CELL DYNAMICS IN MICROFLUIDIC DEVICES UNDER HETEROGENEOUS CHEMOTAXIS AND GROWTH CONDITIONS A MATHEMATICAL STUDY

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CELL DYNAMICS IN MICROFLUIDIC DEVICES UNDER

HETEROGENEOUS CHEMOTAXIS AND GROWTH CONDITIONS:A

MATHEMATICAL STUDY

Jacobo Ayensa-Jiménez

Aragon Institute of Engineering Research

University of Zaragoza

Mariano Esquillor, s.n. 50018, Zaragoza

jacoboaj@unizar.es

Mohamed H. Doweidar

Mechanical Eng. Department,

School of Engineering and Architecture (EINA)

University of Zaragoza

María de Luna s/n, Ediﬁcio Betancourt, 50018, Zaragoza

mohamed@unizar.es

Manuel Doblaré

Aragon Institute of Engineering Research

University of Zaragoza

Mariano Esquillor, s.n. 50018, Zaragoza

mdoblare@unizar.es

Eamonn A. Gaffney

Wolfson Centre for Mathematical Biology

Mathematical Institute

University of Oxford

Woodstock Road, Oxford OX2 6GG, UK

gaffney@maths.ox.ac.uk

November 9, 2022

ABSTRACT

As motivated by studies of cellular motility driven by spatiotemporal chemotactic gradients in mi-

crodevices, we develop a framework for constructing approximate analytical solutions for the location,

speed and cellular densities for cell chemotaxis waves in heterogeneous ﬁelds of chemoattractant

from the underlying partial differential equation models. In particular, such chemotactic waves are

not in general translationally invariant travelling waves, but possess a spatial variation that evolves in

time, and may even may oscillate back and forth in time, according to the details of the chemotactic

gradients. The analytical framework exploits the observation that unbiased cellular diffusive ﬂux is

typically small compared to chemotactic ﬂuxes and is ﬁrst developed and validated for a range of

exemplar scenarios. The framework is subsequently applied to more complex models considering the

full dynamics of the chemoattractant and how this may be driven and controlled within a microde-

vice by considering a range of boundary conditions. In particular, even though solutions cannot be

constructed in all cases, a wide variety of scenarios can be considered analytically, ﬁrstly providing

global insight into the important mechanisms and features of cell motility in complex spatiotemporal

ﬁelds of chemoattractant. Such analytical solutions also provide a means of rapid evaluation of model

predictions, with the prospect of application in computationally demanding investigations relating

theoretical models and experimental observation, such as Bayesian parameter estimation.

1 Introduction

Most biological processes integrate different cell populations, extracellular matrix (ECM) properties, chemotactic

gradients and physical cues, constituting a complex, dynamic and interactive microenvironment [

]. Cells

are also constantly adjusting to accommodate their surroundings, particularly for homeostatically maintaining the

intracellular and extracellular environment within physiological constraints [

]. In response to the different chemical and

physical external stimuli, cells can modify their shape, location, internal structure and genomic expression, as well as

their capacity to proliferate, migrate, differentiate, produce ECM or other biochemical substances, changing, in turn, the

surrounding medium as well as sending new signals to other cells [

]. This two-way interaction between cells and

arXiv:2210.14217v2 [math.AP] 8 Nov 2022

APREPRINT - NOVEMBER 9, 2022

their environment is crucial in physiological processes such as embryogenesis, organ development, homeostasis, repair,

and long-term evolution of tissues and organs among others, as well as in pathological processes such as atherosclerosis

or cancer [

]. Furthermore, developing novel frameworks to investigate and elucidate these mechanisms

and interactions is key to developing novel therapeutic strategies aiming at promoting (blocking) desirable (undesirable)

cellular behaviours [14].

In particular, due to the underlying complexity, in vivo research – both in humans and in animals – is impeded by the

fact it is difﬁcult to control and isolate effects. Thus, a simpler alternative is using in vitro experiments. Nevertheless,

the predictive power currently available, whether in vivo or in vitro, is still poor, as demonstrated the continuous

drop in the number of new drugs appearing annually, despite billion-dollar investments [

]. Indeed, structural

three-dimensionality is one of the most important characteristics of biological processes [

], but in vitro cells are

mostly cultured in a traditional Petri dish (2D culture), where cell behaviour is dramatically different from real tissues

[

]. Recently, microﬂuidics has arisen as a powerful tool to recreate the complex microenvironment that governs

tumour dynamics [

]. This technique allows the reproduction of numerous important features that are lost in 2D

cultures, as well as testing drugs in a much more reliable and efﬁcient way [21, 22, 23, 24, 25].

