Article pp.409-426 du Vol.26 n°5 (2006)

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ARTICLE ORIGINAL ORIGINAL PAPER

Some considerations for a theory of plant tissue mechanics

A.-C. Roudot

^*

RÉSUMÉ

Vers une théorie mécanique des tissus végétaux.

L’analyse de texture des fruits et légumes est une activité très commune.

Cependant, du fait de l’absence de connaissance dans le domaine de la méca- nique des tissus cellulaires, l’interprétation physique de ces analyses est sou- vent très difficile. L’amélioration de ces analyses nécessite donc la création d’une telle théorie mécanique. Après une description critique des principaux modèles utilisés aujourd’hui, nous présentons une nouvelle approche basée sur les automates cellulaires et les systèmes de particules. Les principaux aspects de cette théorie reposent sur l’utilisation d’une méthode de création tissulaire basée sur son développement physique, sur l’importance de la géométrie tissulaire et de l’état de contraintes mécaniques initial, sur la séparation d’échelle entre la création tissulaire et l’application des contraintes en cours de test, et sur l’importance du choix des constantes de temps du tissu et des contraintes appliquées. Quelques résultats obtenus en simulation histologique, tirée de cette théorie, sont présentés permettant la validation de cette nouvelle méthode d’évaluation.

Mots clés

modélisation histologique, théorie tissulaire, comportement des tissus cellulaire, modélisation numérique.

SUMMARY

Analysis of fruit and vegetable texture is very common, but due to a lack of knowledge about cell tissue mechanics, its physical interpretation is very difficult. The improvement of these analyses necessitates the definition of a mechanical theory of such tissues. After a critical description of the various models used nowadays, a new theory is proposed based on cellular automata and particle systems. The main particularities occurring in this theory are the use of a developmental method to create the tissue, the importance of the tissue geometry and initial physical state, the spatial scale separation between tissue creation (microscopic level) and test simulation (cell cluster level) and the

Université de Bretagne Occidentale, UFR Sciences, Laboratoire de Toxicologie Alimentaire, 6, avenue Le Gorgeu, CS 93837, 29238 Brest cedex 3.

* Correspondance : alain-claude.roudot@univ-brest.fr

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time scale separation between the time constant of the cells and test duration.

Some results of the histological simulation method, drawn from this theory, are discussed.

Keywords

histological modelling, cell tissue theory, cell tissue behaviour, integrative model.

1 – INTRODUCTION

For many years, research on food quality improvement has required a better knowledge of food products and a definition of new equipment for food texture analysis permitting objective evaluation of rheological characteristics. Besides these experimental methods, attempts have been made in order to get a better theoretical background. Different kinds of modelling have been undertaken: mathematical and/

or numerical ones. These models can be grouped into a few categories, each one with its advantages and drawbacks. Thus, the older ones try to define constitutive equations of products taken in their globality. Attempts have also been made to understand product behaviour by taking into account cell behaviour. More compli- cated methods have also been used, based on the mathematical theory of homogenization, or organ growth. At least one new method has been proposed derived from cellular automata: histological modelling. The final purpose of all these methods is to have a theoretical background in order to interpret experimental data, and they therefore have to be evaluated taking into account this purpose. However, although all these methods can give more or less good results when simulating some particular mechanical tests on plant tissues, they are somewhere empirical and from time to time seem to use ad hoc hypotheses to extend their area of validity. So, even if it seems possible to consider a modelling program as a theory in complex situations (PARTRIDGE and LOPEZ, 1984), it could be more appropriate to clearly define the basis of plant tissue mechanics in order to get clearer interpretations of experimental tests. From another point of view, a modelling program only uses some elements of a theory and is generally not an exhaustive description of it.

Plant tissue is an aggregate of cells acting in unison. The structure of these cells determines the shape of the plant and also influences the cell division or enlarge- ment processes. A complete theory of plant tissue mechanics would have to specify the essential processes and structures in the proper time sequence at all levels of organisation. This goal does not seem attainable because of the complexity of the system (LINDENMAYER, 1975). However, it should be possible to attempt a simplified theory, focusing on tissue behaviour, leaving aside subcellular micromechanics.

