Constitutive model for the rheology of biological tissue Suzanne M. Fielding1James O. Cochran1Junxiang Huang2Dapeng Bi2and M. Cristina Marchetti3 1Department of Physics Durham University Science Laboratories South Road Durham DH1 3LE UK

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Constitutive model for the rheology of biological tissue

Suzanne M. Fielding,1James O. Cochran,1Junxiang Huang,2Dapeng Bi,2and M. Cristina Marchetti3

1Department of Physics, Durham University, Science Laboratories, South Road, Durham DH1 3LE, UK

2Department of Physics, Northeastern University, MA 02115, USA

3Department of Physics, University of California, Santa Barbara, CA, USA

The rheology of biological tissue is key to processes such as embryo development, wound healing

and cancer metastasis. Vertex models of conﬂuent tissue monolayers have uncovered a spontaneous

liquid-solid transition tuned by cell shape; and a shear-induced solidiﬁcation transition of an initially

liquid-like tissue. Alongside this jamming/unjamming behaviour, biological tissue also displays an

inherent viscoelasticity, with a slow time and rate dependent mechanics. With this motivation, we

combine simulations and continuum theory to examine the rheology of the vertex model in nonlinear

shear across a full range of shear rates from quastistatic to fast, elucidating its nonlinear stress-strain

curves after the inception of shear of ﬁnite rate, and its steady state ﬂow curves of stress as a function

of strain rate. We formulate a rheological constitutive model that couples cell shape to ﬂow and

captures both the tissue solid-liquid transition and its rich linear and nonlinear rheology.

The rheology of biological tissue is crucial to processes

such as morphogenesis, wound healing and cancer metas-

tasis. On short timescales, tissues withstand stress in a

solid-like way. On longer timescales, they reshape via

internally active processes such as cell shape change, re-

arrangement, division and death [1,2]. Tissues are thus

viscoelastic [3]. Power law stress relaxation [4,5] and

slow oscillatory cell displacements [6] after straining un-

derline their rate dependent mechanics. Tissues further-

more undergo spontaneous solid-liquid transitions [7–11]

driven by both active processes, such as ﬂuctuations of

cell-edge tensions, motility and alignment, and geometric

constraints [12], with important implications for morpho-

genesis and cancer progression. Nonlinear rheological re-

sponse to tensile stretching includes stiﬀening [13] or ﬂu-

idization [14] of single cells, and stiﬀening then rupture

of tissue monolayers [15]. Internal activity can likewise

induce nonlinear phenomena such as superelasticity [16]

and fracture [17].

Understanding tissue rheology theoretically is thus of

major importance. Well studied vertex and Voronoi mod-

els [9,18,19] of conﬂuent tissue, with no gaps between

cells, represent a 2D tissue monolayer as a tiling of polyg-

onal cells. They capture a density-independent solid-

liquid transition tuned by a parameter characterising the

target cell shape, which in turn embodies the competition

between cortex contractility and cell-cell adhesion [7–9].

Vertex models have also been used to study the linear

mechanics of tissues [20–22], and their response to non-

linear stretch [23] and shear [24–27]. Recently, vertex

model simulations of a tissue that is ﬂuid-like in zero

shear demonstrated a shear-induced rigidity transition

above a critical strain, applied quasistatically [27].

While vertex models and other mesoscopic models have

played an important role in advancing our understand-

ing of tissue mechanics, it is also helpful to develop

coarse grained continuum rheological constitutive mod-

els. Early work formulated a continuum model that cou-

ples cell shape and cell motility, capturing some of the

glassy dynamics of tissue [28]. Inspired by early hydro-

dynamic theories of active ﬂuids and gels [29,30], con-

tinuum constitutive models have been developed to char-

acterize the role of cell shape change, rearrangements,

division and death in morphogenesis [2,25,31–35].

Still lacking, however, is a continuum hydrodynamic

constitutive model capable of describing both the spon-

taneous solid-liquid transition of conﬂuent tissues and its

rheological response to external deformation and ﬂow.

Inspired by mean-ﬁeld theories of cell-shape driven tran-

sitions [22,27,28] and by ﬂuidity models of the rheology

of dense soft suspensions [36], we introduce such a model.

The key new insights of our approach are as follows.