In addition to such in vitro models, mathematical in silico models are a powerful tool for dealing with many problems in

physics, engineering, and biology. In particular, cell population evolution models based on transport partial differential

equations (PDEs) have been widely used to study many biological processes, including cancer [

]. For instance,

tumour development is a key example of a highly dynamic and complex biological process that originates from

external signals or stimuli modulated by the particular microenvironment. Furthermore, when a given treatment is

applied (surgery, chemotherapy, radiotherapy, immunotherapy, hormones or a combination thereof), the tumour and its

microenvironment undergo signiﬁcant alterations. This leads tumour cells to proliferate and generate microenvironments

that promote the death of surrounding cell types and the survival of tumour cells that are more adaptable and resistant.

That is why, when modelling the enormous variety and complexity of a tumour and its microenvironment, the resulting

differential equations are highly non-linear and strongly coupled [

]. The numerical resolution of the equations,

especially in the era of high performance computing, has been extensively utilised in the simulation of “what if”

scenarios and the study of effects and hypotheses in isolation, something that is often impossible to do with in vivo and

in vitro models [

]. In turn, the construction and exploitation of in silico experiments is thus being increasingly

used in the early stages of designing drugs and therapies against tumours.

A particular niche of interest is Glioblastoma (GBM), the most common and aggressive primary brain tumour [

], with

extensive studies dedicated to mathematical modelling its evolution [

], reproducing aspects of GBM histopathology

[

] and incorporating the inﬂuence of tumour microenvironment (TME) chemical and mechanical cues [

]. It has

been demonstrated that GBM progression is extensively controlled by the local oxygen concentrations and gradients

[

], motivating many studies to incorporate the role of oxygen gradients and hypoxia in tumour progression [

Some models of GBM have reproduced cell culture evolution under different experimental conﬁgurations [

], using a

go-or-grow transition switch, governed by nonlinear activation functions for the chemotaxis and growth. Such studies

therefore implicate the balance between cell migratory and proliferative activity, together with their relation to the

different TME stimuli, as playing a key role in GBM evolution.

Nevertheless, the complexity of the equations to be solved often require numerical simulations that are impractical, due

to the high computational cost, especially in the resolution of inverse problems such as parameter estimation, model

selection, the design of experiments, sensitivity studies, model structural analysis and Uncertainty Quantiﬁcation (UQ).

Although many modern techniques as Reduced Order Models (ROM) and metamodels using Artiﬁcial Intelligence

(AI) have been developed in recent years [

], the existence of analytical solutions, although approximate, provides

key information to test and validate numerical algorithms, inform a mechanism based understanding across parameter

space and to allow initial predictions of Quantities of Interest (QoI), such as travelling fronts, equilibria, the ranges of

variation in the solution across parameter space and parameter sensitivities, among others. Indeed, some works in the

last years have focused on the use of these techniques for analysing GBM progression [42, 43, 44].

The interaction of cells with a chemoattractant leads to a type of Keller - Segel (K-S) model [

], which generally

have a rich structure as reviewed by G. Arumugam and J. Tyagi [

]. One of the main interests concerning the K-S

model is the existence and characteristics of travelling waves (see for instance [

]). In this work, we explore the

dynamics of cell populations under gradients of a chemotactic agent for one-dimensional problems. We move beyond

travelling waves to investigate evolving wave solutions in the heterogeneous environments that are often found in

microdevices and physiology. This general class of problems allows the treatment of a wide variety of situations related

to the evolution of tumours, while the analysis of the associated PDEs enables the quantiﬁcation of histopathological

characteristics, such as the spread of pseudopalisades and the response of the population to oscillatory stimuli. This

knowledge can be used for the design of experiments, to speed up the characterisation processes of cell populations and

to validate or rule out possible models.

APREPRINT - NOVEMBER 9, 2022

In particular, we are interested in modelling cell motility dynamics in microﬂuidic devices, which are experimental

platforms that have been demonstrated to accurately recreate biomimetic physiological conditions [

], with application

in bioengineering and biomedical research [

]. In many situations, the chemotactic agent concentration may be

assumed as known, either because it can be directly measured, or because its concentration can be computed by solving

a diffusion problem that is, to good approximation, decoupled from the cell population ﬁeld.

