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## D’Arcy Thompson: a rst mathematical approach to mor- phogenesis

In 1917, D’Arcy Thompson published his only scientic book, entitled \On growth and form » [17]. While other contemporary scientists focused on experimental analysis, D’Arcy Thompson’s investigations were based on a mathematical approach. As Wilhelm His, D’Arcy Thompson was skeptical about evolution theories and natural selection, which were dominant during the 19th century. His idea of morphogenesis was based on the role of physical forces in shaping organisms. Therefore, the book has a purely mechanical approach where living organisms are treated as material bodies subjected to physical forces, obeying to simple physical and geometrical laws. An example of D’Arcy Thompson’s approach can be found in the second chapter of his book, where he focused on the eect of external forces on animals of dierent sizes. He stated that big animals are subjected to inertial forces while small animals are subjected to surface tension. It follows that big animals have strong and heavy structures in order to support the gravitational force, while small animals need lighter structures in response to the weaker surface tension.

### Alan Turing: chemical basis of morphogenesis

In 1952 Alan Turing, published his paper on the chemical bases of morphogenesis, which later became another milestone in biomathematics [21]. This was his only contribution on morphogenesis, but it represents the rst reaction-diusion model for pattern formation. The main ingredients of Turing’s model are:

The presence of at least two chemical species which undergo chemical reaction. Turing called them morphogens, in order to underline their role in generating a new pattern. The concentration of the two morphogens in the cell drives the cell activity. (In this sense genes can be considered indirectly as morphogens).

In absence of diusion, the system is in a stable state, dened by homogeneous concentrations of the two reactants. Under certain conditions, diusion can destabilize the homogeneous state and a new non-homogenous pattern arises. This is counterintuitive because generally diusion would rather be thought to introduce chaos in a system, instead of generating an organized pattern [22]. Turing’s model predicted the existence of six possible steady-states as shown in Figure 1.5. The uniform stationary (I) and oscillatory (II) states, the short wavelengths stationary (III) and oscillatory (IV) states and the nite wavelength stationary (V) and oscillatory (VI) states. Of particular interest is the Case VI, which occurs when the diusion coef- cients of the two morphogens dier substantially and initiate the so-called short range activation long range inhibition [23] mechanism. The two morphogens are seen as an activator and an inhibitor, respectively, which can act on themselves as well as on the other. A small perturbation in the homogeneous concentration can induce an increase in the activator concentration and initiate the feedback which lead to the formation of one

of the Turing’s patterns in Figure 1.5 (b).

#### Positional Information: molecular basis of morphogenesis

In the years right after its publication, Turing’s work didn’t receive a great attention. The problem of emergence of organized cellular patterns in the tissue was brought back to the attention of developmental biologists some years later in the 70s. In fact, Lewis Wolpert introduced the concept of Positional Information (PI) in order to explain how complex patterns could arise from initial asymmetries in the tissue [28]. The main idea of Wolpert is that the position of a cell in the tissue species the information about the molecular changes the cell will undergo. The key elements of Wolpert’s theory can be summarized in the denition of:

A mechanism which species the polarity in the tissue. Polarity is the direction in which PI is specied and is dened with respect to one or more reference points.

A mechanism for specifying the dierent responses of the cell.

PI can be specied by a quantitative variation of some factor such as the concentration, or a qualitative variation of some cell parameters such as a combination of genes or enzymes. A set of cells which have their PI specied with respect to the same reference points constitutes a eld. Interpretation of PI is the process by which PI species the cell state and conversion is the mechanism by which PI is translated in a particular cellular activity. Furthermore, PI is universal in organisms and size invariant, meaning that if a part of the tissue is removed, the tissue is still able to pattern and interpret the PI.

The concept of positional information is well claried in the French Flag Model depicted in Figure 1.6. In this example, the mechanism which species polarity is the monotonic variation of the morphogen concentration C, in respect to the reference values C0 and CF . The thresholds C1 and C2 identify the mechanism for the dierential response of the cells. The interpretation acts according to the following rule: cells with position in the region where C0 < C < C1 express the blue pigment, a cell in the region where C1 < C < C2 expresses the white pigment and cells in the region where C2 < C < CF express the red pigment. The molecular patterning in the early Drosophila embryo has been explained using positional information [29].

The models proposed by Turing and Wolpert oer two dierent points of view on pattern formation. A rst dierence comes from the fact that Turing aimed at modeling spontaneous formation of a pattern, while Wolpert asked how a more complex pattern can arise from an asymmetry (polarity) in the tissue. Furthermore, in Turing’s model the concentration of morphogens is directly related to the spatial pattern, in this sense it is a \pre-pattern ». Conversely, Wolpert introduced an interpretation step where the cell activity is specied by the concentration gradient. On the other hand, in their recent paper Green and Sharpe have proposed a dierent mechanism through which the two models can cooperate in pattern formation [30]. For example, in the mouse limb bud a periodic pattern develops with dierent wavelengths depending on the position along a polarity gradient.

**Mathematical theory of growth and remodeling**

The aim of this section is to give a mathematical description of growth and remodeling in soft tissues.

The mathematical theory for volumetric growth and remodeling adopted in this thesis, has been rst proposed by Skalak [31] and later formalized by Rodriguez et al. [37]. As noted in the rst chapter, it has its origins in the plasticity theory.

