On the morphological stability of multicellular tumour ...ciarletta/GiversoCiarletta_EPJE-final draft.pdf · of avascular tumour growth and to assess the e cacy of anti-cancer drugs

EPJ manuscript No.(will be inserted by the editor)

On the morphological stability of multicellular tumour spheroidsgrowing in porous media

Chiara Giverso1 and Pasquale Ciarletta1,2a

1 Dipartimento di Matematica - MOX, Politecnico di Milano and Fondazione CEN, Piazza Leonardo da Vinci, 32 - 20133Milano, Italy;

2 CNRS and Sorbonne Universites, UPMC Univ Paris 06, UMR 7190, Institut Jean le Rond d’Alembert, 4 place Jussieu case162, 75005 Paris, France

Received: date / Revised version: date

Abstract Multicellular tumour spheroids (MCTSs) are extensively used as in-vitro system models forinvestigating the avascular growth phase of solid tumours. In this work, we propose a continuous growthmodel of heterogeneous MCTSs within a porous material, taking into account a diffusing nutrient fromthe surrounding material directing both the proliferation rate and the mobility of tumour cells. At thetime scale of interest, the MCTS behaves as an incompressible viscous fluid expanding inside a porousmedium. The cell motion and proliferation rate are modelled using a non-convective chemotactic massflux, driving the cell expansion in the direction of the external nutrients’ source. At the early stages,the growth dynamics is derived by solving the quasi-stationary problem, obtaining an initial exponentialgrowth followed by an almost linear regime, in accordance with experimental observations. We also performa linear-stability analysis of the quasi-static solution in order to investigate the morphological stability ofthe radially symmetric growth pattern. We show that mechano-biological cues, as well as geometric effectsrelated to the size of the MCTS subdomains with respect to the diffusion length of the nutrient, candrive a morphological transition to fingered structures, thus triggering the formation of complex shapesthat might promote tumour invasiveness. The results also point out the formation of a retrograde flow inMCTS close to the regions where protrusions form, that could describe the initial dynamics of metastasisdetachment from the in-vivo tumour mass. In conclusion, the results of the proposed model demonstratethat the integration of mathematical tools in biological research could be crucial for better understandingthe tumour’s ability to invade its host environment.

PACS. 87.18.Hf Spatiotemporal pattern formation in cellular populations – 87.18.Gh Cell-cell communi-cation; collective behavior of motile cells – 46.32.+x Static buckling and instability

1 Introduction

A multicellular tumour spheroid (MCTS) is an ensembleof tumour cells organized in a multi-layered structure1,2.In general, MCTS consists of a central core of necroticcells, surrounded by a layer of quiescent (i.e. dormant)cells and an outer rim of proliferating cells1–4.MCTSs are widely used in vitro to study the early stagesof avascular tumour growth and to assess the efficacy ofanti-cancer drugs and therapies, since their growth andstructure resemble the in vivo avascular phase of solidtumour invasion. Such an early growth phase is charac-terized by diffusion-limited growth, since the tumour ab-sorbs vital nutrients via diffusion from the external envi-ronment1,3,5. Thus, diffusion may become suddenly inef-fective in the center of the tumour mass, forming a charac-teristic necrotic core (see Fig. 1-a). At later stages, a solid

a e-mail: [email protected]

tumour is characterized by the occurrence of angiogenesis(i.e. the process by which the tumour induces new bloodvessels formation from the nearby existing vasculature),thus switching to a vascular growth phase6,7.The analysis of the avascular growth phase in tumours hasattracted a lot of interest in the mathematical and physi-cal research communities, and a large number of in-silicomathematical models has been proposed2,8–16. Thanks tothe controllability and the reproducibility of the experi-mental setting, MCTS has become a widely used systemmodel for the development of theoretical models.The classical approach of deterministic tumour model com-prised an ordinary differential equation (ODE), derivedfrom either mass conservation or population dynamics,coupled with at least one reaction-diffusion equation, rep-resenting the spatio-temporal distribution of vital nutri-ents or chemical signals inside the tumour2,9–12,14,15,17.Only recently, many authors have extended such modelsincluding the pivotal role of mechanics in tumour growth.

2 Chiara Giverso, Pasquale Ciarletta: On the morphological stability of multicellular tumour spheroids

In most cases, fluid-like constitutive equations have beenused to model the tumour mass18–26.This choice is obvi-ously only an approximation of the by far more complexbehaviour of cellular aggregates, that also display solid-like properties related to the adhesive characteristics ofcells27 and to the mechanical properties of the single cellin the cluster. Thus, in some limiting cases, cell aggre-gates are better described as solids with linear or eventu-ally nonlinear elasticity, in which compressive and shearloads are balanced by the solid stress in the body, depend-ing on the strain of its material points28–33. A solid-likeconstitutive equation has been advocated for its suitabil-ity of accounting for both residual stresses29,32,34 and theplastic behaviour of cellular aggregates35–37. Even thoughthese considerations support the idea that a cellular ag-gregate can behave as a solid at some extent, experimen-tal evidences26,38,39 have shown that aggregates behaveas elastic solids on short timescales (of the order of a fewminutes) but display a fluid-like behaviour at longer times.Furthermore, it was shown that cellular aggregates behaveas a elastic solids at time scales short compared to that ofcell division and apoptosis, and as a fluid (with the trace-less stress that relaxes to zero) for long times40. Thus thedescription of MCTS as a liquid is widespread.Even though the existing mathematical models on bothsolid tumours and MCTS successfully reproduce the ex-perimentally observed growth dynamics2,9–12,14,15,17,41,42,they poorly consider the mechanical and chemical inter-action with the surrounding environment. Furthermore,most approaches assume that the initial spherical symme-try is preserved during the growth of the aggregate28–30,whilst only in few cases11,12,15 the development of tumourirregular contours has been taken into account. Indeed, itis known that some solid tumours, e.g. carcinomas, growalmost spherically only in the first stages of their progres-sion1,3,5, while they might show a less defined and evenasymmetric outer boundary43 (see Fig. 1-b). Since higherirregular contours usually indicate aggressive tumours, thecapability to undergo a morphological transition mightpromote tumour infiltration and invasion within the sur-rounding tissue2,11,12,15,44–46. Thus, it has been proposedthat some measure of the irregularity of a tumour bound-ary (e.g. its fractal index measured via particular medi-cal imaging techniques such as computerized tomographyscans), may provide clinicians with useful information forits prognosis and treatment44–46, being potentially usefulin predicting the efficacy of drug treatment or chemother-apy47,48.

