Active viscoelastic models for cell and tissue mechanics

Abstract

Living cells are out of equilibrium active materials. Cell-generated forces are transmitted across the cytoskeleton network and to the extracellular environment. These active force interactions shape cellular mechanical behaviour, trigger mechano-sensing, regulate cell adaptation to the microenvironment and can affect disease outcomes. In recent years, the mechanobiology community has witnessed the emergence of many experimental and theoretical approaches to study cells as mechanically active materials. In this review, we highlight recent advancements in incorporating active characteristics of cellular behaviour at different length scales into classic viscoelastic models by either adding an active tension-generating element or adjusting the resting length of an elastic element in the model. Summarizing the two groups of approaches, we will review the formulation and application of these models to understand cellular adaptation mechanisms in response to various types of mechanical stimuli, such as the effect of extracellular matrix properties and external loadings or deformations.

Full text

Active viscoelastic models for
cell and tissuemechanics
Bahareh Tajvidi Safa1, Changjin Huang2, Alexandre
Kabla3 and Ruiguo Yang1,4,5
1Department of Mechanical and Materials Engineering, University of Nebraska-Lincoln,
Lincoln, NE 68588, USA
2School of Mechanical & Aerospace Engineering, Nanyang Technological University, Singapore
639798, Singapore
3Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, UK
4Department of Biomedical Engineering, and 5Institute for Quantitative Health Science and
Engineering (IQ), Michigan State University, East Lansing, MI 48824, USA
BTS,0000-0003-1280-5319; AK,0000-0002-0280-3531;
RY,0000-0002-1361-4277
Living cells are out of equilibrium active materials. Cell-
generated forces are transmied across the cytoskeleton
network and to the extracellular environment. These active
force interactions shape cellular mechanical behaviour,
trigger mechano-sensing, regulate cell adaptation to the
microenvironment and can aect disease outcomes. In recent
years, the mechanobiology community has witnessed the
emergence of many experimental and theoretical approaches
to study cells as mechanically active materials. In this review,
we highlight recent advancements in incorporating active
characteristics of cellular behaviour at dierent length scales
into classic viscoelastic models by either adding an active
tension-generating element or adjusting the resting length of
an elastic element in the model. Summarizing the two groups
of approaches, we will review the formulation and application
of these models to understand cellular adaptation mechanisms
in response to various types of mechanical stimuli, such as the
eect of extracellular matrix properties and external loadings
or deformations.
1. Introduction
Living cells, their surrounding extracellular matrices (ECM) and
tissues as a whole exhibit viscoelastic properties, that is, having
both an elastic and a viscous behaviour. Mechanical tests are
used to characterize the mechanical properties of cells and tissues
and help us understand and predict cellular behaviour in healthy
or pathological conditions at dierent time scales [1–4]. For
example, experimental studies on cell monolayers devoid of
substrate have reported a viscoelastic solid-like behaviour under
© 2024 The Authors. Published by the Royal Society under the terms of the Creative
Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits
unrestricted use, provided the original author and source are credited.
Review
Cite this article: Tajvidi Safa B, Huang C, Kabla
A, Yang R. 2024 Active viscoelastic models for cell
and tissue mechanics. R. Soc. Open Sci. 11:
231074.
https://doi.org/10.1098/rsos.231074
Received: 24 July 2023
Accepted: 25 February 2024
Subject Category:
Physics and biophysics
Subject Areas:
biomechanics, biomaterials
Keywords:
active model, viscoelasticity, cell modeling, cell
mechanics, tissue mechanics
Authors for correspondence:
Alexandre Kabla
e-mail: ajk61@cam.ac.uk
Ruiguo Yang
e-mail: ryang6@unl.edu

constant strains [5]. In other words, on the time scale of seconds, cell monolayers behave like a
viscous uid and dissipate stress, then reach a plateau in stress on the time scale of minutes, which
is a characteristic of elastic solid materials [5]. On the other hand, cell aggregates exhibit solid-like
behaviour on short time scales, for example, in a few seconds, and uid-like behaviour at time scales
of the order of minutes to hours [4,6,7]. In this case, the behaviour of cell aggregates is similar to
a viscoelastic uid material. Additionally, the mechanisms governing the viscous-like behaviour can
also dier depending on the time scale of load application [8]. For example, on the time scale of tens
of seconds to minutes, stress dissipation occurs owing to the turnover of actin laments and reorgan-
ization of the actomyosin network [4,8,9]. On the time scale of minutes to hours, stress dissipation
mechanisms at the cellular scale, such as oriented cell division and cell rearrangements, can start to
inuence the response to mechanical loadings [8,10–12]. Transitions between uid-like and solid-like
behaviours in living cells can also occur in response to mechanical stimuli. For instance, short-term
uidization has been reported immediately upon strain application, which is usually followed by
stiening [13–16].
The experimental results are often analysed by mathematical models to capture the important
features of the material response, presented in terms of model parameters. These parameters can
then be used for classication, comparison and prediction of the mechanical behaviour of cells and
tissues subjected to other loading conditions [17]. By relating the model parameters to the underlying
biological processes on the molecular and cellular scale, the physical meaning of the parameters can
sometimes be assigned. Since tissues exhibit time-dependent mechanical behaviours [5,18], a common
modelling approach is to consider the cells and tissues as a viscoelastic continuum and to describe their
mechanical response from quantitative mechanical interrogations, often stress–strain relationships, in
terms of a combination of stiness and viscosity, or elastic and loss moduli under dynamic loadings.
