Open access peer-reviewed chapter

Improved Reduced Order Mechanical Model and Finite Element Analysis of Three-Dimensional Deformations of Epithelial Tissues

By Ara S. Avetisyan, Asatur Zh. Khurshudyan and Sergey K. Ohanyan

Submitted: March 9th 2016Reviewed: July 26th 2016Published: December 14th 2016

DOI: 10.5772/65027

Downloaded: 940

Abstract

In this chapter, we analyse non-uniform bending of single-layer cell tissues—epithelia, surrounding organs throughout the body. Dimensionally reduced model is suggested, which is equivalent to membranes with bending stiffness: the total elastic energy of the tissue is a combination of stretching and bending energies. The energy, suggested in this chapter, is a piecewise function, the branches of which correspond to a specific deformation regime: compression, pure bending and stretch.

Keywords

  • morphogenesis
  • embryonic tissue
  • adherens junctions
  • relaxation
  • quasi-convex energy densities
  • isometric deformations
  • short maps
  • tissue mechanics
  • Γ-convergence
  • lower semi-continuity
  • hexagonal elements

1. Introduction

Morphogenesis, the evolutionary process of shape and structure development of organisms or their parts, is driven by certain types of cell shape changes or, in other words, deformations. One type of such deformations is the apical constriction—visible shrinkage of the apical side of cells, leading to bending of epithelia—cell sheets, which surround organs throughout the body. First occurring at early stages of embryogenesis, the apical constriction results initially flat epithelia to be bent obtaining three-dimensional form, which, depending on physiological context and morphogenetic stage, leads to different consequences. It has been first hypothesized in [1] that in various developmental systems the apical constriction may drive the bending of epithelia. Using bulky mechanical construction, the hypothesis of Rhumbler is first practically tested in [2]. The testing mechanism consists of 13 identical bars (cell walls) that are kept apart by means of stiff tubes connecting their centres (cell kernels), and are held in a row by rubber bands connecting their ends (cell membrane). In the first stage, the rubber bands at both sides are stretched equally and the mechanism is in straight equilibrium. Shortening the rubber bands at, for instance, the upper side uniformly in each segment (cell), the mechanism is bent on the side of the greater tension, as it could be expected. Thus, Rhumbler’s hypothesis could take place.

A more illustrative, computer model for verifying Rhumbler’s hypothesis is suggested in [3], where bars and bands are replaced by virtual cells. Prescribing certain mechanical properties to the cell membrane and cytoplasmic components, and assuming that as a result of deformations the volume of a single cell is preserved, which is implicitly done by Rhumbler and Lewis, a model of cuboidal epithelia folding is suggested, based on the local behaviour of individual cells. The model is demonstrated on the example of ventral furrow formation in Drosophila. A sequence of cells in the form of a cylindrical shell, representing the cross section of ventral furrow, is deformed such that apical constriction occurs in its lower row of cells. Increasing the applied stresses, different steps of the furrow formation are illustrated. Figure 1 expresses how much Odell’s model is close to the real microscopic picture [4].

Figure 1.

Ventral furrow formation inDrosophilaaccording to [3] (upper) and [4] (lower).

Much is known about the causes of the apical constriction, but some issues still remain unexplored [58]. The most studied causes include contraction of actin filaments—fibrous network localized at the cortex of the cell, by interacting with motor protein myosin—the most known converter of chemical energy into mechanical work. Under influence of myosin, actin fibres contract leading to shrinkage of the cell in the area of their localization [9, 10]. The contraction size and principal direction of the fibres, that is, the microscopical deformation of a single cell, mainly depend on the tissue type: actin-myosin network contraction may deform columnar (occurring, for instance, in digestive tract and female reproductive system) or cuboid (resp. in kidney tubules) cells into trapezoidal-, wedge- or bottle-shaped cells. The deformation size strongly depends also on the emplacement of deforming cells: cells with different placement are constricted differently, so that macroscopically one observes localized wrinkles (see Figure 2).

Figure 2.

Apical constriction of ventral furrow at different times [10].