First, we distinguish the role of geometric frustration (en-

coded in the cell perimeter p), from that of T1 topological

rearrangements (encoded in our ﬂuidity variable a). The

former is key to the zero-shear liquid-solid transition and

(when coupled to our orientation tensor σij ) strain stiﬀ-

ening at small to modest imposed strains [27]. The latter

cause the plasticity associated with the stress overshoot

at imposed strains O(1), and the ultimate steady ﬂowing

state. Second, in modeling the geometric frustration, we

distinguish a tensor characterizing individual cell shape

(of which pis the trace), and a tensor characterizing the

average cell orientation at the tissue scale [28].

We furthermore submit this new continuum model to

stringent comparison with simulations across a full range

of shear rates from quasi-static to fast. We demonstrate

our continuum model to capture both the zero-shear

solid-liquid transition and strain stiﬀening transitions re-

ported in Ref. [27], the full nonlinear stress vs. strain be-

havior after the inception of shear, and the steady state

ﬂow curves of stress vs. shear rate.

Vertex model simulations — The vertex model [18,19]

represents the tightly packed conﬂuent cells of a 2D tissue

monolayer as c= 1 ···Ncpolygons that tile the plane.

Each cell is deﬁned by the location of its nc= 1 ···νc

vertices, with any two neighbouring vertices αand βcon-

nected by an edge of length ℓαβ . The elastic energy of

the tissue is controlled by the interplay of pressure within

each cell and tension along the cell edges. Assuming the

cell-edge tension per unit length is uniform across the

arXiv:2210.02893v2 [cond-mat.soft] 29 Sep 2023

tissue, the energy can be written as

E=1

cκA(Ac−Ac0)2+κP(Pc−Pc0)2,(1)

where each cell experiences an energy cost for deviation

of its area Acand perimeter Pcfrom target values Ac0

and Pc0, with area and perimeter stiﬀness κAand κP.

The ﬁrst term on the RHS models 3D cell volume incom-

pressibility via an eﬀective 2D area elasticity [19,37].

The second describes the competition between cell corti-

cal contractility and adhesion between neighbouring cells

in controlling cell-edge tension and perimeter [7,19,37].

We denote by ⃗

Fn=−δE

δ⃗xnthe total force on the nth

vertex of the tiling at position ⃗xndue to interactions with

all other vertices. In an applied shear of rate ˙γ, with ﬂow

direction xand shear gradient y, we assume over-damped

dynamics with drag ζ,d⃗xn

dt =ζ−1⃗

Fn+ ˙γyn⃗

ˆx, with Lees-

Edwards periodic boundary conditions. The cells also

undergo T1 topological neighbor exchanges that allow

the tissue to plastically relax stresses [9,38–40].

To focus on amorphous tissue structures, we simulate a

50 : 50 bidisperse tiling of Nc= 4096 cells of target areas

A0= 1,1.4, which sets our length unit. We adjust Pc0for

the two cell populations to maintain the target cell shape

p0=Pc0/√Ac0the same for all cells. We choose units

in which κA= 1 and ζ= 1 and set κp= 1.0 throughout.

We vary p0and the imposed shear rate ˙γ. As an initial

condition, we seed a planar Voronoi tiling then evolve

the above dynamics to steady state in zero shear. At

time t= 0, we switch on shear and measure the shear

stress Σij (t) = 1

NPN

n=1 Fnixnj , where the sum is over

all Nvertices in the tiling, and the mean cell perimeter

p(t) = 1

NcPNc

c=1 pc. Denoting by ⃗

tncthe unit vector along

the edge of length lncbetween the ncth and (nc+ 1)th

vertices of cell c, we deﬁne a single-cell shape tensor σc

ij =

νcPνc

n=1 lnctnc

itnc

j, where the sum is over the νcvertices

of the c-th cell, and the tissue-scale averaged orientation

tensor σij =1

NcPNc

c=1 σc

ij . We use the same notation

Σij , σij , p for the counterpart coarse-grained quantities

in our constitutive model below.

In the absence of external stress, the vertex model ex-

hibits a liquid solid transition as a function of the target

shape p0[7,41]. For p0< p∗

0the energy barriers to T1

transitions are ﬁnite and the system is a solid with a

ﬁnite zero-frequency linear shear modulus. At the criti-

cal value p∗

0, the mean energy barrier for T1 transitions

vanishes, giving liquid response for p0> p∗

0. For our

bidisperse tiling, p∗

0= 3.85. For monodisperse disordered

polygons p∗

0≃3.81, a value close to that of a regular

pentagon [7]. This value is renormalized by motility [9]

and by cell alignment with local spontaneous shear [42].

