Trends in Food Science & Technology 19 (2008) 59e66
Review
Modelling fruit
(micro)structures,
why and how?
H.K. Mebatsiona,*,
P. Verbovena, Q.T. Hoa, B.E.
Verlindenb and B.M. Nicola€ıa,b
a
Postharvest Research Group, BIOSYST-MeBioS,
Katholieke Universiteit Leuven, W. De Croylaan 42,
B-3001 Leuven, Belgium (Tel.: D32 16 32 05 90; fax:
D32 16 32 29 55; e-mail: hibru.mebatsion@biw.
kuleuven.be)
b
Flanders Center of Postharvest Technology, W. De
Croylaan 42, B-3001 Leuven, Belgium
The relationships between fruit structure and the material
properties affecting fruit quality are not well understood to
date. One reason is that the effect of fruit structure is difficult
to investigate due to the presence of important structural features at all spatial scales. Multiscale modelling offers a framework in which the relevant transport processes are studied at
microscopic scale and the resulting information is transferred
to the global scale by homogenization procedures. In this respect, modelling the geometry at the smaller and larger scales
is an essential aspect of study. This paper presents the advances
that have been made on geometrical modelling of fruit at different scales.
Introduction
The physical properties of biological materials, such as
fruit, are important for the control of their metabolism
and quality. Most biological materials are living, i.e., they
maintain metabolic processes in an attempt to preserve their
natural state. If the metabolism cannot be continued, the biological material quickly changes its structure, in most
cases degrades, finally resulting in death. Fruit, after harvest, continues its respiratory activity to preserve the integrity of the cellular microstructure. The microstructure
* Corresponding author.
0924-2244/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved.
doi:10.1016/j.tifs.2007.10.003
determines the mechanical properties of the fruit responsible for texture (e.g., firmness, crunchiness) as well as how
they perform or fail when a load is applied during postharvest handling. The material structure is also important to
supply and remove the required gasses (O2 and CO2) for intracellular respiration. Here, the intercellular air spaces are
the main pathways to bring or remove them to or from the
center of the biological material. In this respect, both the
microstructure (the layout of cells and intercellular spaces)
and the macrostructure (shape and size of the material) are
important.
Biological materials appear continuous when viewed at
the macroscopic scale. It is therefore often assumed that
these materials behave as a (non)linear, (visco)elastic continuum and predictive models are developed based on
macroscopic continuum physics (Ho, Verlinden, Verboven,
Vandewalle, & Nicolai, 2006; Nguyen et al., 2006). However, the macroscopic properties of the fruit likely depend
on various microscopic histological and cellular features
such as the types of tissue, the geometric properties of
the cell, the presence of an adhesive middle lamella between individual cells, the cellular water potential, the
mechanical properties of the cell wall, the presence of intercellular spaces, and many more (Bao & Suresh, 2003).
These features cover a wide range of spatial scales, from
nanoscopic (plasmodesmata, plasma membranes), over
microscopic (cell wallemiddle lamella complex, cell geometry), to macroscopic (actual geometry of the material).
The material properties of the continuum model, such as
the elasticity modulus and the diffusion properties of the
tissue should therefore be considered as apparent material
parameters which incorporate not only actual physical material constants such as the compressibility of water and
air, but also the microscale geometry of the tissue and
the intracellular space. The relationship between the macroscopic apparent properties and the microscopic features
is not understood well to date (Ghosh, Lee, & Moorthy,
1996; Wood & Whitaker, 1998). As a consequence, the
available continuum models have a limited range of
validity.