These extracted mechanical properties have long been regarded as disease biomarkers [19,20]. For
instance, metastatic cancerous cells exhibit lower stiness than benign cells [21].
Perhaps more importantly, cell mechanics is not only a by-product of the underlying molecular
structure but also a means for cells to actively adapt to environmental cues, in service to a preferred
cellular function, such as cell migration in wound healing [22] and tissue morphogenesis in develop-
ment [23]. Being viscoelastic in nature, cells can dissipate the imposed stress owing to external strains.
However, sometimes this passive response is insucient to maintain the mechanical integrity of the
cell. For instance, cells need to use eective stress relaxation mechanisms such as actin polymerization
to prevent tissue fracture. In other instances, under rapid strain applications, cells may need to stien
or actively pull back to make further deformation dicult, thus preventing further damage to their
cytoskeleton. From the adaptation perspective, the evolving elastic and viscous properties owing to
the active adjustments of cytoskeleton tension can be considered as a way to facilitate the response
and adaptation to external stimuli. Probing active cellular behaviour through the lens of mechanics
is particularly intriguing because it oers a window from which the adaptation can be quantitatively
examined up close with dened mechanical stimuli.
In the cell cortex as well as at the cell population scale, living maer not only responds to exter-
nal forces or deformation as any traditional material would but also often exhibits force-generating
mechanisms emerging from actin polymerization, adhesion dynamics and actomyosin contractility.
This process is referred to as being ‘active’ in this review. Active maer theory is another continuum-
based model used to describe the dynamics of cell cytoskeleton and cell monolayers (reviewed in
[9,24–26]). This type of model is based on the theory of liquid crystals and can be employed to describe
the mechanics of the actomyosin cortex for time scales longer than the turnover time of actin laments
[27]. However, these models rarely capture the way such materials respond to external deformations or
stresses; it remains challenging to capture, both experimentally and theoretically, the impact of active
processes on the mechanical state of living maer.
Experimentally, a common approach is to subject the cellular materials to tension or compression
and then assess their response using stress–strain curves. At the tissue scale, mechanical testing
machines can be used to investigate the macroscopic behaviour of tissue samples (gure 1a) [28,29].
A similar technique has been developed to study the response of cell monolayers when subjected to
compression or tension (gure 1b) [30]. In addition, to examine the microscale characteristics of living
tissues, atomic force microscopy (AFM) and nanoindentation techniques can be employed, where a
probe tip is used to apply forces in the pico-nano Newton range [31,32].
The AFM technique can also be used to study the active and passive behaviour of individual
cells [33–35] and can be adapted for cell pair studies in single-cell force spectroscopy (SCFS), where
a single cell adhered to the cantilever beam serves as the probe tip (gure 1c) [36]. Furthermore,
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various methods are developed to study the response of single cells and cell doublets in tension and
compression tests. For instance, in microplate assays, cells are compressed or stretched between a xed
and a deformable plate [37,38], microbead assays use microbeads to apply force via magnetic [39,40]
or optical tweezers [41–43] (gure 1d), and the micropipee aspiration method involves subjecting
cells to negative pressures and measuring their deformation [44–46] (gure 1d). This method can
also be used for cell pair studies by bringing two cells into contact via micropipees, that is, dual
pipee aspiration (DPA) (gure 1c) [47], or it can be combined with other techniques such as optical
tweezers [48]. Recently, a new micromanipulator device has been introduced that can directly measure
forces in cell pairs under controlled loading conditions, leading to advancements in the precision of
the interrogation of cell pair mechanics (gure 1c) [49]. Dierent techniques have varied ranges of
resolution and loading rates; thus, one has to consider the application requirements when choosing an
experimental technique. For instance, AFM techniques benet from higher spatial resolution and force
sensitivity compared with methods employing micropipee aspiration [50,51].
These experimental techniques are employed to probe cellular response at various temporal and
spatial scales. In addition to measuring the global mechanical behaviour of cells and tissues, these
techniques can also be employed to examine the mechanical properties of specic components of
(a)
(c)
(d)
(b)Tissue
Cell pair
Single Cell
Cell layer
Figure 1. Experimental techniques used to probe cell mechanics in various spatial scales. (a) Mechanical testing machine used to
study the response of tissue samples in compression and tension tests. (b) The recent technique developed for investigating the
mechanical response of cell layers to uniaxial stretch and compression. (c) In cell pair studies, mechanical stimulus is applied through
various methods such as a single cell adhered to a tipless atomic force microscopy (AFM) cantilever in single-cell force spectroscopy
(SCFS), two micropipettes in dual pipette aspiration (DPA) and a microstructure fabricated using two-photon polymerization technique
in single-cell adhesion micro tensile tester (SCATT). (d) Microplate assay is a commonly used method to monitor the force generated
by individual cells when exposed to variations in substrate stiness. Other techniques, such as micropipette aspiration, AFM and optical
and magnetic tweezers, involve applying controlled displacement as the mechanical stimulus and monitoring the forces within cells or
investigating changes in displacement in response to controlled forces.