It was recently realized that this transition has a geo-

metric origin associated with the underconstrained na-

ture of the energy in Eq. 1[20,22,43]. For regular

hexagons the transition occurs at the isoperimetric value

piso =p8√3≃3.722. Below this value it is not possible

to satisfy both target area and perimeter and the ground

state has p=p∗

0and ﬁnite energy. This is the solid or

incompatible state. For p0> piso there is a family of

zero energy area and perimeter preserving ground states,

with p=p0. The system can accommodate an exter-

nally applied linear shear by adjusting its shape within

this degenerate manifold [22]. The compatible system is

therefore a liquid with zero shear modulus, although it

stiﬀens and acquires rigidity at ﬁnite strains [27].

Constitutive model — We now construct a continuum

model that accounts for the mean-ﬁeld liquid-solid transi-

tion, and also captures the key rheological features of the

vertex model: (i) reversibility of linear response to small

strains, (ii) strain stiﬀening at intermediate strains, (iii)

plastic relaxation at larger strains, due to T1 cell rear-

rangements, and (iv) a yield stress in the steady state

ﬂow curve Σ( ˙γ), as obtained in Ref. [27]. Although our

model below is cast in frame invariant form, capable of

addressing any ﬂow, we focus on response to simple shear,

to compare with our vertex model simulations.

We assume dynamics of the cell perimeter governed by:

˙p+vk∇kp= ˙γ−1

τp

(p−p0)(p−p∗

0−ασij σij ),(2)

with αand τpconstants and invariant strain rate ˙γ=

p2Dij Dij . In the absence of shear, prelaxes on a

timescale τpto a steady state that displays a transcriti-

cal bifurcation as a function of the target cell perimeter

p0, with p=p∗

0in the solid phase p0< p∗

0and p=p0

in the liquid phase p0> p∗

0, capturing the liquid-solid

transition [7]. The same transcritical structure emerges

by writing exact equations for the relaxation of a single

cell modeled as a regular n−sided polygon according to

the vertex model dynamics prescribed above.

In shear, the perimeter is advected by ﬂow and

stretched by the shear rate ˙γ. In addition, the coupling

ασij σij captures a key intuition of our approach: that a

shear-induced global cell orientation σij provides an ef-

fective mean ﬁeld that distorts the individual cell’s shape

paway from its zero-shear value. As a result, in the solid

phase pincreases relative to its zero shear value p=p∗

from the outset of straining. In the liquid phase, pin-

creases relative to its zero shear value p=p0only after a

critical strain amplitude γc, capturing the strain-induced

stiﬀening transition [27]. The behavior introduced by the

coupling of single-cell shape, as quantiﬁed by the mean

perimeter p, to the tissue-scale cell shape σij is analogous

to the inﬂuence of cell alignment due to internally gener-

ated stresses in Drosophila germband extension [42]. In-

deed, the form of coupling of pto σij in Eqn. 2is justiﬁed

both by experiment [42] and mean ﬁeld theory [22,27].

The cell orientation tensor is taken to obey an evolu-

tion equation of the widely used Maxwellian form,

˙σij +vk∇kσij =σikKkj +Kkiσkj + 2Dij −aσij ,(3)

where Kij =∂jviis the strain rate tensor and Dij =

2(Kij +Kji). The last term in Eq. 3describes plastic

relaxation. It vanishes in linear response (small strains),

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ConstitutivemodelfortherheologyofbiologicaltissueSuzanneM.Fielding,1JamesO.Cochran,1JunxiangHuang,2DapengBi,2andM.CristinaMarchetti31DepartmentofPhysics,DurhamUniversity,ScienceLaboratories,SouthRoad,DurhamDH13LE,UK2DepartmentofPhysics,NortheasternUniversity,MA02115,USA3DepartmentofPhysics,Universit...

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Constitutive model for the rheology of biological tissue Suzanne M. Fielding1James O. Cochran1Junxiang Huang2Dapeng Bi2and M. Cristina Marchetti3 1Department of Physics Durham University Science Laboratories South Road Durham DH1 3LE UK

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