Because of the mechanical nature of cell shape change, in particular, apical constriction, its theoretical study first of all must rely on mechanical principles and constitutive laws. In early mechanical models, the tissue is modelled as a continuum material, so that the position and behaviour of individual cells are unimportant, that is, only macroscopic deformation of the tissue is studied. Moreover, the height of the cells (viz. the thickness of the tissue) is supposed to be negligible with respect to other measures, such that the deformation of the tissue can be described in terms of its mid-surface. In such models, the actin network is not explicitly accounted and the contraction forces are modelled as a force term acting on the outer surface of the epithelium. The actin network is explicitly accounted in [11], where thin elastic shell model based on linear Cauchy relations is derived to describe apical constriction in initially non-flat epithelia. The model is tested to simulate apical constriction in initially flat, cylindrical and spherical tissues. Ventral furrow formation in Drosophilais simulated (see Figure 3), and it becomes evident from comparison of Figures 1 and 3 that the model, being three-dimensional, is in good correspondence with that from [3]: Figure 1 corresponds to cross section of cylindrical shell from Figure 3. However, it does not involve the behaviour of individual cells and describes only macroscopic deformation. For further introduction into other mechanical models, we refer to [1126].

Figure 3.

Ventral furrow formation inDrosophilaaccording to [11].

The physical and geometrical characteristics of individual cells are taken into account by mechanical model suggested in [27] describing three-dimensional deformations of cell sheet. Both actin-myosin network contractile tensions and cell-cell adhesion stresses are involved to contribute in epithelial tissue bending. Prescribing specific mechanical characteristics to cytoplasmic components and to the kernel, the effective energy of a cell, which is assumed to be a hexagonal prism with constant volume, is expressed in terms of its basal length. As a result of simulations, it is particularly established that when adhesion stresses, distributed on lateral sides of the cell, are large enough, the effective energy has two local minima, that is, there are two equilibrium cell shapes. More particular, depending on position, cell shape may be discontinuously transited from squamous to columnar forms. Figure 4 shows that there is a whole domain in the plane of lateral adhesion and actin-myosin tensions for which the effective energy has two local minima.

Figure 4.

Epithelial cell aspect ratio as a bistable phenomenon [27]:αlis the contractile force andΛais the actin-myosin tension.

In mathematical terms, it means that the total energy of the tissue is not lower semi-continuous, which implies that there is no convergence of its minimizers, and, therefore, it becomes impossible to derive real deformation of the tissue [28, 29]. To be more illustrative, consider a membrane, that is, a tissue without bending stiffness, which is compressed by boundary stresses. The only way that the membrane can accommodate a compressed state is due to wrinkling[30]. Wrinkles may occur more and more finely, so the limiting deformation will be smooth, but evidently it will not minimize the membrane energy.

To overcome such difficulties, the membrane energy density is usually substituted by its quasi-convex envelope[28], which ensures the lower semi-continuity of its total energy. In calculus of variations, there are several ways to construct or even compute convex envelopes for functionals, depending on first- and second-order derivatives [28, 3136].

Another possibility for incorporating the lack of lower semi-continuity of membrane energy functionals is accounting the bending stiffness of the membrane and adding to the energy the bending contribution [37]. So, it is established in [33, 34] that the total energy of a membrane with some bending stiffness and thickness h, that is, the functional

E[r]=hEm[r]+h3Eb[r]E1

of deformation rconsisting of membrane (Em)and bending (Eb)contributions is lower semi-continuous for arbitrary Emas far as Ebis lower semi-continuous. Explicit forms for Emand Ebas well as sufficient conditions on material constants for which Eb(and therefore E) is lower semi-continuous are derived ibid. Such materials are called stable.

However, the most rigorous and general approach seems to be the derivation of low-order energies from general three-dimensional non-linear elasticity by means of Γ-convergence. The concept was originally developed by De Giorgi, see [29]. It turns out that the Γ-limit of three-dimensional elastic energy functional, when the thickness of the body goes to zero, is always lower semi-continuous and since most commonly used two-dimensional energy functionals, that is, membrane energy, pure bending (Kirchhoff) energy and higher-order terms (von Kármán–like energies), are already derived as such limits (see Section 3), it becomes possible to analyze real deformations of particular tissues.