In this work we go beyond the state-of-the art in thefield2,49,50 by proposing a mathematical model that ac-counts for the presence of a surrounding porous mediawith a finite thickness. Thus, nutrient diffusion from theexternal environment creates a chemical gradient that di-rects both the proliferation rate and the motility of thetumour cells. MCTS is modelled as a viscous fluid withadhesive interactions at the border, expanding inside aporous material.This work is organized as follows. First, we introduce inSection 2 the mathematical model describing the expan-

(a)

(b)

Figure 1. (a) Morphological evolution of a multicellular tu-mour spheroid of HeLa cells, showing the development of anundulated contour and a necrotic core (reproduced with per-mission from51). (b) Solid tumours extracted from mice afterorthotopic implant of MCF10CA1a cell lines in the mammaryfat pad of the nude mice (courtesy of T. Stylianopoulos (CancerBiophysics Laboratory, University of Cyprus).

sion of an initially spherical tumour. In Section 3, we de-rive the radially-symmetric solution of the quasi-stationaryproblem. Then, we perform a linear stability on the quasi-static tumour growth. Finally, in Section 4, we discuss themodelling results with respect to the key chemo-mechanicaland geometric parameters that govern the mathemati-cal problems, highlighting the key mechano-biology effectsthat promote a morphological transition during tumourinvasion.

2 Mathematical Model

MCTS is modelled as a three dimensional continuum grow-ing inside a rigid porous structure, representing the sur-rounding environment, usually extracellular matrix (ECM)or matrigel. The outer boundary of the tumour is consid-ered as a freely moving material interface separating thetumour cells from the surrounding medium.In particular, we account for the presence of a central re-gion of necrotic cells, surrounded by a layer of quiescentand proliferating cells. Thus, the whole domain Ω is di-vided in different regions, depending on the residing cel-lular population (see Fig. 2):

– the necrotic cells are located in the central core of thespheroid, in a region called ΩN (t), with

ΩN (t) = (r, θ) : r < RN (t), 0 < ϕ ≤ π, 0 < θ ≤ 2π ,

Chiara Giverso, Pasquale Ciarletta: On the morphological stability of multicellular tumour spheroids 3

N

T

H

RN

RT

Rout

Figure 2. Representation of the domain used for the analyticalanalysis. At time t = 0, the three domains ΩN , ΩT and ΩHare concentric spherical shells, with radius RN , RT and Rout,respectively. In this work, we consider that only the tumourboundary ∂ΩT evolves in time.

where RN is the radius of the necrotic core, that mightevolve in time;

– the proliferative and quiescent tumour cells are locatedin the region

ΩT (t) = (r, θ) : RN (t) < r < RT (t), 0 < ϕ ≤ π, 0 < θ ≤ 2π ,

where RT is the radius of the spheroid, whose evolutionin time represents the growth of the MCTS;

– the healthy cells fill the remaining space,

ΩH(t) = (r, θ) : RT (t) < r < Rout, 0 < ϕ ≤ π, 0 < θ ≤ 2π ,

being Rout the outer boundary of the whole domain.

The boundary between the necrotic core and the prolifer-ative region is called ∂ΩN (t), whereas the moving inter-face between the tumour region and the healthy spaceis denoted with ∂ΩT (t). In the following we will con-sider that the interior boundary between the necrotic coreand the quiescent-proliferative region does not evolve intime, since we are interested only in the evolution of theMCTS boundary, which is related to tumour infiltrationinside the healthy region. Furthermore, we assume thatthe porous material is homogeneously distributed in thewhole region Ω = ΩN ∪ ΩT (t) ∪ ΩH(t) and it is neitherproduced/degraded (i.e. behaves as inert matter), nor de-formed (i.e. structurally rigid) by the moving tumour cells.We will consider a single nutrient species (e.g. oxygen)with volume concentration n(x, t), diffusing from the fixedouter boundary ∂Ω through the porous material. Thus,we assume that the vascular network providing the sourceof nutrients is outside the modelled domain, and can berepresented by a boundary term at ∂Ω. The diffusion coef-ficient is a constant value Dn everywhere, but the nutrientis only consumed, with an uptake rate γn, in the regionoccupied by the proliferative and quiescent cells. Thus,the 3D homogenized concentration per unit volume of this

generic chemical species, indicated with n(x, t), obeys thefollowing reaction-diffusion equation

n(x, t) =

Dn∇2n(x, t) in ΩN ,

Dn∇2n(x, t)− γnn(x, t) in ΩT (t) ,

Dn∇2n(x, t) in ΩH(t) .

(1)

We remark that, in principle, the uptake rate γn shoulddepend on the tumour cell density, although, in the fol-lowing, it will be considered homogeneous and constantover time.The diffusing nutrient notably not only affects the growthof single individuals in the tumour but also directs cellmovements, e.g. through chemotaxis52,53. Therefore, weconsider a non-convective mass flux term, m, taking intoaccount both tumour proliferation and chemotactic mo-tion, differently from the standard volumetric productionrate considered in literature2,11–15. Accordingly, the massbalance inside ΩT (t) reads

dρ

dt+ ρ∇ · v = ∇ ·m in ΩT (t). (2)

where ρ is the tumour cell density, which is approximatelythe same of water. Since mass transport phenomena inMCTSs are driven by the local concentration of chemicals,the mass flux vector appearing in Eq. (2) should dependon nutrient availability. A simple constitutive law for mcan be taken in the form of a chemotactic term52,54, i.e.m = χρ∇n, where χ is the chemotactic coefficient, hereconsidered constant. Consequently, the mass flux m de-scribes the expansion of the tumour due to proliferationand driven by chemotaxis towards higher concentration ofnutrients.Assuming that the living aggregate can be macroscopicallymodelled as a Newtonian fluid, Darcy’s law describes itsmotion inside the inert, porous surrounding medium2,18.Thus, the cell velocity v is related to the pressure field pthrough

v = −Kp∇p , (3)

where Kp is related to the permeability of the medium, k,and the viscosity of the cellular material, µ, by Kp = k/µ.Assuming the incompressibility of the cellular spheroid,which is mostly composed by water, we impose dρ/dt = 0in Eq. (2), so that the relation between the pressure p andthe nutrient concentration n reads