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cells. For example, micropipee aspiration techniques have been used to investigate the microrheology
of the cell nuclei [52–54]. The monitored stress/strain response of the cellular materials will be the
outcome of both active force-generating and passive mechanisms operating at those scales. From a
modelling perspective, capturing active behaviour requires a mechanism of introducing change to the
otherwise xed viscoelastic models. Researchers over the years have incorporated active adaptation
mechanisms by introducing active empirical mathematical models to link mechanical characteristics to
the underlying biological processes without simulating the details of the underlying chemical signals.
Rheological models for describing the mechanical behaviour of cellularized materials across various
length scales have previously been reviewed [4,15,17,55]. In this article, our primary goal is to focus
on dierent forms of integrating activity into rheological models, emphasizing variations in their
denitions. We here summarized these reported mechanisms into two broad categories. The rst group
integrates a force/stress (force divided by cross-sectional area)-generating element to classic viscoelas-
tic models. This active element can be constant or time-dependent. The second group introduced
mathematical methods that adjust the resting length of an elastic spring or the reference stress-free
shape in two dimensions/three dimensions in classic viscoelastic models. Mathematical models in both
categories are developed in conjunction with specic techniques that probe the active cell response to
various types of mechanical stimuli. In some cases, the active models can be mathematically equiva-
lent, but dierent parameters may oer a beer link with the underlying biology. In this article, we
present a selection of such models, introducing rst the experimental ndings and then focusing on
the formulation and application of each type of active viscoelastic model. We will begin with a short
introduction to the biology of active cell behaviour.
2. Biology of cellular active behaviour
Cell cytoskeleton mainly comprises lamentous proteins that preserve the cell structure, arrange
organelles, and resist, transmit and generate forces [56,57]. These proteins can be categorized into
three groups: microtubules, intermediate laments and actin laments [58]. Active force generation
is realized by actin laments. They are constructed by assembling monomeric actin. Actin-binding
proteins bind to actin laments and form dierent structures, such as the lamellipodium network,
contractile bundles of stress bres and the contractile network of the cell cortex [57,59]. Actin la-
ments are engaged in active processes such as actin treadmilling and force generation by consuming
the energy provided by adenosine triphosphate (ATP) hydrolysis [60–62]. ATP molecules aach to
ATP binding sites on actin monomers, and the ATP-bound actin monomers will be assembled at
the plus/barbed end of the actin laments leading to lament growth [60,63]. ATP molecules will
slowly hydrolyse to adenosine diphosphate (ADP) and the ADP-bound actin monomers will start to
disassemble from the minus/pointed end of the actin laments [60,64]. The process of assembling and
disassembling actin monomers is referred to as actin treadmilling [65]. Actin laments use the energy
from ATP hydrolysis to generate protrusion forces to help cells in spreading and migration [66,67].
Actin treadmilling is also crucial in endocytosis, exocytosis and phagocytosis to engulf large particles
[57].
In addition, ATP hydrolysis provides the energy for contractile stress generation in the actomyosin
network. Myosin motors convert chemical energy from ATP hydrolysis to mechanical energy and slide
actin laments past one another to produce force [68–70]. This process is similar to the shortening of
sarcomeres in muscle cells [37,38,64,70]. The contractile forces generated from this process will then be
transmied to neighbouring cells and the ECM via cell–cell junctions and focal adhesions, respectively
[66,71,72]. At the cell and tissue level, contractile force production controls cellular activities, such as
cell migration [67,73–75], proliferation [76,77], stem cell lineage determination [78,79], tissue regener-
ation [80] and morphogenesis [81,82]. Important for cellular mechanical characterization, contractile
forces also regulate cellular response to substrate stiness and mechanosensing [83]. The active
ATP-dependent processes that are at play in subcellular scales can also regulate cellular response
under various loading conditions at cell and tissue scales [5,59,84,85].
3. Active contractile elements
Rheological models are useful tools to quantitatively analyse the results of mechanical tests. However,
associating the molecular scale origin with the model parameters is challenging, in particular, where
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cell behaviour deviates from the response of traditional passive materials [17]. Thus, adding elements
that represent the active behaviour of cells can equip classic models with the ability to empirically link
the mechanical response to the underlying biological processes.
To represent active cellular dynamics from intracellular contractility, contractile/force-generating
elements (CEs) can be added to standard viscoelastic models. In this section, we will discuss
active CEs that can mimic cell behaviour when they are exposed to dierent types of mechanical
stimuli. We begin with the rst active model, which was presented by Hill to elucidate muscle
contraction and its application in studying mechanosensing in single cells. In the following section,
we will explore an active uid model that considers the transient nature of the actin network to
describe characteristics of the single-cell response to the substrate stiness. In this model frame-
work, tension generation via myosin activity will be denoted by , and regulating mechanisms
of the dynamics of tension build-up will be discussed. Additionally, we will also provide some
examples for both models.
3.1. Muscle contraction dened by the Hill model
Muscles function as intricate biochemical mechanisms that transform chemical energy into mechanical
energy through actomyosin interactions to provide movement in our bodies. Hill proposed the rst
theory to describe muscle contraction back in 1938 [29]. He conducted experiments using the sartorius
muscle of a frog and explained the macro properties of muscles. In these experiments, Hill quantied
both force generation and velocity of length changes in muscles under various loading conditions. The
two endpoints of the curve representing the force–velocity relationship in muscles (gure 2a(i)) were
generated in two extreme cases: zero force (isotonic condition) and zero velocity (isometric condition).