By suggested improvements, it is supposed to get rid of disadvantages of previous models and take into account the fact that the total energy of the tissue can have two local minima. Thus, we are intended

  • to account the height of each individual cell, so the model is fully three-dimensional,

  • to pick up strongly localized internal forces to model the phenomenon of apical constriction more precisely,

  • to introduce stretching and bending contributions into the total elastic energy of the epithelial tissue, thus making the model more realistic and convenient not only for qualitative but also for quantitative analysis,

  • to use the quasi-convexification of the total energy, so the real three-dimensional deformations of the epithelial tissues can be identified as minimizers of the total energy by a gradient flow technique.

The former allows making simplified toolboxes for analysing three-dimensional deformations of epithelial tissues. The rough diagram of the improved model looks like in Figure 5, so that the characteristics of each individual cell are important, as well as the actin belt is explicitly involved in the model.

Figure 5.

Diagrams of a single-cell layer: cross section (left) and 3D shape (right).

The chapter is organized as follows: In Section 2, some preliminary definitions and main notations are brought to make the chapter independent of outer sources. In Section 3, known results from Γ-convergent energies are shortly described. In Section 4, the quasi-convex envelope of membrane energy densities of general form is derived explicitly and sufficient conditions for their convexity are derived. Moreover, a specific energy is suggested for different deformation regimes that fully satisfies the needs of basic mechanical theory suggestion. Finally, in Section 5 main results of finite element analysis of tissues of particular initial shapes (flat, cylindrical, spherical) are discussed. Overall conclusion completes the body of the chapter.

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2. Preliminaries

We begin with the definition of concepts used throughout the paper, which and much more on this topic can be found in [28, 29].

There are different concepts of convexity in higher dimensions, such as quasi-, poly-, rank-oneand separately convexity. Here, we use all but poly-convexity, so it is of no use to enlarge the chapter by its definition.

Definition 1(quasi-convexity). A function

is said to be quasi-convex if

W(G)1μ(Ω)ΩW(G+r0(x))dx,E2

for every bounded open setΩ, for every

and for every

Definition 2(rank-one convexity). A functionis said to be rank-one convex if

W(G+θF)(1θ)W(G)+θW(G+F),E3

for everyθ[0,1],

withrank F=1.

Definition 3(separately convexity). A function

is said to be separately convex if

W(x1,,xi1,xi,xi+1,,xN)E4

is convex for everyi=1,,Nand for every fixed

In other words, a function

is separately convex if it is convex on every line parallel to a coordinate axis.

In general, convexity implies quasi-convexity, which implies rank-one convexity, and finally rank-one convexity implies separately convexity.

The quasi-convexity notion is generalized by Meyers [38] for functionals depending on higher-order derivatives. We refer to [32, 35, 36] for practical use of the definition.

Definition 4(k-quasi convexity). A functionis said to bekquasi-convex if

W(G)1μ(Ω)ΩW(G+kr0(x))dx,E5

for every open-bounded setΩ, for every

and for every

In particular, 2-quasi-convex functions, which we use here, are the restriction of quasi-convex functions to symmetric matrices [39] (in the case of k-quasi convexity, see [40]).

We widely use the quasi-convex envelope of W(G)or apparent energy density, Wq(G), which can be defined in terms of deformations of the form r(x)=Gx+r0(x)with

constant,
and r0(x)=0on the boundary Ω[41, 42]:

Wq(G)=1μ(Ω)infr0|Ω=0ΩW(G+r0)dx.E6

It is evident that for any

and when W=0, Wq=0as well.

Deformation is always denoted by

with open-bounded
We introduce the first and second fundamental forms of a deformation r

I=(r)Tr,II={2rxixjn}i,j=1,2,E7

respectively, in which

is the unit of the outer normal to Ω. The first fundamental form quantitatively characterizes the stretch or compression of the membrane, while the second fundamental form characterizes its curvature.