In biological materials, the shape and size of components at all scales show considerable variability, connecting
passages are tortuous, connectivity is random and above all,
many length scales come into play (Mendoza et al., 2007;
Sen, 2004). Due to this complexity of the geometric and
spatial arrangement of structures, investigation into the
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H.K. Mebatsion et al. / Trends in Food Science & Technology 19 (2008) 59e66
microstructural geometry of fruit is evident (Mebatsion,
Verboven, Verlinden, et al., 2006). In this respect, this paper
reviews the merits and demerits of geometrical models that
can be used to study biological material behaviour by
means of computer based simulations. We will concentrate
on simulation models that take into account the spatial dimensions of the material. These models are most often
solved by means of the finite element method (Yue,
Chen, & Tham, 2003). The models are illustrated for fruit
tissues. The spatial scales that are considered in this paper
are:
The macroscale addresses the fruit as a whole. At this
scale the fruit is considered as a continuum, and may
consist of different connected tissues, all with homogeneous properties (Ho et al., 2006). At this scale, one
can also consider samples or parts of the material, consisting of different tissues with important detailed subtissue components (e.g., the layers of the cuticle) that
are still modelled as different but connected homogeneous components (Veraverbeke, Verboven, Oostveldt,
& Nicola€ı, 2003).
At the mesoscale, the actual topology of the individual
fruit tissues is considered, incorporating the layout of
the intercellular space, cell walls and individual cells
as building blocks. The different tissues are different
compositions of these ‘basic’ units. The spatial relationship of the structures and biophysical processes is only
starting to be investigated (Aalto & Juurola, 2002; Ivanova, Petrov, & Kadushnikov, 2006; Mendoza et al.,
2007).
At the microscale, single cells are distinguished in the
tissue and the physics of the microscale features (cells,
cell walls, cell membranes) is studied. Each of the microscopic components is subject to active biophysics research. Research is conducted to determine functional
properties of the cell wall (Juge, 2007; Klis, Mol, Hellingwerf, & Brul, 2002; Lee et al., 2004; McCann
et al., 2001) and the cell membrane and its components
(Murata et al., 2000; Rustom, Saffrich, Markovic,
Walther, & Gerdes, 2004; Tyerman, Niemietz, & Bramley, 2002).
From the above, it is clear that to bridge the gap between
the existing knowledge of microscale processes and the
macroscale behaviour of biological materials, the mesoscale needs to be resolved. This means that the microstructural topology of tissues must be measured and used in an
appropriate framework that combines the information on all
scales. Multiscale modelling is a new paradigm to resolve
this issue. The content of this paper proceeds as follows.
In the next section, multiscale modelling methods that
link the finer microstructure scale to a coarser macroscopic
scale are introduced, which is followed by the geometric requirements in multiscale modelling. Further, geometrical
modelling approaches at different spatial scales of fruit
are reviewed. The final section presents conclusions and future directions.
Multiscale modelling
Multiscale models are basically a hierarchy of submodels which describe the material behaviour at different
spatial scales in such a way that the sub-models are interconnected. Microstructures are much larger than the molecular dimension to justify a continuum approach for
modelling, but they are much smaller than the characteristic
length of the macroscale (Kouznetsova, Brekelmans, &
Baaijens, 2001). As a result, investigation of the microstructure becomes a prerequisite to understand the transitional theoretical frameworks and modelling techniques to
bridge the gap between length scale extremes (Ghoniem,
Busso, Kioussis, & Haung, 2003).
Multiscale modelling may involve challenging physical
processes such as transport phenomena, which are prohibitively expensive to solve at different length scales. Sometimes it is sufficient to find the solution of the coarser
scale by including procedures to construct the equations
on the coarser scale that account for the contribution of
finer scales (Hou, 2005). However, this amounts to writing
effective equations for the macroscale that account for
lower scales, which is a difficult task. For example, the
skin mass transfer resistance to water transport of fruit to
the ambient environment can be well approximated by
the summation of the resistances of the different tissues
(cuticle, epidermis, and hypodermis) of the skin (Nguyen
et al., 2006). Alternatively, equations for the fine scale itself
can be solved. The up-scaling of fine scale solutions to
a macroscale solution is known as homogenization. Homogenization has been defined as a collection of methods
for extracting or constructing equations for the coarse scale
(macroscale) behaviour of materials and systems, which incorporate many smaller (nano-, micro-, meso-) scales. The
main objective of such an approach is constructing simpler
fine scale equations that are considerably less expensive to
solve, and whose solutions have the same coarse scale
properties (Brewster & Beylkin, 1995; Mehraeen & Chen,
2006).
