# Hydraulic and electric control of cell spheroids

Charlie Duclut<EMAIL_ADDRESS>Max-Planck-Institut für Physik Komplexer
Systeme, Nöthnitzer Str. 38, 01187 Dresden, Germany.  Jacques Prost
<EMAIL_ADDRESS>Laboratoire Physico Chimie Curie, UMR 168, Institut
Curie, PSL Research University, CNRS, Sorbonne Université, 75005 Paris,
France.  Mechanobiology Institute, National University of Singapore, 117411
Singapore.  Frank Jülicher<EMAIL_ADDRESS>Max-Planck Institut für Physik
Komplexer Systeme, Nöthnitzer Str. 38, 01187 Dresden, Germany.  Center for
Systems Biology Dresden, Pfotenhauerstr. 108, 01307 Dresden, Germany.  Cluster
of Excellence Physics of Life, TU Dresden, 01062 Dresden, Germany.

###### Abstract

We use a theoretical approach to examine the effect of a radial fluid flow or
electric current on the growth and homeostasis of a cell spheroid. Such
conditions may be generated by a drain of micrometric diameter. To perform
this analysis, we describe the tissue as a continuum. We include active
mechanical, electric, and hydraulic components in the tissue material
properties. We consider a spherical geometry and study the effect of the drain
on the dynamics of the cell aggregate. We show that a steady fluid flow or
electric current imposed by the drain could be able to significantly change
the spheroid long-time state. In particular, our work suggests that a growing
spheroid can systematically be driven to a shrinking state if an appropriate
external field is applied. Order-of-magnitude estimates suggest that such
fields are of the order of the indigenous ones. Similarities and differences
with the case of tumors and embryo development are briefly discussed.

Understanding how cells collectively organize to form complex structures and
organs is the fundamental question raised in morphogenesis. This self-
organization stems from the interplay of biochemical Green and Sharpe (2015);
Miller and Davidson (2013), mechanical Miller and Davidson (2013); Mammoto
_et al._ (2013); Ladoux and Mège (2017), but also hydraulic Ruiz-Herrero  _et
al._ (2017); Dumortier  _et al._ (2019) and electrical processes Levin and
Martyniuk (2018); Silver and Nelson (2018). A long-standing paradigm in
developmental biology is that cell chemical signals, in the form of
morphogens, control cell growth and differentiation leading to tissue
patterning Green and Sharpe (2015). Although this biochemical signaling is of
paramount importance in developmental control, it is now well-established that
mechanical forces between cells or mediated by the extra-cellular matrix can
also provide regulatory cues that are equally important Mammoto  _et al._
(2013).

The crucial importance of hydraulics in morphogenesis, which should not come
as a surprise given the large water content in tissues, has also been
highlighted in multiple experiments. Hydraulic oscillations have for instance
been shown to provide a robust mechanism for size control of the mouse embryo
Chan  _et al._ (2019) and during the Hydra regeneration Fütterer  _et al._
(2003). The role of electrical signals in tissue patterning, although already
studied by Roux Roux (1892) at the end of the 19th century, has gained a new
interest only recently McCaig  _et al._ (2005); Levin and Martyniuk (2018);
McLaughlin and Levin (2018). In addition to its key function in receiving and
relaying sensory information in the nervous system Watanabe  _et al._ (2009),
bioelectricity has been shown to have a dramatic importance in large-scale
patterning: an alteration of the electrical signaling in Planaria regeneration
causes for instance the emergence of animals with multiple heads Levin and
Martyniuk (2018). Similarly, it has recently been observed that an external
electric field can be used to reverse the morphogenetic fate of Hydra Braun
and Ori (2019).

In the quest of understanding how these different mechanisms come together to
shape tissues and organs, simple cell aggregates such as spheroids have
offered an appealing territory to observe tissue development and to formulate
hypotheses on the underlying mechanisms. Remarkably, a rich behavior is
observed even in single-cell type spheroids, as for instance their ability to
pump fluid and form liquid-filled lumens Martín-Belmonte  _et al._ (2008);
Debnath  _et al._ (2002). More recently, organoids have focused an increasing
attention as they can recapitulate complex morphogenetic processes in a
relatively simple and controllable environment Huch and Koo (2015); Dahl-
Jensen and Grapin-Botton (2017).

To unravel the connections between biochemical signals and tissue mechanics,
mechanical perturbations of organoids and cell spheroids can be performed. For
instance, atomic force microscopy has been used to probe the mechanical
properties of mammary organoids Alcaraz  _et al._ (2008) and revealed the
importance of both extracellular matrix stiffness and laminin signaling to
maintain tissue integrity. Perturbation of the osmotic pressure around cell
spheroids using large molecules of Dextran has also highlighted the importance
of isotropic stress in tissue growth Montel  _et al._ (2011); Delarue  _et
al._ (2013).

A broad understanding of tissue mechanics therefore requires to consider
tissue electrohydraulic properties and to be able to perturb tissues by
electric or hydraulic means. In this theoretical work, we study the response
of a cell spheroid to electric and hydraulic perturbations. We propose an
experimental setup where a pipette or drain is used to impose a fluid flow or
an external electric current through a micrometric drain. Our work shows that
one could control the size of a cellular assembly using such a setup.

Our theoretical approach relies on generic features of the physics of the
electro-hydrodynamic phenomena at play within tissues. We adopt a coarse-
grained approach of tissues, which describes many cells and their micro-
environments as a continuum with active material properties Kruse  _et al._
(2005); Marchetti  _et al._ (2013); Ranft  _et al._ (2010). Cell mechanical
characteristics, fluid pumping, ion transport and electrical properties are
thus considered in a unified framework Sarkar  _et al._ (2019); Duclut  _et
al._ (2019) that we use to analyze spheroid response to an external
perturbation.

In particular, we highlight in the following that steady external flow or
electric current imposed by a drain can systematically drive a proliferating
spheroid to degeneracy. Such technique could be relevant in a medical context
where it could be used to suppress cancerous tumors.

Methods for suppressing malignant tumors are numerous: a first path to control
tumor size is to use drugs to disturb the chemical regulation of cancerous
cells to prevent them from proliferating (chemotherapy). Immunotherapy has
offered a strong alternative in mitigating cancer by stimulating the immune
system to suppress tumors Scott  _et al._ (2012). In addition, radiotherapy –
which allows the suppression of the tumor by damaging the genetic material of
cancerous cells through radiations – has also proved effective for tumor
removal Baskar  _et al._ (2012).

Physics-based methods can also be used to provide new techniques for directly
suppressing the cancerous tissues. High-intensity focused ultrasounds can for
instance be used to locally overheat the cancerous cells Al-Bataineh  _et al._
(2012). Recent experiments also suggest that lower intensity ultrasound waves
could be used to strain and suppress mechanically cancerous tissues
Mittelstein  _et al._ (2020); Tijore  _et al._ (2020), and low-frequency
ultrasounds could be used to increase selectivity Tijore  _et al._ (2020).
Electrical perturbations have also been used: cancerous tissues can for
instance be successfully suppressed using irreversible electroporation Miller
_et al._ (2005) by applying large voltage pulses in the tissue. More recently,
micro-electrodes have been used in electrolytic ablation methods to locally
change pH and kill a cancerous mass Perkons  _et al._ (2018).

Compared to other treatments and ablation techniques Knavel and Brace (2013),
direct electrohydraulic perturbations of cancerous tissues have however
remained largely unexplored. With the theoretical study that we present here,
we aim at highlighting potential novel methods for controlling size of cell
spheroids, that could also be used to suppress cancerous tumors.

##  I Continuum model of a spheroid with a drain

(a) (b)

Figure 1: Sketch of a spheroid with drain. The drain is used to impose an
electric current $I^{\rm ext}$ or a volumetric flow rate $Q^{\rm ext}$,
driving a fluid flow or current density throughout the cell spheroid. (a)
Flows from inside of the spheroid to outside correspond to $Q^{\rm ext},I^{\rm
ext}>0$. (b) Flows from outside to inside correspond to $Q^{\rm ext},I^{\rm
ext}<0$. The pressure inside the lumen, outside the spheroid and at the outer
end of the drain are denoted $P_{1}^{\rm ext},P_{2}^{\rm ext}$ and $P^{\rm
d}$, respectively, and correspondingly for the electric potential: $U_{1}^{\rm
ext},U_{2}^{\rm ext}$ and $U^{\rm d}$.

Following Refs. Sarkar  _et al._ (2019); Duclut  _et al._ (2019), we consider
the tissue at a coarse-grained level, such that individual cells are not
described but the tissue as a whole is studied as a continuum material with
active electrical, hydraulic and mechanical properties. To capture the
hydraulic properties of the tissue, we also adopt a two-fluid description,
where the cells, which form the first fluid, are permeated by the interstitial
fluid Ranft  _et al._ (2012). In the long-time limit (several days or weeks)
that we consider here, cells are able to reorganize and to relax the internal
stresses within the tissue, such that an effective description of the tissue
as a viscous active fluid at long times is used Ranft  _et al._ (2010).

To analyze specifically the effect of a drain on a cell spheroid, we consider
a spherical tissue of radius $R_{2}$ enclosing a spherical lumen of radius
$R_{1}$ (see Fig. 1). A drain of inner radius $R_{\rm d}$ is inserted inside
the spheroid and can be used to impose an external flow or an external
current. We consider, despite the presence of the drain, a system with
spherical symmetry. The description we propose in the following is therefore
effectively one dimensional and depends only on the distance $r$ to the
spheroid center.

Directional ion pumping through the spheroid is achieved if cells have a
polarity. We thus define a cell polarity field $\bm{p}$ with unit norm that we
assume, for simplicity, to be oriented along the radial direction:
$\bm{p}=\bm{e}_{r}$. Cells also display a nematic ordering – due for instance
to an anisotropy in their shape – that we describe with the nematic tensor
$q_{\alpha\beta}$ (Greek indices indicate Cartesian coordinates). We assume in
the following that the nematic ordering is defined by the same preferred axis
as the polarity of the cells:
$q_{\alpha\beta}=p_{\alpha}p_{\beta}-(1/3)\delta_{\alpha\beta}$. This
situation is for instance observed in colon carcinoma cell spheroids Dolega
_et al._ (2017). Finally, the consideration of ion pumping within the tissue
requires the introduction of an electric field $\bm{E}$ and electric current
density $\bm{j}$, which obey a generalized Ohm law as we discuss in the
following.

###  I.1 Tissue mechanical, hydraulic and electrical properties

The total tissue stress is decomposed as $\sigma_{\alpha\beta}=\sigma^{\rm
c}_{\alpha\beta}+\sigma^{\rm f}_{\alpha\beta}$ where here and in the following
the superscripts $\rm c,f$ stand for the cells and interstitial fluid
contributions, respectively. Neglecting inertia and in the absence of external
bulk forces, force balance within the tissue reads:

$\displaystyle\partial_{\beta}\sigma^{\rm c}_{\alpha\beta}+f_{\alpha}$
$\displaystyle=0\,,$ (1a) $\displaystyle\partial_{\beta}\sigma^{\rm
f}_{\alpha\beta}-f_{\alpha}$ $\displaystyle=0\,,$ (1b)

where $f_{\alpha}$ represents internal forces between the cells and the
interstitial fluid, and where summation over repeated indices is implied.

