cancer Center and Research Institute,
Tampa, FL 33612, USA  Philipp M. Altrock  Department of Integrated
Mathematical Oncology, H. Lee Moffitt Cancer Center and Research Institute,
Tampa, FL 33612, USA  Department of Evolutionary Theory, Max Planck Institute
for Evolutionary Biology, 24306 Ploen, Germany  Corresponding author:
<EMAIL_ADDRESS>
###### Abstract

Cancer cell population dynamics often exhibit remarkably replicable, universal
laws despite their underlying heterogeneity. Mechanistic explanations of
universal cell population growth remain partly unresolved to this day, whereby
population feedback between the microscopic and mesoscopic configurations can
lead to macroscopic growth laws. We here present a unification under density-
dependent birth events via contact inhibition. We consider five classical
tumor growth laws: exponential, generalized logistic, Gompertz, radial growth,
and fractal growth, which can be seen as manifestations of a single
microscopic model. Our theory is substantiated by agent based simulations and
can explain growth curve differences in experimental data from in vitro cancer
cell population growth. Thus, our framework offers a possible explanation for
the large number of mean-field laws that can adequately capture seemingly
unrelated cancer or microbial growth dynamics.

Keywords: population growth, cancer growth curves, cell motility, contact
inhibition

##  1 Introduction

Cancer cell population dynamics often exhibit remarkably replicable, universal
laws (Wodarz and Komarova, 2005; Pérez-García et al., 2020) despite their
genetic, epigenetic, and phenotypic heterogeneity (Reiter et al., 2019;
Hausser and Alon, 2020). The derivation of a universal growth law for tumors
has been the subject of investigations for decades (Mombach et al., 2002;
Guiot et al., 2003; Rodriguez-Brenes et al., 2013; Benzekry et al., 2014). The
winding search for a comprehensive mathematical characterization of tumor
growth has belied two seemingly opposing concepts. First, a growth model
should accurately describe the empirical data (Altrock et al., 2015; Brady and
Enderling, 2019). Second, a growth model should be based on and derived from a
biological mechanistic framework (Cheng et al., 2009; Gerlee, 2013; West and
Newton, 2019). Cell population growth models with underlying mechanisms should
be preferred because of their utility in comparative studies, model selection,
and hypothesis generation.

Models with a strong biological motivation often receive less attention due to
their lack of data-fitting potential, and mechanistic approaches might be
preferably used (Gerlee, 2013). Exponential growth often fits well to data,
but dynamics over longer timescales are better approximated by logistic,
Gompertz, or power law models (Benzekry et al., 2014). Intriguingly, the
Gompertz model was originally employed as the best match to observed
exponential decay of net cancer growth rates (Laird, 1964). Attempts at
mechanistic explanations of growth with time-dependent rates remain partly
unresolved to this day.

Gompertzian models find wide application, e.g. in finance (Islam et al.,
2002), population regrowth through niche occupation (Kimmel et al., 2021),
microbial dynamics (Zwietering et al., 1990), and tumors (Steel, 1977a; Spratt
et al., 1993). Its use in cancer modeling, even without biological mechanistic
foundation, is indicative of its predictive and descriptive potential
(Benzekry et al., 2014). To reconcile the disconnect between descriptive power
and mechanistic insight, several studies have proposed a deeper understanding.
Recent work by West & Newton derives several related growth laws
mechanistically by assuming gradual occupation of cellular microstates (West
and Newton, 2019; West et al., 2016). This approach indicates that population
feedback between the populations’s microscopic and mesoscopic configurations
can lead to macroscopic growth laws.

In cancer cells, increased motility allows for continued movement after
contact with neighbors (Ribatti, 2017). In normal tissues, high cell density
leads to inhibition of cellular proliferation upon reaching maximal capacity
(Eagle and Levine, 1967; Pavel et al., 2018). Loss of contact inhibition is
common in solid tumors and increases invasion (Mayor and Carmona-Fontaine,
2010; McClatchey and Yap, 2012). Cell density is also impacted by cell
migration. There is often assumed to be a dichotomy between migration (go) and
proliferation (grow) (Hatzikirou et al., 2012; Gallaher and Anderson, 2013;
Kimmel et al., 2020)). The go-or-grow hypothesis remains controversial
(Vittadello et al., 2020), yet might also arise from DNA replication and
genomic instability (Aguilera and Gómez-González, 2008; Kimmel et al., 2020).

Here, we present a unifying framework that leads to multiple classical growth
laws based on the mechanism of local contact inhibition. Three major factors
regulate contact inhibition and tumor cell density dynamics: proliferation,
migration, and the size of the local interaction region. Each of these
microscopic factors may be subject to heterogeneity and temporal variations.
Inevitably, these variations lead to diverse macroscopic growth dynamics. Key
properties of these macroscopic dynamics could still be explained using simple
bio-physical mechanisms. The theory we introduce here considers five classical
tumor growth laws: exponential, generalized logistic, Gompertz, radial growth,
fractal growth. Corroborated with agent based simulations and statistical
analysis of data from in vitro cancer cell population growth experiments, we
show that these five growth laws are manifestations of a single microscopic
model under varied assumptions of contact inhibition.

