Prediction of water loss and viscoelastic deformation of apple tissue using amultiscale model
Wondwosen A. Aregawi 1, a, Metadel K. Abera 1, a, Solomon W. Fanta 1,Pieter Verboven 1, Bart Nicolai 1,2,*
1 MeBioS, Department of Biosystems, University of Leuven, Willem de Croylaan 42, 3001 Heverlee, Belgium
2 VCBT, Flanders Centre of Postharvest Technology, Willem de Croylaan 42, 3001 Heverlee, Belgium
a Joint first authors
* Corresponding author: Bart Nicolai
MeBioS, University of Leuven
Willem de Croylaan 42
B-3001 Heverlee
BELGIUM
Tel. +32 (0)16 322375
Email:
1
Abstract
A two-dimensional multiscale water transport and mechanical model was developed to predict the water loss and deformation of apple tissue (Malus × domestica Borkh. cv. ‘Jonagold’) during dehydration. At the macroscopic level, a continuum approach was used to construct a coupled water transport and mechanical model. Water transport in the tissue was simulated using a phenomenological approach using Fick’s second law of diffusion. Mechanical deformation due to shrinkage was based on a structural mechanics model consisting of two parts: Yeoh strain energy functions to account for non-linearity and Maxwell’s rheological model of visco-elasticity. Apparent parameters of the macroscale model were computed from a microscale model. The latter accounted for water exchange between different microscopic structures of the tissue (intercellular space, the cell wall network and cytoplasm) using transport laws with the water potential as the driving force for water exchange between different compartments of tissue. The microscale deformation mechanics were computed using a model where the cells were represented as a closed thin walled structure. The predicted apparent water transport properties of apple cortex tissue from the microscale model showed a good agreement with the experimentally measured values.Deviations between calculated and measured mechanical properties of apple tissue were observed at strains larger than 3%, and were attributed to differences in water transport behavior between the experimental compression tests and the simulated dehydration-deformation behavior. Tissue dehydration and deformation in the high relative humidity range (> 97% RH) could, however, be accurately predicted by the multiscale model. The multiscale model helped to understand the dynamics of the dehydration process and the importance of the different microstructural compartments (intercellular space, cell wall, membrane and cytoplasm) for water transport and mechanical deformation.
Keywords:plant tissue, turgor, nonlinear mechanics, diffusion
1.Introduction
Fresh fruit such as apples are mostly composed of water[1].Water loss of fruit during storage is the main cause for the reduction of fruit quality after harvest. At the microscopic level, fruit tissue undergoes large deformations as a result of hygrostresses associated with dehydration[2–4]. At the microscopic level, deformation of cells occurs as a result of the loss of turgor acting on the cell walls surrounding the cell during the dehydration process [5]. Firmness is an important quality attribute critical to consumer acceptance. Turgor pressure, cell size and shape, presence of intercellular spaces, and chemical composition have a major influence on tissue strength and macroscopic fruit firmness[6].To better understandwater loss and large deformations during dehydrationand how they are affected by microscale features of the tissue,a modelling approach is most appropriate.
In the macroscopic, continuum approach, the tissue is considered as homogeneous; lumped parameters are used as apparent material properties, and incorporate the effect of microstructural features such as the ensemble of cell membranes, cell vacuoles pores and cell walls. Macroscopic water transport models in food have been developed by, amongst others,[7–9], which all adopted a continuum approach based on Fick’s second law of diffusion.Biological materials such as fruit can be modeled as a nonlinear viscoelastic continuum [10], when viewed at the macroscopic scale. Different approaches to model water transport and shrinkage have been reported. In the simplest approach,the volume change due to the shrinkage of the tissue is equal to the volume of water removed from the material[11,12]. The principle of virtual work has also been applied to hygrostress formation, where the deformation is based on linear elasticity theory [13–16].Recent work has allowed better understanding and modeling of coupled water transport and large deformation of fruit tissue during dehydration at the macroscale[2].
