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AIP ADVANCES 2, 011210 (2012)

Dynamic density functional theory of solid tumor growth:

Preliminary models

Arnaud Chauviere,1,aHaralambos Hatzikirou,1,bIoannis G. Kevrekidis,2

John S. Lowengrub,3,cand Vittorio Cristini1

1Department of Pathology, University of New Mexico, Albuquerque, New Mexico 87131, USA

2Department of Chemical and Biological Engineering and the Program in Applied and

Computational Mathematics, Princeton University, Princeton, New Jersey 0854, USA

3Department of Mathematics, University of California at Irvine, Irvine, California 92697,

USA

(Received 5 September 2011; accepted 11 February 2012; published online 22 March 2012)

Cancer is a disease that can be seen as a complex system whose dynamics and growth

resultfromnonlinearprocessescoupledacrosswiderangesofspatio-temporalscales.

The current mathematical modeling literature addresses issues at various scales but

the development of theoretical methodologies capable of bridging gaps across scales

needs further study. We present a new theoretical framework based on Dynamic

Density Functional Theory (DDFT) extended, for the first time, to the dynamics of

living tissues by accounting for cell density correlations, different cell types, pheno-

typesandcellbirth/deathprocesses,inordertoprovide abiophysically consistentde-

scription of processes across the scales. We present an application of this approach to

tumor growth. Copyright 2012 Author(s). This article is distributed under a Creative

Commons Attribution 3.0 Unported License. [http://dx.doi.org/10.1063/1.3699065]

I. INTRODUCTION AND MOTIVATION

In oncology and tumor biology, a wealth of genomic, proteomic and pathology-based cell

phenotypic and microenvironmental data are collected at the microscale (i.e., cell scale and below).

Treatment strategies, on the other hand, are determined by tumor response at the tissue-scale in

space over long scales in time. Thus, a fundamental gap that is currently hampering progress in

individualizedcancercareistheextrapolationofpatient-specificmicroscalequantitativeinformation

into formulations of therapeutic and intervention strategies at clinically relevant scales (e.g., organ

scales). What is urgently needed is the development of biophysically sound mechanistic connections

among the multi-modal, multi-dimensional and multiscale phenomena and variables involved in

tumor progression. Such multiscale models would enable the integration of abundant microscale

phenotypedata(cellarchitecture,mitoticrates,etc)intoacomprehensivepictureofindividualtumor

behavior and response to treatment at clinically relevant scales that emerge from the underlying

microscopic processes.

We recognize that oncologists and tumor biologists focus almost exclusively on processes

at the sub-cellular and cellular domains (e.g., signaling pathways), and largely overlook physical

phenomena associated with transport of cells and chemicals (e.g., oxygen diffusion) and momentum

(e.g., mechanical forces among cells and other stromal components), which occur at the mesoscopic

scale that spans the distance over which transport gradients are established (i.e., 100 − 200μm of

tissue for the establishment of oxygen gradients). These mesoscopic phenomena are the missing link

to bridge the current gap between the scales.

aindicates co-first and co-corresponding author; Electronic mail: achauviere@salud.unm.edu.

bindicates co-first author.

cindicates co-corresponding author; Electronic mail: lowengrb@math.uci.edu.

2158-3226/2012/2(1)/011210/13

C ?Author(s) 2012

2, 011210-1

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011210-2 Chauviere et al. AIP Advances 2, 011210 (2012)

To help bridge this gap, we propose a mathematical approach based on Dynamic Density

Functional Theory1(DDFT), which provides a mesoscale continuum framework directly derived

from a stochastic, discrete model. Unlike standard mean-field models, the DDFT approach accounts

for correlations.1,2We extend the approach to model the spatio-temporal dynamics of multicellular

systems by including cell type, phenotype and birth/death processes, which are major features of

biological tissues.

The DDFT, and its stochastic, discrete counterpart, can be used to develop a hybrid continuum-

discrete multiscale model that satisfies the following properties we deem essential for a multiscale

model: 1) different representations to describe the same quantity of interest at different scales as

needed (e.g., discrete representations are used in regions where the continuum model is not valid

due, for example, to fluctuations); 2) direct calibration/validation of the cell-scale parameters and

equationsfromindividualmicroscalemeasurements;andfinally3)upscalingtechniquestoformulate

(and close) the continuum equations at the larger scales. In contrast to current approaches, such a

multiscale model would provide an accurate description of the feedback and interactions among

processes across different scales and enable the model components at the meso- and tissue scales to

be precisely determined from cell scale modeling, and underlying biological measurements, without

“fitting” them directly.

II. CURRENT APPROACHES: “MULTIPLE-SCALE” MODELS

Due to the intrinsic multiscale nature of cancer, a deeper understanding requires the develop-

ment of models that integrate and combine the phenomena spanning the multiple scales involved.

Hybrid continuum-discrete implementations,3which typically seek to combine the best of the tissue

(continuum) and cellular (discrete) scale approaches while minimizing their limitations, are a very

promising modeling approach. Many published methods claim to be multiscale because they are

based on a hybrid description of the tumor components, typically by using a discrete representation

ofthevariouscellpopulationsandcontinuumfieldstodescribecellsubstrates(e.g.,nutrients,oxygen

and diffusible factors). These models incorporate processes at multiple scales but have difficulties in

capturing the non-trivial interactions among scales that are responsible for the growth of malignant

tissue and do not satisfy the definition of multiscale model introduced above. Here, we concisely

review two promising recent approaches to the development of functional multiscale models for

solid tumor growth. A more complete review may be found in Refs. 4–6.