To obtain the dynamics of the spheroid, that is, the time evolution of the
inner and outer radii $R_{1}$ and $R_{2}$, we now need to specify the material
properties (or constitutive equations) of the tissue. For simplicity, we
assume that the interstitial fluid is only driven by pressure gradients, such
that the fluid stress is $\sigma^{\rm f}_{\alpha\beta}=-P^{\rm
f}\delta_{\alpha\beta}$. For the cell stress, the description of the material
properties is made easier by decomposing the stress into a traceless symmetric
part $\tilde{\sigma}^{\rm c}_{\alpha\beta}$ and an isotropic part $\sigma^{\rm
c}\delta_{\alpha\beta}$. The constitutive equation for the isotropic part then
reads Sarkar  _et al._ (2019); Duclut  _et al._ (2019):

$\displaystyle\sigma^{\rm c}+P^{\rm c}_{\rm h}=\bar{\eta}v^{\rm
c}_{\gamma\gamma}-\nu_{0}\tilde{\sigma}^{\rm
c}_{\alpha\beta}q_{\alpha\beta}-\nu_{1}p_{\alpha}E_{\alpha}-\nu_{2}p_{\alpha}V_{\alpha}\,,$
(2a) where $-\sigma^{\rm c}$ is the local pressure and we have introduced the
cell strain-rate tensor $v^{\rm c}_{\alpha\beta}=(\partial_{\alpha}v^{\rm
c}_{\beta}+\partial_{\beta}v^{\rm c}_{\alpha})/2$ and where $V_{\alpha}=v^{\rm
c}_{\alpha}-v^{\rm f}_{\alpha}$. The bulk viscosity $\bar{\eta}$ is a familiar
term that is also present for a passive fluid, except that its microscopic
origin is different for a tissue. The homeostatic pressure $P^{\rm c}_{\rm h}$
is a property specific for tissues that results from a balance of cell growth
and cell death Basan  _et al._ (2009). In addition to these terms, we have
included additional terms in the isotropic stress: cell anisotropies can
couple to the anisotropic stress, resulting in the term proportional to
$\nu_{0}$. The terms proportional to $\nu_{1}$ and $\nu_{2}$ represent
bioelectric and biohydraulic stresses induced by a coupling to the electric
field or by a (relative) fluid flow, respectively. A similar expansion for the
traceless anisotropic part of the stress tensor reads:
$\displaystyle\tilde{\sigma}^{\rm c}_{\alpha\beta}=2\eta\tilde{v}^{\rm
c}_{\alpha\beta}+\zeta q_{\alpha\beta}-\nu_{3}[E_{\alpha}p_{\beta}]_{\rm
st}-\nu_{4}[V_{\alpha}p_{\beta}]_{\rm st}\,.$ (2b) where $\tilde{v}^{\rm
c}_{\alpha\beta}$ is the traceless part of the cell strain-rate tensor, and we
have moreover defined the symmetric traceless part of a dyadic product of
vectors: $[A_{\alpha}B_{\beta}]_{\rm st}\equiv
A_{\alpha}B_{\beta}+A_{\beta}B_{\alpha}-(2/3)A_{\gamma}B_{\gamma}\delta_{\alpha\beta}$.
The coefficient $\eta$ is the usual shear viscosity while the other terms
represent active couplings, and the terms proportional to $\nu_{3}$ and
$\nu_{4}$ are the anisotropic counterpart of the terms proportional to
$\nu_{1}$ and $\nu_{2}$ in Eq. (2a). The active stress $\zeta q_{\alpha\beta}$
is a hallmark of active systems and shows the ability of cells to generate
anisotropic stresses due to cell division or contraction of their
cytoskeleton. This active stress can be regulated by the cells and we
therefore consider that it depends on the local pressure at linear order as:
$\displaystyle\zeta=\zeta_{0}-\zeta_{1}(\sigma^{\rm c}+P^{\rm c}_{\rm h})\,.$
(2c)

Including all the linear terms allowed by symmetry, we now write the
constitutive equations for the internal force density between cells and
interstitial fluid:

$\displaystyle f_{\alpha}\\!=\\!-\kappa(v^{\rm c}_{\alpha}\\!-\\!v^{\rm
f}_{\alpha})+\lambda_{1}p_{\alpha}\\!+\lambda_{2}E_{\alpha}\\!+\lambda_{3}q_{\alpha\beta}E_{\beta}\\!+\lambda_{4}\partial_{\beta}q_{\alpha\beta}\,.$
(3)

The first term $-\kappa(v^{\rm c}_{\alpha}\\!-\\!v^{\rm f}_{\alpha})$ accounts
for the friction between the interstitial fluid and the cells and leads to
Darcy’s law Darcy (1856) in the description of porous materials. The term
$\lambda_{1}p_{\alpha}$ accounts for active fluid pumping by the cells, while
the terms proportional to $\lambda_{2}$ and $\lambda_{3}$ correspond to
isotropic and anisotropic electroosmotic contributions. The last term,
proportional to $\lambda_{4}$, characterizes a differential pumping term due
to the bending of the cells.

To complete the tissue properties description, one finally needs to specify
the constitutive equation for the electric current density:

$\displaystyle j_{\alpha}\\!=\\!-\bar{\kappa}(v^{\rm c}_{\alpha}\\!-\\!v^{\rm
f}_{\alpha})+\Lambda_{1}p_{\alpha}\\!+\Lambda_{2}E_{\alpha}\\!+\Lambda_{3}q_{\alpha\beta}E_{\beta}\\!+\Lambda_{4}\partial_{\beta}q_{\alpha\beta}\,,$
(4)

where $\Lambda_{2}$ and $\Lambda_{3}$ are the isotropic and anisotropic
conductivities of the tissue. The term proportional to $\bar{\kappa}$
characterizes the current due to the (relative) flow of ions between cells as
a consequence of a reverse electroosmotic effect Kirby (2013). The coefficient
$\Lambda_{1}$ characterizes the contribution of ion pumping to the electric
current, while the coefficient $\Lambda_{4}$ is an active flexoelectric
coefficient. It indicates that a spatially nonuniform cell polarity
orientation is obtained in response to an electric field, and has been shown
to play a crucial role in the nucleation of a lumen in spherical cell
aggregates Duclut  _et al._ (2019).

###  I.2 Drain description

The drain can be used to impose an external fluid flow or an external electric
current in two equivalent ways: either by imposing directly a volumetric flow
rate $Q^{\rm ext}$ and electric current $I^{\rm ext}$ through the drain, or,
alternatively, a pressure difference $\Delta P=P^{\rm d}-P^{\rm ext}_{2}$
(with $P^{\rm ext}_{2}$ the pressure at the outer spheroid boundary and
$P^{\rm d}$ the pressure at the outer end of the drain, see Fig. 1) and an
electric potential difference $\Delta U=U^{\rm d}-U^{\rm ext}_{2}$ (with
$U^{\rm ext}_{2}$ the electric potential at outer spheroid boundary and
$U^{\rm d}$ the electric potential at the outer end of the drain) can be
applied.

In both cases, the imposed external fluid flow and external electric current
density at the lumen boundary read:

$\displaystyle Q^{\rm ext}=4\pi R_{1}^{2}v^{\rm
ext}_{1}\,,\quad\text{and}\quad I^{\rm ext}=4\pi R_{1}^{2}j^{\rm ext}_{1}\,,$
(5)

where $v^{\rm ext}_{1}$ and $j^{\rm ext}_{1}$ are the fluid velocity and
electric current density at the boundary between the spheroid and the lumen.

In the following, we focus on the case where $Q^{\rm ext}$ and $I^{\rm ext}$
are imposed externally. The case of a pressure difference or electric
potential difference imposed by the drain are discussed in App. B.

###  I.3 Continuity equations and boundary conditions

If cell density and interstitial fluid density are equal and constant, which
we assume in the following, then the total volume flux $v_{\alpha}=\phi v^{\rm
c}_{\alpha}+(1-\phi)v^{\rm f}_{\alpha}$ is divergence-free:
$\partial_{\alpha}v_{\alpha}=0$ Ranft  _et al._ (2012). In the presence of the
drain, which imposes a nonvanishing fluid velocity at the inner boundary of
the spheroid, the integration of the total flow incompressibility in spherical
coordinates yields a relation between the fluid velocity $v_{r}^{\rm f}$ and
cell velocity $v_{r}^{\rm c}$ in the tissue, which reads:

$\displaystyle v_{r}^{\rm f}$ $\displaystyle=\frac{v^{\rm
ext}_{1}}{1-\phi}\left(\frac{R_{1}}{r}\right)^{2}-\frac{\phi}{1-\phi}v^{c}_{r}\,.$
(6)

Similarly, charge conservation in the quasistatic limit
$\partial_{\alpha}j_{\alpha}=0$ can be integrated in the case where an
external current density $j^{\rm ext}_{1}$ is imposed on the inner boundary,
yielding the current density $j_{r}=j^{\rm ext}_{1}\left(R_{1}/r\right)^{2}$
throughout the tissue.

The spheroid is surrounded by an external fluid both inside (in the lumen),
and outside. This fluid exerts a hydrostatic pressure on the tissue that is
balanced by the tissue surface tension and by the total normal stress at the
boundaries:

$\displaystyle-\sigma^{\rm c}_{rr}(R_{1})+P^{\rm f}(R_{1})$
$\displaystyle=P^{\mathrm{ext}}_{1}-2\gamma_{1}/R_{1}\,,$ (7a)
$\displaystyle-\sigma^{\rm c}_{rr}(R_{2})+P^{\rm f}(R_{2})$
$\displaystyle=P^{\mathrm{ext}}_{2}+2\gamma_{2}/R_{2}\,,$ (7b)

where we have introduced the inner and outer tissue surface tensions
$\gamma_{1}$ and $\gamma_{2}$. Fluid exchange between the spheroid and the
outside is driven by osmotic conditions:

$\displaystyle v^{\rm ext}_{1}\\!\\!-\mathrm{d}R_{1}/\mathrm{d}t$
$\displaystyle=\\!+\Lambda^{\rm f}_{1}\left[(P^{\mathrm{ext}}_{1}-P^{\rm
f}(R_{1}))-\Pi^{\mathrm{ext}}_{1}\right]+J_{{\rm p},1}\,,$ (8a) $\displaystyle
v^{\rm ext}_{2}\\!\\!-\mathrm{d}R_{2}/\mathrm{d}t$
$\displaystyle=\\!-\Lambda^{\rm f}_{2}\left[(P^{\mathrm{ext}}_{2}-P^{\rm
f}(R_{2}))-\Pi^{\mathrm{ext}}_{2}\right]-J_{{\rm p},2}\,.$ (8b)

Here, $\Lambda^{\rm f}_{i}$ is the permeability of the interface to water
flow. The fluxes $J_{{\rm p},i}$ can be nonzero as a result of active pumps
and transporters that maintain an osmotic pressure difference, and act
effectively as water pumps. We have also introduced $v^{\rm ext}_{2}$, the
external flow imposed at outer spheroid boundary. Conservation of the
volumetric flow directly yields $v^{\rm ext}_{2}=v^{\rm
ext}_{1}(R_{1}/R_{2})^{2}$.