Specically, when force is kept at 0, the muscle can reach its maximum shortening velocity, max.
Similarly, when the muscle is restricted from changing its length, the maximum level of force, that is,
the stall force max, can be generated at the steady-state condition.
Hill proposed an empirical function to describe the correlation between the active force generation,
, and the shortening velocity, , as follows [29]:
(3.1)(+)(+) = (max +)⋅,
where a, b and max +⋅= are constants specic to each muscle type. The Hill model was rst
discovered on frog skeletal muscles. However, later studies have shown that the dimensionless form of
this model with a shape factor =/max =/max ≈0.25 can be used to describe the behaviour of other
muscle types,
(3.2)(+)(+) = (1 + ),
where =
max and =
max [88]. Hill also proposed a phenomenological model to describe muscle
mechanics. The original model consists of a contractile element (CE) that generates force and an elastic
element (SE), as shown in gure 2a. CE is governed by equation (3.2) and SE aects the length and rate
of change in the length of CE during contractions [29,86]. Other forms of this model have also been
introduced by integrating more elastic and viscous elements to account for the viscoelastic properties
of the muscle and its interactions with the connective tissue surrounding muscle bres [89]. The Hill
model has enabled researchers to explore the mechanics of muscles using only a few rheological
parameters. However, the Hill model falls short of elucidating the underlying biological mechanisms
of force generation in muscles [89–91]. Another shortcoming of this model is that it fails to consider
variations in the contractile characteristics of various bre types within muscles and the dependence of
muscle tension on the movement history [89,91]. Modications aimed at enhancing the accuracy of the
Hill-type model predictions are reviewed in [91].
This model is extremely versatile and not only can it be used to describe the behaviour of various
types of muscles, but also it can be adapted and employed to describe force generation in non-muscle
cells and their response to the physical and mechanical properties of their microenvironment [92–94].
Mitrossilis et al. [37] studied the response of a single C2.7 myoblast cell and a 3T3 broblast mounted
between two microplates and showed a time-dependent force generation in single cells in response
to the microplate stiness (gure 2b(iii)). The force–velocity curves for single cells have a similar
shape as the force–velocity curves reported for muscle bres. Moreover, they showed that the reduced
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Hill equation (3.2), with the same shape factor = 0.25 as for muscles, can describe the normalized
shortening speed, =/max versus the normalized force, =/max, as shown in gure 2a(ii). In
these experiments, the maximum force and velocity are measured at innite microplate stiness and
very low stinesses, respectively. Therefore, showing that the coupling between force generation in
cells in contact with their surroundings still follows Hill’s model is quite remarkable, as it implies that
structured muscle actomyosin and cytoskeletal actomyosin exhibit similar behaviours independent of
the network architecture.
In addition, since the same function could explain the experimental results of dierent muscle
types, Huxley suggested that the force–velocity relationship is generic in muscles and proposed a
molecular explanation [95]. This molecular model incorporates the dynamics of the interaction between
actin laments and myosin motors, that is, the number of myosin heads connected to actin laments
and the formation of temporary connections between actin and myosin heads (i.e. cross bridges).
Understanding these dynamics brings valuable insight into the biological processes regulating the
predictions of the Hill model. For instance, cross-bridge-type models have shown that the maximum
Schematic Diagram
of the Model
(a)
(b)
(i)
(i)
(iii)
(ii)
(ii) (iii)
(i) (ii)
(iv)
(c)
Constitutive Equations
The Hill model
Contractile element including the dynamics of cytoskeleton turnover
Time-dependent contractile element
Experimental Results
Figure 2. Models of active contractility relating force/tension generation and the rate of contraction. (a) Model proposed by Hill [29]
to describe muscle contraction [86]. CE serves as a force-generating contractile element and SE is an elastic spring that aects the
dynamics of contraction. (i) Force–velocity curve obtained in Hill’s experiments. (ii) Normalized speed of shortening =/max
and normalized force generation =/max in single cells in response to various substrate stinesses can be represented by the
reduced Hill equation (3.2) [37]. (b) Active uid model proposed to study mechanotransduction in cells [38]. (i) This model accounts
for the transiently cross-linked nature of the actomyosin network and force generation via myosin motor activity. (ii) Equivalent of
the model presented in (i), which has the same constitutive equation. The net eect of two dashpots is similar to one dashpot. (iii)
Evolution of traction force generation over time for low (grey) and high (black) substrate stiness. The two main characteristics of force
generation–time curves, that is, the plateau force  and the rate of force generation / are dependent on the substrate stiness
[37]. (iv) The plateau force has a linear relationship with substrate stinesses when the stiness is below 60 nN μm−1 and for higher
values of stiness,  saturates [37]. (c) Active Maxwell model that assumes a time-dependent increase in cellular contractile force
with the same prole as actin and myosin accumulation in the RhoA-activated areas [87]. (i) Changes in the local intensity of actin
and myosin in the regions of RhoA activation. (ii) The plateauing exponential function used in the model to represent the stress prole
during activation of RhoA. (iii) Experimental (black) and theoretical (red) evolution of strain energy during intervals of RhoA activation
and relaxation.
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speed of muscle shortening happens owing to the rate of myosin aachment and detachment [89,95]
rather than the extent of lament overlap [96,97]. This model has its limitations as well. For example,
this model does not include the eect of power stroke (i.e. a crucial step in the force generation cycle in
muscles where myosin heads pull the actin laments and generate force) [89,91].