Thus, the aim of this work is to propose such a theory able to explain most mechanical particularities of plant tissue during the growing process or physical testing.

2 – PLANT TISSUE DESCRIPTION

The definition of a mechanical theory of plant tissue needs a good knowledge of plant tissue organization because the mechanical properties of tissues are based on the mechanical properties of cells, cell walls and so on.

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The plant body consists of structural units: the cells. Each one is enclosed in its cell wall and touches other ones through a cementing, mainly pectic, substance.

Cells are then grouped together forming tissues. The basic geometry of a cell is tridi- mensional: its average shape being somewhat like a polyhedron with about 14 faces (MATZKE, 1946). However, the cells have different aspects depending on their functions and positions inside the tissue (JEWELL, 1979). In our work, the two main cell categories which will be taken into account are the protective cells which are in the epidermis, and the parenchyma cells. The former are elongated and quite small compared with the others and very tightly pressed together leaving few spaces between them. The latter are rounder, larger, and form the bulk of the primary tissues in plant materials. They are not as pressed together as the protective cells and, depending on the plant varieties, are partly separated by intercellular spaces. Most of the mechanical strength of plant tissue is assumed to be linked to the mechanical resistance of the network formed by the cell walls. Elasticity is controlled by that network bending, then some buckling phenomena appear and cell wall collapse can be observed if the incident stress is high enough. Another part of its strength is due to cells' elasticity depending on their turgor pressure (NILSSON et al., 1958). This pressure corresponds to the osmotic pressure generated within the cell sap. In plants subjected to stress from water loss, tissues become flaccid, related to a lack of pressure due to plasmolysis, which means water transfer through the cell wall. In fact, a third element of plant tissue is involved in its strength: the intercellular voids (RUESS and STÖSSER, 1993). This last element has very often been forgotten, because of the difficulty of measuring it. However, in plant tissues, it is very important from a volume point of view (more than 25% of the total volume of some fruits can be made up of such voids), and from a mechanical point of view: these zones are the damping parts of tissues, allowing some high stress to be applied on tissues without destruction (WENIAN, 1991). These voids are also important in wave propagation creating impedance variations.

Intracellular elements are of no importance for a mechanical theory, because it has never been demonstrated that they have a direct visible influence on the mechanical properties of plant tissues.

An important element of tissue description is its heterogeneity. As mentioned above, cells have different sizes and shapes depending on their location. But even in a small area, they are all different and it is impossible to find a homogeneous tissue at the cell scale, either for cell shape, or for void volume or spatial distribution. This is a very important fact to keep in mind when interpreting mechanical tests on plant tissues.

3 – TYPICAL MECHANICAL TESTING OF PLANT TISSUES

Mechanical tests are mostly used on plant tissues in order to evaluate their texture. This last notion is assumed to be a complex mix of different parameters, mostly mechanical. Then crushing, indentation or fracture tests are applied to tissues and their results compared to organoleptic results. The final aim is to understand the links between human appreciation and mechanical characterisation (HARKERet al., 1998; SZCZESNIAK, 2002). Until recently, the only results were correlations between these two kinds of testing (human and mechanical). The main reasons for that were the need to find testing methods more objective and more repetitive than human

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experiments, and the lack of knowledge about the physical behaviour of cellular tissue. So during the last thirty years many studies have been made on plant tissues and researchers have begun to feel the necessity for a theoretical background, in order to interpret these experiments in a better way. In fact, comparisons with other areas show the complexity of the structure of plant tissue. One can find some attempt at theorization in areas like rock or sand mechanics. The latter are probably the nearest domains to our own (ROUDOT, 1996). However great difficulties occur even if sand grains cannot be deformed, there are two physical phases (solid, and gas or liquid) and so on. In plant tissue the problem is more complex because cells can have their shape and size modified, their rigidity can change and there are at least, three phases in a tissue (solid, liquid and gas). These facts can explain the difficulties occurring when attempting to explain texture tests.

One can look at some typical results obtained in the texture analysis of plant products to better understand the particularity of living cellular tissues.