The normal velocity of the cells at the boundaries has to match the growth of
the spheroid radii. An increased cell proliferation in a thin surface layer
has been observed in growing spheroids Delarue  _et al._ (2014); Montel  _et
al._ (2011); Delarue  _et al._ (2013). We thus allow for a thin surface layer
of cells, both facing outside and to the lumen, to have a growth rate that
differs from the bulk. The cell velocity boundary conditions then read:

$\displaystyle v_{r}^{\rm c}(R_{1})=\mathrm{d}R_{1}/\mathrm{d}t+v_{1}\,,$ (9a)
$\displaystyle v_{r}^{\rm c}(R_{2})=\mathrm{d}R_{2}/\mathrm{d}t-v_{2}\,,$ (9b)

where $v_{i}=\delta k_{i}n^{\rm c}_{i}h/n^{\rm c}$ with $h$ the thickness of
the boundary layers, $n^{\rm c}_{i}$ and $\delta k_{i}$ the cell number
density and the cell growth rate in the surface layers, respectively.

##  II Dynamics of the spheroid growth and order of magnitudes

(a) (b) (c) (d) (e) (f)

Figure 2: Spheroid size $R_{2}$ (red curve) and lumen size $R_{1}$ (blue
curve) normalized to a characteristic size $R_{0}$ as a function of the
normalized time $\hat{t}=t/\tau_{0}$ for different external flow protocols
$\hat{Q}(\hat{t})=Q^{\rm ext}(t)/Q_{0}$ (dashed black curve; its magnitude is
not at scale). The definitions of $\tau_{0},R_{0}$ and $Q_{0}$ are given in
Eq. (12). (a) Spheroid growth without external intervention. (b)-(d)
Successful protocols: the spheroid is suppressed. (e)-(f) Unsuccessful
protocols: the spheroid continues to grow. These plots are obtained by solving
the dynamics equation (21a) and (21b) with the parameter values given in Table
2.

We have introduced in the previous section a model for a spherical spheroid
with a drain. Solving force balance (1) together with the boundary conditions
(7)-(9) then allow us to obtain the dynamics of the inner $R_{1}(t)$ and outer
$R_{2}(t)$ radii of the spheroid in the quasistatic limit. We obtain two
coupled nonlinear differential equations for the spheroid radii, Eqs. (21a)
and (21b) (see App. A), which have been studied in Ref. Duclut  _et al._
(2019) in the absence of a drain. As we will see in the following, imposing an
external fluid flow or electric current have dramatic consequences for the
spheroid growth and can be used to control its size.

Before discussing how the presence of external fields modify the dynamics of a
spheroid and which protocols can be used to control its growth, we first use
our model to discuss the order of magnitudes of the external flux and electric
current for which we expect a significant change in the spheroid dynamics. To
obtain these estimates, we assume that the lumen size is small and use
$R_{1}=0$ in Eq. (21b) that describes the dynamics of the spheroid. We then
compare in this equation the stresses generated by imposed flows or electric
currents to the stresses stemming from internal activity in the absence of
flows and currents. We find that the external volumetric flow required to
significantly perturb the spheroid is of the order:

$\displaystyle Q^{\rm ext}\simeq\sigma^{\rm
typ}\frac{4\pi(1-\phi)}{\kappa^{\rm eff}}R_{2}\,,$ (10)

where $\kappa^{\rm eff}=\kappa-\bar{\kappa}\lambda/\Lambda$ (with
$\lambda=\lambda_{2}+2\lambda_{3}/3$ and $\Lambda=\Lambda_{2}+2\Lambda_{3}/3$)
is an effective friction coefficient and $\sigma^{\rm typ}$ is a typical scale
of tissue stress in the absence of flows and currents. Note that the
appearance of $\kappa^{\rm eff}$ in Eq. (10) indicates that the finite bulk
permeability of the spheroid governs the effects of the external flow imposed
via the drain. Note however that for smaller spheroids of size
$R_{2}\lesssim(1-\phi)/\kappa^{\rm eff}\Lambda_{2}^{\rm f}\simeq
10-50~{}\mu$m, the flow-induced effects are not dominated by bulk permeation
but rather by surface permeation and we have: $Q^{\rm ext}\simeq
4\pi\Lambda_{2}^{\rm f}\sigma^{\rm typ}R_{2}^{2}$. We can estimate the
external electric current required to perturb the spheroid:

$\displaystyle I^{\rm ext}\simeq 4\pi\sigma^{\rm typ}\Lambda R_{2}/\lambda\,.$
(11)

One can use experimental values and order-of-magnitude estimates of the
parameters that appear in Eqs. (10) and (11) (see Table 1 in App. D). We
estimate that a volumetric flow $Q^{\rm ext}\simeq 10^{3}-10^{5}~{}\mu$m3/s is
sufficient to observe a significant change in the dynamics of a millimetre-
sized spheroid, and this volumetric flow scales linearly with the size of the
spheroid. Similarly, electric current of the order $I^{\rm ext}\simeq
10^{0}-10^{2}$ nA can perturb the dynamics of a millimetre-sized spheroid (see
also Fig. 6 in App. D).

Externally imposed flows and electric currents could therefore be used to
induce a change in the spheroid behavior. In the following sections, we focus
in more details on how these external perturbations can be used to control the
size of the spheroid.

##  III Hydraulic and electric control of the size of a spheroid

###  III.1 Examples of protocols for spheroid suppression

Using our model and solving numerically Eqs. (21a) and (21b) (see App. C), we
can analyse various protocols for the suppression of a spheroid. The dynamics
of the spheroid and its lumen, $R_{2}(t)$ and $R_{1}(t)$, is therefore studied
for different external flow protocols $Q^{\rm ext}(t)$ and electric current
protocols $I^{\rm ext}(t)$. To keep the discussion as general as possible, we
introduce dimensionless quantities: a dimensionless radius
$r_{2}(\hat{t})=R_{2}(t)/R_{0}$, time $\hat{t}=t/\tau_{0}$, external
volumetric flow $\hat{Q}=Q^{\rm ext}/Q_{0}$ and electric current
$\hat{I}=I^{\rm ext}/I_{0}$, where we have defined:

$\displaystyle\begin{split}R_{0}=\Lambda^{\rm
f}_{2}\bar{\eta}\,,&\quad\tau_{0}=\bar{\eta}/|P^{\rm eff}_{2}|\,,\quad\\\
Q_{0}=4\pi\bar{\eta}^{2}(\Lambda^{\rm f}_{2})^{3}|P^{\rm eff}_{2}|\,,&\quad
I_{0}=4\pi\Lambda\bar{\eta}\Lambda^{\rm f}_{2}|P^{\rm
eff}_{2}|/|\lambda|\,.\end{split}$ (12)

Note that the effective pressure $P^{\rm eff}_{2}$ introduced above is a
modification of the homeostatic pressure $P^{\rm c}_{\rm h}$ by the external
osmotic pressure and by electric and active contributions. Its expression can
be found in App. A.

We display in Fig. 2 different protocols that can be applied to a growing
spheroid in order to suppress it. If the external flow magnitude is
sufficiently large and if it is applied long enough, the interventions are
successful and lead to the spheroid suppression. Figure 2(b) shows an example
of a successful suppression of the spheroid as a sufficiently strong flow has
been applied until the spheroid is suppressed. A larger flow magnitude leads
to a faster spheroid suppression (Fig. 2(c)). We also show in Fig. 2(d) an
example of successful suppression protocol for which the magnitude of the flow
is lowered as the spheroid size decreases.

(a) (b) (c)

Figure 3: Dimensionless spheroid size $R_{2}/R_{0}$ (red curve) and lumen size
$R_{1}/R_{0}$ (blue curve) as a function of dimensionless time
$\hat{t}=t/\tau_{0}$ in the presence of an imposed electric current
$\hat{I}(\hat{t})=I^{\rm ext}(t)/I_{0}$ and imposed flow
$\hat{Q}(\hat{t})=Q^{\rm ext}(t)/Q_{0}$ (dashed brown and black curve; their
magnitude is not at scale). The definitions of $\tau_{0},R_{0},Q_{0}$ and
$I_{0}$ are given in Eq. (12). (a) Spheroid growth without external
intervention. (b) Successful protocol with external electric field. (c)
Successful protocol with external electric field and flow. Parameter values
used for these plots can be found in Table 2.

Conversely, if the magnitude of the imposed external flow is not sufficient,
or if the duration of the flow is too short, the protocol can be unsuccessful
in suppressing the spheroid. The spheroid may only have a slower growth rate
(Fig. 2(e)) or shrink significantly but resume growing as soon as the external
flow is turned off (Fig. 2(f)). Note that for any spheroid size, there exists
a critical flow magnitude $|Q_{c}|$ such that imposing a steady flow at
magnitude $|Q^{\rm ext}|>|Q_{c}|$ will eventually lead to the spheroid
suppression, while $|Q^{\rm ext}|<|Q_{c}|$ will be unsuccessful. Figure 2(b)
shows a protocol with $|Q^{\rm ext}|\simeq|Q_{c}|$.

In the examples above we have focused on the case where an external volumetric
flow is imposed. The same analysis and similar procedures can be used in the
case where an external current $I^{\rm ext}(t)$ is imposed (see Fig. 3). We
show in Fig. 3(b) a protocol that leads to the suppression of the spheroid.
Note that an increased shrinking is obtained if one applies simultaneously an
electric current and an external flow, as both effects are additive. Such
intervention is displayed in Fig. 3(c) where we observe a faster shrinking due
to external flow in addition to the application of an electric current.

Importantly, we emphasize that the protocols we are discussing here are slow
and take place on long time scales: using parameters values shown in Table 2,
the examples shown in Figs. 2 and 3 correspond to $R_{0}\simeq 1$ cm and
$\tau_{0}\simeq 10$ days. This shows that the suppression of the spheroid
requires a slow and steady flow or the application of a small electric current
for several weeks.

(a) (b) (c)

Figure 4: Phase portraits $({\rm d}r_{2}/{\rm d}\hat{t},r_{2})$ of a spheroid
as given by Eq. (13). (a) Example of phase portraits of spheroids that can
grow in the absence of a drain. (b) Example of the effect of a negative value
of the parameter $w_{0}$ on the phase portraits. (c) Example of a negative
value of the parameter $w_{1}$ on the phase portraits. Parameters (in the
order green, blue, orange): $\hat{v}_{2}=\\{0.5,0.45,-0.05\\}$,
$\delta_{2}=\\{-1,1,1\\}$ and $\hat{\lambda}=\\{2,1.92,2.36\\}$. Drain
parameters: plot (a), $w_{0}=w_{1}=0$; plot (b) $w_{0}=-0.2,w_{1}=0$; and plot
(c) $w_{0}=0,w_{1}=-0.1$. Note that we have considered only the numerator of
Eq. (13) to draw these phase portraits as the denominator is always positive.