Further aempts have been made to incorporate various aspects of molecular biology with the Hill
model. An example involves a model that integrated the dynamics of actin lament slippage during
the process of force generation, allowing the prediction of the relationship between force generation
and stiness of micropillars [98]. This model was further rened to account for the temporal evolution
of the force over time by integrating an internal variable, specically representing the progression of
myosin motor stalling over time [99]. Despite these improvements, this model falls short in explaining
the limiting factors for the maximum force generation and maximum speed of shortening of the cells.
A model that addresses these shortcomings is discussed in the next section.
3.2. Active element that includes the dynamic turnover of the actomyosin network
Étienne et al. developed an active uid model to link the fundamental features of cellular molecular
mechanics with a passive phenomenological model [38]. These features encompass the transient nature
of the actomyosin network, force generation via myosin motor activity and actin polymerization. The
transiently cross-linked actomyosin network is modelled as a Maxwell uid in series with an active
element , mimicking force generation via myosin motors (gure 2b). The constitutive equation of the
network is shown as follows:
(3.3)
˙+−2
˙=.
Here,  is the characteristic time scale of cross-linker unbinding (i.e. elastic-like in short time scales
(<) and viscous over longer periods of time (>)), and  is the elastic modulus of the cell.
 represents the maximum value of contractility or stall force that could be generated in cells,
determined by two factors. First, it is inuenced by the rate at which myosin motors can generate
stress (1/) and contract the cell with an elastic modulus of . Second, it is aected by the rate of
cross-linker unbinding (1/), which counteracts the increase of stress in the system. Consequently,
 is proportional to (/). This model eectively captures the evolution of force generation
in cells leading to the establishment of tension in the steady state. Using a single dashpot (gure
2b(i)), instead of the proposed two (gure 2b(ii)), results in a similar constitutive equation. However,
using two dashpots highlights the loss of force generation in the steady-state condition when the net
displacement of the microplates is 0, indicating internal creep.
This model can predict the critical stiness over which the plateau force () remains constant,
the response of cells to step changes in substrate stiness, and the rate at which force is generated
across various substrate stinesses. Interestingly, the constitutive equation (3.3) can also be wrien in a
similar form as the Hill model of muscle contraction,
(3.4)

+  +=,
where =,= 2+, and =+ −(
˙)/(2) . Here, =
˙ is the speed of shortening, 
represents the rate of actin polymerization and =/2 is the internal creep. This equation allows the
examination of non-muscle cell behaviour in two extreme cases: = 0 (stiness of the substrate equal
to 0) and = 0 (stiness of the substrate equal to innity). These extreme scenarios illustrate the role
of the molecular mechanisms that both govern and limit cell responses. Basically, cells initiate force
generation upon aachment to the substrate. If the resistance of the external environment is lower
than the force generated via myosin motors (e.g. when = 0), cells will start to contract the microplate,
which would, in turn, increase its resistance against the cell. This increased external resistance leads to
a reduction in the rate of retrograde ow. Moreover, cross-linker unbinding and actin polymerization
are two internal mechanisms that antagonize the rate of retrograde ow. This explanation can also be
shown according to equation (3.4); when  is 0, the maximum shortening length is max =
2−2.
These two internal factors also determine the maximum force generated in the system in the case
of very high stinesses. As demonstrated by max =1−+

2
+ 2, the maximum force in the
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cell is not equal to . In this scenario, actin polymerization requires extra work by myosin motors,
which will be lost as a boundary creep. Furthermore, cross-linker unbinding will also result in force
dissipation, leading to internal creep.
An analogous active Maxwell uid model was employed to understand the mechanical character-
istics underlying the observed cellular response to local RhoA activation [87]. In their experiments,
Oakes et al. used optogenetic probes to recruit a cytosolic photo-recruitable protein RhoA-specic
guanine exchange factor (prGEF) to the plasma membrane and activated RhoA over periods of
15 minutes. During local activation of RhoA, the uorescent intensity of both actin and myosin II
increased exponentially, plateaued and then decreased during the relaxation period (gure 2c(i)). A
similar response has been observed while measuring traction forces and strain energy in cells (gure
2c(iii)). Besides, RhoA activation resulted in a sudden enhancement of traction force generation in the
cell borders, whereas in the activation region traction forces did not change.
In the model used to describe this behaviour, contractile stress is introduced as an internal
boundary condition, that is, 0<, = ± 0,=∓0, to represent RhoA activation in the cell area.
The prole of 0 follows a similar paern as the actin and myosin accumulation in the activation area
(gure 2c(ii)). In addition, a two-dimensional model of stress bres embedded in a passive viscoelastic
environment was used to estimate the direction and magnitude of the actomyosin ow towards the
activation region. Finally, through the application of this model, they showcased the role of Zyxin in
regulating the time scale of the initial elastic behaviour.
Active CEs are employed in chemomechanical models as well. These models integrate the eect of
the interplay between the mechanical characteristics of cells, external and internal mechanical stimuli
and the cascade of biochemical signals to simulate cell behaviour in various scenarios. For example,
a chemomechanical model illustrated the growth dynamics of cell–ECM adhesion structures and
highlighted the regulatory inuence of the stiness of the nucleus and ECM [100]. Another model
demonstrated how the interaction between cells and ECM aects both gene expression and nuclear
architecture [101]. This group of models is reviewed in [102,103].