##  2 Methods

Here, we describe the experimental (in vitro) procedures, analytical modeling,
and computational (agent based) modeling, and data fitting procedures. The
quantities and parameters of the model are summarized in Table 1.

Parameter name | Symbol | Typical values | Range
---|---|---|---
Number of cells at time $t$ | $n(t)$ | $0,\dots,10^{8}$ | $[0,\infty)$
System domain length | $L$ | $100,\dots,1000$ | $[0,\infty)$
Birth neighborhood domain/set for cell $i$ | $\Omega_{i}$ | – | –
Birth neighborhood size for cell $i$ | $\omega_{i}$ | $4,\dots,8$ | $[0,24)$
Mean birth neighborhood | $\mu_{\omega}$ | $4,\dots,8$ | $[0,24)$
Density-dependent birth rate at site $i$ with population size $n$ | $\lambda_{i,n}$ | – | –
Density-dependent death rate at site $i$ with population size $n$ | $\delta_{i,n}$ | – | –
Cell motility coefficient | $\Gamma$ | $0,\dots,\infty$ | $[0,\infty)$
Cell migration rate (ABM) | $m$ | $0,\dots,1$ | $[0,\infty)$
Migration neighborhood set (ABM) | $\Phi$ | – | –
Migration neighborhood size (ABM) | $\phi$ | 4,…,8 | $[0,\infty)$
Table 1: Parameter table.

###  2.1 Cell culture and immunofluorescence experiments

For cancer cell culture experiments, we used the MCF-7, MDA-mb-231 breast
cancer cell lines (Figure 1), as well as ovarian cancer cell lines OVCA3,
A2780s, TOV112D, which were acquired from American Type Culture Collection
(ATCC, Manassas, VA, 2007-2010) and were maintained in RPMI 1640 (Life
Technologies, Cat# 11875-093) supplemented with 10% fetal bovine serum
(HyClone Laboratories). Cells were tested for mycoplasma contamination and
authenticated using short tandem repeat DNA typing according to ATCC
guidelines.

Figure 1:  Experimental observations of cancer cells distributed at low and
high density. A, B: Breast cancer cell line MCF-7 at low and high density. C,
D: Breast cancer cell line MDA-mb-231 at low and high density.

Cells were seeded on an 8 chamber microscopy slide overnight. They were then
rinsed with phosphate-buffered saline (PBS), fixed in cold Methanol:Acetone
(1:1) for 10 minutes and further permeabilized by 0.5% triton 100 and then
blocked with 5% bovine serum albumin in PBS. Samples were incubated with LAMP2
rabbit primary antibody (1:100; ab 218529) from abcam and secondary Alexa-
Fluor 488 antirabbit (1:1000) antibody. WGA was used to stain the membrane and
DAPI for the Nucleous. Coverslips were mounted using ProLong Gold Antifade
Reagent (Life Technologies) and images were captured with a Leica TCS SP5
(Leica) confocal microscope. Cell count measurements were performed with an
oil-immersion x63 objective.

###  2.2 Agent based simulations of tumor growth

Simulations of a stochastic agent based model were performed using the Hybrid
Automata Library software package, which allows for fast simulation of tumor
growth models in spatial domains with real-time visualization. The model is
implemented using Java JDK Version 8. Each stochastic simulation is
initialized with a single tumor cell placed in the center of an $L\times L$
two-dimensional domain. The model is a birth-death-migration process, carried
out on a two-dimensional lattice where each grid point contains at most one
single tumor cell. In each generation, all cells in the domain follow the
birth-death-migration cell step flowchart (Figure 2A). While iterating through
all cells, the current focal cell may migrate to a random neighboring grid
point at rate $m$. The migration neighborhood is a precisely defined set of
lattice locations defined in relation to the focal cell’s location, defined as
$\Phi=1,2,\dots,\phi$ ($\phi\in L\times L$) with example neighborhoods given
in Figure 2B. Next, the focal cell undergoes division at a rate $b$, where the
daughter cell is placed in a random neighboring grid point and the parent cell
remains on the original lattice point. Similarly, the lattice locations within
the birth neighborhood are precisely defined as $\Omega=1,2,\dots,\omega$,
which may be identical, larger, or smaller than $\Phi$. If a birth event does
not occur, the cell may undergo apoptosis (death) and is removed from the
domain. After each generation, the update order in which cells iterate is
shuffled.

Figure 2:  Overview of computational modeling and theoretical approaches. A:
agent based model (ABM) cell step flowchart. At each time step, the cell
checks if a migration event occurs (at rate $m$). Next, the cell checks if a
birth even occurs, and divides if there is a free lattice point to place the
new daughter cell within the birth neighborhood. If no free space is
available, a birth does not occur. In the case of no birth, the cell may
undergo death (at rate $d$). B: Protypical neighborhood paradigms for
migration or birth for the individual-based model simulations. C: Schematic of
the theory hierarchy that describes the general model (red) and the limiting
regimes we consider. High migration ($\Gamma$ large) leads to well-mixed
behavior, while low migration ($\Gamma$ small) leads to radial growth.