###  III.2 Spheroid dynamics with external flows and currents

We now discuss quantitatively how a drain can be used to control the size of a
spheroid. To keep the discussion as simple as possible, we consider here the
limit of a spheroid enclosing a small lumen (or lumenless). In the limit of a
small lumen compared to spheroid size $R_{1}\ll R_{2}$, the differential
equation that describes the dynamics of the spheroid shrinks to (see App. A):

$\displaystyle\frac{\mathrm{d}r_{2}}{\mathrm{d}\hat{t}}$
$\displaystyle=\frac{w_{0}+(\hat{v}_{2}+w_{1}/2)r_{2}-\delta_{2}\,r_{2}^{2}+\hat{\lambda}\,r_{2}^{3}/4}{r_{2}(3+r_{2})}\,,$
(13)

where we use the dimensionless radius $r_{2}$ and time $\hat{t}$ introduced in
the previous section. The dimensionless parameters $w_{0,1}$ describe the
effects of an externally-imposed flow or electric current. If they are set to
zero, Eq. (13) reduces to the dynamics of a lumenless spheroid without drain
as studied in Ref. Duclut  _et al._ (2019). They are defined as
$w_{n}=j_{n}\hat{I}+u_{n}\hat{Q}$ with $n=0,1$ and where $\hat{Q}$ and
$\hat{I}$ are the dimensionless volumetric flow and electric current,
respectively, as defined above. The effects of an imposed external flow are
captured by the coefficients:

$\displaystyle u_{1}\\!=\\!\frac{\kappa^{\rm eff}\bar{\eta}(\Lambda^{\rm
f}_{2})^{2}}{(1-\phi)}\,,\\!\quad u_{0}\\!=\\!1-\beta_{\rm u}\,.$ (14)

Here, $u_{1}$ describes a permeation effect due to the finite permeability of
the spheroid to fluid flows and involves the ratio of the surface permeability
$\Lambda^{\rm f}_{2}$ and of the (effective) bulk permeability $\propto
1/\kappa^{\rm eff}$. The parameter $u_{0}$ includes direct effects of the
external flow – proportional to $Q^{\rm ext}/\Lambda^{\rm f}_{2}$ in the
dimensional equation corresponding to $u_{0}=1$ – which reflects the spheroid
volume change imposed by the external flow due to tissue incompressibility
(see also App. A). We have furthermore defined

$\displaystyle\beta_{{\rm u}}=\frac{\Lambda^{\rm
f}_{2}}{1-\phi}\left(3\nu_{2}-4\nu_{0}\nu_{4}+\frac{\bar{\kappa}}{\Lambda}(3\nu_{1}-4\nu_{0}\nu_{3})\right)\,,$
(15)

which represents the bioelectric and biohydraulic contribution. Indeed, this
term is a sum of terms proportional to the parameters $\nu_{i}$, and thus
stems from the coupling between the electric field (or the interstitial fluid
flow) and the cell polarity that appears in the cell stress.

Imposing an external electric current $I^{\rm ext}$ also contributes to the
spheroid growth control via the parameters:

$\displaystyle j_{1}=1\,,\quad j_{0}=\beta_{\rm j}\,.$ (16)

The coefficient $j_{1}=1$ corresponds to electroosmotic flow due to the
imposed current. The coefficient $j_{0}$ accounts for bioelectric and
biohydraulic contributions where

$\displaystyle\beta_{{\rm
j}}=\frac{3\nu_{1}-4\nu_{0}\nu_{3}}{\lambda\bar{\eta}\Lambda^{\rm f}_{2}}\,.$
(17)

Note that since the coefficients $\beta_{\rm u,j}$ are a sum of
phenomenological parameters for which we only have order of magnitude
estimates, it is difficult to obtain a reliable estimate of their magnitude
and even of their sign. However, estimating upper bounds for $\beta_{\rm u,j}$
suggests that these bioelectric and biohydraulic contributions are small
compared to the other effects. A further discussion of these contributions
requires precise estimates of the bioelectric and biohydraulic couplings that
could be obtained from experiments on spheroids in presence of a drain.

In equation (13), we have used the definitions Duclut  _et al._ (2019)

$\displaystyle\begin{split}&\hat{\lambda}\\!=\\!\frac{\lambda^{\rm
eff}\bar{\eta}\Lambda^{\rm f}_{2}}{|P^{\rm
eff}_{2}|}\,,\quad\delta_{2}=\frac{P^{\rm eff}_{2}}{|P^{\rm
eff}_{2}|}\,,\quad\hat{v}_{2}=\frac{v^{\rm eff}_{2}}{\Lambda^{\rm
f}_{2}|P^{\rm eff}_{2}|}\,,\\\ &v^{\rm eff}_{2}=3v_{2}-\frac{2(\gamma^{\rm
app}_{2}+\gamma^{\rm app}_{0})}{\bar{\eta}}\,.\end{split}$ (18)

Here, $\lambda^{\rm eff}$ is an an effective pumping coefficient and
$\gamma^{\rm app}_{0,2}$ are apparent tensions defined in App. A. The apparent
surface tension $\gamma^{\rm app}_{2}$ is a modification of the tissue surface
tension $\gamma_{2}$ stemming from the flexoelectric term proportional to
$\Lambda_{4}$. This flexoelectric contribution plays a crucial role in lumen
nucleation Duclut  _et al._ (2019). In the dynamics of the outer radius of the
spheroid, its effect is however minor. Finally, the effective pumping
$\lambda^{\rm eff}$ combines the active pumping $\lambda_{1}$ and an electric
contribution $\Lambda_{1}\lambda/\Lambda$ due to electroosmosis.

(a) (b) (c)

Figure 5: State diagrams of the long-time growth behavior of the spheroid as a
function of its initial radius $R_{2}(t=0)=R_{2}^{0}$ and of the imposed
volumetric flux $Q^{\rm ext}$. Shades of blue indicate the time for
suppression $t_{\rm s}$ (see text). Note that all parameters are made
dimensionless using the normalization given in Eq. (12). These plots are
obtained by solving the dynamics equations (21b) with the parameter values
given in Table 2 and with a fixed value of $R_{1}=2R_{\rm d}$.

###  III.3 Control of the spheroid dynamics

We use Eq. (13) to discuss how external currents and flows can be used to
control the growth and shrinkage of a spheroid. Note that whether the spheroid
grows (${\rm d}r_{2}/{\rm d}t>0$) or shrinks (${\rm d}r_{2}/{\rm d}t<0$) only
depends on the numerator of Eq. (13).

Figure 4 shows phase-space trajectories of the outer radius $r_{2}$ for
spheroids that are able to grow in the absence of a drain. Figure 4(a)
displays the case without external flow or current. The following scenarios
are possible: the spheroid may be growing with an unstable fixed point at
$r_{2}=0$ (green curve); alternatively, there might be an additional stable
fixed point corresponding to a steady-state, either with finite radius
$r_{2}^{*}\neq 0$ (blue curve), or with vanishing radius $r_{2}^{*}$ (orange
curve). Imposing an external flow implies nonzero values of $w_{0}$ and
$w_{1}$, which are negative in the case where fluid flows out of the lumen (as
depicted in Fig. 1(b)). The case of a nonvanishing value of $w_{0}$ is
displayed in Fig. 4(b): in this case, the trajectories are shifted downward by
an amount $w_{0}$. In this case, even the the growing spheroid (green curve)
has a critical radius below which it shrinks (empty green circle). The stable
steady-state (blue curve) has moved to $r_{2}^{*}=0$. Figure 4(c) shows the
effect of a nonvanishing value of $w_{1}$: the slope of the phase-space
trajectories is modified. Note that this perturbation favors shrinking of the
spheroid: the unstable critical radius below which the spheroid shrinks (empty
circles along the $x$ axis) is shifted to larger values compared to the case
without external perturbation shown in Fig. 4(a).

We can estimate the critical volumetric flow $Q_{c}$ needed to change a
growing spheroid of size $R_{2}$ to a shrinking one. From Eq. (13), we obtain

$\displaystyle Q_{c}=-\frac{4\pi\Lambda^{\rm f}_{2}R_{2}(\bar{\eta}v^{\rm
eff}_{2}-P^{\rm eff}_{2}R_{2}+\lambda^{\rm
eff}R_{2}^{2}/4)}{1+R_{2}\kappa^{\rm eff}\Lambda^{\rm
f}_{2}/2(1-\phi)-\beta_{\rm u}}\,,$ (19)

where the minus sign indicates that $Q^{\rm ext}<0$ is required to induce
shrinkage (corresponding to flow out of the drain, see Fig. 1(b)). We obtain a
similar expression for the critical electric current that is required to
induce shrinkage:

$\displaystyle I_{c}=\frac{4\pi R_{2}(\bar{\eta}v^{\rm eff}_{2}-P^{\rm
eff}_{2}R_{2}+\lambda^{\rm
eff}R_{2}^{2}/4)}{-R_{2}\lambda/(2\Lambda)+\bar{\eta}\Lambda^{\rm
f}_{2}\beta_{\rm j}}\,.$ (20)

Note that $\lambda<0$ and therefore $I_{c}$ is positive (corresponding to an
electric current into the drain, see Fig. 1(a)). One can therefore always find
a value of the externally applied flow or current to turn a growing spheroid
into a shrinking one.

###  III.4 State diagrams of hydraulic control of spheroid growth

We conclude this section by presenting state diagrams for a spheroid of
initial size $R_{2}(t=0)=R_{2}^{0}$ to which a steady external flow $Q^{\rm
ext}$ is imposed, see Fig. 5. We consider a spheroid with a drain inserted to
the center but without lumen. This corresponds to choosing $R_{1}$ equal to
the outer radius of the drain. For simplicity, we fix $R_{1}=2R_{\rm d}$,
where $R_{\rm d}$ is the inner drain radius. We again consider spheroids that
can grow in the absence of a drain. Figure 5(a) displays the state diagram for
the case where the unperturbed spheroid is growing at all radii (corresponding
to the green curve in Fig. 4(a)). Figure 5(b) shows the state diagram for a
spheroid which is growing above a critical radius (corresponding to the yellow
curve in Fig. 4(a)). Figure 5(c) is the state diagram for spheroids that, in
the absence of a drain, either reach a steady-state radius or grow without
bounds (corresponding to the blue curve in Fig. 4(a)).

Three different regions exist in the diagrams shown in Fig. 5: (i) a growth
region (green), where the imposed flow is not sufficient to arrest growth.
Note that for increasing values of $Q^{\rm ext}$, growth slows within this
region; (ii) a steady-state region (in orange), in which the spheroid reaches
a finite size. Within this region a larger imposed flow drives the spheroid to
smaller steady-state sizes; (iii) a suppression region (in blue) where the
spheroid shrinks until it is suppressed. We define that a spheroid has been
suppressed when its size becomes smaller than a cut-off size
$R_{2}=R_{1}+h_{\rm c}$, with $h_{\rm c}=10~{}\mu$m a typical cell thickness.
In the region of spheroid suppression, the time $t_{\rm s}$ that it takes for
the spheroid to be suppressed is indicated by shades of blue. For fixed
initial spheroid size, the time for suppression decreases as the magnitude of
the imposed flow increases.

The solid black lines indicate boundaries between the different regions. Note
that the line separating the steady-state region and the suppression region
depends on the cut-off $h_{\rm c}$. The dashed white line indicates the
nullcline ${\rm d}r_{2}/{\rm d}\hat{t}=0$ and gives the critical value of the
external flow $Q_{c}$ required to switch from a growing to a shrinking
spheroid (given by Eq. (19) for $R_{1}=0$). Note that below the nullcline the
spheroids grow and above the nullcline they shrink. Note that a similar
analysis can be performed in the case of an external current $I^{\rm ext}$. In
this case the same behaviors are found: growth, suppression and steady-state
regions which depends on the initial spheroid size and on the magnitude of the
imposed electric current.