4. Integrating activity in viscoelastic solid models
The previous section demonstrated how active tension originating from actomyosin dynamics can be
integrated within constitutive equations that relate force and rate of contraction. Étienne et al.’s model
provides, in particular, a detailed description of the transient regimes leading to the establishment
of steady active stress [38]. These descriptions treat active materials as uid without a reference
to an intrinsic shape. It is, however, common for tissues to exhibit solid-like characteristics, with a
well-dened reference shape, and the role of active stresses may then be interpreted as an apparent
tension or a change in reference shape. In this section, we review these dierent approaches, highlight-
ing their similarities and dierences.
4.1. Constant active element
A simple way to introduce active contraction across a material is to include a stress-generating unit of
constant value. Such an approach was used by Wya et al., when they investigated the short-time-scale
response of Madin–Darby canine kidney (MDCK) monolayers to in-plane compressions [18], a process
observed during morphogenetic processes [104,105] and the normal physiological function of many
epithelial tissues [106,107]. The MDCK monolayers were placed between two rods and compressed
at dierent rates. Quickly compressing the monolayer to strains below a threshold level (ε ~ 33%)
resulted in transient folds that disappeared in time scales of the order of seconds, while the folds
created owing to higher strains were permanent. The same buckling threshold of ε ~ 33% was observed
when the monolayers were compressed both rapidly and at a low rate. In addition, using actomyosin
inhibitors, they have demonstrated that actomyosin activity regulates the rate of tissue aening, the
buckling threshold, pretension and the long-time-scale stiness of the monolayers.
A simple active rheological model could reproduce the results of their studies conducted under
dierent loading conditions (gure 3a). The model consists of a constant active element , which
brings the system to a tensile state even at 0 external load, in parallel to a standard linear solid
model. The MDCK monolayer buckles when the stress in the monolayer approaches the compression
range. Therefore, under compressive strains that would normally cause compressive stress, the model
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assumes that stress in the monolayer remains 0. The constitutive equation under tensile and compres-
sive stresses is dened as
(4.1)
= ++1and =under tensile stress,
= 0 and =−
−
1under compressive stress,
where  is the pretension in the monolayer,  is the device strain,  characterizes the long-term
stiness of the tissue,  and  describe the short-time-scale response, 1 is the strain in the spring
with stiness  satisfying 
˙1+1
=
˙, and =/ is the characteristic time scale. Based on the model
predictions for the steady state, when the applied strain is larger than the buckling threshold  , >,
(a) Constant contractile element
(b)
(c)
(i) (ii)
(i)(ii)
Maxwell model
Considering cellular contractility
(d) Considering local actin concentration at the wound edge
The Munoz model
Figure 3. Active models for solid-like materials. (a). Active rheological model that includes a constant CE, , in parallel to a standard
solid model to account for the prestress in cell monolayers [18]. Temporal evolution of strain (i) and tissue stress (ii) as a function
of device strain presented for epithelial monolayers undergoing compression at a low strain rate (0.5% s−1). (b) The active element
proposed by Muñoz et al., which adapts its resting length in response to deformations [108]. The evolution of stress over time in stress
relaxation tests predicted by the Maxwell model and active models with two denitions of elastic strain is presented in a table. Stress
in the models is dened using the elastic strain, that is, =. Changing the denition of elastic strain can result in an equivalent
evolution of stress in the active element and the Maxwell model for ≈−1. When the strain is very small, the dierence between
the evolution of stresses in all three models will be negligible. However, at larger strains, the stress in the system would be dependent
on the way that the strain is dened. (c) The modied version of the active element proposed by Muñoz et al., which also considers
the eect of cellular contractility [109]. (i) The average curves representing the evolution of stress over time for MDCK monolayers
stretched at 75% s−1 to various amplitudes of strain plotted on a semi-log scale. (ii) Demonstration of the correlation between the
model calculated from the active rheological model and the characteristic time  calculated from tting the stress–time curves with
an empirical function, that is, =−/−+, for dierent loading conditions and actomyosin treatments. (d) An example of a
hybrid vertex model employed to study wound healing. The top view of the epithelial tissue with ablated cells is shown along with the
rheological models of nodal segments (cell-centre connections) in black and vertex segments (cell boundaries) in red [110].
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tissue strain will be equal to the device strain, = and the stress in the tissue will follow =+ .
However, when the monolayer is compressed past the model buckling threshold, stress levels fall to 0
and tissue strain plateaus at =−/. These model predictions are consistent with the experimental
results shown in gure 3a(i) and (ii). The model can also predict the response of the monolayer to
a step of compressive strain of dierent magnitudes. Besides, the model provides a simple way to
capture how treatments aecting actomyosin control the amount of active tension revealing that the
buckling threshold observed in the experiments agrees with model predictions −/ in all conditions.
4.2. Active behaviour modelled as change in the resting length of a spring
The previous approach focused on how active stress, combined with external perturbations, would set
the mechanical evolution of a viscoelastic tissue. The Maxwell branch and its dashpot, in particular,
account for the remodelling and plasticity of the material. Another way to represent this is through the
explicit evolution of the rest shape, or stress-free state, of the material. Both remodelling and tensioning
can be accounted for through the evolution of the resting length of cells and tissues over time, as
proposed by Muñoz et al. [108]. We rst describe how a Maxwell-like behaviour can emerge from this
strategy and then how activity can be added to this class of models.