Definition 5. Energy density function

is called frame indifferent if for all rotations R3SO(3)

W(R3G)=W(G).E8

If for allR2SO(2),

W(GR2)=W(G),E9

then it is called isotropic.

In other words, frame indifferent energy densities are invariant against three-dimensional rotations before a deformation is applied and that isotropic energy densities are invariant against two-dimensional rotations after a deformation is applied.

Definition 6(short (δ-short) deformation). A deformation

is called short(δ-short) ifI(r)Id2×2(δId2×2) a.e. inΩ.

Definition 7(isometric deformation). A deformation

is called isometric ifI(r)=Id2×2a.e. onΩ.

Short deformations result compressive stresses, while isometric deformations do not allow stretching or compression. The case IId2×2corresponds to tensile or, in general, non-compressive stresses. It is known that for non-compressive stresses, the membrane energy density function is already quasi-convex [41, 42].

3. Γ-convergence and Γ-limits of three-dimensional non-linear elastic energy

Rigorous derivation of two-dimensional energy functionals for thin bodies from three-dimensional functionals is of particular interest. There are several ways for dimension reduction, such as asymptotic expansion, Γ-convergence technique, and so on (for general survey, see [43]). In [44], a hierarchy of plate models is derived as Γ-limit of three-dimensional elastic energy functional when the thickness of the body h0. It turns out that the scaling of external force in hplays an important role for Γ-limit derivation. Suppose a body occupies the domain Ω×[h2,h2], and the applied forces

active on the outer surface of the body satisfy

1hαf(h)finL2(Ω,R3),E10

and

Eh[r]=1hE[r]=Ω×(12,12)W(hr)dxE11

denotes the rescaled elastic energy of the body, with

hr=(r1x1,r2x2,1hr3x3).E12

The total energy of the body will be

Eh[r]=Eh[r]Ω×(12,12)f(h)(x)rdx.E13

The Γ-limit

Γlimh01hβEh[r]E14

for β0will provide constitutive mechanical models to analyse deformed three-dimensional shape of initially flat thin bodies.

We combine the results of [4448] in Theorem 1.

Theorem 1. Let the stored energy density function is frame indifferent and satisfy

W=0onSO(3),E15
W(F)cdist2(F,SO(3)),c>0,E16

and in a neighbourhood ofSO(3), WC2.

Then

(i) (Membrane theory [45]). Supposeα=β=0and

W(F)C(1+|F|p),W(F)c|F|pC,p2,c>0.E17

Then

is aβ-minimizing sequence, that is, if

limsuph01hβ[Eh[r]infEh]=0,E18

then

(for a subsequence). The limiting deformationr¯is independent ofx3and minimizes

E00[r]=Ω[Wq(r)fr]dxE19

among all

. For derivation ofWqfrom givenWsee Section 4 subsequently.

(ii) (Constrained membrane theory [46]). Suppose that0<α<53andβ=α. Then

and everyβ-minimizing sequencer(h)has a subsequence with
in. The limitr¯is independent ofx3and minimizes

Eα0[r]=Ω[fr]dxE20

among all short deformations

.

(iii) (Non-linear bending theory [47]). Supposeα=β=2. Then|infεh|Ch2and ifr(h)is aβ-minimizing sequence then there is strong convergencer(h)r¯in

(for a subsequence). The limiting deformationr¯is isometric, independent ofx3, belongs toand minimizes

E20[r]=Ω[124Q2(II)fr]dxE21

among all isometric deformations, which belong toThe non-linear strain satisfies

1h[(r(h))Tr(h)Id]x3(II+symamine3),E22

where2symG=GT+G,aminis the solution of

Q2(G)=minaR3Q3(G+ae3+e3a),Q3(G)=2WG2|GId=0.E23

In all cases, there is convergence of energy, that is,

limh01hβEh[r(h)]=limh01hβinfEh=Eα0[r¯]=minEα0.E24

For α>0, the limiting deformation r¯is not only isometric but is close to a rigid motion. For that reason in [44], for deformation

the deformation

r˜(h)=(R(h))Tr(h)c(h)E25

with R˜(h)SO(3)and

is introduced and the averaged in-plane and out-of-plane displacements

U(h)(x1,x2)=1212[(r˜1(h)r˜2(h))(x1,x2,x3)(x1x2)]dx3,V(h)(x1,x2)=1212r˜3(h)dx3E26

are rescaled by

u(h)=1hγU(h),v(h)=1hδV(h),E27

respectively.