A. Decoupled multiple-scale approach

A mechanistic agent-based model (ABM) of tumor growth was recently introduced7,8and

applied to simulating the progression of ductal carcinoma in situ (DCIS) of the breast (Fig. 1). Cell

motion is determined from an overdamped balance of adhesive, repulsive, and motile forces exerted

among the cells and by the extracellular matrix (ECM). Cells (which may have different sizes)

are associated to individual phenotypic states, e.g., proliferative, motile. Transition rates between

the states are governed by exponentially distributed random variables. As illustrated in Fig. 1,

a “bottom-up” approach was used for ABM parameter calibration from experimental data at the

cell-scale (immunohistochemistry of pathology specimens). In particular, experimentally measured

indexesofproliferationandapoptosiswereusedtoobtainaverageratesofindividualcellproliferation

and apoptosis rates in the ABM. Note that such a calibration has also been successfully performed in

the context of liver regeneration.9However, in Refs. 7,8, an upscaling calibration protocol based on

volume-averaging was additionally used with the ABM to calculate the coarse-grained proliferation

and apoptotic volume rate parameters at the tissue scale, which provided input to a continuum

model10thus avoiding fitting. A “top-down” approach was used to compare the model results to

tumor volumes measured in patients. This led to the first accurate, patient-specific assessment of

DCIS tumor volumes – a key translational issue in current surgical planning.11

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011210-3 Chauviere et al.AIP Advances 2, 011210 (2012)

FIG. 1. Flow of information across the formulations at the cellular and tissue scales in a decoupled multiple-scale approach

for DCIS.7,11Bottom images: (left) Bottom-up calibration: immunohistochemistry measurements from patient tissue used

for parameter calibration7(proliferation and apoptotic indices PI and AI); (center) ABM simulation of DCIS;7(right) Top-

down validation: comparison of model-predicted tumor volume (dashed curve) to the corresponding volume measured from

histology of the excised tumors from patients (symbols). Reprinted with permission from Ref. 8. Copyright 2010, Cambridge

University Press.

B. Coupled multiple-scale approach: Hybrid models

Inastudyoftheeffectofstressonthegrowthofavasculartumorspheroidsinanagarosegel12,13

a hybrid model was developed where different descriptions of tumor cells were used in different

regions of the spheroids. In the thin proliferating zone at the spheroid edge, an ABM was used

where the discrete cells were represented as deformable ellipsoids, enabling a detailed description

of intercellular dynamics, cell sizes and shapes, and cell-cell interactions. In the larger quiescent

and necrotic regions in the spheroid interior, a continuum viscoelastic description was used. The

coupling among the discrete and continuum components was limited to the incorporation of discrete

cells into the continuum regions through mass and momentum fluxes.

Another hybrid approach has been used to simulate the growth of invasive glioma tumors14

where invading tumor cells were modeled using an ABM as discrete points (i.e., zero size) and a

continuum model was used for the tumor bulk, assuming spherical symmetry. The coupling among

thediscreteandcontinuummodelswaslimitedtotheproductionofdiscretecellsfromthecontinuum

field through mass fluxes at the tumor edge.

In a recent study of a hypoxia-induced epithelial-to-mesenchymal transition during tumor

growth,15an ABM for tumor cells was coupled directly to a continuum model for tumor volume

fractions through mass and momentum exchanges between the two representations. Rules were

posed to describe the conditions under which the discrete and continuum representations were

used. Briefly, discrete cells were released in hypoxic regions, to model the epithelial-mesenchymal

transition, and discrete cells were converted back to the continuum description at locations where

their population exceeds a threshold (Fig. 2).

C. Assessment of current approaches

The decoupled two-step approach does not allow for any dynamical exchange between cell- and

tissue-scale models, which limits its applicability. The hybrid approaches improve over the decou-

pled methods in that there are interactions among the continuum and discrete cell representations.

However, in both multiple-scale approaches the variables and fluxes considered at the macroscale

(continuum) are pre-assumed and are not obtained by upscaling the cell scale (discrete) dynamics.

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FIG. 2. Hybrid multiple-scale simulation of vascularized tumor growth.15Discrete tumor cells (blue dots) are released from

hypoxic areas of the continuum tumor regions (gray volumes). Vessel sprouts (light lines) and newly formed vessels (dark

lines) releasing oxygen and nutrients are shown. Time is in proliferation time units.

Multiscale modeling frameworks, such as the heterogeneous multiscale method16and the

equation-free approach,17can be used to provide effective macroscale fluxes by numerically up-

scaling the results from models at the microscale. However, these methods do not provide analytical

forms for these fluxes even if the macroscale continuum description is valid and the fluxes can be

definedsolelyintermsofcontinuumvariables.Inthenextsection,wepresentanovelapproachusing

dynamical density functional theory for modeling tissue and tumor growth, which provides a model-

ing framework for obtaining analytical expressions for the macroscale fluxes from the microscopic

processes.

