Dynamic density functional theory of solid tumor growth: Preliminary models.

Vittorio Cristini
The Victor and Ruby Hansen Surface Professor of Molecular Modeling of Cancer, University of New Mexico
ABSTRACT Cancer is a disease that can be seen as a complex system whose dynamics and growth result from nonlinear processes coupled across wide ranges of spatiotemporal scales. 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, phenotypes and cell birth/death processes, in order to provide a biophysically consistent description of processes across the scales. We present an application of this approach to tumor growth.
 [Show abstract] [Hide abstract]
ABSTRACT: In our previous work [J. Chem. Phys. 136, 024502 (2012)], we reported a demixing phase transition of a quasitwodimensional, binary Heisenberg fluid mixture driven by the ferromagnetic interactions of the magnetic species. Here, we present a theoretical study for the timedependent coarsening occurring within the twophase region in the densityconcentration plane, also known as spinodal decomposition. Our investigations are based on dynamical density functional theory (DDFT). The particles in the mixture are modeled as Gaussian soft spheres on a twodimensional surface, where one component carries a classical spin of Heisenberg type. To investigate the twophase region, we first present a linear stability analysis with respect to small, harmonic density perturbations. Second, to capture nonlinear effects, we calculate timedependent structure factors by combining DDFT with Percus' test particle method. For the growth of the average domain size l during spinodal decomposition with time t, we observe a powerlaw behavior l∝t^{δ_{α}} with δ_{m}≃0.333 for the magnetic species and δ_{n}≃0.323 for the nonmagnetic species.Physical Review E 09/2013; 88(31):032301. · 2.31 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: There have been many techniques developed in recent years to in silico model a variety of cancer behaviors. Agentbased modeling is a specific discretebased hybrid modeling approach that allows simulating the role of diversity in cell populations as well as within each individual cell; it has therefore become a powerful modeling method widely used by computational cancer researchers. Many aspects of tumor morphology including phenotypechanging mutations, the adaptation to microenvironment, the process of angiogenesis, the influence of extracellular matrix, reactions to chemotherapy or surgical intervention, the effects of oxygen and nutrient availability, and metastasis and invasion of healthy tissues have been incorporated and investigated in agentbased models. In this review, we introduce some of the most recent agentbased models that have provided insight into the understanding of cancer growth and invasion, spanning multiple biological scales in time and space, and we further describe several experimentally testable hypotheses generated by those models. We also discuss some of the current challenges of multiscale agentbased cancer models.Seminars in Cancer Biology 05/2014; · 7.44 Impact Factor  SourceAvailable from: Tamal Das[Show abstract] [Hide abstract]
ABSTRACT: According to the World Health Organization, cancer is one of the leading causes of death worldwide. Cancer research, in its all facets, is truly interdisciplinary in nature, cutting across the fields of fundamental and applied sciences, as well as biomedical engineering. In recent years, microfluidics has been applied successfully in cancer research. There remain, however, many elusive features of this disease, where microfluidic systems could throw new lights. In addition, some inherent features of microfluidic systems remain unexploited in cancer research. In this article, we first briefly review the advancement of microfluidics in cancer biology. We then describe the biophysical aspects of cancer and outline how microfluidic system could be useful in developing a deeper understanding on the underlying mechanisms. We next illustrate the effects of the confined environment of microchannel on cellular dynamics and argue that the tissue microconfinement could be a crucial facet in tumor development. Lastly, we attempt to highlight some of the most important problems in cancer biology, to inspire next level of microfluidic applications in cancer research.Biomicrofluidics 01/2013; 7(1). · 3.39 Impact Factor
Page 1
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
resultfromnonlinearprocessescoupledacrosswiderangesofspatiotemporalscales.
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 pathologybased 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 tissuescale in
space over long scales in time. Thus, a fundamental gap that is currently hampering progress in
individualizedcancercareistheextrapolationofpatientspecificmicroscalequantitativeinformation
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 multimodal, multidimensional 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 subcellular 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 cofirst and cocorresponding author; Electronic mail: achauviere@salud.unm.edu.
bindicates cofirst author.
cindicates cocorresponding author; Electronic mail: lowengrb@math.uci.edu.
21583226/2012/2(1)/011210/13
C ?Author(s) 2012
2, 0112101
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0112102 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 meanfield models, the DDFT approach accounts
for correlations.1,2We extend the approach to model the spatiotemporal 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 cellscale 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: “MULTIPLESCALE” 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 continuumdiscrete 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 nontrivial 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 multiplescale approach
A mechanistic agentbased 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 “bottomup” approach was used for ABM parameter calibration from experimental data at the
cellscale (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
volumeaveraging was additionally used with the ABM to calculate the coarsegrained proliferation
and apoptotic volume rate parameters at the tissue scale, which provided input to a continuum
model10thus avoiding fitting. A “topdown” approach was used to compare the model results to
tumor volumes measured in patients. This led to the first accurate, patientspecific assessment of
DCIS tumor volumes – a key translational issue in current surgical planning.11
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0112103 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 multiplescale approach
for DCIS.7,11Bottom images: (left) Bottomup 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 modelpredicted 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 multiplescale 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 cellcell 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 hypoxiainduced epithelialtomesenchymal 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 epithelialmesenchymal
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 twostep approach does not allow for any dynamical exchange between cell and
tissuescale 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 multiplescale approaches the variables and fluxes considered at the macroscale
(continuum) are preassumed and are not obtained by upscaling the cell scale (discrete) dynamics.