3.1 Compression and impact tests

After a compression or an impact the plant product shows a clear bruised semi- hemispherical area. A better analysis through fine cutting of that part permits the vis- ualisation of continuous lines of destroyed cells, separated by intact tissue (WENIAN, 1991). The difference between compression and impact (with the same energy deformation) appears in the density of bruised cells (greater in impact) and the size of the deformed area (greater in compression) (figure 1). On some samples, like Asian Pears for instance, the deformed zone is star-shaped, and not semi-hemispherical. These particularities are typically consequences of the cell structure of the samples, and have to be taken into account in the final interpretations, which is almost never done.

3.2 Puncture test

This is one of the most often used tests and its measurements are not understood very well. The test involves plunging a small cylinder in the tissue and measuring the stress-strain evolutions. It is assumed to measure a mix of compression (under the plunger) and shearing. In fact, it can easily be seen that in front of the crushing area, there is a small quantity of destroyed cells (dead zone) which is the

a

b

Figure 1

Visual aspect of the result of an impact (a) and a compression (b) of the same energy on an apple. One can note the black lines (destroyed cells) surrounded by intact tissue.

During an impact, contact time is shorter than tissue time constant and many cells are destroyed, while during a compression, tissue may rearrange and better support

the energy.

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actual testing apparatus (ROUDOT et al., 1990). This is continuously reinforced by new crushed cells, and some are pseudo continuously extruded along the plunger (figure 2). This phenomenon obviously influences the measurements which are thus far more difficult to interpret.

3.3 Fracture test

Another test, among many others, can be used: the fracture test. The aim is to provoke a fracture in the tissue and to measure different stresses during the evolu- tive part of the phenomenon (VINCENT et al., 1991; FAHLOUL and SCANLON, 1996). In this case, the fracture initiation takes place in the intercellular voids and the direction of these fractures is driven by void geometry. So, it is impossible to attempt the analysis of such a test without a theoretical background in cellular tissue mechanics.

These few examples clearly show that interpreting texture tests on plants, with only a background in continuous mechanics is of little interest. As seen above, plant tissue is very complex to analyse from the geometrical and mechanical points of view, and so different types of research were undertaken in order to obtain a model able to explain (and foresee) tissue deformation under mechanical stress.

4 – MODELLING METHODS

4.1 Constitutive methods

The aim of these models is to obtain an equation able to explain the mechanical behaviour of the tissue (figure 3). This kind of model can be interesting because it can enable the prediction of the bulk deformation of a tissue submitted to particular stresses. In order to obtain this equation, different methods may be used (TRIPODI et al., 1992):

– an empirical one is based on the analysis of experimental data of the material under specific loading conditions (SHARMA and RAFIE, 1983; CASAS-ALENCASTER

and GOMEZ-CHAVEZ, 2000). Then statistical calculations are made in order to Figure 2

Photograph of the area just below the cylindrical plunger (in white) during a puncture test. The actual shape of the plunger is hemispherical

because lots of crushed cells (black) are lying in front of the plunger modifying its shape.

These cells are evacuated from time to time along the vertical parts of the plunger.

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obtain the best equations matching these data. However, its main problem is that it often gives good results for one type of loading and is unable to predict any others. Another problem occurring with this method, is that there is no link between equation parameters and material properties or geometry (TOKUMOTO

et al., 1999). In this case it is impossible to interpret the results in terms of the physical or physiological characteristics of the tissue;

– the macroscopic way of modelling considers the bulk solid as a continuum and describes the load-deformation characteristics of the material as a whole (KAJUNA et al., 1996; LINARES and DAL FABBRO, 2001). The theoretical basis is then continuous mechanics. The main problem is to know whether there is any reason to accept that a cellular medium can be analysed as a continuous one?

Nevertheless, this kind of model has given some good results in the cohesive soil area. Another problem is the necessity of assuming an isotropic behaviour and a global homogeneity of the tissue, because no information is given about the actual geometry of cells and voids. This creates some difficulties in interpreting results and in comparing experiments with simulated results.