##  IV Conclusion

Using a coarse-grained description of a tissue as an active material capable
of exerting mechanical stresses, transporting ions and pumping fluid, we have
proposed a technique to perturb tissues electrically and hydraulically. This
technique relies on an imposed fluid flow or electric current source using a
micrometric drain or electrode which can turn a growing spheroid to shrinkage.

Our theory allows us to estimate orders of magnitudes of the external flow and
current that are required to suppress an initially growing spheroid. For small
spheroids with radius of about a hundred micrometers, we show for instance
that an electric current of a few nanoamperes or a volumetric flow of about
$10^{3}~{}\mu$m3/s could have a significant impact on the spheroid state (see
Fig. 6 in App. D). It means that even a passive drain connecting the inner
part of the spheroid to the surrounding medium could already alter growth of
small spheroids. For larger aggregates, connecting the drain to a pump leads
to spheroid suppression for any size, provided that the imposed external flow
is sufficiently strong and maintained long enough (for instance, for spheroids
of a few millimeters in radius an electric current of tens to hundreds of
nanoamperes or a volumetric flow of about $10^{4}-10^{5}~{}\mu$m3/s could be
sufficient, see Fig. 6). Spheroid suppression can be obtained also by imposing
an electric current, and both fluid flow and electric current application
could be combined to accelerate the process.

Our approach has also allowed us to characterize the different long-time
states of a growing spheroid subject to an external perturbation. Depending on
the magnitude of the hydraulic or electrical field, we have shown that the
growth can be either slowed down or arrested (Fig. 5). In the latter case, the
arrest of growth can lead to a steady state, or even to the shrinking and
eventually the suppression of the spheroid. If the magnitude of the external
field is above a threshold value, the suppression of the spheroid is always
achieved, and stronger magnitudes then accelerate the process. Our coarse-
grained description is generic, as it depends on effective tissue parameters
but is insensitive to many details. Our approach does not distinguish between
a suppressed state where all cells have disappeared and a state where a few
cells still remain. This distinction depends on cellular details that we are
not considering here.

We have also shown how different protocols can be used to obtain this
suppression: for instance, a longer perturbation at a weaker magnitude is
slower but as efficient as a shorter but stronger one; decreasing the
magnitude of the flow or electric current as the size of the spheroid
decreases can also be used to suppress a spheroid (see Fig. 2 for examples).
Interestingly, protocols that could lead to spheroid suppression need to be
carried out over a sufficiently long time period. We estimate that the slow,
steady flow or electric current that mediates the progressive suppression of a
spheroid has to be maintained over days or weeks. This is in contrast with
typical cancer ablation techniques Knavel and Brace (2013), for which
treatments are brief and intense. Radiofrequency ablation for instance
requires the application of an alternating current with high frequency (up to
500 kHz) and high voltage (up to several kV) in order to heat the tissue Erez
and Shitzer (1980). Similarly, irreversible electroporation ablations use
microsecond pulses of high electric potential (up to 3 kV) Miller  _et al._
(2005).

At the time scales considered in this manuscript, the case of a tumor is more
complex than that of a spheroid. First, one has to compare the effect of the
flux on the tumor with that on the surrounding healthy tissue. This question
will be addressed in future work. Second, the geometry is in general more
complex than a simple sphere, a difficulty which will have to be dealt with.
Last, during the suppression process, the escape of metastatic cells should be
prevented. The signs of stresses that we find here corresponds to flows
towards the sphere center. This suggests that escape of material might be
hampered by imposed flows or currents, thereby providing a barrier against the
escape of cells.

Our work is based on generic mechanical, hydraulic and electrical properties
of tissues. It thus paves the way for novel methods to control spheroid size.
If applied to cancerous tissues, it could provide a means to influence
cancerous tumor growth, arrest their proliferation and even suppress these
tumors. Indeed, the hydraulic and electrical perturbations we have proposed
here do not rely on identifying and hindering specific chemical pathways that
are characteristic of cancerous tissue Hanahan and Weinberg (2011), but rather
on general, physical responses to an external field.

### Acknowledgments

We thank C.T. Lim and T.B. Saw for useful discussions.

##  Appendix A Dynamics of a spheroid in the presence of a drain

The dynamical equations for the spheroid and lumen radii are obtained by first
considering force balance (1) in spherical coordinates with boundary
conditions (9). This yields the velocity profiles of cells and interstial
fluid. A differential equation for the radii dynamics $R_{1,2}(t)$ is then
obtained from the normal stress balance (7) together with the permeation
conditions at the boundary given by Eq. (8). Details on this computation in
the case without a drain can be found in Ref. Duclut  _et al._ (2019).
Including the external fields due to the presence of the drain, we obtain the
following coupled differential equations for the dynamics of the radii $r_{1}$
and $r_{2}$ of the lumen and of the spheroid:

$\displaystyle\begin{split}&\chi\frac{\mathrm{d}r_{1}}{\mathrm{d}\hat{t}}+\frac{3(\hat{V}_{1}+\mathrm{d}r_{1}/\mathrm{d}\hat{t})r_{1}^{2}}{r_{2}^{3}-r_{1}^{3}}+\frac{3(\hat{V}_{2}-\mathrm{d}r_{2}/\mathrm{d}\hat{t})r_{2}^{2}}{r_{2}^{3}-r_{1}^{3}}=\delta_{1}\\\
&-\frac{2}{r_{1}}\left(\hat{\gamma}_{1}-\hat{\gamma}_{0}\frac{r_{1}(r_{1}+r_{2})}{r_{1}^{2}+r_{2}^{2}+r_{1}r_{2}}\right)+(u_{1}\hat{Q}+\hat{I})\frac{1}{r_{1}}\frac{(r_{2}-r_{1})(r_{1}+2r_{2})}{2(r_{1}^{2}+r_{2}^{2}+r_{1}r_{2})}\\\
&+\frac{1}{r_{1}^{2}}\left(\chi\hat{Q}+u_{2}\hat{Q}+j_{2}\hat{I}-(u_{3}\hat{Q}+j_{3}\hat{I})\frac{(r_{2}-r_{1})(2r_{1}+r_{2})}{r_{1}^{2}+r_{2}^{2}+r_{1}r_{2}}\right)\\\
&+\hat{\lambda}(r_{2}-r_{1})\frac{r_{1}^{2}+2r_{1}r_{2}+3r_{2}^{2}}{4(r_{1}^{2}+r_{2}^{2}+r_{1}r_{2})}\,,\end{split}$
(21a)
$\displaystyle\begin{split}&-\frac{\mathrm{d}r_{2}}{\mathrm{d}\hat{t}}+\frac{3(\hat{V}_{1}+\mathrm{d}r_{1}/\mathrm{d}\hat{t})r_{1}^{2}}{r_{2}^{3}-r_{1}^{3}}+\frac{3(\hat{V}_{2}-\mathrm{d}r_{2}/\mathrm{d}\hat{t})r_{2}^{2}}{r_{2}^{3}-r_{1}^{3}}=\delta_{2}\\\
&+\frac{2}{r_{2}}\left(\hat{\gamma}_{2}+\hat{\gamma}_{0}\frac{r_{2}(r_{1}+r_{2})}{r_{1}^{2}+r_{2}^{2}+r_{1}r_{2}}\right)-(u_{1}\hat{Q}+\hat{I})\frac{1}{r_{2}}\frac{(r_{2}-r_{1})(2r_{1}+r_{2})}{2(r_{1}^{2}+r_{2}^{2}+r_{1}r_{2})}\\\
&+\frac{1}{r_{2}^{2}}\left(-\hat{Q}+u_{2}\hat{Q}+j_{2}\hat{I}+(u_{3}\hat{Q}+j_{3}\hat{I})\frac{(r_{2}-r_{1})(r_{1}+2r_{2})}{r_{1}^{2}+r_{2}^{2}+r_{1}r_{2}}\right)\\\
&-\hat{\lambda}(r_{2}-r_{1})\frac{3r_{1}^{2}+2r_{1}r_{2}+r_{2}^{2}}{4(r_{1}^{2}+r_{2}^{2}+r_{1}r_{2})}\,,\end{split}$
(21b)

where we have introduced dimensionless radii $r_{i}(\hat{t})=R_{i}(t)/R_{0}$
with $R_{0}=\Lambda^{\rm f}_{2}\bar{\eta}$ and a dimensionless time
$\hat{t}=t/\tau_{0}$ with $\tau_{0}=\bar{\eta}/|P^{\rm eff}_{2}|$. We have
also introduced the dimensionless parameters:

$\displaystyle\begin{split}&\hat{\gamma}_{0,1,2}\\!=\\!\frac{\gamma^{\rm
app}_{0,1,2}}{\bar{\eta}\Lambda^{\rm f}_{2}|P^{\rm
eff}_{2}|}\,,\quad\hat{V}_{1,2}=\frac{v_{1,2}}{\Lambda^{\rm f}_{2}|P^{\rm
eff}_{2}|}\,,\quad\delta_{1,2}\\!=\\!\frac{P^{\rm eff}_{1,2}}{|P^{\rm
eff}_{2}|}\\\ &\hat{\lambda}\\!=\\!\frac{\lambda^{\rm
eff}\bar{\eta}\Lambda^{\rm f}_{2}}{|P^{\rm
eff}_{2}|}\,,\quad\chi\\!=\\!\frac{\Lambda^{\rm f}_{2}}{\Lambda^{\rm
f}_{1}}\,,\end{split}$ (22)

where the effective parameters are defined as:

$\displaystyle\begin{split}&P^{\rm eff}_{1,2}=\Pi_{1,2}^{\rm ext}-P_{\rm
h}^{\rm c}-\frac{J_{{\rm p},1,2}}{\Lambda^{\rm f}_{1,2}}\\\
&-\frac{2}{3}\left(\zeta_{0}\nu_{0}+\lambda_{4}+\frac{\Lambda_{4}\lambda+(3\nu_{1}/2-2\nu_{0}\nu_{3})\Lambda_{1}}{\Lambda}\right),\end{split}$
(23a) $\displaystyle\gamma^{\rm app}_{1,2}=\gamma_{1,2}\mp
4\nu_{3}\Lambda_{4}/\Lambda\,,\quad\gamma^{\rm
app}_{0}=\left(3\nu_{1}/2-2(2+\nu_{0})\nu_{3}\right)(\Lambda_{4}/\Lambda)\,,$
(23b) $\displaystyle\lambda^{\rm
eff}=\lambda_{1}-\Lambda_{1}\lambda/\Lambda\,.$ (23c)

Here $\Lambda=\Lambda_{2}+2\Lambda_{3}/3$ is an effective conductivity and
where $\lambda=\lambda_{2}+2\lambda_{3}/3$. In addition to these terms, the
presence of external flows and currents gives new dimensionless contributions
which are defined as:

$\displaystyle\hat{Q}=\frac{Q^{\rm ext}}{4\pi\bar{\eta}^{2}(\Lambda^{\rm
f}_{2})^{3}|P^{\rm eff}_{2}|}\,,\quad\hat{I}=\frac{I^{\rm
ext}\lambda}{4\pi\Lambda\bar{\eta}\Lambda^{\rm f}_{2}|P^{\rm
eff}_{2}|}\,,\quad u_{1}\\!=\\!\frac{\kappa^{\rm eff}\bar{\eta}(\Lambda^{\rm
f}_{2})^{2}}{(1-\phi)}\,,$ (24a) $\displaystyle u_{2}\\!=\\!\frac{\Lambda^{\rm
f}_{2}}{(1-\phi)}\left(\nu_{2}-4(\nu_{0}-1)\nu_{4}/3-\frac{\bar{\kappa}}{\Lambda}(4(\nu_{0}-1)\nu_{3}/3-\nu_{1})\right)\,,$
(24b) $\displaystyle u_{3}\\!=\\!\frac{\Lambda^{\rm
f}_{2}}{(1-\phi)}\left(\nu_{2}-2(1+2\nu_{0})\nu_{4}/3-\frac{\bar{\kappa}}{\Lambda}(2(1+2\nu_{0})\nu_{3}/3-\nu_{1})\right)\,,$
(24c) $\displaystyle
j_{2}\\!=\\!\frac{4(\nu_{0}-1)\nu_{3}/3-\nu_{1}}{\lambda\bar{\eta}\Lambda^{\rm
f}_{2}}\,,\quad
j_{3}\\!=\\!\frac{2(1+2\nu_{0})\nu_{3}/3-\nu_{1}}{\lambda\bar{\eta}\Lambda^{\rm
f}_{2}}\,,\quad$ (24d)

where $\kappa^{\rm eff}=\kappa-\bar{\kappa}\lambda/\Lambda$ is an effective
friction coefficient.

To understand the effect of an external flux and an external electric current
on the spheroid size, we can rewrite Eq. (21b) in the form of an equation for
the dynamics of the spheroid volume $V=4\pi R_{2}^{3}/3$. To simplify the
analysis, we take a vanishing lumen size ($r_{1}=0$) and we consider only the
terms that comes from the externally imposed current and flux. It yields:

$\displaystyle\begin{split}\frac{\mathrm{d}V}{\mathrm{d}t}&=Q^{\rm ext}+Q^{\rm
ext}\frac{\Lambda^{\rm f}_{2}\kappa^{\rm eff}R_{2}}{1-\phi}+I^{\rm
ext}\frac{\Lambda^{\rm f}_{2}\lambda R_{2}}{\Lambda}-Q^{\rm
ext}(u_{2}+2u_{3})\\\ &-I^{\rm ext}\frac{\lambda\Lambda^{\rm
f}_{2}}{\Lambda}(j_{2}+2j_{3})\,,\end{split}$ (25)

where we have reintroduced dimensional quantities to simplify the following
discussion. The first term on the right-hand side of this equation is a direct
consequence of imposing an external flow: since the tissue is incompressible,
an imposed flow from the outside to the inside of the spheroid ($Q^{\rm
ext}>0$) provokes an increase in volume, while an imposed flow from the inside
of the spheroid to the outside ($Q^{\rm ext}<0$) shrinks its size. The second
term is a signature of the finite permeability of the tissue, which acts as a
porous medium, and the effect is therefore proportional to the ratio of the
surface permeability $\Lambda^{\rm f}_{2}$ and the bulk permeability
$(\kappa^{\rm eff}R_{2}/(1-\phi))^{-1}$. The third term accounts for a similar
phenomenon as the second one but due to the electroosmotically generated flow
due to the imposed current. The two last terms correspond to bioelectric and
biohydraulic contributions: indeed, the terms $\beta_{\rm v,1,2}$ and
$\beta_{\rm j,1,2}$ involve the parameters $\nu_{i}$ and thus stem from the
coupling between the electric field (or the interstitial fluid flow) and the
cell polarity that appears in the cell stress.

In the case of a small lumen compared to the spheroid size, one can consider
the limit $R_{1}\ll R_{2}$ to describe the spheroid dynamics. In this limit,
we can in particular obtain from Eq. (21b) an equation for the dynamics of
$r_{2}$ only, which is given by Eq. (13) in the main text. The parameters
$\beta_{\rm u,j}$ introduced in the main text read $\beta_{\rm
u}=u_{2}+2u_{3}$ and $\beta_{\rm j}=j_{2}+2j_{3}$.

##  Appendix B Imposed pressure and electric potential difference

The main text describes the spheroid size control when an external volumetric
flow $Q^{\rm ext}$ or an external current $I^{\rm ext}$ is imposed.
Alternatively, a pressure difference or an electric potential difference can
be imposed between the outer part of the drain and the outer layer of the
spheroid, which can be considered as the conjugate ensemble. In this appendix,
we provide the additional equations required to discuss this situation.

### Imposed pressure difference

In order to compute the pressure difference $\Delta P=P^{\rm d}-P^{\rm
ext}_{2}$, we decompose it as $\Delta P=\Delta P_{\rm d1}+\Delta P_{12}$ where
$\Delta P_{12}=P^{\rm ext}_{1}-P^{\rm ext}_{2}$ is the pressure difference
across the spheroid and $\Delta P_{\rm d1}=P^{\rm d}-P^{\rm ext}_{1}$ is the
pressure difference between the outer part of the drain and the lumen and
reads:

$\displaystyle\Delta P_{\rm d1}=Q^{\rm ext}(K_{\rm d}+K_{\rm lum})\,,$ (26)

where $K_{\rm d,lum}$ are the hydraulic resistances of the drain and of the
lumen, respectively. The drain hydraulic resistance is given by $K_{\rm
d}=\frac{8\eta^{\rm f}L_{\rm d}}{\pi R_{\rm d}^{4}}$, where $\eta^{\rm
f}\simeq 10$ mPa.s is the viscosity of the interstitial fluid, $L_{\rm
d},R_{\rm d}$ the width and radius of the drain.

The pressure difference $\Delta P_{12}=P^{\rm ext}_{1}-P^{\rm ext}_{2}$ can
then be computed using Eq. (8). In dimensionless form, it reads:

$\displaystyle\Delta\hat{P}_{12}=\Delta\tilde{\Pi}+\hat{Q}\left(\frac{\chi}{r_{1}^{2}}+\frac{1}{r_{2}^{2}}\right)-(\chi\dot{r}_{1}+\dot{r}_{2})-\Delta\hat{P}^{\rm
f}\,,$ (27)

where we use the short-hand notation
$\dot{r}_{1,2}=\mathrm{d}r_{1,2}/\mathrm{d}\hat{t}$ and we have defined:

$\displaystyle\begin{split}&\Delta\hat{P}_{12}=\frac{P^{\rm ext}_{1}-P^{\rm
ext}_{2}}{|P^{\rm eff}_{2}|}\,,\quad\Delta\hat{P}^{\rm f}=\frac{P^{\rm
f}(r_{1})-P^{\rm f}(r_{2})}{|P^{\rm eff}_{2}|}\\\
&\Delta\tilde{\Pi}=\frac{\Pi_{1}-\Pi_{2}-(J_{1}/\Lambda^{\rm
f}_{1}-J_{2}/\Lambda^{\rm f}_{2})}{|P^{\rm eff}_{2}|}\,.\end{split}$ (28)

The dimensionless quantities $P_{2}^{\rm eff}$, $r_{1,2}$, $\chi$ and
$\hat{Q}$ have been introduced in App. A. To compute the interstitial pressure
difference $\Delta P^{\rm f}=P^{\rm f}(R_{1})-P^{\rm f}(R_{2})$, one can use
the identity $\Delta P^{\rm
f}=-\int_{R_{1}}^{R_{2}}\mathrm{d}r\,\partial_{r}P^{\rm f}(r)$. Indeed, the
gradient of the interstitial fluid pressure can be obtained using force
balance (1b) and the constitutive equations (3) and (4). It reads:

$\displaystyle\partial_{\hat{r}}\hat{P}^{\rm
f}(\hat{r})=u_{1}\hat{v}(\hat{r})-\hat{\lambda}-\frac{2\hat{\lambda}_{4}}{\hat{r}}-\frac{u_{1}\hat{Q}+\hat{I}}{\hat{r}^{2}}\,,$
(29)

where we have introduced, in addition to the dimensionless quantities already
introduced in App. A, the dimensionless radius $\hat{r}=r/R_{0}$, cell
velocity $\hat{v}=v^{\rm c}_{r}/(\Lambda^{\rm f}_{2}|P^{\rm eff}_{2}|)$, and
interstitial fluid pressure $\hat{P}^{\rm f}=P^{\rm f}/|P^{\rm eff}_{2}|$ and
the dimensionless parameter
$\hat{\lambda}_{4}=(\lambda_{4}-\Lambda_{4}\lambda/\Lambda)/|P^{\rm
eff}_{2}|$. The dimensionless cell velocity appearing in Eq. (29) can be cast
into the following form:

$\displaystyle\hat{v}(\hat{r})=\frac{a}{\hat{r}^{2}}+\frac{b}{\hat{r}}+c+d\hat{r}+e\hat{r}^{2}\,,$
(30)

where we have introduced the following $\hat{r}$-independent factors:

$\displaystyle\begin{split}a&=\frac{r_{1}^{2}r_{2}^{2}}{r_{12}^{2}}\left(r_{1}(\tilde{v}_{2}-\dot{r}_{2})+r_{2}(\tilde{v}_{1}+\dot{r}_{1})\right)-\frac{\hat{\lambda}r_{1}^{3}r_{2}^{3}}{4r_{12}^{2}}\\\
&\quad+\frac{\hat{Q}r_{1}r_{2}}{r_{12}^{2}}\left(u_{3}(r_{1}+r_{2})-\frac{u_{1}r_{1}r_{2}}{2}\right)+\frac{\hat{I}r_{1}r_{2}}{r_{12}^{2}}\left(j_{3}(r_{1}+r_{2})-\frac{r_{1}r_{2}}{2}\right)\,,\end{split}$
(31a) $\displaystyle b$ $\displaystyle=-u_{3}\hat{Q}-j_{3}\hat{I}\,,$ (31b)
$\displaystyle c$
$\displaystyle=\frac{u_{1}\hat{Q}}{2}+\frac{\hat{I}}{2}-\frac{2\hat{\gamma}_{0}}{3}\,,$
(31c)
$\displaystyle\begin{split}d&=\frac{\hat{\lambda}}{4}\left(r_{2}+\frac{r_{1}^{3}}{r_{12}^{2}}\right)-\frac{(\tilde{v}_{1}+\dot{r}_{1})r_{1}^{2}+(\tilde{v}_{2}-\dot{r}_{2})r_{2}^{2}}{r_{2}^{3}-r_{1}^{3}}\\\
&\quad+\frac{\hat{Q}}{r_{12}^{2}}\left(u_{3}-\frac{u_{1}(r_{1}+r_{2})}{2}\right)+\frac{\hat{I}}{r_{12}^{2}}\left(j_{3}-\frac{r_{1}+r_{2}}{2}\right)\,,\end{split}$
(31d) $\displaystyle e$ $\displaystyle=-\frac{\hat{\lambda}}{4}\,,$ (31e)

and the following dimensionless quantities (in addition to those already
introduced in App. A):