III. DYNAMIC DENSITY FUNCTIONAL THEORY FOR MULTICELLULAR SYSTEMS

Dynamic density functional theory (DDFT) provides a macroscopic continuum framework di-

rectly derived from the microscopic scale, where processes occurring at the microscale (i.e., in an

interacting particle system) are integrated within the mesoscale description and incorporate corre-

lations among the discrete components.1,2DDFT has been successfully used to describe Brownian

dynamics of non-equilibrium colloidal particles, e.g., applied to crystal growth in single and bi-

nary component materials.18–20The theory naturally incorporates elastic, plastic and viscoelastic

deformations as well as defects.

As far as we are aware, DDFT has not been used in the context of multicellular systems.

However, recent experiments on confluent epithelial cell sheets show that there are intriguing

similarities between the dynamics of collective cell motion at high cell densities and the dynamics of

supercooled colloidal and molecular fluids approaching a glass transition.21In particular, at low cell

densities it is found that the cells flow like a fluid. However, as the cells proliferate and compress,

a critical density emerges above which solid-like behavior is observed and structural relaxations

occur only over long time scales such as the mitosis time scale or longer (e.g., on the order of days).

Accordingly, dynamic heterogeneities are seen to develop concomitant with a decrease in diffusive

motion and the development of peaks in the dynamic structure factor. Together, these results suggest

the importance of accounting for correlations among cells at the mesoscale, which makes DDFT a

natural modeling choice. However, to model physiological tissues as multicellular systems, DDFT

must be extended to account for cell proliferation (mitosis) and death (apoptosis/necrosis), as well

as multiple cell types and phenotypes, which are also the main features needed to model malignant

tissues.

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A. Classical Dynamic Density Functional Theory

We briefly outline DDFT for the dynamics of N interacting (tumor) cells of a single species and

outline the steps that allow for the derivation of equations describing the spatio-temporal evolution

of the tissue-scale description. We describe the motion of each cell by the overdamped Langevin

equation:

dri

dt

= ?

?

N

?

j=1

Fint

ij+ Fext

i

?

+

√2D ηi(t),

(1)

where ri= (rα

physical domain ?dand ? is the cell motility coefficient linked to the diffusion coefficient D by

the Einstein relation D = ? T (T is an effective temperature). To simplify the presentation, we

assume no hydrodynamic interactions between the cells, i.e. no cell-cell friction, which could be

included by adding a viscous drag force in the Langevin equations via a microscopic friction tensor

actingontherelativevelocitywithneighboringcells.22Althoughneglectingcell-cellfrictionmaybe

more appropriate in the absence of active cell motion, cell-cell friction could easily be incorporated

using a similar modeling framework. The forces?N

betweentwocells.Wemodelmicroenvironmentalinfluences(e.g.,chemotaxis,haptotaxis,cell-ECM

adhesion, etc) on cell movement by forces Fext

i

= −∇Vext(ri,t), where Vextis an external potential

depending on the microenvironment. The vector ηi(t) is a Gaussian noise satisfying?ηα

α,β are coordinate indices. From Eq. (1) one can derive the equation for the N-particle position

probability distribution P(rN,t) (or N-body distribution function) by using the multivariable Ito

formula of stochastic calculus23to obtain the Smoluchowski equation

i)α=1...d∈ ?dis the position of the center of mass of the i-th cell in the d-dimensional

j=1Fint

ij= −?N

j=1∇riVint(|ri− rj|) model

cell-cell interactions (e.g., adhesion, repulsion) via a pair potential Vintthat depends on the distance

i(t)?= 0 and

?ηα

i(t)ηβ

j(t?)?= δijδαβδ(t − t?), where <· > is the averaging over different noise realizations, and

∂P(rN,t)

∂t

=

N

?

i=1

∇ri·?−? Fi(rN,t) + D ∇ri

?

P(rN,t),

(2)

where rN= {ri}i=1...N is the N-dimensional position vector of cells and Fi=?N

Summarizing the approach in Ref. 24, the discrete density field of the N-cell system is defined

as ¯ ρ(r,t;rN) =?N

domain ?dis split into small subdomains (nodes) where a local density is defined with fluctuations

around an equilibrium value. The cell density is evaluated around the position r ∈ ?dof a node of

size l. The corresponding coarse-grained density ρ(r,t) is the limit of ¯ ρ(r,t;rN) for large N and

small l. Let P[ρ, t] denote the probability of finding a specific realization of function ρ over the

domain r ∈ ?d:

P[ρ,t]=

j=1Fint

ij+ Fext

i

denotes the sum of forces exerted on a single cell.

i=1¯δ(ri(t) − r), where¯δ is a coarse-grained delta function over a domain of char-

acteristic length scale l, which lies between the cell size and the cell-density correlation length. The

?

δ?ρ − ¯ ρ(·;rN)?P(rN,t)drN=

Using the Smoluchowski equation (2) and the above definition, we can evaluate the temporal

dynamics of the cell density distribution P[ρ, t]. The derivation involves the technique of projection

operators and is based on the essential assumption that the density ρ(r,t) evolves with a time scale

well-separated from the time scale corresponding to the evolution of the generalized position vector

rN. Finally, the time evolution of P[ρ, t] is described by the following functional Fokker-Planck

equation:24

∂P[ρ,t]

∂t

Eq. (4) is a mesoscopic description of the initial system (1) and includes fluctuations that can be

averaged out only for larger systems, where the first moment of P[ρ, t] (the cell density) is assumed

to dominate the deterministic large-scale dynamics. In Eq. (4) the functional operator LM[ρ,t]

? ?