Then, we have

Theorem 2. (iv) (Linearized isometry constraint). Suppose2<α<3and setβ=2α2,γ=2(α2),2δ=γ. If2<α<52, suppose in addition thatΩis simply connected. Then,Chβinfεh0. Ifr(h)is aβ-minimizing sequence then there exist constantR(h)SO(3)andc(h)∈ ℝ3 such that

R(h)R¯,E28
hr˜IdinL2(Ω×(12,12);R3×3),E29
u(h)u¯,v(h)v¯,inW1,2(Ω;R2),v¯W2,2(Ω;R2),E30
u¯+(u¯)T+v¯v¯=0.E31

Moreover, the pair(v¯,R¯)minimizes the functional

ElinvK(v,R)=Ω[124Q2(2v)R33f3v]dx,E32

subject todet2v=0.

(v) (von Kármán theory). Suppose thatα=3and setβ=4,γ=2,δ=1. Thenand for a subsequence of aβ-minimizing sequence, (1) and (2) hold and the limit triple(u¯,v¯,R¯)minimizes von Kármán functional

EvK(u,v,R)=12Ω[Q2(12[u+(u)T+vv])+124Q2(2v)]dxR33Ωf3vdx.E33

(vi) (Linearized von Kármán theory). Supposeα>3and setβ=2α2,γ=α1andδ=α2. ThenChβinfεh0and for a sequence of aβ-minimizing sequence. (2) holds withu¯=0and the pair(v¯,R¯)minimizes the linearized von Kármán functional

EvK(v,R)=Ω[124Q2(2v)R33f3v]dx.E34

Moreover, the non-linear strain satisfies

1h1α([(hr(h))Thr(h)]12Id)symG.E35

In all cases, we have convergence of the rescaled energy

to the minimum of the limit functional
. Moreover, forf30we haveR¯33=1or1.

Remark 1. (i) Practically, Theorems 1 and 2 establishΓ-convergence and give theΓ-limits of three-dimensional non-linear energy functional in the rangeβ[0,53)[2,). The rangeβ[53,2)remains unexplored.

(ii) Even in the caseβ=53, it is still possible to find a sequencerk(h)for which the functional

is bounded whenh0. The limiting deformationsrare composed of finitely many affine isometries separated by sharp folds and are called origami deformations [46]. It is proved ibid that the class of origami deformations is the closure of the class of short deformations with respect to the uniform convergence.

(iii) In particular, whenα=β=2, for isotropic homogeneous bodies we obtain

Q2(II)=2μ|II|2+λ2μλ+2μtr2II,E36

in whichλandμare Lamé constants, coinciding with plate energy derived by Kirchhoff much earlier in [49].

4. Relaxation of epithelial elastic energy by Pipkin’s procedure and by adding the bending contribution

As was mentioned in Section 1, there are two ways to relax the epithelial elastic energy. The first way is to relax the stretching energy for deformation regimes corresponding to compressive stresses. For that reason, Pipkin‘s procedure [41] seems to be the most common tool.

Suppose that the stretching energy density function W=W(r)is frame indifferent, isotropic and positive except the cases

r=(100100)andr=(δ00δ00),1<δR,E37

where it turns to zero, that is, attains its minimum. On elastic energies with two minima, see [50]. This assumption is motivated by bistability of cell shapes during apical constriction (see Figure 4).