$\displaystyle r_{12}^{2}$ $\displaystyle=r_{1}^{2}+r_{2}^{2}+r_{1}r_{2}\,,$
(32a) $\displaystyle\tilde{v}_{1,2}$ $\displaystyle=\hat{V}_{1,2}\mp
2\hat{\gamma}_{0}/3\,.$ (32b)

The dimensionless pressure difference (27) can therefore be computed in terms
of the dimensionless parameters of the model, although its expression is
lengthy. One can however expand this expression in the limit of a thin
spheroid (i.e $R_{1}\sim R_{2}$), yielding the expression:

$\displaystyle\Delta\hat{P}_{12}=\frac{1}{|P^{\rm
eff}_{2}|}\frac{2(\gamma_{1}+\gamma_{2})}{R_{2}}+(r_{2}-r_{1})A+\mathcal{O}\left((r_{2}-r_{1})^{2}\right)\,,$
(33)

where the first term on the right-hand side is the Laplace contribution in the
limit $R_{1}\to R_{2}$ and the dimensionless coefficient $A$ represents the
first correction in term of the thickness $R_{2}-R_{1}$ and reads:

$\displaystyle\begin{split}A&=\frac{u_{1}\left(\hat{V}_{2}-\chi\hat{V}_{1}-\Delta\tilde{\Pi}\right)}{1+\chi}+\frac{2}{r_{2}}\left(\frac{u_{1}(\hat{\gamma}_{1}+\hat{\gamma}_{2})}{1+\chi}+\hat{\lambda}_{4}\right)+\frac{2\hat{\gamma}_{1}}{r_{2}^{2}}\\\
&\quad+\frac{\hat{Q}}{r_{2}^{2}}\left(\frac{2(u_{2}-u_{3})}{r_{2}}-u_{1}\right)+\frac{2\hat{I}(j_{3}-j_{2})}{r_{2}^{3}}\,.\end{split}$
(34)

### Imposed electric potential difference

Similarly to the pressure, one can compute the electric potential difference
$\Delta U=U^{\rm d}-U^{\rm ext}_{2}$ by decomposing it as $\Delta U=\Delta
U_{\rm d1}+\Delta U_{12}$ with $\Delta U_{12}=U^{\rm ext}_{1}-U^{\rm ext}_{2}$
the electric potential difference across the spheroid and with $\Delta U_{\rm
d1}=U^{\rm d}-U^{\rm ext}_{1}$ the electric potential difference across the
drain. This difference reads

$\displaystyle\Delta U_{\rm d1}=I^{\rm ext}(1/G_{\rm d}+1/G_{\rm lum})\,,$
(35)

where $G_{\rm d,lum}$ are the electrical conductances of the drain and of the
lumen. The electrical conductance of the drain is computed as $1/G_{\rm
d}=\frac{\rho^{\rm f}L_{\rm d}}{\pi R_{\rm d}^{2}}$, where $\rho^{\rm f}\simeq
1~{}{\rm\Omega}$.m is the resistivity of the interstitial fluid (taken to be
that of salted water, whose conductivity is about $G\simeq 1$ S/m).

The electric potential difference across the spheroid $\Delta U_{12}$ can then
be computed in the following way. From Eq. (4), an expression for the electric
field can be obtained, which can then be used to compute the electric
potential difference using the identity $\Delta
U_{12}=\int_{R_{1}}^{R_{2}}\mathrm{d}r\,E(r)$. Defining the dimensionless
electric field $\hat{E}=E_{r}\,\bar{\eta}\Lambda^{\rm
f}_{2}\lambda/|P_{2}^{\rm eff}|$, we have:

$\displaystyle\hat{E}(\hat{r})=-\hat{\Lambda}_{1}-\bar{u}_{1}\hat{v}(\hat{r})-\frac{2\hat{\Lambda}_{4}}{\hat{r}}-\frac{\bar{u}_{1}\hat{Q}-\hat{I}}{\hat{r}^{2}}\,,$
(36)

where we have defined the dimensionless parameters:

$\displaystyle\bar{u}_{1}=\frac{\bar{\eta}(\Lambda^{\rm
f}_{2})^{2}\bar{\kappa}\lambda}{(1-\phi)\Lambda}\,,\quad\hat{\Lambda}_{1}=\frac{\bar{\eta}\Lambda^{\rm
f}_{2}\lambda\Lambda_{1}}{\Lambda|P_{2}^{\rm
eff}|}\,,\quad\hat{\Lambda}_{4}=\frac{\lambda\Lambda_{4}}{\Lambda|P_{2}^{\rm
eff}|}\,,$ (37)

and where the expression for the dimensionless cell velocity $\hat{v}$ is
displayed in Eq. (30).

Figure 6: Order of magnitudes of the external fields required to observe a
significant change in the spheroid dynamics as a function of its size as given
by Eqs. (10)-(11). (a) Volumetric flow $Q^{\rm ext}$. (b) Electric current
$I^{\rm ext}$.

##  Appendix C Numerical solution to the dynamics equations

Figures 2 and 3 of the main text have been obtained by solving the coupled
equations (21a) and (21b) using the dimensionless parameters given in Table 2.
The presence of the drain provides a natural cut-off for the (dimensionless)
lumen radius $r_{1}$: if at a time $t_{0}$ the inner radius $R_{1}(t)$ reaches
the size of the outer drain radius, which we assume to be equal to $2R_{\rm
d}=10~{}\mu$m, we fix its (dimensionless) value to be $r_{1}(t>t_{0})=2R_{\rm
d}/R_{0}$ and solve only Eq. (21b) with this fixed value for $r_{1}$. If the
external flow is stopped at a subsequent time $t_{1}>t_{0}$ (see Fig. 2(f) of
the main text for instance), we restart solving simultaneously Eqs. (21a) and
(21b), taking $r_{1}(t_{1})=2R_{\rm d}/R_{0}$ as the initial condition.

Figure 5 of the main text has been obtained by taking $r_{1}=2R_{\rm d}/R_{0}$
and solving numerically equation (21b) using the dimensionless parameters
given in Table 2, with the initial condition $R_{2}(t=0)=R_{2}^{0}$ and a
given imposed external flux $Q^{\rm ext}$. If a spheroid reaches a cut-off
size $R_{2}=2R_{\rm d}+h_{\rm c}$ after a time $t=t_{\rm s}$, it is classified
as “suppressed” and its suppression time $t_{\rm s}$ is used to produce the
color coding.

##  Appendix D Estimates for parameter values

Estimation of the phenomenological parameters appearing in the continuum model
is crucial for our analysis. Some of these parameters, such as the cell shear
and bulk viscosities, the tissue surface tension and the bulk permeability for
instance, have already been estimated in experiments. For most of the
remaining parameters, experimental values are not yet available and we have
used order-of-magnitude estimations (see Refs. Sarkar  _et al._ (2019) and
Duclut  _et al._ (2019) for details). We provide parameters values in Table 1
and references for these values.

Parameters | Exp. values | Parameters | Estimations Sarkar  _et al._ (2019); Duclut  _et al._ (2019)
---|---|---|---
$\eta$ Forgacs  _et al._ (1998) |  $10^{4}$ Pa.s | $\bar{\kappa}$ |  $10^{3}$ A.s$/$m3
$\bar{\eta}$ Montel  _et al._ (2011) |  $10^{9}$ Pa.s | $\lambda_{1}$ |  $-10^{8}$ N$/$m3
$\gamma_{1,2}$ Forgacs  _et al._ (1998) |  $10^{-3}$ N/m | $\lambda_{2},\,\lambda_{3}$ |  $-10^{6}$ N$/$m${}^{2}/$V
$\kappa^{-1}$ Netti  _et al._ (2000) |  $10^{-13}$ m2/Pa/s | $\lambda_{4}$ |  $-10^{3}$ N$/$m2
$\Pi_{1,2}^{\rm ext}$ Prather  _et al._ (1968); *brace1977interaction |  $10^{3}$ Pa | $\Lambda_{1}$ |  $10^{0}$ A$/$m2
$v_{1,2}$ Delarue  _et al._ (2014) |  $10^{-10}$ m/s | $\Lambda_{2},\,\Lambda_{3}$ |  $10^{-2}$ A$/$V/m
$P^{\rm c}_{\rm h}$ Montel  _et al._ (2011) |  $-10^{3}$ Pa | $\Lambda_{4}$ |  $10^{-5}$ A$/$m
$\zeta_{0}$ Delarue  _et al._ (2014) |  $10^{3}$ Pa | $\nu_{1}$ |  $10^{1}$ N/m/V
$\zeta_{1}$ Delarue  _et al._ (2014) | $-10^{-1}$ | $\nu_{2}$ |  $10^{8}$ Pa.s/m
$R_{\rm d}$ |  $5\,10^{-6}$ m | $\nu_{3}$ |  $10^{0}$ N/m/V
|  | $\nu_{4}$ |  $10^{7}$ Pa.s/m
|  | $\Lambda_{1,2}^{\rm f}$ |  $10^{-10}$ m/Pa/s
|  | $J_{{\rm p},1,2}$ |  $10^{-11}$ m/s
|  | $1-\phi$ | $10^{-1}$
|  | $\nu_{0}$ | $1$
Table 1: Experimental values and references (left columns) and estimated values (right columns) of the phenomenological parameters of the model appearing in the constitutive equations. Figure | Parameters values
---|---
$\delta_{1}$ | $\delta_{2}$ | $\chi$ | $\hat{\gamma}_{0}$ | $\hat{\gamma}_{1}$ | $\hat{\gamma}_{2}$ | $\hat{\lambda}$ | $\hat{v}_{1}$ | $\hat{v}_{2}$ | $u_{1}$ | $u_{2}$ | $u_{3}$ | $j_{2}$ | $j_{3}$
2 | 0.1 | -1 | 1 | 0 | -0.05 | 0.15 | 0 | 1 | 0.1 | $10^{2}$ | 0 | 0 | 0 | 0
3 | 0.3 | -1 | 1 | 0 | 0.15 | 0.15 | 0.2 | 0.1 | 0.5 | $10^{2}$ | 0 | 0 | 0 | 0
5(a) | - | -1 | 1 | 0 | - | 0.15 | 0.5 | - | 0.3 | $10^{2}$ | 0 | 0 | 0 | 0
5(b) | - | 1 | 1 | -0.15 | - | 0.15 | 5.5 | - | 0 | $10^{2}$ | 0 | 0 | 0 | 0
5(c) | - | 1 | 1 | 0 | - | 0.15 | 4 | - | 0.15 | $10^{2}$ | 0 | 0 | 0 | 0
Table 2: Dimensionless parameter values used for plotting the figures.