δ?ρ(r?,t) − ¯ ρ(r?,t;rN)?δ(r?− r)P(rN,t)drNdr?.

(3)

= LM[ρ,t] P[ρ,t].

(4)

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describes the dynamics of the coarse-grained density:24

?

whereδ/δρ(r,t)isafunctionalderivativewithrespecttoρ andFC[ρ]isacoarse-grainedfree-energy

functional, which is specific to the biophysical processes modeled and will be described in detail

later. In order to derive a deterministic equation for the density, we multiply Eq. (4) by the density

field and integrate, thereby performing an averaging procedure over the possible realizations of the

density field. For appropriate boundary conditions, we integrate by parts and assume vanishing local

fluctuations of the density field to obtain:

?

Alternatively, one could derive the above result, starting again from Eq. (4), by using a free-energy

minimization principle, as in Ref. 25. We remark that another option to derive Eq. (6) directly from

Eq. (4) is proposed in Ref. 1, by defining ¯ ρ as the sum of noise-averaged delta functions without

appealing to a free-energy minimization principle. However, we do not take this alternate approach

here as it is more difficult to account for cell-proliferation as we will propose further in our paper.

The deterministic equation (6) captures the spatio-temporal dynamics of the cell density

of a single cell type at the tissue level in the absence of cell proliferation, is physically-

based, and takes the form of a continuity equation with the flux proportional to the gradient

δFC[ρ]

δρ

of the coarse-grained free-energy functional. In general, this functional takes the form

FC[ρ] = Fid[ρ] + Fext[ρ] + Fex[ρ]. The first term corresponds to random cell motion and is

given by Fid[ρ] = T?drρ(r)?ln(Kρ(r)) − 1?, where the prefactor K is the analogue of the ther-

fluctuations). The second term describes the response of cells to an external field, e.g. active cell

movement, and is given by Fext[ρ] =?drρ(r)Vext(r,t). The third term characterizes the excess

by26,27

?

where ρ(2)(r,r?,t) is the two-particle density distribution function. Since ρ(2)depends on ρ(3), the

three-particle distribution function, and so on, Eq. (6) is not closed. To close Eq. (6), the system

needstobetruncatedandρ(2)needstobeapproximated. Severalapproximations asdescribedbriefly

below are used.

LM[ρ,t]P[ρ,t] = −

dr

δ

δρ(r,t)

?

∇ ·

?

ρ(r,t)∇

?

D

δ

δρ(r,t)+ ?δFC[ρ(r,t)]

δρ(r,t)

??

P[ρ,t]

?

,

(5)

∂ρ

∂t(r,t) = ?∇ ·

ρ(r,t)∇

?δFC[ρ(r,t)]

δρ(r,t)

??

.

(6)

mal de Broglie wavelength (i.e., the average distance between cells under the influence of stochastic

free energy that arises from cell-cell interactions, which for pair potentials as used above, is given

ρ(r,t)∇δFex[ρ(r,t)]

δρ(r,t)

=

dr?ρ(2)(r,r?,t)∇rVint(|r − r?|),

(7)

1. Uncorrelated mean-field approximation

The simplest way to model ρ(2)and close Eq. (6) is to neglect density correlations and take

ρ(2)(r,r?,t) ≈ ρ(r,t)ρ(r?,t). This is known as the mean-field approximation.28In this case, Eq. (6)

becomes

∂ρ

∂t(r,t) = D∇2ρ(r,t) + ?∇ · [ρ(r,t)∇Vext(r,t)]

?

We note that similar formulations have been previously used to model cell-cell adhesion via the

nonlocal operator.29Further, the mean field approximation fails when either the density is low, so

thatfluctuations arenon-negligible, orwhencorrelationsareimportant,e.g.,arisingfromlong-range

interactions. Interestingly, for a repulsive interaction potential, this leads to a version of Darcy’s law

since for a uniform density ρ = ρ∞, the pressure is p(r) = ρ∞??dr?Vint(|r − r?|).

+?∇ ·

ρ(r,t)

?

dr?ρ(r?,t)∇rVint(|r − r?|)

?

.

(8)

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2. Correlated mean-field approximation

In the DDFT framework rather than assuming that there are no correlations, the adiabatic

approximation1ρ(2)(r,r?,t) ≈ ρ(2)

function evaluated for a system with equilibrium density ρ0(r) ≡ ρ(r,t). Although ρ(2)

known in general, the corresponding excess energy Fex[ρ] can be expanded around a constant

reference density ρ∞to yield30

D

2?

eq(r,r?) is used, where ρ(2)

eqis a two-particle density distribution

eqis still not

Fex[ρ(r,t)] ? Fex(ρ∞) −

?

dr?ρ(r,t)

?

dr??ρ(r?,t)c(2)

eq(r − r?;ρ∞),

(9)

where ?ρ(r,t) = ρ(r,t) − ρ∞and c(2)

relation function of the uniform system, which is related to the total correlation function h(r), with

r = |r|, by the Ornstein-Zernike (OZ) equation31

h(r) = c(2)

where spherical symmetry is assumed. At this point, the system is still not closed and an additional

closure relation needs to be imposed between h and c(2)

equation (10). There are numerous examples of physically-based closure relations in the literature,31

which relate h, c(2)

closure relation. With these approximations, Eq. (6) becomes

eq(r − r?;ρ∞) = −δ2Fex[ρ]