The disadvantage of continuum models is that they do not allow to investigate the effect of microstructural features of the tissue on macroscopic processes such as water loss and mechanical deformation. In contrast, in microscale models the heterogeneous structure of the tissue is taken into accountby representing the complex cellular structure through a geometrical model [17–19].A microscale water transport model was developed to describe water transport in fruit at the cellular level through the inter-cellular space and cells [20].Different approaches to model the mechanical behavior of cellular tissues have been reported. A finite element model was developed to simulate the compression of a single suspension-cultured tomato cell [21]. However, the model was limited to mechanical loading and did not account for the deformation due to water loss. Other authors have used smoothed particle hydrodynamics (SPH) for the cytoplasm and a discrete element model (DEM) for the cell wall to simulate mechanical behavior cellular tissue in response to dynamic stimuli [22–24]. The model incorporated the mechanics during and after cell failure and cell wall rupture. However, the model predicts water efflux due to mechanical loading but not water transport as such between the cells.A microscale water transport model coupled with a mechanical deformation model has been developed to describe water transport and mechanical properties of pores, cell walls, cells and cell membranes[5]. Although this model provided detailed predictions of the local microscale water distribution and mechanical deformation, the extension of the microscale modelling techniques to predict the dehydration process of an entire fruit would require far more computational power than is currently available.
The multiscale modelling paradigm offers an alternative approach to combine continuum type models defined at the macroscale level with the level of detail models incorporating the microscale features. A multiscale water transport and large deformation model is basically a hierarchy of models which describe the water transport and large deformation at different spatial scales as shown in figure 1. The coupling of the hierarchy of models is via computational analysis, i.e., model results relevant to a particular spatial scale are linked to simulations at a different spatial scale. Multiscale modelling has been successfully applied in soft matter systems [25,26], in material science and geoscience [27–33]. It has also been used as a useful tool for biological applications[34–36].
In this study, a 2D multiscale model is introduced and validated to perform a computational analysis of water transport and large deformation of apple (cv. Jonagold) cortex tissue at different scales(illustrated in figure 2). On the microscale, water transport and deformation of tissue are computed for the actual cell assembly of apple tissue. With these simulations, the apparent water diffusion and mechanical properties of tissues are calculated from a homogenization procedure. These parameters are consequently used to perform simulations of the dehydration and shrinkage of apple tissue samples using a macroscale model.
2.Multiscale model
2.1.Microscale model
2.1.1.Microscale water transport model
The transport of water in the intercellular space, the cell wall network and cytoplasm were modeled using diffusion laws and irreversible thermodynamics[37]. For the cells, the unsteady-state model of water transport reads:
/ (1)where ρw,c is wet base cell density (kg m-3), ρc is cell density (kg m-3), Dc is the water diffusion coefficient inside cells (m2s-1), and cw, cis water content on wet base (kg kg-1).
The wet base cell density and dry matter density inside cells are related by the following equation:
/ (2)where Xcis the dry matter base water content (kgkg dm-1).
Using the relationship between water content on wet and dry matter base the following equation can be derived:
/ (3)The water capacity cψ, c (kgkg dm -1 Pa-1) is given by:
/ (4)where ψc the water potential of the cell (Pa).
Substituting equations (2), (3) and (4) into equation (1) leads to:
/ (5)For the cell wall, the unsteady-state diffusion model is given by:
/ (6)For the air, unsteady-state diffusion is modeled by:
/ (7)A simple flux law was applied to describe water transport through the cell membrane [37]:
/ (8)2.1.2.Microscale mechanics model
A cell micromechanics model was recently developed [38], and the basic characteristics are summarized below. In this model, the cell is represented as a closed thin walled structure, maintained in tension by turgor pressure. The cell boundary is represented as a set of walls (modeled as springs) connected at points called vertices(see figure 3).
Newton’s second law was considered to model the shrinkage mechanics. The following system of equations is solved for the velocity and position of the vertices i of the cell wall network:
/ (9)/ (10)
where is the mass of the vertex (kg) which is assumed to be unity in order to simplify the model, which makes the rate of change of velocity (acceleration) of the vertices equal to the net force acting on the vertex; (m) and (ms-1) are the position and velocity of node i, respectively, and is the total force acting upon this node (N).