An intermediate method between global and cellular ones is based on finite element method (DEWULF et al., 1999). A discretisation of the product is made taking into account some irregularities or discontinuities. Anyway, if this method gives good results in analysing global responses to mechanical stresses (CHERNG, 2000;

LU and ABBOTT, 1997), it cannot give any information on the relative influence (or importance) of local parameters.

Another approach in this kind of modelling is the analysis of plant tissue as a cellular medium. Cellular media have been theorised for twenty years (GIBSON and ASHBY, 1988). The application of this theory to plants is due to the obvious links between industrial cellular solids and living tissues: the stiffening agents are the same, namely the cell wall structure and the cells' inner pressure (NIKLAS, 1989). This method gives good agreement with experimental data when looking at the variations of Young’s modulus with turgor pressure, or the link between stiffness and the

F

E₁ E₂ E₃

η1 η2 η3

E₄

Figure 3

Example of a constitutive model used in order to explain the physical behaviour of plant tissue under mechanical stresses. Here, a Maxwell generalised model.

The unsolved problem is the link between the parameters E_i and ηi and the physical parameters of the tissue (pressure, size of cells, and so on).

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number of cells. The theoretical problem encountered with this method is the same as above: globally it is a macroscopic modelling method based on cellular mechanics. So, the only possibility of interpreting results in terms of actual parameters is statistical. (ROJAS et al., 2002)

However, this last method is a great advance in the understanding of plant mechanics compared with other constitutive methods because it clearly distin- guishes two important aspects of this kind of tissue: the cell wall network and the inner pressure of cells. Unfortunately, these methods cannot explain such problems as fracture initiation or the actual shape of bruised areas.

4.2 Microscopic approach

One can see that the main problem with constitutive models is the impossibility of interpreting results in terms of a tissue’s physical parameters. This is the reason why some researchers prefer microscopic methods of modelling. In these methods, each particle in a bulk solid is considered as a distinct entity and then mechanical behaviour is evaluated from the “integration” of all cell behaviour (TRIPODI et al., 1992). In texture analysis of plant food, particles are equivalent to cells which have traditionnally been considered as disks (in 2D modelling) or spheres (in 3D modelling). Progress in computer science has permitted the development of models with more realistic cell geometry (GAO and PITT, 1991). But the isotropic assumption always occurs. Then the tissue is made up of an array of homogeneous cells of identical size, shape and mechanical properties. More recently some attempts in modelling more realistic cells have been made in determining membrane behaviour (DAVIES et al., 1998; KUNZEK et al., 1999) or in using a finite element model (WU and PITTS, 1999) (figure 4). In this case, the cell shape is calculated using actual images of apple cells and then the cell is assumed to be pressurised. During the compression simulation, the plasmolysis phenomenon is taken into account. Finally, results from volumetric elastic modulus and cell wall stresses agree with those expected.

However, even if it seems possible to model one cell and apply external stresses on it with results corresponding to those expected, the aim of tissue mechanics modelling has not yet been attained. Cellular tissue behaviour is not only the summation of cell behaviour, and the possibility of calculating tissue behaviour from cell one is not studied at all. The problem of taking into account cell shape, spatial distribution and mechanical behaviour heterogeneity is no longer evaluated even if some some statistical integration methods exist (ROUDOT, 2000).

In fact, the only positive element is the possibility of creating a cell geometry not far from the real one, and calculating its inner pressure. Nevertheless, experimental knowledge is lacking concerning actual contact phenomena (stress distribution, plasmolysis, void and cell deformations, possibilities of cell debonding, and so on), and no method exists for taking into account this complexity in the model.

So, it seems that microscopic methods have reached their limits and are unable to give an efficient basis to a tissue theory. They can be used as simplified methods of physical interpretations but their inability to integrate tissue bulk inhomogeneity and anisotropy is far too limiting.