To proceed, we ﬁrst describe the structure of the mathematical problem associated with the response of cell populations

to chemotactic gradients, together with the general assumptions and hypotheses about the underlying mechanisms. We

derive pertinent features about the solution ﬁeld, for instance that it possesses a migratory structure with a transition

zone wavefront. We are also able to estimate the wave front evolution and the shape of the solution proﬁle. In particular,

we compute an analytical solution for speciﬁc exemplar cases associated with speciﬁc relevant experimental situations.

These include a constant spatial gradient of chemoattractant, together with temporal oscillations associated with a

ﬂuctuating source, quadratic proﬁles of chemoattractant, offering additional nonlinear features, and an exponential

proﬁle of chemoattractants corresponding to the diffusion from a localised source. We further apply the general results

to the analysis of a cell culture microﬂuidic experiment, representing a slightly simpliﬁed version for an in vitro model

of GBM progression, as developed in [

], showing how the methods presented here can generate analytical results for

the simulation of microdevice representations for migratory tumour cell dynamics.

2 Methods

2.1 The model

We study a broad class of problems that are related to the dynamics of a cell culture in microﬂuidic devices under the

inﬂuence of a chemotattractant, when the concentration of the agent can be computed or measured. A schematic view

of this situation is represented in Fig. 1.

(a) Scheme of the experimental conﬁguration. (b) 1D approximation of the cell culture.

Figure 1:

Typical experimental conﬁguration for modelling cell cultures.

Due to the much larger length of the

lateral channels relative to the width of the chamber, the domain geometry of the model is assumed one-dimensional,

with length given by the length of the chamber,

. The cell concentrations are associated with a continuum ﬁeld

u=u(x, t)

, with

denoting time and

the spatial coordinate along the chamber, as indicated. At the lateral edges of

the channel, that is

x= 0, L

, zero ﬂux boundary conditions are imposed, corresponding to the inability of cells to pass

through these boundaries. Image created with BioRender.com.

The non-dimensional equation for the cell population concentration,

, represents cellular diffusion and chemotaxis in

response to a heterogeneous ﬁeld of chemoattractant that generates an advective ﬂux

α(t, x)u

, and also impacts cellular

proliferation via β(t, x)so as to generate the governing equation

ut+ (α(t, x)u)x=Duxx +β(t, x)u(1 −u),(1)

with

D > 0

the non-dimensional cellular diffusion coefﬁcient. This governing equation is also supplemented by

zero-ﬂux boundary conditions, given by

Dux−α(t, x)u|x=0 = 0,(2a)

Dux−α(t, x)u|x=L= 0,(2b)

and initial conditions

u(t= 0, x) = u0(x).(3)

APREPRINT - NOVEMBER 9, 2022

2.2 Computation of the general solution for small diffusion

2.2.1 Outer solution

The main hypothesis, which is usually true for cellular motility due to weak cell-based random motility, is that the

non-dimensional diffusion coefﬁcient satisﬁes

D1

, as we will verify below in the case of GBM cells in microdevices.

Hence, away from boundary layers, diffusion may be neglected compared to the inﬂuence of growth and chemoattractant

driven migration. Then, Eq. (1) may be approximated by:

ut+ (α(t, x)u)x=β(t, x)u(1 −u).(4)

This is a ﬁrst-order hyperbolic PDE, amenable to the method of characteristics. If we know the initial data

u(0, x) =

u0(x)

, we can parameterise the initial data via

with the relation

(t, x, u) = (0, s, u0(s))

. Also, there is another

family of characteristic curves, emerging from the

(t, x)

points where

x= 0

and

t > 0

, with the imposition of the

x= 0

boundary condition of no ﬂux, Eq. (2). Assuming that

α(0, t)6= 0

, and that the boundary is away from the

transition region of the cellular wavefront, so that to excellent approximation ux= 0 since spatial gradients are small,

we conclude from the boundary contition, Eq. (2), that

u(0, t) = 0

to the same level of approximation. Therefore, this

boundary condition can be parameterised via

with the relation

(t, x, u)=(s, 0,0)

. Hence, at

t= 0

and

x= 0

there is

an emerging singular characteristic that splits the domain in two regions. The geometric interpretation of the method of

characteristics is shown in Fig. 2, which shows the projection of the characteristic curves onto the plane (t, x).

t∗

Figure 2:

Projection of the characteristic curves.

The two families of characteristic curves are shown in blue and red.

