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Early mechanistic vision
The rst speculations on morphogenesis date back to ancient Greece time. In order to explain how shapes and structures originate in living organisms, in his book Historia Animalium, Aristotle introduced the epigenetic hypothesis. Although his approach was mainly philosophical, it was based on the idea that structures and shapes gradually de-velop and are not pre-existent from the beginning. By contrast, this latter vision called preformationism, became predominant in the years when theology was having a strong in-uence on the society and persisted until the 17th Century. Malpighi even claimed that a miniature of the human body (a homunculus) was present in the head of each sperm, completely avoiding the problem of generation of shapes in developing organisms. Besides these very early attempts, mostly based on intuition and theological preconceptions, the rst approach to morphogenesis based on mechanical and geometrical arguments came from Galileo Galilei. After having started his studies in medicine, Galileo followed his passion for mathematics and physics and he devoted his entire life to investigate mechan-ical problems. His method was based on the idea that \mathematics is the best way of supplying physics with the nest rules of logics » [1]. The wide interests of Galileo cov-ered some aspects of biomechanics such as allometric studies on vertebrates. He showed that most organisms change their shape in function of the loads that they have to hold, according to a scaling law in the form:
where a0 is a constant depending on the organism, is the scaling exponent, x the reference variable and y is the dependent variable. If an animal increases its length l, then its mass M, being proportional to the volume, would increase by a power = 3 and its strength S, being proportional to the area, by = 2. Therefore, in the case of isometric growth, its mass would have increased more than its strength and thus the animal would have been subjected to a higher load while having a lower strength to support it. Thus, Galileo proved that the changes in size are actually governed by allometric scaling (see Figure 1.1).
The 19th century was the era of Darwinism and evolutionary theories, but also of the discovery of the cell and of the birth of experimental embryology. In particular, a school of thought which established itself in the second half of the 19th century was the Recapitulation theory of Ernst Haeckel [12]. Supported by Darwinism, he claimed that steps of embryonic development of a species correspond to adult stages of their ancestors. Haeckel’s school was very in uential in the scienti c community of the time and every attempt to show that the theory was unfounded and not able to provide a through description of morphogenesis was brutally suppressed.
Wilhelm His and the \constrained expansion » model
The recapitulation theory was particularly rejected by the German scientist Wil-helm His. He rather supported a mechanical causation behind embryonic development. He performed several experiments to prove the constrained expansion principle, i.e. expansion of a spherical or cylindrical tissue surrounded by inelastic tissue [13]. He per-formed several experiments on rubber tubes under constrained compression and tension, in order to model the gut tube, the brain formation and other changes of shape during embryogenesis. Among the most relevant results, His showed that the gut tube mor-phogenesis could be modeled by a rubber tube under tension. In Figure 1.2 a sketch of His’s mechanical explanation for gut shaping is depicted. Because of Haeckel scienti c in uence, His’s work had to wait some decades to be di used. The input came right from one of Haeckel’s students, Wilhelm Roux.
Wilhelm Roux and developmental mechanics
At the end of the 19th century, Wilhelm Roux shifted the attention from evolution ( nal purpose) to mechanisms (cause) in developmental biology. As well as being a promoter of His’s work, he was interested into two aspects of development: rst, the role of self-di erentiation and second, the role of dependent-di erentiation. He investigated self-di erentiation by performing a lot of experiments on embryos, which eventually opened the way to the discovery of regulation and induction. Regulation is the property by which the embryo develops normally even if a part of it is removed. Induction is the ability of a cell or a tissue to in uence the development of another.
The experiments to test the role of self-di erentiation in the embryo where based on the analysis of the rst cleavage of the embryo. Roux separated the two blastomeres which resulted after the rst cleavage, killing one of the two, and he observed that from the surviving blastomere, a half-embryo still developed. He erroneously explained the result in the context of the mosaic theory, rst introduced by August Weismann in the 1880s [3]. According to Weismann’s theory, the nucleus of the egg contains a number of factors which determinate the cell fate, as shown in Figure 1.3. During cleavage these determinants are asymmetrically distributed between the daughter cells in the embryo. As cleavage proceeds, the potentiality of each daughter cell decreases while the cells become more and more specialized. Roux thought that killing one of the two blastomeres corresponded to remove part of the cell determinants, which resulted in the development of a half-embryo.