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4.3 Integrative approaches

As neither the constitutive methods nor the microscopic ones can be chosen as a possible basis for plant tissue theory, the only possibility is to stay at the tissue level, understood as a macroscopic view of a microscopic structure. Then, the first problem to be solved is to discover whether it is possible to find a macroscopic equivalent of the tissue (AURIAULT, 1991; MEI and AURIAULT, 1996; HORI and NEMAT- NASSER, 1999). The main condition for this is that it will be possible to separate the scales, this separation concerning both the structure of the medium and the excitation itself. What we want to obtain is therefore an equivalent macroscopic problem, i.e. relations between “averaged” quantities and effective parameters. The macro- scopic description will thus be a continuous description as opposed to the microscopic one which is discontinuous. If this is possible it will be said that we have homogenised the cellular tissue (figure 5). For that, it is necessary for the macroscopic domain to contain a large number of microscopic elements or cells. In other words if L is a characteristic length of the macroscopic domain (for instance in our work, the vegetable length), and l a characteristic of the cells (for instance a cell diameter): L/l>>1.

The second point is the separation of scale of the excitation to be simulated. For instance, if the excitation frequency is high, its wavelength is long and can be of the same order of size as l. Then the separation does not exist, and homogenisation is not possible. For instance homogenisation is not possible in a study of ultrasound transfer in a cell, but possible for sound transfer, because of the frequency level (some megaHertz against some Hertz).

It can be seen that homogenisation process is not always possible from a physical point of view. For instance, modelling ultrasound transfer in a plant tissue is

Figure 4

Example of an attempt to modelise a cell using finite element method showing a polyhedral shape and the different forces applied (from WU and PITTS, 1999).

The problem of this method is to define and calculate tissue organisation, which is not a physical characteristic.

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impossible because of this separation condition. The difficulty in evaluating the contact stresses concentrations is also a problem in using these methods (LIAO et al., 2000). The last problem encountered with this method is the same as described above, i.e. the difficulty of relations between the simulation parameters and the geometrical characterization of the tissue components.

Cellular automata correspond to another approach to the problem, analysing macroscopic consequences of microscopic elementary variations (WU and DAVID, 2002). Applying this method to cell mechanics may seem to be obvious because both automata and tissue are based on the variations of individual cells and on the consequences of their interactions on a macroscopic element (STARK and HUGHES, 2000). In fact, asynchronous automata appear to be of great interest in biological modelling. Their main advantage for our work is that each cell is individualised and can be known, independently of its neighbourhood, although it works in a network having its own behaviour. An extension of automata is known as “particle systems”, in which each cell has its own geometrical and physical parameters capable of changing (REEVES, 1983; FLEISCHER, 1995). Each parameter can fluctuate between two frontiers, depending on cell external (or internal) conditions. There is also a possibility of creating different kinds of cells, each kind having globally the same behaviour, which in our domain means creating epidermal cells with their own living conditions, parenchyma cells with their own conditions, and so on. This latter model does not clearly take into account the cell geometry, and that is the reason why we have defined histological modelling (ROUDOT et al.,1991). This last method will be described below because it is an application of tissue theory obtained from the analysis of the models above and of the experimental data, which is explained in the next paragraph.

Figure 5

Example of an homogenisation problem. The modelled tissue is a periodic arrangement of a shape. In order to evaluate its behaviour, one isolates one part (low) and study its limits conditions. Then, the homogenisation theorem allows tissue reconstruction.

In a plant tissue different problems occur: lack of periodicity, lack of knowledge to evaluate the limits conditions, difficulties to separate microscopic and macroscopic domains.

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5 – PLANT TISSUE THEORY

5.1 Description

We have seen that all the proposed models have limitations and cannot be selected as a possible theory of tissue mechanics. In most cases (namely in microscopic and constitutive approaches), this fact is a consequence of analysing a complex system as a simple subsystem, excluding most of the actual one. So the behaviour of this subsystem deviates from what can be expected from experimental tests on the actual tissue. These deviations seem unexplained because much of the actual tissue organization is abstracted and invisible. These reductive methods based on the Newtonian paradigm (classical mechanics not taking into account the problems of complexity), cannot explain systems in which phenomena such as

“emergence” (which is a consequence of complexity and means that some effects appear which were not foreseeable) occur (YATES, 1994). It appears that the best approach is the integrative one. It is not by chance that modern methods used in creating numerical landscapes or growth of plants are based on integrative methods (REEVES, 1983). It is because they permit this kind of unpredicted fact linked with their high degree of recursivity (KIM and AIZAWA, 2000).