δρ(r)δρ(r?)|ρ(·)=ρ∞is the direct two-point cor-

eq(r;ρ∞) + ρ∞

?

dr?h(r?)c(2)

eq(|r − r?|;ρ∞),

(10)

eqto be able to solve the Ornstein-Zernike

eqand the interaction potential Vint; see Sec. III C for an example of a specific

∂ρ

∂t(r,t) = D∇2ρ(r,t) + ?∇ · [ρ(r,t)∇Vext(r,t)]

?

eqremainstobeevaluatedviatheOZequationandaclosureequation.Wenotethatinboththe

uncorrelated and correlated mean-field models, the nonlocal terms may be further approximated and

localized in space using gradient expansions. Next, we extend the DDFT framework to multicellular

systems.

−D∇ ·

ρ(r,t)

?

dr??ρ(r?,t) − ρ∞

?∇rc(2)

eq(|r − r?|;ρ∞)

?

,

(11)

wherec(2)

B. Multicellular Dynamic Density Functional Theory

AsafirststeptowardsthedevelopmentofDDFTformulticellularsystems,weextendtheDDFT

framework to account for different types of cells and cell phenotypes. We then develop an extension

to birth/death processes.

1. Multiple cell types

We consider Mσdifferent cell species, identified by the index k, with each species having Nk

cells with a common set of biophysical properties. The Langevin equation describing the motion of

the i-th cell in population k, via its center of mass located at rk

i, reads

drk

dt

i

= ?k?Mσ

?

l=1

Nk

?

j=1

Fkl,int

ij

+ Fk,ext

i

?

+

?

2Dkηk

i(t),

(12)

where all the notation used in the previous section is generalized to account for the properties

associated with a species k. The multispecies Eq. (12) is a straightforward generalization of the

single species Eq. (1) where there are now cell-cell interactions among the different species k and l

(e.g.,tomodeldifferentialadhesion)viatheforcesFkl,int

Fk,ext

i

, which act on each species individually (e.g., only a subset of the cell species may respond to

particular microenvironmental forces).

The derivation of the deterministic counterpart (i.e., the equation for the continuous cell density

ρkof each species) of the multispecies Langevin equations (12) is similar to the single species case

ij

andexternalforcesinthemicroenvironment

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presented in Section III A. We use multivariable stochastic Ito calculus for each species to derive

the corresponding Smoluchowski equation for each Nk-body distribution function. Then, we use

Kawasaki’s approach24and introduce the probability Pk[ρk, t] of finding a specic realization of ρk,

whose dynamics is given by a functional Fokker-Planck equation. By averaging over all possible

realizations, we obtain a deterministic continuity equation for the cell density ρkof each species k,

which reads

?

where FC[ρ] denotes the coarse-grained free-energy functional of the entire multispecies system of

interacting cells, i.e., FC[ρ]=FC[ρ1,...,ρMσ]. The energy associated with random cell motion and

the external field are simply sums of the contributions from each species. The excess part of the free

energy is the generalization of Eq. (7) and is given by

∂ρk

∂t(r,t) = ?k∇ ·

ρk(r,t)∇

?δFC[ρ(r,t)]

δρk(r,t)

??

,

(13)

ρk(r,t)∇δFex[ρ(r,t)]

δρk(r,t)

=

Mσ

?

l=1

?

dr?ρ(2)

kl(r,r?,t)∇rVkl,int(|r − r?|),

(14)

where ρ(2)

relations analogous to those described in Secs. III A 1andIII A 2 can be used to complete and close

the system of equations, yielding a system of coupled integrodifferential equations.

kl(r,r?,t) describes cross-correlations between species k and l. Approximations and closure

2. Multiple cell phenotypes

We next consider the case in which cells have different phenotypes. Recent experimental

evidence suggests that phenotypic traits, such as the state of differentiation, are not necessarily

fully hierarchical but rather a stochastically evolving, potentially reversible continuum.32–36Similar

to Refs. 36,37, where Markov processes have been used to model differentiation dynamics, we

introduce a stochastic differential equation that governs the temporal evolution of a phenotype

vector σiof cell i where each component of σidescribes a cell phenotypic trait (e.g., differentiation,

proliferation, etc):

dσi

dt

= Ai(rN,σN) +

?

2Dσησ

i(t),

(15)

where Ai corresponds to evolutionary and external pressures and models drift forces driving the

evolution of (the possibly interdependent) phenotypic traits of cell i. Similar to the definition of rN

(the generalized N-dimensional position vector of the whole set of cells), σN= {σi}i = 1. . . Nis the

generalized N-dimensional phenotype vector of the whole set of cells. The second term in Eq. (15)

accounts for uncorrelated random fluctuations of the phenotype (it is straightforward to extend the

model to include correlated fluctuations).