We restrict attention to deformations rwith

r=(σ100σ200),E38

and denote Wq(r):=W˜q(σ1,σ2). Here |σj|=λj, j{1;2}, are the principal stretches, the squares of which are the eigenvalues of the first fundamental form. Therefore

λ1,2(r)=12[tr(I(r))±|I(r)|22det(I(r))].E39

In view of frame indifference of W, and thence, W˜, we conclude that W˜q(±σ1,±σ2)=W˜q(λ1,λ2).

From quasi-convexity of Wq, in particular, follows its rank-one convexity, therefore since the matrix

A=(δ1100000),δ110,E40

is rank one, we have

therefore W˜qis convex with respect to λ1. Repeating the argument when

A=(000δ2200),δ220,E41

we see that W˜q(λ1,λ2)is separately convex.

Under assumptions made, we have W˜q(λ1,λ2)=0if λ1=λ2=1and λ1=λ2=δ. Since δ>1and 0W˜qare convex along lines parallel to λ1=0and λ2=0, therefore W˜q=0on edges connecting the vertexes (±δ,±δ)in the σ1σ2plane. Repeating the argument, we conclude that W˜q=0inside the square Ω0={(λ1,λ2);λ1δ,λ2δ}. So, for δ-short maps Wq0.

Since W˜q(λ1,λ2)is even and convex in λ2, it will attain its minimum with respect to λ2only when λ2=0. Denote by W˜0(λ1)=minλ2W˜(λ1,λ2)when λ1>δ. Suppose that λ0(λ1)is the largest value of λ2for which W˜q(λ1,λ0)=W˜0(λ1), λ1>δ. The convexity of W˜qin λ2implies that W˜q(λ1,λ2)=W˜0(λ1)for λ2λ0(λ1), λ1>δ. In the same way, using the symmetry of W˜qin its arguments, we will arrive at W˜q(λ1,λ2)=W˜0(λ2)for λ1λ0(λ2), λ2>δ. Evidently, for the rest of deformation regimes,Wis convex.

If we denote by Ω1={(λ1,λ2);λ1>δ,λ2λ0(λ1)}and Ω2={(λ1,λ2);λ1λ0(λ2),λ2>δ}, the ranges of principal stretches, corresponding to uniaxial tensions in directions of those stretches, respectively [51], we finally arrive at

W˜q(λ1,λ2)=(0,inΩ0,W˜0(λ1),inΩ1,W˜0(λ2),inΩ2,W˜(λ1,λ2),else.E42

Besides opportunity of explicit relaxation, in [41] the derivation of criteria for convexity and quasi-convexity is described. In view of rank-one convexity, W˜qsatisfies Legendre-Hadamard inequality from which it follows

W˜q10,W˜q20,W˜q110,W˜q220,E43
G0,W˜q11W˜q22W˜q12HG,W˜q11W˜q22+W˜q12HG.E44

For convexity of W˜qin corresponding domain instead of the last two equalities, the stronger conditions

W˜q11W˜q22(W˜q12)20,G|H|,E45

must be satisfied. Above

W˜qj=W˜qλj,W˜qjk=2W˜qλjλk,G=λ1W˜q1λ2W˜q2λ12λ22,H=λ2W˜q1λ1W˜q2λ12λ22.E46

The other way is the adding of pure bending contribution to the elastic energy of the tissue. Since actin-myosin network contraction leads to compressive or non-stretching stresses, we have to incorporate the elastic energy mainly for such deformations. According to Section 3 for non-stretching stresses, we have

  1. (short deformations and compressive stresses)

    Γlimh01hβE(h)=0in the rangeβ[0,53),E47

  2. (isometric deformations, neither stretch nor compressive stresses)

    Γlimh01h2E(h)=Eb.E48

The deformation regime IIdcorresponding to tensile stresses leads to Em[r], which is already lower semi-continuous. In the case of uniaxial tension, there is no bending and the energy expression will look like the middle rows of Wq(42). Thence, the total energy might be given as follows:

E[r]=Ωfrdx+(0,IδId,12ΩW˜0(λ1(r))dx,for uni-axial tension in the direction ofλ1,12ΩW˜0(λ2(r))dx,for uni-axial tension in the direction ofλ2,124ΩQ2(II)dx,I=Id,12ΩWq(r)dx,otherwise.E49