4.2.1. Model proposed by Muñoz et al.
Muñoz et al. proposed a model to account for the inuence of cytoskeleton activity on cell shape by
representing plasticity as a change in the rest length of a spring [108]. A material’s constitutive equation
therefore takes the form of a relationship between the rate of change of the rest length and the stress (or
equivalently some metric of elastic strain) in the material. In this model, the current resting length of
the material (lament, cell or tissue) , that is, the total length of the material when no external load is
applied to it, is proportional to the elastic strain . The rate of changing the resting length under strain
is dened as
(4.2)

˙
=,
where  is the remodelling rate of the network dened as the network resistance to adjusting its
conguration to the applied deformation,  is the current elastic strain, =−
 and  is the current total
length of the network. According to Muñoz’s model, the denition of current elastic strain is dierent
from the apparent strain =−0
0 , where 0 is the initial length and the resting length of the network.
For small deformations, Muñoz’s approach is mathematically equivalent to a linear Maxwell model
with a characteristic time of =
 [108]. However, at large deformation, the models dier, leading to
rather complex relaxation dynamics for Muñoz’s model (see gure 3b). This dierence results from
the particular denitions of strains and the resulting nonlinearities emerging from them. For instance,
using a logarithmic form for the elastic strain = ln 
 (i.e. using the true strain denition rather than
the engineering strain), Muñoz’s model would match the Maxwell model up to large deformations,
with a relaxation time scale independent of the strain amplitude, as shown in the table in gure 3b. At
a mathematical level, controlling the rest length of a spring or having a dashpot in series is therefore
largely equivalent.
However, introducing a dynamic rest length enables a slightly dierent interpretation of the
physiological mechanisms [108]. The dashpot element in the Maxwell model is often associated with
remodelling the system but could also account for the resistance of the cytoplasmic uid  to the
applied strain rate and dissipate power. On the other hand, for Muñoz’s active element, the inelastic
part of the external power will be used to overcome the resistance of the cytoskeleton laments 
to adapt to the new conguration imposed by the external strain. The active model proposed by
Muñoz has been generalized to two-dimensional/three-dimensional continuum models [111] and also
integrated into discrete models such as cell-centred [112], vertex [113] and cell-centred/vertex hybrid
[109,110] approaches. By incorporating a porosity parameter representing the density of polymers
in the cell cytoskeleton, the continuum model proposed by Asadipour et al. can also replicate the
immediate uidization in cells in response to transient strains and the subsequent gradual stiening
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[111]. These adaptations have facilitated the study of epithelial tissue behaviour in both two and three
dimensions. A modelling approach similar to the Muñoz model proposed by Esfahani et al. could also
demonstrate stiening in response to high strain rates applied to epithelial cell pairs [49]. A few of the
modications made to the Muñoz model are presented in the following sections.
4.2.2. Active element considering cellular contractility
Mosaa et al. [109] modied the evolution law of the resting length of the material proposed by Muñoz
et al. [108] by introducing a contractility parameter  to account for the inherent contractility of the
cells,
(4.3)

˙
= −.
In this model, when the elastic strain  reaches to , the resting length will not change any more,
and as previously stated [108], when  is zero, the model behaves similarly to the Maxwell model.
Mosaa et al. implemented the modied active element in a hybrid cell-centred/vertex model where
cells interact through both cell centrs, presented by nodes and cell–cell junctions, presented by the
connection between vertices [109]. This model could successfully simulate tissue extension and wound
healing.
Khalilgharibi et al. have used a similar approach for ing the results of stress relaxation tests
conducted on MDCK monolayers [5]. Their studies have shown that stress in the MDCK monolayer
increases promptly after strain application. Then, the stress will gradually relax along with an increase
in the monolayer length, which is regulated by actomyosin activity. Moreover, they have noticed a
strain-dependent characteristic time  for monolayers stretched at 75% −1 strain rate (gure 3c(i)),
which cannot be explained by standard linear viscoelastic models. Therefore, they proposed a model
that consists of an elastic spring in parallel to an active element that sustains a constant pre-strain 
and changes its resting length   to relax the imposed stress and return its strain to  . The changing
of the resting length of the monolayer   in response to an applied strain 0 is dened as
(4.4)

˙
= −
= 0−.
Here,  is the eective strain =−  /  with  representing the actual length of the
monolayer after applying the deformation and  is the rate of changing the resting length. The
characteristic time predicted by this model,  =0/1+0, increases with the applied strain,
which is consistent with their experimental observations, as shown in (gure 3c(ii)).
Another example of the active element that accounts for cellular contractility involves integrating
this model into a hybrid two-dimensional cell-centred/vertex model to analyse the wound healing
process [110]. In this model, the vertex segments (cell boundaries) and nodal segments (cell-centre
connections) connect the apical and basal sides. Nodal segments are characterized using an elastic
spring, and the behaviour of vertex segments is described by an active model consisting of two
branches in parallel (gure 3d). The rst branch is an active element that accounts for the changes
in the resting length of the vertex following equation (4.3). The second branch is an elastic spring
with an additional time-varying contractility parameter Υ
^, which accounts for the eect of high actin
concentration at the wound edge (i.e. purse string contractility) and increases the stress in the elastic
spring. The outcomes of their simulations have demonstrated the regulatory mechanism of both purse
string contractility and tissue contractility on wound healing speed.