Whereas Eq. (15) models the dynamics of the phenotype of cell i, the movement of that cell

is still modeled via the Langevin equations. Assuming that the biophysics driving the cell spatio-

temporal dynamics in the Langevin equations may depend on the cell phenotype, we shall consider

?(σi), Fi(rN,σN) =?N

dri

dt

Equations (15)and(16) form a discrete coupled system accounting for the stochastic dynamics of

cell in both the physical and phenotypic spaces. Similar to the derivation performed in Section III A,

we introduce the generalized N-particle probability distribution P(rN,σN,t). Then, assuming that

the interspecies noise terms ησ

to a function of the vector (rN,σN) to derive the Smoluchowski equation governing the spatio-

temporal evolution of P(rN,σN,t). The rest of the derivation of a deterministic equation involves a

new definition of the discrete density field, i.e. ¯ ρ(r,σ,t;rN,σN) =?N

j=1Fint

ij(ri,σi,rj,σj) + Fext

i(ri,σi) and D(σi) in Eq. (1), which we rewrite

for clarity as:

= ?(σi)Fi(rN,σN) +

?

2D(σi)ηi(t).

(16)

i(t) are statistically independent, we apply the multivariate Ito formula

i=1¯δ(ri(t) − r)¯δ(σi(t) − σ),

with¯δ being the coarse-grained delta function already introduced and σ the vector of continuous

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011210-9 Chauviere et al.AIP Advances 2, 011210 (2012)

phenotype traits of cells at position r at time t. The application of Kawasaki’s approach24to define

the generalized coarse-grained density ρ(r,σ,t) as the limit of ¯ ρ(r,σ,t;rN,σN) and to define the

probability P[ρ, t], whose dynamics is governed by a functional Fokker-Planck equation, remains

valid. Here the generalized density ρ depends on both spatial and phenotypic continuous variables

r and σ. The averaging over all possible realizations yields the following deterministic equation for

the generalized density ρ:

?

+Dσ∇2

where ∇σ denotes the gradient with respect to the phenotype vector σ. Here, the functional free

energy now involves integration over σ. In particular, the excess part of the free energy is now given

by

ρ(r,σ,t)∇δFex[ρ(r,σ,t)]

δρ(r,σ,t)

where ρ(2)?(r,σ),(r?,σ?),t?is a generalized two-particle density distribution function.

physical space (i.e., diffusion, advection by an external field, as in Eqs. (8)and(11) with coefficients

depending now on the phenotype σ). In addition, the transport may also be influenced by the nature

of the interactions among cells of different phenotypes via the potential Vint(see Eq. (18)). The

second and third terms on the right hand side of Eq. (17) account for the phenotype dynamics

that includes diffusion (e.g., for random mutations) and a drift term with velocity A (e.g., for

evolutionary forces) within the phenotype space. Remark that those terms are to be compared with

those of population dynamics models developed for cell differentiation37,38in which the level of

differentiation is described by a continuous variable as suggested in Ref. 39.

While a similar approximation procedure as described in Sec. III A 2 can be used to close the

system utilizing a multicomponent Ornstein-Zernike equation, it is not clear that the physically-

based closure relations described in Ref. 31 are appropriate for this context. This is under study, see

Sec. IV.

As an example, suppose that the phenotype is solely described by the state of differentiation

of a cell. Then let σ = s be a scalar variable that denotes the differentiation state of a cell with s

= 0 denoting the undifferentiated state and s = 1 the fully differentiated state. Then, the drift term

A = A(r,s) is also a scalar, with A ≥ 0 describing hierarchical forces driving differentiation and A

≤ 0 describing de-differentiation. Feedback that modifies cell differentiation rates can be captured

usingthisapproachandamoredetailedmodelthatrelatesAtoexternal(e.g.,fromthemicroenviron-

ment)andinteraction(e.g.,amongcellsatdifferentdifferentiationstages)forcesanalogoustothosein

Eq. (1), but now in the phenotype space.

∂ρ

∂t(r,σ,t) = ?(σ)∇r·

ρ(r,σ,t)∇r

?δFC[ρ(r,σ,t)]

δρ(r,σ,t)

??

σρ(r,σ,t) − ∇σ· [A(r,σ)ρ(r,σ,t)],

(17)

=

? ?

dσ?dr?ρ(2)?(r,σ),(r?,σ?),t?∇rVint(|r − r?|,σ,σ?),

(18)

The first term on the right hand side of Eq. (17) corresponds to the transport of cells in the

3. Cell birth/death processes

We next describe the generalization of DDFT to include birth/death (BD) processes. We present

the derivation for a single species, which can be generalized straightforwardly to a multispecies,

multiphenotype system. The derivation of macroscopic descriptions of randomly moving reacting

particles has a long history (see Ref. 23 for example). Only recently, rigorous mathematical results

have been developed to treat stochastic spatially-distributed BD processes, e.g. Ref. 40.

We begin byconsidering onlyBDprocesses.Under theassumption ofalocalstochastic Markov

process, cell birth/death can be classically modeled via the following Fokker-Planck equation:41

∂P[ρ,t]

∂t

= LBD[ρ,t]P[ρ,t],where(19)

LBD[ρ,t]P[ρ,t] =

?

dr

δ

δρ(r)

??

−D(1)

BD[ρ(r)] +1

2

δD(2)

BD[ρ(r)]

δρ(r)

?

P[ρ,t]

?

,

(20)

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011210-10Chauviere et al. AIP Advances 2, 011210 (2012)

and the cell density functionals D(i)

BD process.