###  2.3 Analytical model of tumor growth

Let us define a location with generalized coordinates $\vec{q}$ and suppose
that the probability for finding a cell at a particular site, given $n$ cells
at time $t$, is given by the probability density (mass) function
$f(\vec{q}|n,t)$. We assume a birth-death process, birth events are density-
dependent. An example setup is shown in in Figure 2A. Tumor growth is
constrained to a two-dimensional lattice where each lattice point can contain
at most a single cell.

Contact inhibition may occur within some localized region, or “neighborhood,”
near a cell. Commonly used neighborhoods are shown in Figure 2B, but we can
consider a neighborhood domain of any arbitrary size. The neighbor domain for
the $i$th cell we denote by $\Omega_{i}$ with number of elements $\omega_{i}$.
We can partition the domain into two disjoint groups via the union
$\Omega_{i}\cup L\backslash\Omega_{i}$. Define the random variable
$X_{i}\in\\{0,1,\dots,\omega_{i}\\}$ as the number of filled neighbor sites
seen by cell $i$ and let $\mathbb{P}(X_{i}=x)$ be the probability of observing
$x$ filled sites. In a similar way, we will define
$Y_{i}\in\\{0,1,\dots,l-\omega_{i}-1\\}$ as the number of filled sites not in
the neighborhood. In what follows, we often remove the subscripts for brevity.

Given a focal cell $C$ at position $\vec{q}_{i}$, denoted by $C(\vec{q}_{i})$,
we seek the probability of observing $x$ neighbors and $y$ non-neighbors given
$n$ cells. Formally, we want an expression for the following conditional
probability

$\mathbb{P}(X_{i}=x,Y_{i}=y|n,C(\vec{q}_{i})).$ (2.1)

We can express the function (2.1) in terms of a product, assuming
independence, to gain further insight into the behavior of this system:

$\displaystyle\mathbb{P}(x,y|n,C(\vec{q}_{i}))$
$\displaystyle=\mathbb{P}(y|n,x,C(\vec{q}_{i}))\mathbb{P}(x|n,C(\vec{q}_{i})).$

We note here that since the groups partitioned by neighborhood are disjoint,
we require the condition that $x+y=n-1$ and so
$\mathbb{P}(y|n,x,C(\vec{q}_{i}))=1$ only if $y=n-x-1$ and 0 otherwise. This
allows us to consider simply the conditional probability:

$\mathbb{P}(x|n,C(\vec{q}_{i}))=\frac{\mathbb{P}(x,C(\vec{q}_{i})|n)}{\mathbb{P}(C(\vec{q}_{i})|n)},$
(2.2)

with the restriction that $y=n-x-1$. All of the information for birth is tied
into this distribution. An individual-based birth-death-movement process of
cell $C(\vec{q}_{i})$ with total number of cells $n$ is defined via the
following transition rates:

$\displaystyle C(\vec{q}_{i})$
$\displaystyle\stackrel{{\scriptstyle\lambda_{i,n}}}{{\longrightarrow}}C(\vec{q}_{i})+C(\vec{q}_{j}),$
(2.3a) $\displaystyle C(\vec{q}_{i})$
$\displaystyle\stackrel{{\scriptstyle\delta_{i,n}}}{{\longrightarrow}}\emptyset,$
(2.3b)

where $\vec{q}_{j}\in\Omega_{i}$. Death is density-independent (i.e.
constant), while the birth rate is density-dependent through contact
inhibition. That is, we only allow a birth event to occur for cell at
$\vec{q}_{i}$, if a neighbor site is available:

$\lambda_{i,n}=\lambda[1-\mathbb{P}(\omega_{i}|n,\vec{q}_{i})].$ (2.4)

If all cells are able to divide, we can get the population growth rate via
summing over all cells. The situation in which all cells are able to divide is
analogous to a well-mixed population. In contrast, we can also suppose that
only a subset of the cells are able to divide. A well-known example would be a
radially growing tumor where only surface-dwelling cells are able to divide.
We will look at both of these cases and see how they impact the resulting
growth laws.