We have, therefore, two families of characteristic curves. For the ﬁrst one:

dτ= 1, t(0) = 0,(5a)

dτ=α(t, x), x(0) = s, (5b)

dτ=u(β(t, x)(1 −u)−αx(t, x)) , u(0) = u0(s),(5c)

and for the second:

dτ= 1, t(0) = s, (6a)

dτ=α(t, x), x(0) = 0,(6b)

dτ=u(β(t, x)(1 −u)−αx(t, x)) , u(0) = 0.(6c)

Noting the uniqueness of the solution to Eqs. (6) courtesy of Picard’s theorem, we have by inspection that these

equations only generate the trivial solution u(x, t)=0.

APREPRINT - NOVEMBER 9, 2022

However, the solution to the ﬁrst family, given by Eqs. (5), is typically more complex, though one always has

t=τ. (7)

Progress can be readily made when

•α(t, x)is linear in x, such that α(t, x) = a(t)x+b(t)

•α(t, x)is separable, that is α(t, x) = f(x)g(t).

In particular when α(t, x)is linear in xwe have

x=seRt

0a(η)dη+Zt

b(η)eRt

ηa(ξ)dξdη=: F(t;s),(8)

which deﬁnes

F(t;s)

for

α(t, x) = a(t)x+b(t)

. We generalise this deﬁnition so that

x=F(t;s)

on the characteristic

curve given by the value of s, and whenever this relation can be uniquely inverted for s, we write s=G(t;x).

In contrast when α(t, x) = f(x)g(t)Eqs. (5a) and (5b) may be integrated to obtain

f(z)=Zτ

g(η) dη. (9)

In turn, Eq. (9) generates the relation

x=F(t;s)

on the characteristic curve. Further motivation for examples of linear

and separable chemotactic response functions for α(t, x)are given in Section 3 below.

In both cases, or even in the more general case where

F(t;s)

cannot readily be determined analytically for all relevant

t, s

, the location of the transition from

u= 0

, and thus the location of the transition region for the cellular wavefront,

x∗=x∗(t)

, is given by the characteristic with

s= 0

– hence

x∗(t) = F(t; 0)

. This is particularly informative about

the general behaviour of the solution, for instance in determining the wavespeed. With respect to Eq. (5c), we proceed

using the change of variable r= 1/u, we obtain the ODE in terms of the τ=tvariable:

r0(τ)+(β(τ, x(τ;s)) −H(τ;s)) r=β(τ, x(τ;s)),(10)

where for the linear case H(τ;s) = a(τ)and H(τ;s) = f0(F(τ;s))g(τ)for the separable case.

Noting the integration is along a characteristic, and thus

is ﬁxed, this equation is of the form

r0+p(τ)r=q(τ)

for

p(τ) = β(τ, x(τ;s)) −H(τ;s)and q(τ) = β(τ, x(τ;s)), with sﬁxed, so a general expression is given by

r(τ, s) = exp −Zτ

p(η, s) dη 1

u0(s)+Zτ

q(η, s) exp Zη

p(ξ, s) dξdη,(11)

where u0(s)is indeed the value of u0at the location of the characteristic when τ=t= 0.

Recapping, suppose x=F(t;s)may be inverted to give s=G(t;x). Then, noting

x(0, s) = F(0; s) = s,

by the parameterisation of the initial data, we have

u(x, t) = u(x=F(t;s), t)=1/r(t;s=G(t;x)),

and, in particular

u0(s) = u0(F(0, G(t;x))) = u0(G(t;x)).

Combining these expressions with Eq. (11), we obtain a general expression for rand therefore for u=u(x, t):

u(x, t) = 



u0(s) exp Rt

0p(η, s) dη

1 + u0(s)Rt

0q(η, s) exp Rη

0p(ξ, s) dξdη

s=G(t;x)

,(12)

where s=G(t;x)is ﬁxed on each characteristic curve and

p(η, s) = β(η, F (η;s)) −H(η;s),

q(η, s) = β(η, F (η;s)).

Eq. (12) may be also written in terms of xand tdirectly, obtaining

u(x, t) =

u0(G(t;x)) exp Rt

0p(η, G(t;x)) dη

1 + u0(G(t;x)) Rt

0q(η, G(t;x)) exp Rη

0p(ξ, G(t;x)) dξdη.(13)

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摘要：

CELLDYNAMICSINMICROFLUIDICDEVICESUNDERHETEROGENEOUSCHEMOTAXISANDGROWTHCONDITIONS:AMATHEMATICALSTUDYJacoboAyensa-JiménezAragonInstituteofEngineeringResearchUniversityofZaragozaMarianoEsquillor,s.n.50018,Zaragozajacoboaj@unizar.esMohamedH.DoweidarMechanicalEng.Department,SchoolofEngineeringandArchitec...

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