Some years later, Hans Driesch, another of Haeckel’s students, explained Roux’s results with the discovery of regulation. Driesch repeated the experiment done by Roux using frog embryos. Unlike in Roux’s experiment, after having killed one of the two blastomeres, he removed the killed cell. As a result, he observed that a whole, but smaller embryo developed from the one-cell fertilized egg because the killed cell didn’t prevent the other cell from regulating and forming a complete embryo [14].
The discovery of induction had to wait some more years. Following the methodic experimental approach of Roux, in 1924 Spemann and Mangold transplanted part of the embryonic tissue from a amphibian into an embryo of a di erent amphibian species [15]. They observed that a partial second embryo developed from the transplanted tissue: the hosting tissue induced the development of the hosted tissue.
Roux is also remembered for establishing the concept of functional adaptation as a principle for dependent-di erentiation: cells and tissues respond to change in external conditions in order to preserve their global organization and functions. He applied the principle of functional adaptation to the study of development of bone, cartilage and muscle. Pauwels [16] summarized Roux principle of functional adaptation as follows:
The formation of the di erent types of tissue should be considered as an adaptation of the formative tissue to the function demanded of it.
According to Roux, the bone would be built with the minimum amount of tissue, in order to support the stresses which it undergoes. Moreover, it would adapt under the action of external forces, through growth in width in order to preserve the lightness.
The 20th century
The 20th century has been a century rich in crucial discoveries and scienti c contri-butions to the understanding of morphogenesis. This period can be split into two phases. The rst half of the century was characterized by the statical point of view of D’Arcy Thompson on the relation between physical forces and generation of shapes. The second half was the era of genetics and molecular biology which shifted the focus to the dynamical aspects of morphogenesis.
Figure 1.4: Left: Transformation grid applied to the transformation of the shape of a small amphiopod (a) Harpinia Plumosa into the shapes of two other genera belonging to the same family (b) Stegocephalus In atus, and (c) Hyperia Galba), adapted from [4]. Right: Transformation grid applied to the growth of a skull in human foetus. In both examples the transformation is achieved by applying physical forces on the considered structure, during evolution and growth respectively.
Furthermore, he proposed a grid transformation method, aimed at showing that physi-cal forces can shape a living organism during growth and even evolution, see Figure 1.4. As highlighted by Ulett [18], \On growth and form » had a great in uence on modern biomechanics and is still read and published in reduced and revised versions. D’Arcy Thompson’s book inspired the work of Julian Huxley on allometric growth [19] and of Gould in his attempt to deal with such a mechanistic theory of shape in evolutionary theories [20].
Genetics
The origins of genetics date back to the second half of the 19th Century when Gregor Mendel discovered the inheritance of biological traits [3]. His theory was based on the idea that the hereditary package is transmitted from parents to o spring through a set of discrete hereditary factors. Each factor is a potential expression of a biological trait. For any biological feature, a child receives one factor from the mother and one from the father. The e ective expression of a biological trait is given by the combined action of the two related factors. When the two factors combine, they are not a ected by each other, but they just in uence the potentiality of the fertilized egg. Mendel proved his theory by performing experiments on pea plants, but his results remained mostly unappreciated until the rst half of the 20th century. One of the main problems of Mendel’s theory was that it couldn’t give a concrete description of the nature of these factors. At that time, genetics was considered the study of transmission of heredity properties in the embryo, while embryology was seen as the science investigating how the organisms develop. But there wasn’t any link between the two disciplines, which nally merged only in the second half of the 20th Century. The main steps leading to the fusion of the two elds are basically the discovery that the DNA carries the genetic information and that genes encode proteins. The properties of a cell are determined by the proteins they contain, genes control and act on proteins in order to drive the cell fate and consequently, the development of the embryo. Right after these discoveries, a lot of experiments focused on nding the genes responsible for several morphogenetic events in the embryo. The strategy adopted was to introduce changes in the molecular organization of the DNA in order to observe abnormal changes in a nal structure. In this way, it has been possible to identify which genes play an active role in the formation of an emerging pattern.
Pattern Formation
The fast development of genetics in the years following the 50s also promoted an increasing attention to the chemical and molecular mechanisms which might underpin morphogenetic processes. The concept of pattern formation in morphogenesis originated from that period. Pattern formation can be de ned as the emergence of organized struc-tures in space and time. The main contributions to the development of this concept were given by Alan Turing and Lewis Wolpert.