4.3. Active element considering a time delay in the active rest length changes
One of the potential reasons for the oscillatory response observed in tissues during various processes,
such as morphogenesis, could be the delay between the signal and the response, as stated by Muñoz
et al. [113]. These delays in the responses can be owing to the distance between the sender and
receiver of the biochemical signals or the time necessary for signal processing [113,114]. To study
this phenomenon, they modied the active element that was previously proposed by Muñoz et al. by
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considering the eect of a time delay between the mechanical signals and the active rest length changes
[108],
(4.5)
˙=   −−  −.
Analysing the stability of the delay dierential equation resulted in the limits of oscillation and
stability as follows:
(4.6)oscil =1
 and stab =π
2.
oscil is the time beyond which the rest length of the element oscillates,  is the exponential constant,
and for time scales above stab the value of the rest length is unstable, and its oscillation amplitude will
rise over time. These values might trigger oscillations during embryogenesis.
In addition, delays can also be dependent on the apparent size of the element   . Muñoz et al.
implemented the eect of the size-dependent delays,   =  , into a vertex model to analyse the
oscillation in the cellular area in biological tissues [113]. In the vertex model, the rest length of the
nodal elements was maintained constant, and changes in the resting length of the vertex elements
were dened using equation (4.5). For constant delays, oscillations in the cell area were periodic and
synchronous. However, for size-dependent delays, oscillations started to get increasingly out of sync.
It is argued that this model demonstrated the role of delay in the mechanical response in inducing
oscillations even in the absence of external sources.
5. Summary and future perspectives
The active viscoelastic models outlined here are capable of capturing numerous aspects of cell
behaviour at multiple spatial and temporal scales with a small number of model parameters without
considering the details of the structural components and dynamics of cell–cell and cell–ECM adhesion
sites. These active models provided insight into the results of experiments and predicted the system
behaviour in other arbitrary conditions. Additionally, despite the diculty in establishing a clear
connection between biological processes and model parameters, researchers have used drug treatments
or targeted mutations to demonstrate correlations between model parameters like Young’s modulus,
viscosity or active pretension and biological processes such as actomyosin activity, even molecules that
regulate these processes.
Each active viscoelastic model is described by a constitutive equation that represents a particular
cellular behaviour, such as sensing changes in the substrate stiness, and response to strains at
dierent magnitudes and rates in cell doublets and cell monolayers. Therefore, the existence and
use of a generalized model that can be employed to describe and predict the response of cells in
dierent scenarios is still an open question. Integrating and bridging the gap between phenomenologi-
cal and biophysical models is an important step to improve our understanding of these systems. For
example, a molecular model of the actomyosin cortex inspired a phenomenological model for cell-scale
mechanosensing [38], and a phenomenological model that included the role of actin polymerization
in changing the resting length of the material was incorporated in vertex models to study dier-
ent aspects of tissue dynamics [113]. Consequently, a comprehensive phenomenological model that
captures all the signicant facets of rheological data might also enrich the ndings of biophysical
models and allow us to improve the precision of simulations of cell activity.
Numerical models are an excellent research tool to complement, analyse and interpret experimental
data in the eld of cell mechanics. Numerous models with varying degrees of complexity and details
of the structural elements involved in the observed phenomena have been presented over the years.
Power law [15], fractional (reviewed in [17]), viscoelastic and active viscoelastic models are included
in this category, where the eects of subcellular microstructures on cell rheology are represented by
model parameters. For example, virtual cell (VCell) is a powerful model that includes details of the
nucleus, cytoskeleton, cytoplasm and chromatin bres [115]. This level of detail is computationally
expensive and might not always be necessary. In other words, based on the research question, the
phenomenon of interest, length and time scales and characteristics of the relevant microstructures
can be incorporated into biophysical models. For example, at the molecular scale, chemomechanical
[103,116] and molecular clutch models [117–119] are used to study cell–cell and cell–ECM adhesions,
at the cell-scale statistical approaches can be employed to study cell mechanics [120–122], and cellular
Pos (CPM) [123–125], vertex [126,127] and self-propelled Voronoi [128–130] models are introduced to
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study the mechanical behaviour of epithelial monolayers in two dimensions and three dimensions. The
level of detail can be further reduced by describing the outcomes of experiments using phenomenolog-
ical models before examining the underlying mechanisms.
Ethics. This work did not require ethical approval from a human subject or animal welfare commiee.
Data accessibility. This article has no additional data.
Declaration of AI use. We have not used AI-assisted technologies in creating this article.
Authors’ contributions. B.T.S.: writing—original draft, writing—review and editing; C.H.: writing—review and editing;
A.K.: writing—review and editing; R.Y.: writing—review and editing.
All authors gave nal approval for publication and agreed to be held accountable for the work performed
therein.
Conict of interest declaration. We declare we have no competing interests.
Funding. This study was supported by the NSF (1826135, 2143997) and the NIH (R35GM150623, P20GM113126).
Acknowledgements. R.Y. acknowledges fundings from the NSF (1826135, 2143997) and the NIH (R35GM150623,
P20GM113126).
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