∇2p = − χ

Kp∇2n in ΩT (t) , (4)

which has been obtained substituting the Darcy’s law (3)and the constitutive relations for m in the mass balanceequation (2). In summary, the coupling of Eq. (1) withEq.(4), complemented by a proper set of boundary con-ditions (BCs), describes the macroscopic evolution of theavascular tumour inside the healthy tissue.In particular, for the pressure we impose the Young-Laplaceequation at the moving boundary ∂ΩT (t) and the null ve-locity of the tumour cells at the fixed boundary ∂ΩN , i.e.

p = p0 − σbC on ∂ΩT (t) , (5)

v∂ΩN· nN = 0→ (∇p)|∂ΩN

· nN = 0 on ∂ΩN , (6)


being nN the normal at the fixed boundary ∂ΩN , C thelocal curvature of the free boundary ∂ΩT (t), σb the surfacetension at the moving interface and p0 the constant pres-sure in the outer healthy domain. The surface tension σbarises from the collective adhesive interaction, primarilymediated by cadherins, among tumour cells at the MCTSboundary55.For what concerns the chemical species, in absence of aninterfacial structure, the continuity for the nutrient con-centration and flux can be assumed (both in ∂ΩT (t) andin ∂ΩN ), and the concentration at the outer boundarycan be assumed constant (to model the source of nutri-ents from the external vascular network), so that

n |∂Ω = nout on ∂Ω , (7)

JnK |∂ΩT= 0 , J∇nK |∂ΩT

· n = 0 on ∂ΩT , (8)

JnK |∂ΩN= 0 , J∇nK |∂ΩN

· nN = 0 on ∂ΩN , (9)

where n is the outward normal vector at the boundary∂ΩT and J(·)K |∂Ωj denotes the jump of the quantity be-tween brackets across the boundary ∂Ωj , with j = N,T .Finally, the compatibility condition at the free interfaceimposes

dx∂ΩT

dt· n = v∂ΩT

· n on ∂ΩT . (10)

In the following we will work with dimensionless equa-tions, obtained writing the system of Eqs. (1)-(4) in termsof the dimensionless chemical concentration, n, and thedimensionless pressure, p and referring to the geometryoutlined in Fig. 2. The dimensionless quantities are ob-tained using the following characteristic time tc, lengthlc, velocity vc, pressure pc and chemical concentration nc:

tc = γ−1n , lc =

√Dnγ

−1n , vc =

√Dnγn , pc = DnK

−1p ,

nc = nout. Finally, the resulting dimensionless systemsof equations reads

˙n =

∇2n for r < RN∇2n− n for RN < r < RT (t)

∇2n for RT (t) < r < Rout(11a)

∇2p = −β∇2n for RN < r < RT (t)(11b)

JnK |RN= 0,

q∇n

y|RN· nN = 0, (∇p) · nN = 0

for r = RN (11c)

JnK |RT= 0 ,

q∇n

y|RT·n = 0 , p = p0 − σC

for r = RT (t)(11d)

n(t, Rout) = 1 for r = Rout(11e)

dxRT

dt· n = vRT

· n = −∇p|RT· n for r = RT (t) .

(11f)

For sake of simplicity, we will omit in the following thebarred notation to denote dimensionless quantities. The

nondimensionalization leads to the definition of five di-mensionless parameters, classified into two broad cate-gories:

– β := χnc/Dn and σ := σbKpγ1/2n D

−3/2n =σbKpl

−1/2c D−1

n

define mechano-biology effect on the aggregate expan-sion, and are called motility parameters;

– RN , RT and Rout (i.e. the dimensionless radii of thenecrotic core, of the MCTS and the whole domain,respectively) define the geometrical properties of thesystem with respect to the diffusive length lc, and aredenoted as size parameters.

In particular, the dimensionless parameter β representsthe ratio between the energy associated to the expansionof the MCTS due to the “growth” cues (i.e. chemotaxisand proliferation ) and the energy associated to the diffu-sion of nutrients. On the other hand, the parameter σ isthe ratio between the energy associated to the expansionrelated to “mechanical” cues (i.e. related to the permeabil-ity of the medium and the surface tension and viscosity ofthe cellular material, described as a newtonian fluid) andthe energy associated to the diffusion of nutrients.

3 Linear stability analysis of the quasi-staticsolution

In this Section, we first derive the quasi-static solution ofthe proposed model in order to mimic the early avasculargrowth. We later perform a linear stability analysis to in-vestigate the occurrence of a morphological instability atlater growth stages.

3.1 Quasi-stationary solution

At early stages of avascular growth MCTS maintain aspherical shape1,3,5. Thus, we look for a radially sym-metric quasi-stationary solution, assuming that the dif-fusive process is much faster than the MCTS expansion,so that it is possible to drop the time derivative in Eq.(11a). This assumption is valid in many biological condi-tions, since a fast-growing tumour may expand at a rate ofup to 0.5 mm/day, whereas a typical diffusion time scaleis about 1 min (considering a typical length scale L ≈10−2cm and a typical diffusion coefficientD ≈ 10−6cm2s−1)11.Thus, it is clear that the diffusion timescale of nutrientsis much shorter than the growth timescale, so that thequasi-stationary assumption can be effectively formulated.Furthermore for such long time scale the MSC can be ac-tually treated as a viscous fluid.Specializing our analysis to the case of a spherical tu-mour of radius RT , we will denote with n∗ = n∗(r, t)the quasi-stationary solution of Eq. 11a and with p∗ =p∗(r, t) the quasi-stationary pressure field satisfying (11b).Given the boundary conditions (11c)-(11d)-(11e) and con-sidering that n∗ and p∗ should be bounded, the quasi-


β=0.1 β=1 β=2

Figure 3. Quasi-stationary solution of the proposed model,depicting the radius of the tumour over time for different val-ues of the motility parameter β. At early stages the growth isexponential, as a consequence of the bulk availability of nutri-ents. At later stages, the growth law is almost linear, reflectingthe higher nutrient concentration on the outer surface of thegrowing spheroid.