Summing over all cells, we have the population birth and death rates given by:

$\displaystyle\lambda_{n}$
$\displaystyle=\sum_{i=1}^{n}\lambda_{i,n}=\lambda\sum_{i=1}^{n}[1-\mathbb{P}(\omega_{i}|n,\vec{q}_{i})],$
(2.5a) $\displaystyle\delta_{n}$ $\displaystyle=\delta n.$ (2.5b)

The process governing the mean number $\langle n\rangle$ of the birth-death
process is given by

$\displaystyle\frac{d\langle n\rangle}{dt}$
$\displaystyle=\bigg{\langle}\lambda\sum_{i=1}^{n}[1-\mathbb{P}(\omega_{i}|n,\vec{q}_{i})]\bigg{\rangle}-\delta\langle
n\rangle.$ (2.6)

No approximations have been made to arrive at equation (2.6). To make a
connection to more familiar growth laws, we require an approximation that is
valid for large $n$, which allows us to replace the expectation of the product
with the product of the expectations. The deterministic analog when $n$ is
large allows us to rewrite this in terms of the density $u=\langle
n\rangle/l$:

$\dot{u}=\Gamma\nabla^{2}u+\lambda
u\left[1-\langle\mathbb{E}[\mathbb{P}(\omega_{i}|n,\vec{q}_{i})]_{i}\rangle\right]-\delta
u,$ (2.7)

where we have added a migration term via diffusion. Note, the expectation,
$\mathbb{E}[]_{i}$ is done over each cell $i$ and the average,
$\langle\cdot\rangle$ is the average of all possible states of $n$ cancer
cells (e.g. in this case $n\in\\{0,\dots,l\\}$).

##  3 Results

In our analytical description, equation (2.7) presents a starting point from
which multiple growth laws governed by contact inhibition at birth can be
obtained. If the birth-neighborhood size is independent of $n$ and spatial
distribution $\vec{q}$, then (2.7) reduces to that of exponential growth. The
functional form of the growth law is highly dependent on the behavior of the
average number of neighborhood sites. In the following sections, we look at
special forms of $\omega_{n}$ to discern some of the well-known growth laws
that can emerge via contact inhibition. A schematic of the theory workflow
along with sufficient conditions is shown in Figure 2C.

###  3.1 Radially expanding population

A natural starting point is to assume a growing solid tumor with no migration.
Using equation (2.7) as our point of departure (with $\Gamma=0$), we assume
the tumor is growing radially where only cells on the outer surface with free
space have the ability to divide. Let $r$ be the length scale for tumor cells,
$R$ be the length scale of the tumor volume and $d$ be the dimension
considered (typically $d=2,3$). The number of tumor cells is then estimated by
$n=(R/r)^{d}$. The number of cells not on the surface is given by
$n_{0}=(R/r-1)^{d}$. We now write out the probability of all sites occupied
explicitly,

$\mathbb{P}(\omega_{i}|n,\vec{q}_{i})=\begin{cases}0&\text{if }n=n_{s},\\\
1&\text{if }n=n_{0}.\end{cases}$ (3.1)

Partitioning our expectation using $n=n_{s}+n_{0}$ leads to

$\frac{1}{n}\left[\sum_{i=1}^{n_{s}}\mathbb{P}(\omega_{i}|n,\vec{q}_{i})+\sum_{i=1}^{n_{0}}\mathbb{P}(\omega_{i}|n,\vec{q}_{i})\right]=\frac{n_{0}}{n}.$
(3.2)

Using the relations between the length scales of tumor volume versus typical
cell size leads to

$\frac{n_{0}}{n}=\left(1-\frac{r}{R}\right)^{d}=\left(1-n^{-1/d}\right)^{d}\approx\left(1-dn^{-1/d}\right),$
(3.3)

the last part, valid when $n\gg 1$.

The deterministic growth law is then given by

$\frac{d{\langle n\rangle}}{dt}=\lambda d\langle
n\rangle^{\frac{d-1}{d}}-\delta\langle n\rangle.$ (3.4)

With $d=3$, we obtain the Von Bertalanffy model

$\frac{d{\langle n\rangle}}{dt}=3\lambda\langle
n\rangle^{\frac{2}{3}}-\delta\langle n\rangle.$ (3.5)

For radial growth on a dish, we have $d=2$ and we obtain

$\frac{d{\langle n\rangle}}{dt}=2\lambda\langle
n\rangle^{\frac{1}{2}}-\delta\langle n\rangle.$ (3.6)

Thus, from a general growth equation (2.7) that includes mechanism of movement
and local contact inhibition, one can derive radial cell population growth
laws based on a length-scale argument.

###  3.2 Fractal growth

The notion of self-similarity has been applied to investigate tumor growth as
its shape is often not a perfectly geometric object in the classic sense.
Although population growth is often approximated as radial expansion, this
approximation might be poor at finer length scales. Fractals and
dimensionality can play a significant role in the governing laws that control
tumor growth. Although these shapes are not fractals in the truest sense, as
the self-similarity does not exist beyond a certain length scale (e.g. the
radii of tumor cells is a natural cutoff), the principles of fractal growth
can be used. In particular this idea was used to successfully show the density
falling off in silica particles described in (Orbach, 1986).

The total number of cells will approximately follow $n\propto R^{d}$ where $d$
is the actual dimension of the system and $R$ is the characteristic length
scale of the growing population. In contrast, the surface dwelling cells
$n_{s}\propto R^{D}$ where $D$ is the fractal dimension. It follows that
$n_{s}\propto n^{D/d}$. This leads to a growth law

$\frac{d\langle n\rangle}{dt}=\lambda a\langle n\rangle^{D/d}-\delta\langle
n\rangle,$ (3.7)

where we have introduced the constant $a$ related to the characteristic length
of the tumor cell (and to the fractal dimension of the tumor). With non-
fractal growth, $D=d-1$ and our model reduces to that of radial growth.
However, fractal growth implies a dimension $d-1<D\leq d$.