Thus, the theorisation of tissue mechanics has to be an integrative theory, taking into account all the elements described above which are limitations of the models already used. This theory has to be as complete as possible and must give real answers (or possible answers) to problems as different as the explanation of the puncture tests or internal firmness variations. So, the basis of this theory is that cer- tain important parameters occur in tissue mechanics: the cell wall network, the inner pressure of cells, the integration of individual cell behaviour and the microscopic tissue geometry. Another important element is the separation of the constitutive structure elements of the tissue and its physical behaviour.

Another parameter not often studied in this area is the dynamics of the test being simulated. Very often one uses only two instants: just before the test, when tissue is assumed to be without any external stress, and just after the end of the test, when it is assumed to be stressed. The dynamics of stress application is forbidden. So new conditions have to be inserted into the theory: the history of the tissue, and the dynamics of the tests. Tissue history is very important because its geometry is dependent on this fact. During tissue growth, the cells have to use the areas free from other cells, and, if necessary, to create their own space. During this growth phenomenon, cells are stressed more or less depending on their position, shape, size, and so on. So the initial tissue mechanics, just before the test, has a long history which can have a major influence on the test results (MOREAU, 2003). The dynamics of the test happen because the tissue is inhomogeneous and discontinuous (SUBHASH et al., 1991). Thus stress propagation is influenced by cell geometry and elastic properties. The tissue can adapt or not to the new mechanical conditions, depending on the duration of the test and on the time constants of the individual cells and global tissue.

Thus a cellular tissue is assumed to be a complex system in which we can find different parts:

– the constitutive physical elements, which are the individual cells;

– the physical organisation of these elements, which creates the physical tissue;

– the intrinsic functions of the cells as parts of the tissue.

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These different elements exist in all complex systems and need to be taken into account in a modelling process if further simulations have to be close to actual experimentations.

In our model we will only consider two independent domains: the domain of operation of the structural components of the tissue (i.e. of the cells), and the domain in which they will be considered as a whole (MATURANA, 1987). The first one will be used during creation of the model, and the second one during further simulations using this created tissue. In this case, what we can call the tissue organisation is the configuration of both static and dynamic relations between its components which specify its identity as a whole. So, tissue organisation takes place partly in the first domain and partly in the second one. In fact it is the link between the two domains (figure 6). During physical tests on the created tissue, interactions occur between external elements, the tissue as a whole and individual cells (COTTAM et al., 2000). It follows that interactions can modify both the first and second domains.

Relations between external events and constitutive elements are directed by classical physical laws, all relations with tissue as a whole being directed by structure organisation, and hence its own internal laws.

Finally, what appears, is that tissue behaviour is not determined by tissue structure or external medium, but by interactions inside the tissue itself and between that tissue and this external medium. This is the reason why theories which consider that tissue behaviour comes from cells’ physical characteristics are not sufficient, as they forget the interactions inside the tissue itself and the high degree of recursivity in such a system.

Organisation

External actions Global domain

(simulation) Structural domain (modelling)

Figure 6

Theoretical aspect of the histological modelling process. The structural domain corresponds to the physical creation of the tissue, from a geometrical point of view.

The global domain is used during the simulation. Between these domains one can note what we called tissue organisation which is the configuration of static

and dynamic relations between tissue components.

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5.2 Practical application

To test its ability to answer the different questions existing in the mechanics of plant tissue, this theory has been used to define a modelling and simulation method called histological modelling. This method takes into account all the parameters discussed above, from creation of the tissue to simulation of the tests. In fact mathematical theorization is not yet done, but numerical modelling and simulation are done in order to show the possibilities offered by this new way of describing a cellular tissue under constraints. The method is based on 3-D graphical analysis, in order to have a better view of the strains and geometry of the tissue.

First of all the tissue must be created. Different solutions were used, but all of them consider its growing history. For instance for vegetable analysis, all the cells are supposed to be pre-existing as small spheres, then they grow. Each cell is known as an entity with its center coordinate, its size (radius), and its stress (strictly depending on its contacts with the other cells in its neighbourhood).