According to these authors, the mapping introduced in Eq.(2.1) can be split into two parts. One component is associated to the growth (remodeling) and it transforms the tissue from its initial stress free conguration B0 into a new stress-free grown (remodeled) state, denoted as Bg in Figure 2.1. This may not be an observable physical state and the tissue can never reach it in vivo. Indeed, the grown (remodeled) state Bg is a collection of local grown (remodeled) states of the body parts, which may not be geometrically compatible with each other, meaning that they can overlap and intersect. The second component restores the global compatibility of the tissue and transforms it from the grown (remodeled) state Bg into the nal compatible and residually stressed conguration Ba. According to this decomposition, the deformation gradient F dened in Eq.(2.2), can be split into two components, as follows: F = FeFg.

**Method of incremental deformations superposed on nite deformations**

An accumulation of residual stresses during growth and remodeling can trigger elastic instabilities in the tissue. Morpho-elasticity investigates the emergence of complex patterns in living matter after the occurrence of an elastic instability. Therefore, the method of incremental deformations superposed on nite deformations will be introduced in the following.

Following Ogden [76], the fundamental idea is to perturb the basic solution x(0) to the elastic problem, with a small incremental deformation so that, the perturbed solution can be written as a series expansion to the rst order around the basic solution. The zeroth order term is in the form of a nite deformation, representing the basic solution with the initial shape of the material. The rst order term is in the form of an incremental deformation, dening the morphology of the material after a possible bifurcation. In the following, the method of incremental deformation will be introduced and the rstorder constitutive and governing equations will be derived.

**Theories and methods for solving the incremental problem**

In the previous section, the rst order incremental equilibrium problem has been derived for a body undergoing volumetric growth and remodeling. As already mentioned, the problem is a system of four PDEs with boundary conditions. In this section, the focus will be on the analytical techniques that can be used in order to transform the problem in a more suitable form for implementing an ecient numerical solution procedure. First, the Stroh formalism will be introduced. It will allow to transform the system of PDEs with boundary conditions into a system of rst order ordinary dierential equations (ODEs) with initial conditions. It will be shown that the Stroh formalism provides an optimal form for building a numerical solution procedure when the problem has a Dirichlet boundary condition. Second, the surface impedance method will be illustrated. It will allow to further transform the Stroh form of the incremental problem into a matrix Riccati equation, which in the case of Neumann boundary conditions, allows to build a more ecient numerical solution algorithm.

**Table of contents :**

List of Figures

List of Tables

Abstract

Resume

Riassunto

**1 Introduction to morphogenetic theories in living matter **

1.1 Early mechanistic vision

1.1.1 Wilhelm His and the \constrained expansion » model

1.1.2 Wilhelm Roux and developmental mechanics

1.2 The 20th century

1.2.1 D’Arcy Thompson: a rst mathematical approach to morphogenesis

1.2.2 Genetics

1.2.3 Pattern Formation

1.3 Modern approaches to morphogenesis

1.3.1 Volumetric Growth and Remodeling

1.3.2 Mixture theory

1.3.3 Morphomechanics: hyper-restoration principle

1.3.4 Mechanotransduction

1.4 Summary and conclusions

**2 Morphoelasticity: theory and methods **

2.1 The thermo-mechanics of open systems

2.1.1 Kinematics

2.1.2 Mathematical theory of growth and remodeling

2.1.3 Governing equations

2.1.4 Boundary conditions

2.1.5 Constitutive relations

2.1.6 Summary of the key equations and some comments

2.2 Method of incremental deformations superposed on nite deformations .

2.2.1 Incremental deformation

2.2.2 Incremental boundary value problem

2.2.3 Summary of the key incremental equations

2.3 Theories and methods for solving the incremental problem

2.3.1 Stroh formulation

2.3.2 The surface impedance method

2.3.3 Mixed boundary conditions

2.3.4 Neumann boundary conditions

2.4 Concluding remarks

**3 Morphoelastic modeling of gastro-intestinal organogenesis **

3.1 Introduction to intestinal morphogenesis

3.2 State of the art of biomechanical modeling

3.2.1 Spatially constrained growth models

3.2.2 Dierential growth models

3.3 Homogeneous growth model with spatial constraints

3.3.1 Kinematics

3.3.2 Constitutive equations

3.3.3 Governing equations and basic axial-symmetric solution

3.3.4 Incremental boundary value problem

3.3.5 Stroh formulation of the BVP and numerical solution

3.3.6 Results

3.3.7 Discussion of the results

3.4 Dierential growth model without spatial constraints

3.4.1 Kinematics

3.4.2 Constitutive equations

3.4.3 Governing equations and axial-symmetric solution

3.4.4 Incremental boundary value problem

3.4.5 Stroh formulation of the BVP

3.4.6 Surface impedance method and numerical solution

3.4.7 Theoretical results of the linear stability analysis

3.4.8 Finite element simulations in the post-buckling regime

3.4.9 Numerical results

3.4.10 Validation of the model with experimental data

3.5 Concluding remarks

**4 Helical buckling of pre-stressed tubular organs **

4.1 Preliminary remarks

4.1.1 Introduction to the anatomy and the physiology of arteries

4.1.2 Principle of homeostasis

4.1.3 Residual stresses and stress-free state

4.1.4 Remodeling process in arteries

4.2 Kinematics of the elastic problem

4.3 Constitutive equations

4.4 Governing equations and basic axial-symmetric solutions

4.4.1 Case (a): stress-free internal and external surfaces

4.4.2 Case (b): Pressure load P at the internal surface

4.4.3 Case (c): Pressure load P at the external surface

4.5 Incremental boundary value problem

4.6 Stroh formulation of the BVP

4.7 Surface impedance method and numerical solution

4.8 Numerical results

4.8.1 Eect of the circumferential pre-stretch

4.8.2 Eect of the axial pre-stretch

4.9 Discussion of the results

4.10 Validation of the model with experimental data

4.11 Concluding remarks

**5 Conclusions and perspectives **

**Bibliography**