stationary fields read

n∗ =

2Route

RT +RN

e2RT (RN+1)w+T−e

2RN (RN−1)w−T

if r ≤ RN− e2RN [e2(r−RN )(RN+1)Rout+(RN−1)Rout]r(RN−1)w−

T er−RT +2RN−r(RN+1)er+RT w+

T

if RN < r ≤ RTRout[e2RT (RN+1)(r−RT +1)−e2RN (RN−1)(r−RT−1)]

r(e2RT (RN+1)w+T−e

2RN (RN−1)w−T )

otherwise

(12)

p∗ = p0 +σ

RT+ β(n∗RT

− n∗) , (13)

where we called n∗RT= n∗(RT ) the concentration of the

nutrient at the boundary of the aggregate and we definedw+T = (Rout −RT + 1) and w−T = (Rout −RT − 1), being

wT = (Rout−RT ) the width of the region occupied by thetumour. Then, using Eq. (11f), It is possible to computethe quasi-stationary velocity of the front, which is directedalong the radial direction for symmetry considerations, i.e.v∗ = v∗rer, with

v∗r (RT ) = βRout(e2RN (RN−1)(RT +1)−(RN+1)e2RT (RT−1))

R2T (e2RN (RN−1)(w−

T )−(RN+1)e2RT (w+T ))

.(14)

Equation (14) can be integrated numerically to determinethe evolution of the spheroid border over time. The result,reported in Fig. 3 for different value of the parameter β,highlights the existence of an initial phase in which thegrowth of the aggregate is nearly exponential and a subse-quent one in which the expansion of the tumour is almostlinear, as observed in32.

3.2 Perturbation of the quasi-stationary solution

In this paragraph, we investigate the stability of the steady,radially-symmetric solution by applying small perturba-tions of the MCTS boundary.

Let R∗T be the unperturbed position of the moving inter-face, we consider a small perturbation (ε 1) of the kind

R(θ, ϕ, t) = R∗T (t) + εeλtRe [Ym` (θ, ϕ)] . (15)

where λ ∈ R is the amplification rate (or time-growth rate)of the perturbation and Y m` (θ, ϕ) is the spherical harmonicof degree ` and order m, with m ∈ N, ` ∈ N+ and |m| ≤ `.The spherical harmonics Y m` (θ, ϕ) form a complete setof orthonormal functions and thus any square-integrablefunction can be expanded as a linear combination of spher-ical harmonics. For physical consistency, the variations ofn and p from the quasi-stationary solutions n∗ and p∗

should be in the form

n(r, θ, ϕ, t) = n∗(r, t) + εn1(r)eλtRe [Ym` (θ, ϕ)] (16)

p(r, θ, ϕ, t) = p∗(r, t) + εp1(r)eλtRe [Ym` (θ, ϕ)] , (17)

Using Eq. (11a) and the relation ∇2ΩY

m` +`(`+1)Y m` = 0,

where we set the angular part of the Laplacian operator as∇2Ω(·) = 1/ sin θ ∂/∂θ(sin θ ∂(·)/∂θ)+ 1/ sin2 θ ∂2(·)/∂φ2,

the term n1 must obey the following ODE

r2n′′1(r) + 2rn′1(r)−(`(`+ 1) + (λ+ 1ΩT

)r2)n1(r) = 0 , (18)

where primes denote derivatives on r and 1ΩT= 1 if

RN < r ≤ R∗T ,1ΩT= 0 otherwise. The solution of Eq.

(18), for λ 6= 0,−1 is

n1(r) =

C1i`(

√λr) if r ≤ RN

B1i`(√λ+ 1r) +B2k`(

√λ+ 1r) if RN < r ≤ R∗T

A1i`(√λ+ 1r) +A2k`(

√λ+ 1r) if R∗T < r ≤ Rout,

(19)

where i`(r) and k`(r) are the modified spherical Besselfunction of the first and second kind, respectively, evalu-ated in r. The coefficients A1, A2, B1, B2, C1 appearing inthe expression of n1(r) can be determined imposing the in-cremental boundary conditions for the concentration field(11c), (11d) and (11e), being

Jn1K |RN= 0 ,

s∂n1

∂r

|RN

= 0 , (20)

Jn1K |R∗T

= 0 ,

s∂n1

∂r

|R∗

T= n0 , n1(Rout) = 0 . (21)

The perturbed pressure field p1 in ΩT is obtained fromEq.(11b) that leads to

p1(r) = Qr` +Wr−`−1 − β(B1i`(

√λ+ 1r) +B2k`(

√λ+ 1r)

),(22)

where the constants Q and W can be determined from theboundary conditions on the pressure field (11c) and (11d),considering only the first order terms, i.e.

p1(R∗T ) = −σ 2

R∗T2 (2− (`+ 1)`)− ∂p∗

∂r|R∗

T,∂p1

∂r|RN

= 0 .(23)

Finally, from Eq. (10) it is possible to obtain the fol-lowing dispersion equation

λ = −p∗′′(R∗T )− p′1(R∗T ) , (24)


which has the same form of the relation found for therectilinear front on an infinite domain56 or an expand-ing circular colony57,58. The dispersion equation (24) isan implicit function of the time-growth mode λ and thespherical harmonic degree `, depending on the five dimen-sionless parameters βi, σ, RN , R∗T and Rout. Interestingly,λ does not depend on the azimuthal component of themodel solutions Y m` (φ, θ), i.e. the solutions are indepen-dent of the order m, as observed also in previous worksbased on different models15,50.

4 Results and Discussion

The dispersion equation (24) has been solved numericallyin order to investigate the global stability of the solutionsdepending on the system parameters. The correspondingdispersion diagrams are reported in Fig. 4 for differentvalues of both the size and the motility parameters. Asin classical perturbation theory59, a positive real part ofthe growth rate λ implies global instability, thus high-light a critical spatial mode of the perturbation defined bythe degree ` associated with the highest positive growthrate. Interestingly, Fig. 4 shows that the spheroid frontis linearly unstable at small `, with ` = 1 being alwaysunstable. Indeed, whilst for a spheroid growing inside aninfinite homogeneous domain with constant chemical con-centration, one would expect to find λ = 0 for ` = 1, dueto translational symmetry11,15,50, we must remind that inour case, due to the presence of the external environmentthe translational symmetry is no longer preserved.Furthermore, the dispersion diagrams in Fig. 4 also indi-cate the emergence of a characteristic mode different from` = 1 in the cases of bigger size parameters (see Fig. 4-a), as well as of small values of the motility parameters σ(see Fig. 4-c) and β (see Fig. 4-d). Interestingly, the char-acteristic mode is not significantly affected by varying onlythe dimension of the external domain, while keeping thenecrotic radius RN and the initial tumour radius RT fixed(Fig. 4-b) Moreover, whether the range of unstable modesis highly influenced by the sizes parameters and by themotility parameter σ (Fig. 4-a-c), it is not deeply influ-enced by variations of Rout and β (Fig. 4-b-d). Indeed aseither the size of the domains decreases (Fig. 4-a) or σincreases, the range of unstable modes decreases, up to arange where only ` = 1 is unstable. The dependency onthe size of the domains states that smaller diffusive lengths(i.e. smaller diffusion coefficient or higher absorption rateof the nutrients) lead to highly irregular contours duringthe growth of the tumour. On the other hand, the effecton the mechanical parameter σ on the dispersion diagramshows that, as expected, the surface tension σb, along withan high permeability of the surrounding porous environ-ment k act a stabilizing effect on the front (see Fig. 4-c),whereas the viscosity of the tumour cluster destabilize theborder of the MCTS leading to more aggressive tumours.As β settles the velocity of the quasi-stationary movingfront (see Eq. (14)), the dispersion diagram in Fig. 4-dshows that the tumour developed highly irregular contour