The cells are defined as : c_i(p_i, σi, r_i, V_i), where:

p_i is the cell spatial position, σi, is its stress

r_i, is its radius

V_i, is its neighbourhood, i.e. the cells touching c_i

Each cell is randomly called in order to process a step of growth (one pixel more on its radius) ; this growth has some consequences for itself :

– it has room enough to grow, and the only consequence is its size modification : c_i(p_i, σi, r_i, V_i) c_i(p_i,σi, r_i+1, V_i)

– it has no room enough and either the cell try to move a little (one pixel) towards a less stressed area [c_i(p_i, σi, r_i, V_i) c_i(p_i+d, σ^’i, r_i+1, V’_i), where d is a small distance (classically one pixel in the simulation), V’_i is its new neighborhood] or a less stressed cell in contact has to move in the same way.

[ then

c_i(p_i, σi, r_i, V_i) c_i(p_i, σ^’i, r_i+1, V_i^’) c_j(p_j, σj, r_j, V_j) c_j(p_j+d, σ^’j, r_j, V_j^’) ]

If the cell touches another one, its stress increases, depending on Hooke’s law.

Two limits exist in this growing process: the maximum size of the cell, and a maximum value for its stress.

Another possibility of creating the tissue was used in order to model an ovarian follicule (ROUDOT, 1995). In that case, the different cells can grow and divide them- selves. In order to be able to divide itself, a cell must encounter two conditions:

– having a minimum size;

– having a stress not too high, and not too low.

The total area of the two new cells is the area of the initial one, and the division orientation is randomly chosen. The stress of the new cells is calculated from Hooke’s law.

These growing methods have important consequences:

– during the process, an unique cell modification can change the tissue geometry and its mechanical behaviour;

a

∃ ∈^.^c^j ^{V c}

(

ⁱ

)

^σ^j ^<^σⁱ^,

a a

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– the tissue is inhomogeneous in term of cell size, cell shape, and intercellular voids;

– the tissue is anisotropic, because of these inhomogeneities and of the variations in cell stresses;

– from a computing point of view, each cell of the tissue is individually known (position, size, stress, neighbourhood).

Different other conditions can be added such as skin elasticity, or local growth impossibility and so on.

Once this tissue obtained, the problem is the simulation. Each kind imposed some new considerations. For instance:

– if the test is based on electromagnetic waves transfer, one has to define the absorption, reflexion and refraction coefficients, for voids, membranes and cell. The wave is considered as the sum of its rays:

Each ray is known from its position pos_ialong a straight line with an angle αi

from a horizontal line, and with an energy E_i.

β is the angle between cell membrane and ray direction and β0, the limit angle for reflection:

If , then , where the new angle is

the refraction one and the energy is attenuated,

If , then

Inside a cell, there are no change for a ray. Inside an intercellular void, E_i = 0, the ray is totally absorbed.

– if the test is a mechanical one, one has to decide the tissue time constant and the mechanical stress duration. By combining these two parameters, one calculate the theoretical basic size of the tissue.

[If t₀ is the application time of the external force, and A the application area of this force, the basic size is defined as an influence area Z as:

, where d(c_i,A) is a distance measure between the cell c_i and A; f(t, t_c, F) is a strictly growing function of t, depending of tissue time constant t_c and of the maximum applied force F.

All cells inside Z will have the same mechanical behaviour.]

For instance for a long stress duration with a short time constant of the tissue, this one has time to adapt itself and then the size of the basic element used in calculation can be big enough (numerous cells are considered to be an homogeneous area for the test, with the same stress, and no influence of voids). In case of a high stress, the basic element can be smaller because of the necessity of absorbing high energy. Calculation of this basic element size is the more important part of the simulation conditions.