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 contri-bution on morphogenesis, but it represents the rst reaction-di usion 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 di usion, the system is in a stable state, de ned by homogeneous con-centrations of the two reactants. Under certain conditions, di usion can destabilize the homogeneous state and a new non-homogenous pattern arises. This is counter-intuitive because generally di usion 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 di usion coef-cients of the two morphogens di er 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).
Figure 1.5: Turing’s reaction-di usion model: (a) Examples of the six stable states so-lutions of Turing’s model. (b) The so-called Turing’s pattern is depicted as the Case VI where a stationary wave of nite wavelength develops, adapted from [5].
Turing’s model has later been largely employed for modeling the emergence of several pat-terns in vertebrates such as the stripes in the zebra- sh pigmentation [24], the branching pattern in feathers [25], but also the fabulous seashell patterns [26] and the mechanism of plant phyllotaxis, i.e. the arrangement of leaves on a plant stem [27].
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 speci es the information about the molecular changes the cell will undergo. The key elements of Wolpert’s theory can be summarized in the de nition of:
PI can be speci ed 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 speci ed with respect to the same reference points constitutes a eld. Interpretation of PI is the process by which PI speci es 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 clari ed in the French Flag Model depicted in Figure 1.6. In this example, the mechanism which speci es 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 di erential 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 o er two di erent points of view on pattern formation. A rst di erence 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 speci ed by the concentration gradient. On the other hand, in their recent paper Green and Sharpe have proposed a di erent mechanism through which the two models can cooperate in pattern formation [30]. For example, in the mouse limb bud a periodic pattern develops with di erent wavelengths depending on the position along a polarity gradient.
Modern approaches to morphogenesis
In the previous sections, the main discoveries and theories which have contributed to our current understanding of morphogenesis have been presented. The research eld rapidly expanded in the last decades. In particular, modern approaches focused on the e ect of external stimuli, such as mechanical, chemical, molecular on the generation of patterns and on the structural organization in living organisms.
In the following, the modern approaches to morphogenesis from a biomechanical viewpoint will be summarized.
Volumetric Growth and Remodeling
At the beginning of the 80s, Skalak and coworkers gave the rst analytical descrip-tion of nite volumetric and surface growth [31,32] in a continuum mechanics framework. They introduced the idea that growth can induce incompatibilities in the geometry of a body. According to the authors, the growing body can pass from an unloaded stress-free con guration to a pre-stressed reference con guration arising due to incompatibilities in the growth process. Conversely, if the growth strains are compatible, then no internal stresses will be generated in the tissue.
The seminal work of Skalak opened the door to a number of experimental studies which aimed at characterizing the residual stress distribution in biological tissues. Residual stresses have been found in blood vessels (arteries [10, 33] and veins [34]), heart [35], airways [36]. On the theoretical side, Skalak’s formulation of volumetric growth pre-pared the ground for the paper of Rodriguez and colleagues [37] in which they formalized, in an elegant mathematical formulation, the relation between growth and remodeling on one side and residual stresses on the other side. In their fundamental contribution, they use the multiplicative decomposition of the deformation gradient associated to the growth/remodeling process, into a purely growth/remodeling and an elastic component.
The growth/remodeling deformation introduces the incompatibility in the tissue and the elastic deformation restores the compatibility, while residual stresses arise. As highlighted by Ambrosi et al. [38], the main advantage of this type of representation is that it allows to account for the e ects of growth and remodeling. On the other hand, the mechanisms underlying these processes are neglected.
The multiplicative decomposition was rst introduced in the theory of elasto-plasticity by Kroner [39] and Lee [40] in order to split the inelastic and the elastic components and it has been widely employed in continuum mechanics models. In this framework, the growth and remodeling processes are basically modeled as deformations from a stress-free con guration to a residually stressed state, where the body has grown or remodeled and changed its natural state. The relation between the variation of mass and changes in shapes is given by a volumetric growth. For remodeling, the micro-structural reorganiza-tion of the tissue can be linked for example to the changes in density, sti ness, orientation of bers. Therefore, the thermo-mechanics of open systems has been used to describe the evolution of growth or remodeling parameters in time, as well as the constitutive relations for the material.