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0112104 Chauviere et al. AIP Advances 2, 011210 (2012)
FIG. 2. Hybrid multiplescale 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
equationfree 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 nonequilibrium 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 solidlike 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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0112105 Chauviere et al. AIP Advances 2, 011210 (2012)
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 spatiotemporal evolution
of the tissuescale 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 cellcell friction, which could be
included by adding a viscous drag force in the Langevin equations via a microscopic friction tensor
actingontherelativevelocitywithneighboringcells.22Althoughneglectingcellcellfrictionmaybe
more appropriate in the absence of active cell motion, cellcell friction could easily be incorporated
using a similar modeling framework. The forces?N
betweentwocells.Wemodelmicroenvironmentalinfluences(e.g.,chemotaxis,haptotaxis,cellECM
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 Nparticle position
probability distribution P(rN,t) (or Nbody 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 ith cell in the ddimensional
j=1Fint
ij= −?N
j=1∇riVint(ri− rj) model
cellcell 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 Ndimensional position vector of cells and Fi=?N
Summarizing the approach in Ref. 24, the discrete density field of the Ncell 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 coarsegrained 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 coarsegrained delta function over a domain of char
acteristic length scale l, which lies between the cell size and the celldensity 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
wellseparated 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 FokkerPlanck
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 largescale 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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0112106 Chauviere et al.AIP Advances 2, 011210 (2012)
describes the dynamics of the coarsegrained density:24
?
whereδ/δρ(r,t)isafunctionalderivativewithrespecttoρ andFC[ρ]isacoarsegrainedfreeenergy
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 freeenergy
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 noiseaveraged delta functions without
appealing to a freeenergy minimization principle. However, we do not take this alternate approach
here as it is more difficult to account for cellproliferation as we will propose further in our paper.
The deterministic equation (6) captures the spatiotemporal 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 coarsegrained freeenergy 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 twoparticle density distribution function. Since ρ(2)depends on ρ(3), the
threeparticle 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 cellcell 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 meanfield 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 meanfield 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 cellcell adhesion via the
nonlocal operator.29Further, the mean field approximation fails when either the density is low, so
thatfluctuations arenonnegligible, orwhencorrelationsareimportant,e.g.,arisingfromlongrange
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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0112107 Chauviere et al.AIP Advances 2, 011210 (2012)
2. Correlated meanfield 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 twoparticle 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 OrnsteinZernike (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 physicallybased 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 twopoint cor
eq(r;ρ∞) + ρ∞
?
dr?h(r?)c(2)
eq(r − r?;ρ∞),
(10)
eqto be able to solve the OrnsteinZernike
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 meanfield 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 ith 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 cellcell 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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0112108Chauviere et al. AIP Advances 2, 011210 (2012)
presented in Section III A. We use multivariable stochastic Ito calculus for each species to derive
the corresponding Smoluchowski equation for each Nkbody 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 FokkerPlanck 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 coarsegrained freeenergy 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 crosscorrelations 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 Ndimensional position vector of the whole set of cells), σN= {σi}i = 1. . . Nis the
generalized Ndimensional 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 Nparticle 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 coarsegrained delta function already introduced and σ the vector of continuous
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0112109 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 coarsegrained 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 FokkerPlanck 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 twoparticle 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 OrnsteinZernike 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 dedifferentiation. 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 spatiallydistributed 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 FokkerPlanck 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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01121010Chauviere et al. AIP Advances 2, 011210 (2012)
and the cell density functionals D(i)
BD process.
Next, we consider combined BD and cellmovement processes. We assume that the time scale
of cell movement (i.e., from cellcell 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 FokkerPlanck 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 FokkerPlanck 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, cellcell interaction forces and cellextracellular
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 cellcell interaction forces, one may use the
Morse potential VM(r) =¯VM
from cellcell compression, and longerrange attraction, e.g., resulting from cellcell adhesion. Here,
¯VMand a are parameters that control the welldepth 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 shortrange 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,weassumethatasinglecellcanundergomitosiswithanutrientdependent
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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01121011Chauviere 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
reactiondiffusion 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 timedependent 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 twopoint 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 twopoint 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 noflux boundary conditions, we obtain
=
1
?
dr[λm(n(r,t)) − λd]ρ(r,t).
(28)
eqthat is determined as the solution of the OrnsteinZernike 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 continuumdiscrete 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 tumorhost interface). Further, feedback from continuumscale to the
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01121012Chauviere 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 DDFTtype 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 DDFTbased
approach accounts for correlations at the continuum scale whereas existing models for multicellular
systems tend to use uncorrelated meanfield 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 fluidlike to solidlike behavior. In growing tumors, where
there is significant cell proliferation and compression, similar behavior may be expected. This
cannot be captured in an uncorrelated meanfield model without phenomenological modifications
of the biophysical fluxes (and constitutive laws) to separate fluidlike and elasticlike regimes using,
for example, preimposed 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 twopoint correlation function can
be determined from the total correlation by solving the OrnsteinZernike 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 DDFTbased model provides a description of processes at the mesoscale, further
coarsegraining is needed to describe processes at the tissue scale. This should be possible by
adapting and extending densityamplitude 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: NIHPSOC grants 1U54CA143837 and 1U54CA143907 (VC);
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01121013 Chauviere et al. AIP Advances 2, 011210 (2012)
NIHgrantP50GM76516(JL);NIHICBPgrant1U54CA149196(VC,AC);NSFDMSgrantsDMS
0818104 (VC), DMS0817891 (IGK), DMS0818126 (JL); DOE grant DESC0002097 (IGK).
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