To illustrate fractal growth, we ran stochastic simulations of an individual-
based model (Figure 2A) on a two-dimensional lattice, subject to varying
neighborhoods (Figure 2B), and migration rates using the Hybrid Automata
Library (Bravo et al., 2020). Cells may migrate at rate $m$ within the
migration neighborhood, and may undergo division at rate $b$ within the birth
neighborhood. Importantly, the birth and migration neighborhoods may be
differing sizes.

Figure 3:  Growth curves of the agent based model (ABM) depend on birth
neighborhood ($\omega$) and migration rate ($m$). A: Simulations with varied
birth neighborhood size, $\omega$. Shown for 201x201 domain. B: The cells
composing the tumor surface, $n_{s}$ as a function of the total population
size. A large value of $\omega$ (red) allows faster tumor growth.
Large-$\omega$ tumors proliferate quickly due to increased surface area, and
less contact inhibition. C: The same analysis for varying cell migration rate,
$m$. D: Tumors with high $m$ proliferate faster due to increased surface area.
Insets (B, D) represent the tumor surface drawn at time $t=100$, colored by
neighborhood size, $\omega$ and migration rate $m$, respectively. Note that
all cells have the same birth neighborhood, thus
$\omega_{i}=\omega=\mu_{\omega}$

Figure 3A shows tumor growth with varying neighborhood sizes. The number of
surface-cells $n_{s}$ is shown as a function of total size $n$ with a snapshot
of the tumor surface area after $t=100$ generations (Figure 3B). We observed a
transition from a linear power law to a square root law indicating that radial
growth transitions towards its characteristic growth law once the tumor
reaches a certain size. Tumors with small birth neighborhoods (e.g. the Von
Neumann neighborhood shown in red for $\omega=4$) remain small due to high
contact inhibition of those neighborhoods. Conversely, large neighborhoods
overcome contact inhibition and limited migration through increased access to
free space to place daughter cells.

The effects of contact inhibition are also mitigated by increased migratory
potential of tumor cells (Figure 3C-D). As the migration rate, $m$, increases,
growth potential (Figure 3C) and surface area (Figure 3D) also increase. Here,
we assume a fixed migration neighborhood (1-Moore), but in the next section we
relax this assumption to explore a well-mixed assumption (i.e. a migration
neighborhood which approximates the entire domain).

###  3.3 Well-mixed population growth

Let us now assume that the probability of observing $x$ filled sites is
independent of the focal cell’s location $\vec{q}$. This approximation is
justified when $\Gamma\gg\lambda,\delta$. Then (2.2) reduces to
$\mathbb{P}(x|n)$. The probability of observing $x$ cells is then found by the
number of ways of placing $x$ cells into the neighborhood and the remaining
$n-x-1$ (recall there are only $n-1$ since the focal one has already been
placed) by

$\mathbb{P}(x|n)=\frac{\binom{\omega}{x}\binom{l-\omega-1}{n-x-1}}{\binom{l-1}{n-1}}.$
(3.8)

To solve for the bracketed term in (2.6), we are interested in the probability
that all neighbor sites are occupied. For $l,n\gg\omega$ this leads to

$\mathbb{P}(\omega|n)=\left(\frac{n-1}{l-1}\right)^{\omega}\approx\left(\frac{n}{l}\right)^{\omega},$
(3.9)

we obtain

$\dot{u}=\frac{1}{l}\lambda\bigg{\langle}\sum_{i=1}^{n}\left[1-\left(\frac{n}{l}\right)^{\omega_{i}}\right]\bigg{\rangle}-\delta
u.$ (3.10)

To evaluate the sum, we consider a Taylor expansion up to second order about
the mean $\mu_{\omega}$ and 0. The first leads to

$\dot{u}=\lambda
u\left\\{1-u^{\mu_{\omega}}\left[1+\frac{1}{2}\sigma_{\omega}^{2}(\ln
u)^{2}\right]\right\\}-\delta u.$ (3.11)

Provided we can neglect the standard deviation ($2\gg(\sigma_{\omega}\ln
u)^{2}$) we arrive at a generalized logistic differential equation,

$\dot{u}=\lambda u\left(1-u^{\mu_{\omega}}\right)-\delta u.$ (3.12)

This particular form is identical to Richards’ differential equation
originally devised empirically as a flexible growth function which encompassed
many growth laws (Richards, 1959).