Next, we consider combined BD and cell-movement processes. We assume that the time scale

of cell movement (i.e., from cell-cell interactions, external and random forces) across the length l is

much smaller than the characteristic time of BD processes. That is, the ratio κ = τM/τBD? 1, where

τMand τBDare the time scales of the movement and BD processes, respectively. This implies that

the system evolves during a time τBDunder the composition of LBDand κ−1times of LM, which

yields:

P[ρ,t + τBD] =?1 + τBDLBD

For times much larger than τBD, and using the separation of BD and movement time scales, we

derive a new functional Fokker-Planck equation that replaces Eq. (4):

∂P[ρ,t]

∂t

Following the steps of the derivation of Eq. (6) from Eq. (4), as described in Section III A, an

averaging procedure of the functional Fokker-Planck equation (22) over the possible realizations of

the density field leads to a deterministic equation for the time evolution of the density:

?

which describes both BD processes and cell movement simultaneously in a multicellular system.

BD[ρ], i = 1,2, represent the first and the second moments of the

??1 + τMLM

?κ−1P[ρ,t].

(21)

=?LM[ρ,t] + LBD[ρ,t]?P[ρ,t].

(22)

∂ρ

∂t(r,t) = ?∇ ·

ρ(r,t)∇δFC[ρ(r,t)]

δρ(r,t)

?

+ D(1)

BD[ρ(r,t)],

(23)

C. Application to tumor growth

We illustrate the DDFT modeling framework by presenting an example that contains the basic

phenomenology of tumor growth. We neglect many biophysical processes4,42and assume that tumor

cell movement is mediated by random motion, cell-cell interaction forces and cell-extracellular

matrix (ECM) interactions through haptotaxis, which can be modeled as an external force. In

particular, let E denote the density of the ECM. Then, the external potential can be taken to be Vext

= −ξEE, which models movement of cells up gradients of ECM. We assume that ECM is degraded

and produced by tumor cells. As a simple model of cell-cell interaction forces, one may use the

Morse potential VM(r) =¯VM

from cell-cell compression, and longer-range attraction, e.g., resulting from cell-cell adhesion. Here,

¯VMand a are parameters that control the well-depth and width, respectively, and reis an equilibrium

distance. Of course, more sophisticated models of interactions may be used.43Accordingly, the

overdamped Langevin equation (1) becomes

?1 − e−a(r−re)?2, which describes short-range repulsion, e.g., resulting

dri

dt

= −?

N

?

j=1

∇VM

?|ri− rj|?+ χE∇E(ri) +

√2D ηi(t),

(24)

where χE= ?ξE. The ECM evolves according to

dE

dt

= λE

N

?

i=1

δ (r − ri(t)),

(25)

where λEis the net rate of ECM production, which is assumed to saturate at a specific value Esat

of ECM density, λE=¯λE,prod(Esat− E)+−¯λE,degradeE, and the subscript + denotes the positive

part.

Theproliferationoftumorcellsisdependentontheavailabilityofnutrientswithconcentrationn.

AsasimplemodelofBD,weassumethatasinglecellcanundergomitosiswithanutrient-dependent

probability λm= λm(n), which we represent by means of the stoichiometric equation X

We model cell death (e.g., apoptosis) as occuring with a constant probability λd, described by

the stoichiometric equation X

−→ ∅. This can be implemented as a Markov process and affects

λm

−→ 2X.

λd

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011210-11Chauviere et al. AIP Advances 2, 011210 (2012)

the number of cells in the population making N = N(t). The nutrient is provided by the vascular

system, diffuses through the microenvironment and is uptaken by tumor cells, and thus satisfies the

reaction-diffusion equation

∂n

∂t

= Dn∇2n + Sn− λn

N(t)

?

i=1

δ (r − ri(t)),

(26)

where Dnis the diffusion coefficient, Snrepresents nutrient sources and λnis the nutrient uptake rate

assumed to be proportional to n, λn=¯λnn.

Putting together the methodology described in the previous subsections, and using the fact

that the first moment of the BD process described above is simply the result of two mass action

laws, that is D(1)

for the cell density ρ. Taking correlations into account, and using the approximations described in

Sec. III A 2, the cell density equation becomes

BD[ρ] = λm(n)ρ − λdρ, we can derive the corresponding deterministic equation

∂ρ

∂t(r,t) = D∇2ρ(r,t) − χE∇ · [ρ(r,t)∇E(r,t)] + [λm(n(r,t)) − λd]ρ(r,t)

?

Due to cell proliferation, the reference density ρ∞should be time-dependent to reflect changes in

correlations due to increased cell densities, as observed in Ref. 21. A simple way of implementing

this is to define the reference density to be ρ∞(t) = 1/|?|?drρ(r,t), where the integral is taken

dρ∞

dt

|?|

As described in Sec. III A 2, Eq. (27) is not closed and requires the direct two-point correlation

function c(2)

clarity)

?

together with a closure rule, which we may take, for example, to be the hypernetted chain closure31

(HNC) from the physical sciences,

?

which relates the total correlation function h with c(2)

the given potential VM, Eqs. (29)and(30) form a system of two nonlinear equations for the two

unknowns h and c(2)

Eq. (28), thus providing the direct two-point correlation function c(2)

Finally, the continuum equations for the ECM and nutrient concentration become

dE

dt

−D∇ ·

ρ(r,t)

?

dr??ρ(r?,t) − ρ∞

?∇rc(2)

eq(|r − r?|;ρ∞)

?