only in the case of slowly-moving front (i.e. small chemo-tactic coefficient and proliferation), since for fast movingfront the characteristic mode decrease, until only ` = 1 isunstable.Moreover, it is interesting to consider the role played bythe radius of the growing tumour in the development ofinstabilities, while keeping all the other parameters fixed(see Fig. 5). Fig. 5-a reports the results for a set of pa-rameters Rout, RN , β and σ for which, independently fromRT , the most unstable mode is always ` = 1. This situ-ation corresponds to a sort of translation of the spheroidinside the domain (see Fig. 5-a on the right). On the otherhand, the characteristic mode depends on the MCTS sizein a certain range of material parameters (see Fig. 5-b ).Indeed, it increases for increasing RT , so that bigger tu-mours show more irregularities at their border. Therefore,a growing MCTS can undergo a morphological transitionthat may significantly affect the invasion pattern towardsthe typical finger-like structures observed for invasive car-cinomas (see Fig. 5-b).

Finally, Figure 6 depicts the perturbed pressure andvelocity fields for a linearly unstable perturbation, givenby a spherical harmonic of the kind Y 6

10(θ, ϕ). The highestvariation of the pressure is located in a thin shell closerto the interface of the tumour, so that in the bulk of thetumour the velocity is almost null. In the region just atthe rear of small protrusions (due to the perturbation ofthe boundary), the pressure field increases, so that thevelocity at the border of the MCTS where a protrusionform, for the unstable modes (such as the one reportedin figure), is higher than the velocity in the invaginationon the contour. Furthermore, from the perturbed field itis possible to appreciate small negative radial velocities inthe bulk, just at the rear of the region where protrusionforms. Thus, while the spheroid surface moves outward,some cells inside the cluster move inward. This result con-firms the existence of a radial convergent flow, in additionto the divergent flow that makes the aggregate expand, aspointed out in60,61. This effect combined with the highervelocity associated to the protrusion border could explainthe possible detachment of carcinoma cells that lead tometastasis and thus the higher invasivity of tumours withirregular contours.

5 Conclusions

In this work we have presented a continuum model fordescribing the avascular growth of a multicellular tumourspheroid, comprising a fixed necrotic core surrounded bya region of proliferative cells, guided by the uptake of adiffusing nutrient. The proposed model encapsulates thediffusion of a chemical species from the vasculature of thehealthy region and the tumour cell response to nutrients,via their proliferation and their chemotactic migration in-side the extracellular space. The proposed model differsfrom existing approaches2,49,50 since it considers a growththough a rigid, porous surrounding material. Moreover,the MCTS expansion is guided not only by cell prolifer-ation as in2,49,50, but also by the chemotactic motion of


Varying σ

Rout=50, RT=10, RN=5, β= 1

Varying β

Rout=50, RT=10, RN=5, σ=0.01

Varying Rout

RT=10, RN=5, σ=0.01, β=1

Varying sizes

RT/RN=2, Rout/RT=5, β= 1, σ=0.01

σ=0.01 σ=0.1 σ=1 σ=10 β=0.1 β=1 β=5 β=10

Rout=25 Rout=50 Rout=250 Rout=10 Rout=25 Rout=50 Rout=100 Rout=500

(a) (b)

(c) (d)

Figure 4. Dispersion diagrams for different values of the model parameters (a) Rout keeping q = Rout/RN constant, (b) Routkeeping RN constant, (c) σ and (d) β. The graphs report the value of the time growth rate of the perturbation, λ, with respectto the values of `. In (a) the shapes of the tumour corresponding to the characteristic mode (` = 1, 2, 3, 6) is reported.

cells, through a non-convective mass flux term. Differentlyfrom2,50, that assumed a Gibbs-Thompson relation62 onthe moving boundary for the chemical potential, we con-sidered a mechanical effect in term of a surface tension atthe MCTS outer boundary, leading to the Young-Laplaceequation at the interface.The proposed model is governed by five dimensionless pa-rameters: two of them, β and σ are called motility pa-rameters and representing the mechano-biology cues, theother three are denoted size parameters and are relatedto the typical sizes of the domains with respect to the dif-fusive length. The analytic results predicted the existenceof a quasi-stationary radially-symmetric tumour configu-ration that is always linearly unstable to asymmetric per-turbations involving spherical harmonics Y m` (θ, φ), with

the range of the unstable modes depending on the dimen-sion of the domain with respect to the diffusive lengthand on the motility parameter β, related to the chemo-tactic growth of the tumour. We remark that, whilst aMCTS growing inside an infinite homogeneous domain ismarginally stable, i.e. λ = 0 for ` = 111,15,50, the proposedmodel is always linearly unstable, since translational sym-metry is broken by considering a finite dimension of thesurrounding media. Furthermore, differently from existingworks2,8,9,14,15,17, the perturbation analysis is conductedhere without neglecting the diffusion timescale in the un-stable growth rate.The analysis of the perturbed field also pointed out apossible mechanism that could lead to the detachment ofmetastasis from the primary tumour mass, based on the


Rout=50, RN=1, β= 1, σ=0.01 Rout=10, RN=1, β= 1, σ=0.01

=2 =4 =6 =8 =1

RTRT

RT

=2 =4 =6 =8 =10 =1

RT

Figure 5. Evolution of the time growth rate of the perturbation λ with respect to the dimensionless tumour radius, RT , fordifferent values of ` (with ` = 1, 2, 4, 6, 8, 10). (a) For the chosen set of parameters, ` = 1 is the most unstable mode, whateverthe tumour radius is. The deformed shapes corresponding to RT = 2, 5, 8 are reported aside (the gray region represents the outerenvironment). (b) The characteristic mode ` changes for different values of the tumour radius. Aside the dispersion curves, thesection of the tumour perturbed shapes are reported for RT = 2.5, 8, 12.5, 20, 26, 35 to which the corresponding characteristicmodes are ` = 1, 2, 4, 6, 8, 10 respectively.