If instead, we Taylor expand near 0, we obtain to second order

$\dot{u}=-\lambda u\left[\mu_{\omega}\ln
u+\frac{1}{2}\left(\sigma_{\omega}^{2}+\mu_{\omega}^{2}\right)(\ln
u)^{2}\right]-\delta u.$ (3.13)

To neglect the higher order term imposes the condition

$\frac{2\mu_{\omega}}{\sigma_{\omega}^{2}+\mu_{\omega}^{2}}\gg|\ln u|.$ (3.14)

This leads to

$\dot{u}=-\lambda\mu_{\omega}u\ln u-\delta u,$ (3.15)

which we recognize as a form of Gompertz’s law. The “Gompertz condition” given
by (3.14) provides some mathematical justification for why Gompertz fits are
notoriously poor (and often incorrect) when the tumor is small. Modifications
to these models to alleviate this issue have been introduced, an example is
the Gompertz-Exponential model proposed by Wheldon, which assumes the tumor
initially grows exponentially before switching to a Gompertz law (Wheldon,
1988). If we stipulate that Gompertz emerges via contact inhibition, then a
necessary condition for this law to be valid is that $u$ is not too small.
This occurs through the $\ln u$ term in (3.14), since a sufficient condition
for this law to be a valid fit is if $u\approx 1$ or if the mean number of
neighbor sites $(\mu_{\omega})$ is very small.

Interestingly, one does not recover logistic growth by assuming that every
tumor cell can place offspring at each site. Indeed, (3.8) shows that when
$x=\omega$ and $\omega=l-1$

$\mathbb{P}(l-1|n)=\frac{\binom{0}{n-l}}{\binom{l-1}{n-1}}=\begin{cases}0&\text{if
}n<l\\\ 1&\text{if }n\geq l\end{cases}:=I_{n\geq l}.$ (3.16)

Here, $I_{n\geq l}$ is an indicator function which is 1 when satisfied and 0
otherwise. This leads to the deterministic growth law

$\dot{u}=\lambda u\left(1-I_{u\geq 1}\right)-\delta u,$ (3.17)

which implies exponential growth until saturation if $\lambda>\delta$.

###  3.4 Integrating the theory with experimental data

We fit the appropriate growth laws derived in the previous sections to
experimental cell count measurements. For cells growing in a two-dimensional
domain, we expected on average that the number of neighbor sites could range
anywhere from 1 to 8. This parameter could be slightly lower overall due to a
reduction near the boundary of the domain, but for sufficiently large domains
relative to cell size, this reductive effect should be small. We calibrated
Gompertz and generalized logistic growth laws using NonlinearModelFit (Wolfram
Mathematica 12.0) with the temporal data obtained from the seven different
cell lines for variable initial seeding conditions.

A prediction obtained from the derivation of Gompertz growth via contact
inhibition was that the approximation to the mean-field growth law would be
poor if the initial confluence was low. This effect has been noted both
phenomologically (Wheldon, 1988) and experimentally (Steel, 1977b). To test
this, we compared relative goodness-of-fit between Gompertz and generalized
logistic as a function of initial confluence for each cell line and each
experimental setup (amount of initial confluence). An AIC score was obtained
for both models and Gompertz’s AIC score was normalized by the generalized
logistic’s AIC score. We plotted this normalized score against the initial
confluence on a log-linear scale and fit a line using LinearModelFit (Wolfram
Mathematica 12.0). The resulting fit and cell line data are plotted in Figure
4A. The dashed line at 1.0 is meant to guide the eye, values above this imply
that Gompertz is improvement over generalized logistic. This is not to say
that the fits are better, but more that no information is gained with
generalized logistic, which carries an extra parameter. With an
$R_{\text{adj}}^{2}=0.24$, the straight-line fit is not strong, but a general
upward trend was observed as a function of initial confluence. We also plotted
each cell line’s individual fit (Figure 4A inset). The positive slope for each
cell line indicates that Gompertz generally improves with higher initial
confluence experiments.

Figure 4: Assessing model-fits for the different cell lines (given above) for
in vitro and in silico ABM experiments, using an Akaike Information Criterion
(AIC) A: Confluency vs. AICGompertz/AICGen. Log. score, along with a best fit
line that shows the general improvement of Gompertz fits as the initial
confluency increases. B: Intrinsic growth rate vs. number of neighbors
$(\lambda,\omega$) on a log-log scale. Different experimental setups (initial
confluence and cell line) are colored and a best fit line is determined. In
Figures S1 and S2 we show individual fits using generalized logistic and
Gompertz growth laws, respectively. C: Confluency vs. AICGompertz/AICGen. Log.
score. D: ABM Intrinsic growth rate vs. Number of neighbors $(\lambda,\omega$)
on a log-log scale. Different birth neighborhood size are color coded and a
best fit line is drawn.

Next, we fit the _in silico_ experiments to the growth laws derived in the
previous sections. For cells growing in a sufficiently large dish, we expect
on average that the number of neighbor sites will be only weakly impacted by
the boundary. This effect is slightly diminished due to the reduction near the
boundary, but for sufficiently large systems relative to cell size the effect
will be small. I’m The derivation of Eq. (3.10) assumes that the probability
of every site being occupied is independent of site location. A natural rule
that justifies this assumption is if the movement of cells occurs at a much
faster rate than cell cycle length. We further assume an inverse relationship
exists between $\lambda$ and $\omega$ of the form $C=\lambda\omega$. This
assumptions is based on the results obtained from the fitting routine on the
aggregate experimental data (Figure 4B).