## References

  * Green and Sharpe (2015) J. B. A. Green and J. Sharpe, “Positional information and reaction-diffusion: Two big ideas in developmental biology combine,” Development 142, 1203–1211 (2015).
  * Miller and Davidson (2013) C. J. Miller and L. A. Davidson, “The interplay between cell signalling and mechanics in developmental processes,” Nat. Rev. Genet. 14, 733–744 (2013).
  * Mammoto  _et al._ (2013) T. Mammoto, A. Mammoto, and D. E. Ingber, “Mechanobiology and Developmental Control,” Annu. Rev. Cell Dev. Biol. 29, 27–61 (2013).
  * Ladoux and Mège (2017) B. Ladoux and R.-M. Mège, “Mechanobiology of collective cell behaviours,” Nat. Rev. Mol. Cell Biol. 18, 743–757 (2017).
  * Ruiz-Herrero  _et al._ (2017) T. Ruiz-Herrero, K. Alessandri, B. V. Gurchenkov, P. Nassoy, and L. Mahadevan, “Organ size control via hydraulically gated oscillations,” Development 144, 4422–4427 (2017).
  * Dumortier  _et al._ (2019) J. G. Dumortier, M. Le Verge-Serandour, A. F. Tortorelli, A. Mielke, L. de Plater, H. Turlier, and J.-L. Maître, “Hydraulic fracturing and active coarsening position the lumen of the mouse blastocyst,” Science 365, 465–468 (2019).
  * Levin and Martyniuk (2018) M. Levin and C. J. Martyniuk, “The bioelectric code: An ancient computational medium for dynamic control of growth and form,” Biosystems 164, 76–93 (2018).
  * Silver and Nelson (2018) B. B. Silver and C. M. Nelson, “The Bioelectric Code: Reprogramming Cancer and Aging From the Interface of Mechanical and Chemical Microenvironments,” Front. Cell Dev. Biol. 6, 21 (2018).
  * Chan  _et al._ (2019) C. J. Chan, M. Costanzo, T. Ruiz-Herrero, G. Mönke, R. J. Petrie, M. Bergert, A. Diz-Muñoz, L. Mahadevan, and T. Hiiragi, “Hydraulic control of mammalian embryo size and cell fate,” Nature 571, 112–116 (2019).
  * Fütterer  _et al._ (2003) C. Fütterer, C. Colombo, F. Jülicher, and A. Ott, “Morphogenetic oscillations during symmetry breaking of regenerating _Hydra_ _vulgaris_ cells,” EPL 64, 137–143 (2003).
  * Roux (1892) W. Roux, “über die morphologische Polarisation von Eiern und Embryonen durch den electrischen Strom,” Math. Naturwiss. Kl. 101, 27–228 (1892).
  * McCaig  _et al._ (2005) C. D. McCaig, A. M. Rajnicek, B. Song, and M. Zhao, “Controlling Cell Behavior Electrically: Current Views and Future Potential,” Physiol. Rev. 85, 943–978 (2005).
  * McLaughlin and Levin (2018) K. A. McLaughlin and M. Levin, “Bioelectric signaling in regeneration: Mechanisms of ionic controls of growth and form,” Dev. Biol. 433, 177–189 (2018).
  * Watanabe  _et al._ (2009) H. Watanabe, T. Fujisawa, and T. W. Holstein, “Cnidarians and the evolutionary origin of the nervous system: Cnidarian nervous system,” Develop. Growth Differ. 51, 167–183 (2009).
  * Braun and Ori (2019) E. Braun and H. Ori, “Electric-Induced Reversal of Morphogenesis in Hydra,” Biophys. J. 117, 1514–1523 (2019).
  * Martín-Belmonte  _et al._ (2008) F. Martín-Belmonte, W. Yu, A. Rodríguez-Fraticelli, A. Ewald, Z. Werb, M. A. Alonso, and K. Mostov, “Cell-Polarity Dynamics Controls the Mechanism of Lumen Formation in Epithelial Morphogenesis,” Curr. Biol. 18, 507–513 (2008).
  * Debnath  _et al._ (2002) J. Debnath, K. R. Mills, N. L. Collins, M. J. Reginato, S. K. Muthuswamy, and J. S. Brugge, “The Role of Apoptosis in Creating and Maintaining Luminal Space within Normal and Oncogene-Expressing Mammary Acini,” Cell 111, 29–40 (2002).
  * Huch and Koo (2015) M. Huch and B.-K. Koo, “Modeling mouse and human development using organoid cultures,” Development 142, 3113–3125 (2015).
  * Dahl-Jensen and Grapin-Botton (2017) S. Dahl-Jensen and A. Grapin-Botton, “The physics of organoids: A biophysical approach to understanding organogenesis,” Development 144, 946–951 (2017).
  * Alcaraz  _et al._ (2008) J. Alcaraz, R. Xu, H. Mori, C. M. Nelson, R. Mroue, V. A. Spencer, D. Brownfield, D. C. Radisky, C. Bustamante, and M. J. Bissell, “Laminin and biomimetic extracellular elasticity enhance functional differentiation in mammary epithelia,” EMBO J. 27, 2829–2838 (2008).
  * Montel  _et al._ (2011) 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 Clamp Experiments on Multicellular Tumor Spheroids,” Phys. Rev. Lett. 107, 188102 (2011).
  * Delarue  _et al._ (2013) M. Delarue, F. Montel, O. Caen, J. Elgeti, J.-M. Siaugue, D. Vignjevic, J. Prost, J.-F. Joanny, and G. Cappello, “Mechanical Control of Cell flow in Multicellular Spheroids,” Phys. Rev. Lett. 110, 138103 (2013).
  * Kruse  _et al._ (2005) K. Kruse, J. F. Joanny, F. Jülicher, J. Prost, and K. Sekimoto, “Generic theory of active polar gels: A paradigm for cytoskeletal dynamics,” Eur. Phys. J. E 16, 5–16 (2005).
  * Marchetti  _et al._ (2013) M. C. Marchetti, J. F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost, M. Rao, and R. A. Simha, “Hydrodynamics of soft active matter,” Rev. Mod. Phys. 85, 1143–1189 (2013).
  * Ranft  _et al._ (2010) J. Ranft, M. Basan, J. Elgeti, J.-F. Joanny, J. Prost, and F. Jülicher, “Fluidization of tissues by cell division and apoptosis,” Proc. Natl. Acad. Sci. U.S.A. 107, 20863–20868 (2010).
  * Sarkar  _et al._ (2019) N. Sarkar, J. Prost, and F. Jülicher, “Field induced cell proliferation and death in a model epithelium,” New J. Phys. 21, 043035 (2019).
  * Duclut  _et al._ (2019) C. Duclut, N. Sarkar, J. Prost, and F. Jülicher, “Fluid pumping and active flexoelectricity can promote lumen nucleation in cell assemblies,” Proc. Natl. Acad. Sci. U.S.A. 116, 19264–19273 (2019).
  * Scott  _et al._ (2012) A. M. Scott, J. D. Wolchok, and L. J. Old, “Antibody therapy of cancer,” Nat. Rev. Cancer 12, 278–287 (2012).
  * Baskar  _et al._ (2012) R. Baskar, K. A. Lee, R. Yeo, and K.-W. Yeoh, “Cancer and Radiation Therapy: Current Advances and Future Directions,” Int. J. Med. Sci. 9, 193–199 (2012).
  * Al-Bataineh  _et al._ (2012) O. Al-Bataineh, J. Jenne, and P. Huber, “Clinical and future applications of high intensity focused ultrasound in cancer,” Cancer Treat. Rev. 38, 346–353 (2012).
  * Mittelstein  _et al._ (2020) D. R. Mittelstein, J. Ye, E. F. Schibber, A. Roychoudhury, L. T. Martinez, M. H. Fekrazad, M. Ortiz, P. P. Lee, M. G. Shapiro, and M. Gharib, “Selective ablation of cancer cells with low intensity pulsed ultrasound,” Appl. Phys. Lett. 116, 013701 (2020).
  * Tijore  _et al._ (2020) A. Tijore, F. Margadant, M. Yao, A. Hariharan, C. A. Z. Chew, S. Powell, G. K. Bonney, and M. Sheetz, “Ultrasound-mediated mechanical forces selectively kill tumor cells,” bioRxiv 10.1101, 2020.10.09.332726 (2020).
  * Miller  _et al._ (2005) L. Miller, J. Leor, and B. Rubinsky, “Cancer Cells Ablation with Irreversible Electroporation,” Technol. Cancer Res. Treat. 4, 699–705 (2005).
  * Perkons  _et al._ (2018) N. R. Perkons, E. J. Stein, C. Nwaezeapu, J. C. Wildenberg, K. Saleh, R. Itkin-Ofer, D. Ackerman, M. C. Soulen, S. J. Hunt, G. J. Nadolski, and T. P. Gade, “Electrolytic ablation enables cancer cell targeting through pH modulation,” Commun. Biol. 1, 48 (2018).
  * Knavel and Brace (2013) E. M. Knavel and C. L. Brace, “Tumor Ablation: Common Modalities and General Practices,” Tech. Vasc. Interv. Radiol. 16, 192–200 (2013).
  * Ranft  _et al._ (2012) J. Ranft, J. Prost, F. Jülicher, and J.-F. Joanny, “Tissue dynamics with permeation,” Eur. Phys. J. E 35, 46 (2012).
  * Dolega  _et al._ (2017) M. E. Dolega, M. Delarue, F. Ingremeau, J. Prost, A. Delon, and G. Cappello, “Cell-like pressure sensors reveal increase of mechanical stress towards the core of multicellular spheroids under compression,” Nat. Commun. 8, 14056 (2017).
  * Basan  _et al._ (2009) M. Basan, T. Risler, J.-F. Joanny, X. Sastre-Garau, and J. Prost, “Homeostatic competition drives tumor growth and metastasis nucleation,” HFSP Journal 3, 265–272 (2009).
  * Darcy (1856) H. P. G. Darcy,  _Les Fontaines Publiques de La Ville de Dijon_ (Dalmont, Paris, France, 1856).
  * Kirby (2013) B. J. Kirby,  _Micro- and Nanoscale Fluid Mechanics_ (Cambridge University Press, Cambridge, England, 2013).
  * Delarue  _et al._ (2014) M. Delarue, J.-F. Joanny, F. Jülicher, and J. Prost, “Stress distributions and cell flows in a growing cell aggregate,” Interface Focus 4, 20140033 (2014).
  * Erez and Shitzer (1980) A. Erez and A. Shitzer, “Controlled Destruction and Temperature Distributions in Biological Tissues Subjected to Monoactive Electrocoagulation,” J. Biomech. Eng. 102, 42–49 (1980).
  * Hanahan and Weinberg (2011) D. Hanahan and R. A. Weinberg, “Hallmarks of Cancer: The Next Generation,” Cell 144, 646–674 (2011).
  * Forgacs  _et al._ (1998) G. Forgacs, R. A. Foty, Y. Shafrir, and M. S. Steinberg, “Viscoelastic Properties of Living Embryonic Tissues: A Quantitative Study,” Biophys. J. 74, 2227–2234 (1998).
  * Netti  _et al._ (2000) P. A. Netti, D. A. Berk, M. A. Swartz, A. J. Grodzinsky, and R. K. Jain, “Role of Extracellular Matrix Assembly in Interstitial Transport in Solid Tumors,” Cancer Res. 60, 2497 (2000).
  * Prather  _et al._ (1968) J. W. Prather, K. A. Gaar, and A. C. Guyton, “Direct continuous recording of plasma colloid osmotic pressure of whole blood.” J. Appl. Physiol. 24, 602–605 (1968).
  * Brace and Guyton (1977) R. A. Brace and A. C. Guyton, “Interaction of transcapillary Starling forces in the isolated dog forelimb,” Am. J. Physiol. 233, H136–H140 (1977).