5.3 Example: modelling a compression on apple flesh

For instance, in order to model apple flesh (ROUDOT et al., 1994), it is considered that the majority of final cells exist at the beginning of the process, but are very

W ray pos E

i i i i

i

=

∑ (

^[ ^{], ,}^α

)

β β< ₀ ray pos E ray pos E

It has been shown that its spatial firmness variations are similar to actual ones (figure 7). At this point there exists the possibility of changing cell shapes into polyhedra, using their centre coordinates and a Voronoi-Delaunay method (BROSTOW et al., 1978), modified in order to obtain intercellular voids (figure 8).

a

b

Figure 7

Comparison between an actual measurement of firmness on a slice of apple (a), and two simulations of the same parameter (b). (The clearer the area is, the softer it is,

except for the core). Simulations are made after a growth process, only taking into account the contact stresses between cells and some possibilities

of cell debonding. Differences between the two simulations are only linked with the initial number of cells, and their initial repartition.

Figure 8

Example of the simulated geometry of a plant tissue (here, apple tissue) using the Voronoi-Delaunay method. The cells may be polyhedral, with different shapes and sizes. The intercellular voids may be regulated in order to obtain a model very similar

to an actual tissue (from a geometrical point of view).

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Using this synthetic tissue, it is possible to simulate lots of actual tests: crushing, impact, puncture, wave transfer, and so on. Each test has to be studied to deter- mine its main mechanical elements. For instance, if an impact has to be simulated, one must clearly define the impact parameters (impact duration, maximum stress, maximum impacted area, force variation curve during impact). Tissue resistance also has to be defined (time constant, maximum shear stress, maximum normal stress, relaxation time). In this example, the link between microscopic and global mechanics depends on the tissue time constant and the impact duration: if the duration is short with a fixed time constant, the basic cell cluster used in stress transfer (mechanically homogeneous) is smaller than with a long duration where tissue has enough time to adapt (figure 9).

Different tests have been simulated using this method that give fairly good results, if compared with experimental ones (ROUDOT et al., 1991; ROUDOT et al., 1994; ROUDOT, 1995) (figure 10). The simulation power of histological modelling is essentially due to its highly recursive functioning in stress propagation during tissue creation and test simulation (figure 11). This fact makes it possible to apply few mechanical laws at the cell level, and obtaining some global tissue behaviour.

a b c

Figure 9

Simulation results of compression (a) impact (c), and intermediate (b) tests.

The energy is the same, the difference is the contact time. One can see two lines of destroyed cells in (a), and a far more destructive result in (c), even if some intact

cells can be observed. Comparisons with figure 1 shows a fairly good agreement between actual and simulated tests (if taken into account the difference in the number

of cells involved).

Displacement

Force Force

Displacement

Figure 10

Comparisons between a simulated puncture test (left) and a real one (right) shown in arbitrary scales. The simulation is based on histological modelling and

it takes into account the crushed cells in front of the plunger.

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6 – CONCLUSION

The histological theory of plant mechanics is a new step in the understanding of plant tissue behaviour. Its main interests come from the link between cell mechanics and tissue behaviour, which is not directly computed but is a consequence of the high degree of recursivity: the modification of one cell’s geometry, size, position or physical parameter influences its neighbourhood, which influences the cell directly or not, and so on. The other important fact is the possibility of having an initial tissue state depending on its growth or prior mechanical effects, which means that the tissue has its own history and is not a cell conglomerate isolated from everything except its test conditions. The results which are obtained with the model defined using this theory show that tissue organisation is more important than individual cell mechanics. Different improvements can be made to this model: considering one plant as made up of different tissues with their own different behaviour, using more cell parameters (cell wall elasticity and permeability, for instance), taking into account the intercellular adhesiveness, and so on. No doubt these modifications will improve the simulations, but the most important element now is to define clearly the interactions between different tissues during growth: how do core rigidity and skin elasticity infuence the flesh? Currently these developmental parameters are the most influential ones for tissue creation and final behaviour.

Figure 11

Visual representation of the important area concerned by the size modification of one cell.

In order to grow, a cell must make its place among the tissue and so must push its neighbours. The growing cell is in white in the middle, the untouched cells are in black.

Between these two limits, the clearer a cell is, the more concerned it is by one step of growth of the white one. This is an explanation of the great heterogeneities obtained

in local firmness at the end of a growing process (figure 7).

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