We ran the ABM with $\lambda\omega=0.1$ and $\delta=0.001$, and with the same
fitting procedure as described for the handling of the in vitro data. We then
computed analogous statistics comparing goodness-of-fit, comparing growth rate
vs. number of neighbors (Figure 4C-D). We found, as predicted, that higher
initial confluency improves Gompertz’s goodness-of-fit (relative to
generalized logistic) and that the fits also improve as $\omega\to 0$.
However, the Gompertz-based procedure never reached the same goodness-of-fit
as generalized logistic (compare this to Figure 4A). These differences in
goodness-of-fit likely stem from the fact that it is difficult to give a
natural rule to drive an ABM framework below a single cell as the birth
neighborhood. Despite this discrepancy, the general trend of improvement of
fit with increasing initial confluence and decreasing number of neighbors is
consistent with the theoretical model.

We then looked at the relationship between growth rate and number of
neighbors; using linear regression, we recaptured the growth law given to the
ABM, by noting that if we take the log of both sides of $\lambda\omega=0.1$
and solve, we obtain $\log\omega=-(\ln\lambda+2.30)$) (see Figure 4D).
However, we also observed an interesting trend within the groups defined by
fixed birth neighborhood $\omega$. Instead of looking at the aggregate, if
consider each value of $\omega$ individually and use linear regression, the
slopes appear to approach a value of of -1, which implies an inverse-power
law. Both the cell lines and _in silico_ ABM simulations show a change in
slope of the population surface growth, trending towards an inverse power law
as $\omega$ decreases. We discuss this relationship in detail in section 4.

##  4 Discussion

The emergence of multiple well-known tumor growth laws from the same
individual based process via contact inhibition is striking. Given the
prominence of cell-based models in cancer biology (Chamseddine and Rejniak,
2020; Metzcar et al., 2019), it is important to consider the implications of
model construction on tumor growth dynamics. If we stipulate that a tumor’s
primary impedance to divide is given by its inability to place offspring in a
nearby location, than the mean-field (deterministic) law that determines the
tumor growth is not necessarily unique. The fact that multiple growth laws can
suffice to adequately model _in vitro_ tumor growth can be attributed to the
fact that these growth laws are approximations to the mean-field dynamics in
different limits and when multiple laws are valid, we are at the intersection
when these approximations all hold simultaneously.

The relationship between the parameters ($\lambda,\omega,\delta$) in the
generalized logistic has interesting implications. Assuming an optimal pair
with a small $\omega$ has been obtained, one can travel along this curve as
$\omega\to 0$. The connection between Gompertz and generalized logistic is
well known and requires $\omega\to 0$ and $r\to r_{0}/\omega$, which implies
that $r\omega=r_{0}$ in the limit of small $\omega$. We thus fit the cell line
data to this power law and obtained an adjusted $R^{2}$ of 0.7. Looking at
Figure 4B, it is clear that cell lines with a larger birth neighborhood size
don’t necessarily follow the same inverse-power law, but as $\omega$ gets
smaller the power seems to approach $-1$. Why do cell lines have different
relationships between (average) birth neighborhood size $\omega$ and net
growth rate $\lambda$? One explanation is surface area. Assuming a cell
requires the ability to see an open area for division, a larger surface area
would give it the ability to sense more open space. Thus, a natural prediction
emerges from this theory. Cells with a large $\omega$ are more likely to be
irregularly shaped. In contrast, an argument can also be made for small
$\omega$ to point to cell shape irregularity, but from an non-optimal
standpoint. A malleable shaped cell could easily see a time-dependent function
of $\omega$. Alternatively, cell adhesion may have an impact on the number of
allowed neighbor sites (Anderson, 2005).

Our model states that when a cell dies the site becomes open. Of note,
necrosis is a well-documented phenomenon of growing tumors (Vakkila and Lotze,
2004). A necrotic core can form under hypoxic conditions. One can introduce a
time scale on which a dead cell remains in place before being decomposed.
During this time, the site would be inaccessible. A natural choice would be an
exponential distribution with mean given by the time it takes for macrophages
or other phagocytes to migrate to the location and engulf the dead cell. Of
course, this introduction also implies that the immune system is a natural
player in controlling the number of accessible sites for tumor cells to
divide, both in their ability to remove debris (e.g. dead cells) and their
physical occupation of space. Thus, a more controlled approach to model active
cell removal could be a future addition to our model.

Cell-to-cell signaling is another possible extension that can be incorporated
via a local dependency on $\omega$. These chemicals are often secreted in a
paracrine fashion and lead to multiple cell interactions. These can promote or
inhibit the ability for a cell to proliferate into new space. The inclusion of
such factors can lead to a spatially-dependent $\omega$ (or $\lambda$),
depending on where and how these factors are produced.