A good example of the homogenization approach in
multiscale simulation is presented by Wood, Quintard,
and Whitaker (2002), where the diffusion and reaction
of multi-component species in biofilms are modelled at
all spatial and temporal length scales. The effective biofilm transfer coefficient is a function of microscopic diffusivity of the polysaccharide matrix and the
microstructure of the biofilm (Wood et al., 2002). In their
analysis, the biofilm structure was divided into three subscales. At level I, the biofilm (with dimensions of hundreds of micrometers to millimeters) was modelled as
a continuum. At level II, with a dimension of hundreds
of micrometers, the biofilm was represented as a twophase system consisting of cells (a phase) and an extracellular polysaccharide matrix (b phase). At level III, a single
H.K. Mebatsion et al. / Trends in Food Science & Technology 19 (2008) 59e66
cell contained in a cell membrane was defined. The homogenization procedure involved in the transformation
of information from level III to level I, through incorporation of the cellular geometry at level II. The effective
macroscale diffusivity of the biofilm could be calculated
based on sub-cellular characteristics of the biofilm and
its geometry.
Geometric requirements for multiscale modelling
The simplest application of multiscale modelling is that
the lumped material properties at the macroscale are determined by fitting the results of the macroscopic phenomenological equations to the detailed modelling results at the
finer scales. A representative volume element (RVE) needs
then be defined as the minimum volume over which the
lumped properties (as they can be integrated over the
RVE) can still be calculated and used on larger scales.
This condition is known as the periodicity requirement.
For volumes smaller than the RVE, the continuum hypothesis fails and integrated properties are not constant as
a function of the spatial scale. The clearest illustration of
this concept is the porosity of a material. Suppose you
can look at the microstructure of a material in a discrete
pixelized manner at different fields of view. At the smallest
field of view (1 pixel), porosity is either 1 (inside a pore) or
0 (inside the material matrix). When you increase the field
of view and integrate the porosity (e.g., summing all the 1
and 0 values and dividing the total by the number of pixels),
the porosity will change. Above some field of view the porosity will no longer change. This is the RVE. A detailed
discussion can be found in Mendoza et al. (2007).
At the other end of the spatial spectrum, the lumped material property may start to change again. This indicates
heterogeneity of the material, e.g., fruit consisting of different tissues with a different characteristic porosity (Mendoza
et al., 2007). For materials having different tissues (layers),
local periodicity is the precondition of multiscale modelling. The microstructure has different morphologies corresponding to different macroscopic points while it repeats
itself in a small vicinity of individual macroscopic points
(Kouznetsova et al., 2001). The concept of local periodicity
is depicted in Fig. 1. The figure shows the presence of spatial and topological variability at different positions
(Fig. 1aec) of an apple fruit. The positions represent the
cortex, the vascular bundle and the transition from the cortex to the vascular bundle, respectively. In this respect,
RVEs are limited to a single heterogeneity implying that
RVEs may be repeated to represent the entire microstructural neighbourhood (Lee & Ghosh, 1999). Yet, the actual
choice of the RVE is a rather delicate task. The RVE should
be large enough to represent the microstructure, without introducing non-existing properties (e.g., undesired anisotropy) and at the same time it should be small enough to
allow efficient computation (Gitman, Askes, & Sluys,
2007; Pellegrino, Galvanetto, & Schrefler , 1999).
61
RVE simulations thus require microscale models that
distinguish the different microstructural geometrical features to develop appropriate microscale models (Gitman
et al., 2007; Lee & Ghosh, 1999; Matsui, Terada, &
Yuge, 2004; Pellegrino et al., 1999; Wood et al., 2002).