We assume that the tissue is homogeneous and isotropic, so according to Remark 1

Q2(II)=2μ|II|2+λ2μλ+2μtr2II.E50

Furthermore, the membrane energy density function we take in the following form [52, 53]:

W(r)=γ1(I13)+γ2(I13)2,E51

which is valid for large deformations of incompressible hyper-elastic membranes. Above, I1=trIis the first strain invariant, γ1>0and γ2>0are material parameters, evaluated experimentally in [53]. For a neo–Hookean material 2γ1=μand 2γ2=κ, in which μis the shear modulus and κis the bulk modulus. W˜qand W˜0are determined according to the procedure described above. For that reason, I1must be represented in terms of the principal stretches λ1and λ2as follows:

I1(λ1,λ2)=λ12+λ22+λ32=λ12+λ22+1λ12λ22,E52

in which the incompressibility condition λ1λ2λ3=1is taken into account.

For any fixed λ1>1(uniaxial tension in the first principal direction), the corresponding W˜(λ1,λ2)attains its minimum with respect to λ2at the maximal positive root of

W˜λ2=2λ2(11(λ12+λ42)2)[γ1+2γ2(I13)]=0.E53

Since in the uniaxial tension regime λ12+λ22>1, we finally get

λ0(λ1)=1λ1.E54

Then,

W0(λ1)=1λ12(λ133λ1+2)[γ1λ1+γ2(λ133λ1+2)].E55

The correspondent tension is defined by

τ1(λ1)=W0(λ1)λ1=2λ131λ13[γ1λ1+2γ2(λ133λ1+2)].E56

Since W˜is obviously symmetric in its arguments, in the case of uniaxial tension in the second principal direction, we would have

W0(λ2)=1λ22(λ233λ2+2)[γ1λ2+γ2(λ233λ2+2)],E57

and

τ2(λ2)=W0(λ2)λ2=2λ231λ23[γ1λ2+2γ2(λ233λ2+2)].E58

It is evident, that when we substitute λ1=1in the first case and λ2=1in the second case, we arrive at undeformed configuration of the membrane, so the relaxed energy is fully consistent.

Forces driving tissue deformation are strongly localized and in general are compressive. Force fmust possess those properties.

5. Finite element analysis of three-dimensional shape of some characteristic deformations of tissues during morphogenesis

In this section, we summarize main results of finite element analysis of a single layer tissue model, the elastic energy of which is given by

E[r]=hEm[r]+h3Eb[r]E59

in which Emand Ebare defined in Eqs. (49)(51). Since the discretization procedure is standard, we omit the details and bring only the main results.

In Figure 6, we bring the model of a single-cell (element) and cell-cell junction (in red). All structures (plate and shell) considered in this section entirely consist of such cell groups. In all tissues considered below, the height of a cell h=4rand h=200d, in which ris its side and dis the diameter of actin fibres. The ratio of Young‘s moduli of a cell and actin fibres is Eact=3Ecell, and of actin fibres and cell-cell links Eact=1.5Elink.

We consider

  1. initially flat rectangular plate (Figure 7 (left)),

  2. cylindrical shell (Figure 9 (left)),

  3. spherical shell (Figure 11 (left)).

Figure 6.

Single-cell model (left) and adherens junction model (right): diagonals of the top hexagon imitate actin fibres, red areas between cells imitate junction bonds.

Figure 7.

Apical constriction of initially flat tissue. It consists from 5050 elements or cells and has 529,245 DOFs.

Figure 8.

von Mises stress distribution in deformed configurations.

Figure 9.

Apical constriction of initially cylindrical tissue. It consists from 5975 elements or cells and has 621,375 DOFs.

Figure 10.

von Mises stress distribution in deformed configurations.

Figure 11.

Apical constriction of initially spherical tissue. It consists from 3126 elements or cells and has 325,245 DOFs.