.

(27)

over the full domain ?. Assuming no-flux boundary conditions, we obtain

=

1

?

dr[λm(n(r,t)) − λd]ρ(r,t).

(28)

eqthat is determined as the solution of the Ornstein-Zernike equation (repeated here for

h(r) = c(2)

eq(r;ρ∞) + ρ∞

dr?h(r?)c(2)

eq(|r − r?|;ρ∞),

(29)

h(r) = exp

−?

DVM(r) + h(r) − c(2)

eq(r;ρ∞)

?

− 1,

(30)

eqand the interaction potential VM. Then, for

eq, parameterized by the reference density ρ∞that changes dynamically due to

eqto close Eq. (27).

=?¯λE,prod(Esat− E)+−¯λE,degradeE?ρ,

(31)

dn

dt

= Dn∇2n + Sn−¯λnnρ

(32)

Thus far, we have outlined a stochastic, discrete model and its deterministic, continuum coun-

terpart, which provide descriptions of the same processes occurring during tumor growth at different

scales. These models can be used together to develop a hybrid continuum-discrete multiscale model

using an adaptive mesh and algorithm refinement approach,44,45where the continuum model is used

at coarse grid scales, while the discrete algorithm is used on the finest mesh where fluctuations

are important (e.g., near the tumor-host interface). Further, feedback from continuum-scale to the

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011210-12Chauviere et al. AIP Advances 2, 011210 (2012)

microscale can be accommodated, for example, by assuming that the discrete forces depend on the

cell density.

IV. DISCUSSION AND OUTLOOK

We have proposed a modeling framework for multicellular systems by extending Dynamic

Density Functional Theory (DDFT) to account for different cell types, phenotypes and birth/death

processes. Analytical expressions were obtained for the biophysical fluxes that incorporate density

correlations,whichweremodeledusingseverallevelsofapproximationfollowingaDDFTapproach.

We then applied these ideas to tumor growth and developed a stochastic, discrete model and a

continuum DDFT-type counterpart that modeled the same biophysical processes at larger scales.

We discussed how these continuum and discrete systems can be used to derive a hybrid continuum-

discrete multiscale model that satisfies the essential properties of a multiscale approach as described

in Sec. I.

The primary difference between our framework and existing models is that the DDFT-based

approach accounts for correlations at the continuum scale whereas existing models for multicellular

systems tend to use uncorrelated mean-field approximations. As observed in recent experiments

of collective cell motion,21correlations become more important as cells proliferate and compress

one another, leading to a transition from fluid-like to solid-like behavior. In growing tumors, where

there is significant cell proliferation and compression, similar behavior may be expected. This

cannot be captured in an uncorrelated mean-field model without phenomenological modifications

of the biophysical fluxes (and constitutive laws) to separate fluid-like and elastic-like regimes using,

for example, pre-imposed yield stresses.46,47In contrast, such transitions should be automatically

captured in our approach, as has been observed in the physical sciences.48–50

In the development of a DDFT for multicellular systems, several levels of approximations were

used to close the system of equations including an adiabatic approximation and explicit closure

relations, which relate the correlations to the microscopic interaction potential. While this approach

has been used successfully in the physical sciences, it remains to be seen how appropriate these

approximations are for biological systems. Alternatively, experimental measurements can be used to

generate correlations by exploiting the relations between the structure factor and the correlations.51

For example, experimental measurements of the structure factor can be used to generate the total

correlation function via the Fourier transform. Then, the direct two-point correlation function can

be determined from the total correlation by solving the Ornstein-Zernike equation.

Further, the experiments discussed above on collective cell motion21also underscore the po-

tential importance of taking into account cell sizes and shapes, which are strongly influenced by

compression and migration. While there is little known about DDFT for active, deformable objects,

it may be possible to describe these effects as components in a phenotype vector, such as we have

introduced here.

Because the DDFT-based model provides a description of processes at the mesoscale, further

coarse-graining is needed to describe processes at the tissue scale. This should be possible by

adapting and extending density-amplitude expansions developed in the physical sciences, where

equations are derived for the amplitudes and phases of the density, which vary on much longer

length scales.52–54

WhilewehavediscussedtheapplicationoftheDDFTframeworktotumorgrowth,itisimportant

to note that the framework can easily be made more general to accommodate more detailed tumor-

stroma interactions. Further, heterogeneity and anisotropy associated with the ECM can be modeled

using an approach analogous to that used in the development of a DDFT model of nematic liquid

crystals.55,56

ACKNOWLEDGMENTS

WethankY.-L.Chuang(UNM)forassemblingFig.1,M.T.Lewis(BaylorCollegeofMedicine)

for discussions on breast cancer and P. Liu (Princeton) for discussions on correlations. We gratefully

acknowledge the following grants: NIH-PSOC grants 1U54CA143837 and 1U54CA143907 (VC);

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011210-13 Chauviere et al. AIP Advances 2, 011210 (2012)

NIHgrantP50GM76516(JL);NIH-ICBPgrant1U54CA149196(VC,AC);NSF-DMSgrantsDMS-

0818104 (VC), DMS-0817891 (IGK), DMS-0818126 (JL); DOE grant DE-SC0002097 (IGK).

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