Quasi-Stationary solution Perturbed solution (Rout=35, = 10, = 6)

Pre

ssure

Vel

oci

ty (

Log s

cale

)

xz-plane

0.08

0.07

0.06

0.04

00.02

0.09

-1.x10-3

-1.x10-5

-1.x10-7

1.x10-6

4.x10-4

1.x10-1

xy-plane yz-plane

0.08

0.06

0.04

0.02

0

1.x10-6

2.5x10-5

4.x10-4

6.3x10-3

1.x10-1

Figure 6. Evolution of the quasi-stationary and perturbed pressure and velocity fields for Rn = 1, R∗T = 35, Rout = 50 for a

perturbation of the kind eλtY 610(θ, ϕ). Since higher changes in the velocity and pressure field occurs only at the interface, we use

a logarithmic scale in the velocity plot in order to show small variations of the perturbed field inside the bulk of the tumour.The perturbed velocity field highlights the existence of negative radial velocities (i.e. radial convergent flow), as pointed outin60,61.

development of higher velocity at the border of the MCTSand a convergent flow inside where protrusions form. Thismechanism could explain why the propensity for asym-metric invasion and the installation of irregular morphol-ogy characterize the growth of aggressive carcinomas invivo. Thus, this approach has the potential to foster ourunderstanding on the process of transition from the be-nign to the aggressive tumour stage and might provide alsosome indications for improving therapeutic treatments. In-deed, more blurred and irregular contours detected in vivo

can be related to more malignant tumour, with respect tosmoother and clearer contours that can be associated tobenign carcinomas.However, the present model considers a really simplifiedgeometry and adopts some simplifications in order to ob-tain a model that can be studied analytically. Thus, fu-ture improvements of the proposed mathematical modelshould focus on the explicit description of quiescent cell re-gion (that in the present model corresponds to the regionof the spheroid in which we have an almost null velocity)


and on tracking the evolution of the inner necrotic core,occurring, for instance, when the nutrients concentrationattains a specific value2,12 (whereas in the present workthe fixed radius RN of the necrotic core is a parameter inthe sensitivity analysis). From the modelling point of view,future studies should consider the effect of solid mechani-cal stresses on the growth dynamics of tumours32,61,63–66.In particular the contribution of the mechanical propertiesof the host tissue to the growth and state of stress of thetumour and to the evolution of necrotic region67 shouldbe analysed.Furthermore, the stability analysis can be enriched by con-sidering the weakly nonlinear interactions of the asym-metric modes, as well as their evolution depending on theorder m of the spherical harmonic perturbation (as donefor example in15 in a simplified case). Finally, numericaltechniques should be developed in order to simulate thefully nonlinear evolution of the morphological transition.In conclusion, this work demonstrates that geometric andmechano-biological cues can determine the occurrence of amorphological transition in growing tumours that can pro-mote invasiveness. The theoretical results push towardsthe developments of further biological experiments for ac-curate measures of the surface tension and interstitial pres-sure within MCTS55,68, as well as growth and mobilityproperties of the tumour cells. Indeed, the integration ofmathematical tools in biological research could be crucialfor estimating the tumour’s ability to invade its host en-vironment.

6 Acknowledgments

This work was funded by the ”Associazione Italiana perla Ricerca sul Cancro” (AIRC) through My First AIRCGrant 17412, partially supported by the ”Start-up Pack-ages and PhD Program” project, co-funded by RegioneLombardia through the ”Fondo per lo sviluppo e la co-esione 2007-2013 -formerly FAS”, and by the ”ProgettoGiovani GNFM 2016”.

References

1. J. Folkman and M. Hochberg, “Self-regulation ofgrowth in three dimensions,” J. Exp. Med., vol. 138,pp. 745–753, 1973.

2. H. Byrne and M. Chaplain, “Free boundary valueproblems associated with the growth and developmentof multicellular spheroids,” European Journal of Ap-plied Mathematics, vol. 6, pp. 639–658, 12 1997.

3. R. Sutherland and R. Durand, “Growth and cellularcharacteristics of multicell spheroids,” in Spheroids inCancer Research (H. Acker, J. Carlsson, R. Durand,and R. M. Sutherland, eds.), vol. 95 of Recent Resultsin Cancer Research, pp. 24–49, Springer Berlin Hei-delberg, 1984.

4. K. Groebe and W. Mueller-Klieser, “Distributionsof oxygen, nutrient and metabolic waste concentra-tions in multicellular spheroids and their dependence

on spheroid parameters.,” Eur. Biophys. J., vol. 19,pp. 169–181, 1991.

5. R. Sutherland, “Cell and environment interactions intumor microregions: the multicell spheroid model.,”Science, vol. 240, no. 4849, pp. 177–184, 1988.

6. J. Folkman, “Tumour angiogenesis.,” Adv. CancerRes., vol. 19, pp. 331–358, 1974.

7. V. R. Muthukarruppan, L. Kubai, and R. Auerbach,“Tumour-induced neovascularisation in the mouseeye.,” J. Natl. Cancer Inst., vol. 69, pp. 699–705, 1982.

8. J. A. Adam, “A simplified mathematical model oftumor growth,” Math. Biosci., vol. 81, pp. 229–244,1986.

9. J. A. Adam, “A mathematical model of tumourgrowth. ii. effects of geometry and spatial uniformityon stability,” Math. Biosci., vol. 86, pp. 183–211, 1987.

10. J. A. Adam and S. A. Maggelakis, “Diffusion regu-lated growth characteristics of a spherical prevascularcarcinoma,” Bull. Mat. Biol., vol. 52, pp. 549 – 582,1990.

11. H. Byrne and M. A. J. Chaplain, “Growth of non-necrotic tumours in the presence and absence of in-hibitors.,” Math. Biosci., vol. 130, pp. 151–181, 1995.

12. H. Byrne and M. A. J. Chaplain, “Growth of necrotictumours in the presence and absence of inhibitors.,”Math. Biosci., vol. 135, pp. 187–216., 1996.

13. M. A. J. Chaplain, Experimental and Theoretical Ad-vances in Biological Pattern Formation, ch. The de-velopment of a spatial pattern in a model for cancergrowth., pp. 45–60. Plenum Press, 1993.