With such large differences in length scales, generating geometries that accurately represent the microstructure and at
the same time allow realistic microscale solutions of the
macroscale behaviour is difficult (Kouznetsova et al.,
2001). Thus, one task of multiscale modelling is constructing model geometries accurate enough to represent the microstructure of the real material and make them available
for multiscale computer simulations.
Modelling fruit geometry at different scales
Macroscale geometrical models
The construction of macroscale geometrical model is
usually based on the reconstruction of scanned images
in the form of 3D points on the surface of photographs,
video recordings, computed tomography (CT) or nuclear
magnetic resonance (NMR) images. In a typical photographic experiment, the object is placed on a rotating
disk to get 2D snapshots differing by small angles (Moustakides, Briassoulis, Psarakis, & Dimas, 2000). Each pair
is used to determine the 3D coordinates of the corresponding
side view. The 3D points are then converted to a mathematically expressed geometrical model using the nonuniform rational B-splines (NURBS) (Barron, Fleet, &
Beauchemin 1994; Dimas & Briassoulis, 1999; Moustakides et al., 2000). Jancsók, Clijmans, Nicola€ı, and Baerdemaeker, 2001 used contours of the images at different
angles to reconstruct the 3D geometrical model. The
drawback of such wire frame modelling is that it cannot
produce a good geometrical model for fruit tissues, which
have concave regions in their macrostructure (e.g., apple).
Fig. 2 represents the pear geometry generated by the wire
frame geometrical modelling approach (Fig. 2a) and the
geometry meshed by finite elements (Fig. 2b).
More detailed macroscale continuum models have also
been developed for restricted parts of fruit. Veraverbeke
et al. (2003) used an explicit modelling approach to simulate moisture loss of apple by measuring the transport
properties of different materials (cutin, wax, parenchyma
tissue). They incorporated the cutin, wax and parenchyma
tissue into a continuum model that took into account epidermal structures such as cracks and lenticels. The detailed geometrical features were measured using confocal
laser scanning microscopy and scanning electron microscopy. The model was used to determine apparent diffusion
properties that could be used in a larger macroscale continuum model that only considered two materials, cortex
and skin.
Microstructural models
Unlike engineered materials, biological microstructures
are beyond human intervention and there is a great deal
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H.K. Mebatsion et al. / Trends in Food Science & Technology 19 (2008) 59e66
Fig. 1. The parenchyma tissue in different regions of an apple: (a) cortex tissue; (b) vascular bundle tissue; (c) tissue in the transition from vascular
bundle to cortex (images are the modelled geometry from TEM micrographs). In different parts, the tissue structure is different, demonstrating lack of
macroscopic homogeneity of the material, while at the microscale periodicity is also lacking.
of variation between fruit species (such as pear, Pyrus communis, and apple, Malus domestica), even down to the level
of cultivars, individuals and between positions within individual tissues. Determining representative microstructures
and corresponding models is difficult due to this variability
and because characterization of such geometries at cellular
and sub-cellular levels is not trivial.
Conceptual geometrical models
Conceptual models are geometrical models thought to
represent the microstructure of the biological materials,
Fig. 2. 3D pear fruit geometry (a) and its finite element mesh (b) (adapted from Nguyen et al., 2006).
but they do not have a direct statistical or spatial relationship with the object they stand for: the (spatio-)statistical
distributions of measurable geometry characteristics such
as volume, surface area, aspect ratio and orientation of cells
are not identical. They are a schematic representation of the
microstructure showing different components that make up
the microstructure. Lee et al. (2004) used the schematic
representation of four adjacent plant cells, depicting also
other microstructural components such as the cell wall,
the middle lamella and the intercellular spaces to conceptualize processes and physiological events that are associated
with the plant cell wall. Yao and Le Maguer (1996) also
used a conceptualized ‘sandwich’ model for mathematical
modelling and simulation of mass transfer in osmotic dehydration process. The parenchyma tissue was divided into
extracellular volume (the voids in between cells), intracellular volume and semi-permeable membrane.