Fractal or radial growth demonstrates how the proper deterministic growth law
can change over time. At low tumor volume, all cells are effectively surface
dwelling cells. Hence, the dynamics of all radial/fractal growing tumors are
initially exponential. The shift from exponential to fractal depends on the
birth neighborhood size. Unless the birth neighborhood size is all sites
$\omega_{i}=l$, the tumor will begin to transition from exponential to
fractal-like growth, which can be most easily seen when tumor growth is
presented on a log-log scale. As the tumor continues to grow, the fractal
dimension approaches $d-1$. A possible explanation for this is to first
recognize that no tumor growth is purely fractal as there is a minimal length
scale. As the tumor gets larger, the local curvature will begin to flatten
relative to the size of the overall tumor and begin to look more and more
symmetric. Alternatively, one can view this as shifting your viewpoint further
and further from a tumor, it will begin to look more regular.

In summary, our theory offers a possible explanation for the large number of
mean-field laws that can adequately capture tumor growth dynamics. We do so
through unification under density-dependent birth events via contact
inhibition. Of note, we present the first mechanistic derivation of
Gompertzian growth based on contact inhibition.

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## Supplementary Materials

Figure S1:  Fitting a generalized logistic growth law of longitudinal in vitro
growth data of the the seven cell lines (see Methods).  Figure S2:  Fitting a
Gompertz growth law of longitudinal in vitro growth data of the the seven cell
lines (see Methods).


$S_{\rho_{A}\otimes\rho_{B}}=-\partial_{n}\left[e^{iq_{\sigma_{n_{o}}(y_{1}-y_{4}-y_{2}+y_{3})}}\left(x_{14}x_{23}\right)^{-2h_{\sigma_{n_{o}}}}\right]_{n_{o}=1}.$
(5.18)

In short, for the holographic WCFT, we find

$S_{o}^{WCFT}\neq S_{\rho_{A}\otimes\rho_{B}}+E_{W}(\rho_{AB}).$ (5.19)

This is a new phenomena in the holographic WCFT.

##  6 Conclusion and Discussion

In this work, we studied the reflected entropy in two-dimensional holographic
warped conformal field theory, and paid special attention to its implication
in holography. We showed that in the AdS3/WCFT correspondence, the reflected
entropy of two disjoint intervals is related to the pre-entanglement wedge
cross-section (pEWCS)

$S_{R}(A:B)=2\frac{\mbox{Area(pEWCS)}}{4G_{N}},$ (6.1)

up to the order ${\cal O}(1)$ quantum correction. This provides a nontrivial
check on the relation between the reflected entropy and EWCS beyond the
AdS/CFT correspondence.

The physical implications of the relation (6.1)) are remarkable. On one hand,
it supports the two-sided dynamical construction of EWCS in [10], as described
in Section 2.4. On the other hand, it provides more evidence on both the
AdS3/WCFT correspondence and the swing surface proposal. Most of the previous
studies focused on the entanglement entropy of single interval and its
holographic description. In the two-interval case, one may compute the Rényi
mutual information on the field theory side, but short of bulk
computation[34]. The present study on the entanglement of double intervals
provides nontrivial check on the swing surface proposal of holographic
entanglement entropy.

Another remarkable point from our study is on the semi-classical gravitational
action in AdS3/WCFT. It is not hard to notice that both the reflected entropy
and logarithmic negativity444The situation in the odd entropy is special. We
have no idea why there is (5.19). in holographic WCFT take half of the values
of the ones in holographic CFT. This does not mean that the semiclassical
action of the AdS3 gravity with the $CSS$ boundary condition is half of the
one of the usual AdS3 gravity. The subtle point is that these two quantities
are computed by taking $n\to 1$ limit, and there could be extra terms in the
gravitational action proportional to $(n-1)$, as shown in the Rényi mutual
information[34].

Although we accumulate some features about pre-entanglement wedge ${\cal
W}_{\xi}[{\cal A}]$ and causal wedge ${\cal W}_{{\cal C}}[{\cal A}]$ related
to single interval and two disconnected intervals in the AdS3/WCFT holography,
the implications of these results are not very clear. We expect further
explorations in this direction.

In our study on the field theory side, we computed the dimensions and charges
of various twist operators in non-Abelian orbifold WCFT, which allow us to
read the reflected entropy, the logarithmic negativity and the odd entropy in
the holographic WCFT. One essential assumption in our study is the block
dominance in the large $c$ limit. One interesting question is to justify this
assumption. Another important question is to find the next-to-leading order
correction to the reflected entropy in the large $c$ limit, which should
correspond to the quantum correction to (6.1). This may require a careful
study of operator product expansion of twist operators, as shown in [65, 66,
34].

## Acknowledgments

We are grateful to Peng-xiang Hao for initial participation in the project. We
would like to thank Wei Song and Peng-xiang Hao for many valuable discussions.
The work is in part supported by NSFC Grant No. 11735001.

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