Extended theories and applications
In the years following Skalak’s and Rodriguez’s papers, a lot of work has been done both at the theoretical level and for experimental purposes. From the theoretical view-point, the attention has been pointed to develop constitutive theories for growth and re-modeling. Accounting for mass volumetric sources and uxes, Epstein and Maugin derived constitutive and evolution laws using the thermomechanical principles [41]. Gangho er et al. developed a constitutive framework for volumetric and surface growth involving the Eshelby stress [42]. Following the work of Di Carlo and Quiligotti [43] where accretive forces for growth are included in the system, Ambrosi and Guana derived the evolution laws for mass production which involves a direct link between stress and growth [44].
In parallel, another part of the biomechanical community has focussed on the applica-tion of the growth and remodeling theory to determine the onset of pattern formation after an elastic instability. This branch is called morpho-elasticity. A number of studies have employed the growth theory of Skalak and of Rodriguez et al. in order to model the emergence of geometrical patterns in tissues as an elastic instability due to excessive accumulation of residual stresses in the tissue. Among the main contributions, Ben Amar and Goriely studied the stability of elastic growing shells of di erent geometries [45], Li et al. modeled the formation of mucosal pattern in the oesophagus of pigs [46], Moulton and Goriely modeled the mucosal folding mechanism in growing airways [47].
Mixture theory
The approaches described so far consider the growing tissue as an open system of a single constituent with an internal source of mass. Soft tissues are composed by several constituents such as collagen, elastin, ground substance. In mixture theory, the tissue is considered as a sum of di erent constituents. Each of them obeys some constitutive laws, and it exchanges mass with the other constituents. The mixture theory allows for a more realistic growth considered as exchange of mass between constituents. But, on the other hand, it introduces some controversial issues, such as the constitutive modeling of the partial stresses which act on each constituent and the de nition of their physical meaningful boundary conditions.
From a theoretical viewpoint, one of the early contributions came at the end of the 70s from Bowen. He proposed a thermo-mechanical theory for mixtures, which gave the bases for modeling di using mixtures of elastic materials [48]. Later in the 90s, Cowin proposed poroelasticity, i.e the theory of interactions between deformation and uid ow in a porous medium, as theoretical framework to study the bone [49]. Humphrey and Rajagopal proposed one of the rst mixture models [50] for living materials. The main idea in their model was that each constituent has its own natural con guration. Therefore, the state of stress of each constituent might not be compatible with that of the surrounding tissue and residual stresses arise.
Loret and Simoes developed a theoretical framework for growth within mixture theory [51]. They considered a biphasic tissue composed by a solid and a liquid phase. Using the multiplicative decomposition on the solid phase they split the growth and the elastic e ects and they derived the constitutive laws for growth from the second law of thermodynamics. A similar approach has been applied by Garikipati et al. to the growth of arti cial tendon [52] and by Baek et al. for modeling aneurysm in brain [53].
Ambrosi et al. used the mixture theory in order to investigate the emergence of resid-ual stresses during growth and remodeling of soft tissues [54]. The mixture framework has also been used for modeling tumor growth. Byrne and Preziosi modeled the tumor as a multicellular spheroid constituted by cells, considered as an elastic uid, and extra-cellular space lled by the organic liquid [55]. Such a biphasic model was later extended by Ambrosi and Preziosi to a triphasic mixture which also accounted for the e ect of the extracellular matrix [56]. Finally, Athesian modeled growth and remodeling of a reac-tive mixture including electrical charged solid and uid constituents which are typical of cartilage tissue [57].
Morphomechanics: hyper-restoration principle
Another modern approach to morphogenesis comes from the work of Lev Beloussov. During the 70s he performed several experiments on embryos in order to investigate the e ects of mechanical stresses on early morphogenetic events [58, 59]. Beloussov in-troduced the idea that the changes in shape during gastrulation occur according to a hyper-restoration principle [6, 60]. Cells tend to restore the original stress distribution in the embryo by responding to the external forces with generation of active forces inside the embryo and which overshoot the original stress. Therefore, gastrulation is seen as a chain of events driven by mechanical feedbacks. In Figure 1.7, a graphical interpretation of Beloussov’s results is depicted. For example, during the early stages of gastrulation (A and B) the passive stretching of the roof of the blastocoel induces an active increase of the blastocoel internal pressure (red curve in Figure 1.7ab), which in turn a ects the stress distribution in the neighboring marginal zone (MZ).
Table of contents :
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: arst 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