In the determination of mesophyll diffusion resistance,
Ivanova et al. (2006) used a model consisting of spheres
and cylinders packed resulting in a 3D leaf cell pack. The
construction of the 3D model was based on an algorithm
for random packing of basic 3D shapes (such as spheres
and cylinders) of fixed size determined from the cell geometrical parameters such as cell length to width ratio, average projection area and average projection perimeter
(Ivanova et al., 2006).
Digitized microscopic images
The geometrical model in this procedure is constructed
from a set of approximated polygonal geometries on the
H.K. Mebatsion et al. / Trends in Food Science & Technology 19 (2008) 59e66
boundary of shapes (e.g., cells and grains) (Espinosa & Zavattieri, 2000; Mebatsion, Verboven, Verlinden, et al.,
2006). The digitization procedure was implemented in biological microstructures in the determination of cell geometrical parameters such as geometrical centers (centroids),
areas, and aspect ratio and orientation (Mebatsion, Verboven, Verlinden, et al., 2006). Fig. 3a and b shows a microscopic image of pear fruit cortex and its equivalent model
geometry, respectively. The model geometry has spatial
and geometrical properties that are similar to that of the microscopic image (Mebatsion, Verboven, Verlinden, et al.,
2006; Yue et al., 2003).
In 3D, geometrical models can also be generated using
X-ray computed tomography. X-ray tomography is a noninvasive image acquisition procedure that avoids the cutting
and fixation procedures and allows visualization and analysis of the architecture of cellular materials with an axial and
lateral resolution down to a few micrometers (Cloetens,
Mache, Schlenker, & Mach, 2006; Mendoza et al., 2007).
Moreover, X-ray tomography gives reliable visualization
of smaller intercellular spaces and their network (Cloetens
et al., 2006). Recently, Mendoza et al. (2007) implemented
X-ray tomography to quantitatively characterize the 3D
pore space morphology of apple tissues. The 3D geometrical model of the apple microstructure was obtained from
the reconstruction of a complete stack of 2D cross sections
of the sample with a voxel resolution of 8.5 mm. Fig. 4 represents the 3D reconstructed image and its equivalent finite
element mesh.
In digitized geometrical models, the presence of sharp
edges at the intersection of two neighbouring cells make
the generation of finite element meshes difficult, resulting
in large number of elements and tedious simulations. Furthermore, the geometrical model does not contain any morphological descriptors to generalize the procedure: every
model geometry needs to be a one to one match of a microscopic image.
63
Tessellation models
A tessellation is a division of some Euclidean space
into a countable number of sets, called tiles, that have
non-overlapping interiors and that cover the whole space
with the union of their closures. An important example,
widely used in applications, is the Voronoi tessellation.
The Voronoi tessellation is defined in terms of a countable
set of center points, known as generators (to distinguish
them from arbitrary points in the space). These generators
divide the space into convex tiles, one per center, and each
consisting of those points of the space nearer to that center
than to any other.
Voronoi tessellations. Voronoi tessellations have been applied to study a broad range of microstructures. In engineering materials, they were used in the study of dynamic damage
initiation, evolution and micromechanical modelling (Espinosa & Zavattieri, 2000; Nygards & Gudmundson, 2002)
and multiscale modelling of materials (Raghavan & Ghosh,
2004). In biological materials, Voronoi tessellations were
used in the study of numerical density and spatial distribution
of neurons (Duyckaerts & Godefroy, 2000) and the study of
protein structures (Poupon, 2004). In fruit science, such tessellations were used in the study of cellular shrinkage and deformation (Mattea, Urbicain, & Rotstein, 1989) and in the
generation of statistically equivalent virtual apple fruit microstructure (Mebatsion, Verboven, Verlinden, et al., 2006).
However, Mebatsion, Verboven, Ho, et al. (2006) proved
the spatial variability of the Centroid based Voronoi diagrams
and Poisson Voronoi diagrams to be very different from that
of the real microstructure. A strong correlation is expected
between the layout of cells and the presence and connectivity
of the pores, which is not taken into account in Voronoi-based
models.