Elements of the middle part of the rectangular tissue are compressed in apical sides to imitate apical constriction in cells. Increasing the compressing stresses, the tissue is bent and a blaster-shaped pattern is formed as shown in Figure 7 (right). The quantitative picture of the stresses arising in the tissue is drawn in Figure 8.

Figure 12.

von Mises stress distribution in deformed configurations.

Next, we consider a cylindrical shell to imitate ventral furrow (generally all tubular patterns) formation. Elements of the top part of the cylinder are constrained in the apical sides by compressing the links standing for apical fibres (see Figure 9). Increasing the compressing stresses, the ventral furrow formation is simulated similar to stages presented in Figure 1. The quantitative picture of the stresses arising in the tissue is drawn in Figure 10.

Finally, a spherical shell is simulated. Cells at the top of a semi-sphere are constrained in apical sides and depending on values of compressing stresses various stages of blastopore formation in archenteron can be described (see Figure 11 (right)). The quantitative picture of the stresses arising in the tissue is drawn in Figure 12.

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6. Conclusions

Analysis based on lower semi-continuous energy functionals reveals real three-dimensional deformations of soft epithelial tissues. Having different forms for different deformation regimes, such as compressive stresses (short deformations), uniaxial tensions collinear to principal directions of the first fundamental form, no stretching or pure bending stresses (isometric deformations) and stretching stresses (large deformations), the resulting total energy is lower semi-continuous, so the existence of its minimizers, that is, real deformations, is ensured. Particular energy density functions are chosen and the explicit form of the total energy functional is obtained, thereby the discretization is made easy.

On the basis of obtained energy functional, a three-dimensional discretized model of epithelial tissues undergoing combined stretching and bending deformations is constructed. Discretization elements correspond to single cells forming the tissue. Actin fibres and cell-cell adhesion links, mainly contributing on the tissue energy, are explicitly embedded in elements. Deformations characteristic to specific embryonic tissues (ventral furrow, neutral tube, neurosphere) observed earlier are described quantitatively increasing contractile stresses in fibres.

Dedication

We thankfully dedicate the chapter to the blessed memory of our good fellow and colleague, a candidate of physical and mathematical sciences, Hamlet V. Hovhannisyan (1956–2016), who unexpectedly died before he could realize his best scientific ideas.

Acknowledgments

The theoretical part of the chapter is investigated under the guidance of Doctor, Professor Benedikt Wirth, Institute for Computational and Applied Mathematics, University of Münster, whom we are heartily thankful. The work of As. Kh. and S. O. was made possible in part by a research grant from the Armenian National Science and Education Fund (ANSEF) based in New York, NY, USA.

Nomenclature

Inner (dot) product
λ,μLamé coefficients
λ1,λ2Principal stretches
EhRescaled three-dimensional total energy
μ(Ω)Measure of bounded open set Ω
Gradient (nabla) operator
ΩMidsurface of the epithelium
rank FRank of the matrix F
Tensor product
distUsual distance in three-dimensional Euclidean space
dDiameter of actin fibres
EThree-dimensional elastic energy
EbBending energy
EhRescaled three-dimensional elastic energy
EmMembrane energy
EactYoung’s modulus of actin fibres
EcellYoung’s modulus of a cell
ElinkYoung’s modulus of cell-cell links
hThickness of the epithelium or a single cell
rSide of a single cell
WMembrane energy density function
WqThe quasi-convex envelope of W
IIThe second fundamental form associated with deformation r
IThe first fundamental form associated with deformation r
nUnit normal vector
rDeformation acting from R2to R3
Id2×22×2identity matrix
SO(3)The group of all rotations about the origin of three-dimensional Euclidean space

© 2016 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Ara S. Avetisyan, Asatur Zh. Khurshudyan and Sergey K. Ohanyan (December 14th 2016). Improved Reduced Order Mechanical Model and Finite Element Analysis of Three-Dimensional Deformations of Epithelial Tissues, Perusal of the Finite Element Method, Radostina Petrova, IntechOpen, DOI: 10.5772/65027. Available from:

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