14. H. P. Greenspan, “Models for the growth of a solidtumour by diffusion.,” Stud. Appl. Math., vol. 52,pp. 317–340, 1972.

15. H. Byrne, “A weakly nonlinear analysis of a modelof avascular solid tumour growth,” J. Math. Biol.,vol. 39, no. 1, pp. 59 – 89, 1999.

16. S. A. Maggelakis and J. A. Adam, “Mathematicalmodel of prevascular growth of a spherical carci-noma.,” Math. Comput. Modelling, vol. 13, pp. 23–38,1990.

17. D. L. S. McElwain and L. E. Morris, “Apoptosis asa volume loss mechanism in mathematical models ofsolid tumour growth,” Math. Biosci., vol. 39, pp. 147–157, 1978.

18. D. Ambrosi and L. Preziosi, “On the closure ofmass balance models for tumor growth,” Mathemati-cal Models and Methods in Applied Sciences, vol. 12,no. 5, pp. 737–754, 2002.

19. R. P. Araujo and D. L. S. McElwain, “A history ofthe study of solid tumour growth: The contributionof mathematical modeling,” Bull. Math. Biol., vol. 66,pp. 1039–1091, 2004.

20. J. S. Lowengrub, H. B. Frieboes, F. Jin, Y. L. Chuang,X. Li, P. Macklin, S. M. Wise, and V. Cristini, “Non-linear modelling of cancer: Bridging the gap betweencells and tumours,” Nonlinearity, vol. 23, pp. R1–R91,2010.

21. D. McElwain and G. Pettet, “Cell migration in mul-ticell spheroids: Swimming against the tide,” Bulletin


of Mathematical Biology, vol. 55, no. 3, pp. 655–674,1993.

22. C. Chen, H. Byrne, and J. King, “The influence ofgrowth-induced stress from the surrounding mediumon the development of multicell spheroids.,” J. Math-emat. Biol., vol. 43, pp. 191–220, 2001.

23. K. A. Landman and C. P. Please, “Tumour dynam-ics and necrosis: surface tension and stability.,” Math.Med. Biol., vol. 18, no. 2, pp. 131–158, 2001.

24. M. Steinberg, “Reconstruction of tissues by dissoci-ated cells. some morphogenetic tissue movements andthe sorting out of embryonic cells may have a commonexplanation.,” Science, vol. 141, no. 3579, pp. 401–408, 1963.

25. R. Foty, G. Forgacs, C. Pflegerand, and M. Stein-berg, “Liquid properties of embryonic tissues: Mea-surement of interfacial tensions,” Phys. Rev. Lett.,vol. 72, no. 14, pp. 2298–2301, 1994.

26. G. Forgacs, R. Foty, Y. Shafrir, and M. Steinberg,“Viscoelastic properties of living embryonic tissues:a quantitative study,” Biophys. J., vol. 74, no. 5,pp. 2227–2234, 1998.

27. G. Vitale and L. Preziosi, “A multhiphase modelof tumor and tissue growth including cell adhesionand plastic reorganization,” Mathematical Models andMethods in Applied Sciences, vol. 21, no. 9, pp. 1901–1932, 2011.

28. M. A. J. Chaplain and B. D. Sleeman, “Modelling thegrowth of solid tumours and incorporating a methodfor their classification using nonlinear elasticity the-ory.,” J. Math. Biol., vol. 31, pp. 431–473, 1993.

29. R. Skalak, S. Zargaryan, R. K. Jain, P. A. Netti, andA. Hoger, “Compatability and the genesis of residualstress by volumetric growth.,” J. Math. Biol., vol. 34,pp. 889–914, 1996.

30. D. Ambrosi and F. Mollica, “The role of stress in thegrowth of a multicell spheroid,” J. Math. Biol., vol. 48,no. 5, pp. 477–499, 2004.

31. T. Roose, P. A. Netti, L. L. Munn, Y. Boucher, andR. K. Jain, “Solid stress generated by spheroid growthestimated using a linear poroelasticity model,” Mi-crovascular Research, vol. 66, no. 3, pp. 204–212, 2003.

32. C. Voutouri, F. Mpekris, P. Papageorgis, A. Odysseos,and T. Stylianopoulos, “Role of constitutive behaviorand tumor-host mechanical interactions in the state ofstress and growth of solid tumors,” Phys. Rev. Lett.,vol. 107, p. e104717, 2011.

33. R. Vandiver and A. Goriely, “Morpho-elastodynamics:The long-time dynamics of elastic growth.,” J. Biol.Dyn., vol. 3, pp. 180–195, 2009.

34. T. Stylianopoulos, J. D. Martin, V. P. Chauhan, S. R.Jain, B. Diop-Frimpong, N. Bardeesy, B. L. Smith,C. R. Ferrone, F. J. Hornicek, Y. Boucher, L. L. Munn,and R. K. Jain, “Causes, consequences, and remediesfor growth-induced solid stress in murine and humantumors,” PNAS, vol. 109, no. 38, pp. 15101–15108,2012.

35. D. Ambrosi and L. Preziosi, “Cell adhesion mech-anisms and stress relaxation in the mechanics of

tumours.,” Biomechan. Model Mechanobiol., vol. 8,pp. 397–413, 2009.

36. D. Ambrosi, L. Preziosi, and G. Vitale, “The inter-play between stress and growth in solid tumors,” Me-chanics Research Communications, vol. 42, pp. 87–91,2012. Recent Advances in the Biomechanics of Growthand Remodeling.

37. C. Giverso, M. Scianna, and A. Grillo, “Growing avas-cular tumours as elasto-plastic bodies by thetheory ofevolving natural configurations,” Mechanics ResearchCommunications, vol. 68, pp. 31–39, 2015.

38. B. Aigouy, R. Farhadifar, D. Staple, A. Sagner,J. Roper, F. Julicher, and E. S., “Cell flow reorientsthe axis of planar polarity in the wing epithelium ofdrosophila,” Cell, vol. 142, no. 5, pp. 773–786, 2010.

39. T. Vasilica Stirbat, S. Tlili, T. Houver, J.-P. Rieu,C. Barentin, and H. Delanoe-Ayari, “Multicellular ag-gregates: a model system for tissue rheology,” The Eu-ropean Physical Journal E, vol. 36, no. 8, pp. 1–14,2013.

40. J. Ranft, M. Basan, J. Elgeti, J. J.F., J. Prost, andJ. F., “Fluidization of tissues by cell division andapoptosis.,” Proc. Natl. Acad. Sci. U.S.A., vol. 107,no. 49, pp. 20863–20868, 2010.