Ellipse tessellations. An ellipse can be fitted to any arbitrary shape using a linear least squares approach applied
Fig. 3. Digitized pear fruit (cv. Conference pear) microstructure. (a) Light microscopy image of pear parenchyma cells in the cortex; (b) digitized and
meshed 2D cellular structure.
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H.K. Mebatsion et al. / Trends in Food Science & Technology 19 (2008) 59e66
Fig. 4. 3D model (top) of the microstructure of apple tissue (cv. Jonagold; 8.5 mm voxel resolution; 256 matrix shown; dark zones are air spaces). (a)
X-ray tomography reconstructed xy-slice; (b) binarized slice using local thresholding to separate air spaces from the cellular phase; (c) finite element
mesh (adapted from Mendoza et al., 2007).
to a boundary data or based on the second moments of the
entire region (Mulchrone & Choudhury, 2004). Zhang,
Jayas, and White (2005) implemented the fitting ellipse algorithm in the separation of touching grain kernels to
identify grain samples and implement automated grain
handling and quality monitoring procedures. Similarly,
a moment based ellipse-fitting algorithm was implemented
for sets of cellular images by taking points on the natural
boundary of the cells to evaluate the aspect ratio and orientation of individual apple cells (Mebatsion, Verboven,
Verlinden, et al., 2006; Mebatsion, Verboven, Ho, et al.,
2006).
The ellipse tessellation is an algorithm based on the fitted ellipses of individual microscopic cells. For every microscopic cell, an ellipse was fitted and for every fitted
ellipse, the algorithm searches a region that is not in the intersection with the rest of the fitted elliptical regions. This
yields a set of truncated ellipses, representing the fruit
microstructure (Mebatsion, Verboven, Ho, et al., 2006).
Fig. 5 shows the ellipse tessellated geometrical model of
the microscopic image shown in Fig. 3a. This approach is
advantageous in generating virtual tissues which have similar geometrical (area, aspect ratio and orientation) and spatial distributions as that of the real microstructure.
Conclusions and future direction
Multiscale analysis is an important tool in areas where
material properties are affected by the microstructure.
The straightforward approach uses simple models at the
microscale to estimate parameters at the coarser scale.
To effect this procedure, representative volume elements
of the material should be defined and imaged, small
enough to reduce computational costs and large enough
to validate the periodicity and homogeneity assumption.
However, we demonstrated that there exists arbitrariness
in the geometry of biological microstructures such as
H.K. Mebatsion et al. / Trends in Food Science & Technology 19 (2008) 59e66
65
Fig. 5. Ellipse tessellation of the pear microstructure displayed in Fig. 3a with a finite element mesh (a) and the magnified view of selected region (b).
The model includes individual cells, cell walls and naturally occurring intercellular spaces.
fruit, making both global and local periodicity assumptions difficult to comply with. For engineered foods, we
expect these assumptions not as restrictive and applicability of multiscale modelling more simple. However, in all
cases accurate 3D modelling of the microstructure is
essential.
Microstructural modelling of food materials is at the
stage of infancy. The more representative geometrical
models available solely depend on tessellation or similar algorithms in 2D but have shown promising results. The 3D
image analysis by means of X-ray tomography showed
a more realistic quantitative distinction between pores and
cells accounting for pore connectivity and pore size distributions. However, there have been little or no studies that
have successfully transformed tomographic images to geometrical models including individual cells. Thus, incorporating tomographic information in the tessellation
algorithms to generate 3D geometries remains a challenge.
Acknowledgments
Financial support by the Flanders Fund for Scientific Research (FWO-Vlaanderen) (project G.0200.02) and the
K.U. Leuven (IRO PhD scholarship for Q.T. Ho, Research
council scholarship for H.K. Mebatsion) are gratefully acknowledged. This research has been carried out in the
framework of EU COST action 924.
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