41. J. J. Casciari, S. V. Sotirchos, and R. M. Sutherland,“Mathematical modelling of microenvironment andgrowth in emt6/r0 multicellular tumour spheroids.,”Cell Prolif., vol. 25, pp. 1–22, 1992.

42. M. Marusic, Z. Bajzer, J. P. Freyer, and S. Vuk-Pavlovic, “Analysis of growth of multicellular tu-mour spheroids by mathematical models.,” Cell Pro-lif., vol. 27, pp. 73–94, 1994.

43. R. Muir, Muir’s Textbook of Pathology. CRC Press,15th edn ed., 2012.

44. S. S. Cross and D. W. K. Cotton, “The fractal di-mension may be a useful morphometric discriminantin histopathology,” J. Pathol., vol. 166, pp. 409–411,1992.

45. S. S. Cross, J. P. Bury, P. B. Silcocks, T. J. Stephen-son, and D. W. K. Cotton, “Fractal geometric analysisof colorectal polyps.,” J. Pathol., vol. 172, pp. 317–323, 1994.

46. G. Landini and J. W. Rippin, “How important is tu-mour shape?,” J. Pathol., vol. 179, pp. 210–17, 1996.

47. R. Sutherland and R. E. Durand, “Radiation responseof multicell spheroids: An in vitro tumour model,”Intemat. J. Radiat. Biol., vol. 23, pp. 235–246, 1973.

48. M. Tubiana, “The kinetics of tumour cell proliferationand radiotherapy,” Br. J. Radiol., vol. 44, pp. 325–347, 1971.

49. H. P. Greenspan, “On the growth and stability of cellcultures and solid tumours,” J. Theor. Biol., vol. 56,pp. 229–242, 1976.

50. H. Byrne and M. Chaplain, “Modelling the role ofcell-cell adhesion in the growth and development ofcarcinomas,” Mathematical and Computer Modelling,vol. 24, no. 12, pp. 1–17, 1996.

51. M. Espinosa, G. Ceballos-Cancino, R. Callaghan,V. Maldonado, N. Patino, V. Ruız, and J. Melendez-


Zajgla, “Survivin isoform delta ex3 regulates tumorspheroid formation,” Cancer Letters, vol. 318, no. 1,pp. 61–67, 2012.

52. K. J. Painter, “Continuous models for cell migrationin tissues and applications to cell sorting via differ-ential chemotaxis,” Bulletin of Mathematical Biology,vol. 71, no. 5, pp. 1117–1147, 2009.

53. E. T. Roussos, J. S. Condeelis, and A. Patsialou,“Chemotaxis in cancer,” Nat Rev Cancer., vol. 11,no. 8, pp. 573–587, 2011.

54. E. Keller and L. Segel, “Model for chemotaxis,” J.Theor. Biol., vol. 30, pp. 225–234, 1971.

55. A. Fathi-Azarbayjani and A. Jouyban, “Surface ten-sion in human pathophysiology and its application asa medical diagnostic tool,” Bioimpacts, vol. 5, no. 1,pp. 29–44, 2015.

56. P. Ciarletta, “Free boundary morphogenesis in livingmatter,” Eur. Biophys. J., vol. 41, pp. 681–686, 2012.

57. C. Giverso, M. Verani, and P. Ciarletta, “Branchinginstability in expanding bacterial colonies,” Journalof The Royal Society Interface, vol. 12, no. 104, 2015.

58. C. Giverso, M. Verani, and P. Ciarletta, “Emerg-ing morphologies in round bacterial colonies: compar-ing volumetric versus chemotactic expansion,” Biome-chanics and Modeling in Mechanobiology, pp. 1–19,2015.

59. A. Nayfeh, Perturbation Methods. John Wiley andSons, 2000.

60. M. Dorie, R. Kallman, D. Rapacchietta,D. Van Antwerp, and Y. Huang, “Migration andinternalization of cells and polystyrene microspherein tumor cell spheroids.,” Exp. Cell. Res., vol. 141,no. 1, pp. 201–209, 1982.

61. M. Delarue, F. Montel, O. Caen, J. Elgeti, J.-M.Siaugue, D. Vignjevic, J. Prost, J.-F. m. c. Joanny,and G. Cappello, “Mechanical control of cell flow inmulticellular spheroids,” Phys. Rev. Lett., vol. 110,p. 138103, Mar 2013.

62. J. Langer, “Instabilities and pattern formation in crys-tal growth,” Rev. Mod. Phys., vol. 52, no. 1, pp. 1–30,1980.

63. G. Cheng, J. Tse, R. Jain, and L. Munn, “Micro-environmental mechanical stress controls tumorspheroid size and morphology by suppressing prolifer-ation and inducing apoptosis in cancer cells.,” PLoSONE, vol. 4, no. 2, p. e4632, 2009.

64. G. Helmlinger, P. A. Netti, H. C. Lichtenbeld,R. J. Melder, and R. K. Jain, “Solid stress inhibitsthe growth of multicellular tumor spheroids,” NatBiotech, vol. 15, no. 8, pp. 778–783, 1997.

65. F. Montel, M. Delarue, J. Elgeti, L. Malaquin,M. Basan, T. Risler, B. Cabane, D. Vignjevic,J. Prost, G. Cappello, and J.-F. Joanny, “Stress clampexperiments on multicellular tumor spheroids,” PLoSONE, vol. 9, no. 8, p. 188102, 2014.

66. K. Alessandri, B. R. Sarangi, V. V. Gurchenkov,B. Sinha, T. R. Kieβling, L. Fetler, F. Rico, S. Scheur-ing, C. Lamaze, A. Simon, S. Geraldo, D. Vign-jevic, H. Domejean, L. Rolland, A. Funfak, J. Bi-

bette, N. Bremond, and P. Nassoy, “Cellular capsulesas a tool for multicellular spheroid production andfor investigating the mechanics of tumor progressionin vitro,” PNAS, vol. 110, no. 37, pp. 14843–14848,2013.

67. R. K. Jain, J. D. Martin, and T. Stylianopoulos, “Therole of mechanical forces in tumor growth and ther-apy,” Annual review of biomedical engineering, vol. 16,pp. 321–346, 2014.

68. R. K. Jain, “Barriers to drug delivery in solid tu-mours.,” Scientific American, vol. 271, no. 1, pp. 58–65, 1994.

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