Cell shrinkage or growth is then the result from the action of forces caused by a decrease or increase, respectively, of turgor pressure acting on the cell wall. The water potential of each cell, obtained from the water transport model outlined in the previous section, can be converted to turgor pressure using the relationship presented in section 2.1.3. The resultant force on each vertex, the position of each vertex, and, thus, the shape of the cells is then computed as follows. The total force acting on a vertex is given by the formula
/ (11)/ (12)
The force contributed by the wall is the resultant of the net turgor pressure force between the two adjacent cells working normal to the wall and the force associated with it:
/ (13)The net turgor force on the vertex is calculated by taking the difference in turgor pressure Pc of the two adjacent cells multiplied by half the length of the wall as it is divided by the two incident vertices defining the wall (see figure 3(b)):
/ (14)Hooke's law was employed to determine the force acts along the wall and its magnitude
/ (15)/ (16)
To find the positions of each vertex of all cell walls of every single cell and, thus, the shape of the cells with time, a system of differential equations (9) and (10) for the positions and velocities of each vertex were established and solved using a Runge-Kutta fourth and fifth order (ODE45) method.
2.1.3.Coupling of water transport and mechanical deformation
The transient water transport model is solved for certain time steps and the water loss results in loss of water potential in the cells. The change in water potential of the cells induces loss of turgor pressure. This is valid for the high range of equilibrium relative humidity values of the cells during dehydration until turgor drops to zero[5]. The dehydration is thus performed in the relative humidity range of 99 to 97.7%. Below this value of relative humidity the turgor pressure is zero and the osmotic potential will be equal to the water potential. The osmotic potential can be obtained from equation (17)
/ (17)The turgor pressure is then equal to
/ (18)The relationship between turgor pressure and water potential is shown in figure 4 for the considered range.
2.1.4.Implementation
Threedifferent tissue samples of 1250×1600 μm, which had 80 cells with average cell diameter of 157.39±22μm and a porosity of 14.97 ± 1%, were generated using random simulation with the microscale mechanics model as described in [37]. Then, the geometric models of apple cortex tissue were imported into Comsol Multiphysics 3.5a (Comsol AB, Stockholm, SE) for numerical computation of the water exchange using the model equations outlined above. Meshing was performed automatically by the Comsol mesh generator and produced more than 350,000 quadratic elements with triangular shape for each tissue geometry.
A numerical experiment was carried out to calculate apparent material properties to be used in the macroscale model. As it was described in section 2.1.3, a relative humidity of 99 and 97.7% was chosen and applied to the top and bottom of the tissue geometry, respectively, while the other two lateral boundaries were defined to be impermeable. Afterwards, a sequence of time steps was considered for solving the coupled moisture transport and mechanical deformation model (see figure 5). The finite element method was used to discretise non-linear coupled model equations. In every time step the water potential and water content distribution in the tissue samples as well as the water flux through the sample were solved for a given water potential gradient across the sample. The initial water transport calculation was performed on the initial apple cortex tissue geometry that was obtained using a virtual fruit tissue generator.Afterwards, the water potential of each cell was obtained and converted into turgor pressure using the relation shown in figure 4. Then this set of turgor pressures was used in the shrinkage mechanics presented in section 2.1.3 to find the new equilibrium configuration of the cells. The mechanical equilibrium was calculated using a dedicated Matlab code (Matlab 7.6.0, The Mathworks, Natick, MA). The whole system of equations was numerically solved using a Runge-Kutta method of order 4 and 5. The simulation was iterated until a mechanical equilibrium state was reached. This equilibrium was assumed once the velocity of all points was below a given threshold, as the velocities would go to zero only when the system would be at a steady state. The resulting tissue geometry was then introduced and meshed again in Comsol and the next time step was initiated. In total eight time steps of 50 s were required to reach equilibrium.
The computation time was 2412 seconds for the unsteady state water transport simulation in each time step on a 8 Gb RAM quad-core PC, and 20 seconds for the mechanical equilibrium calculations.
The numerical values of the water transport and mechanical parameters used in the microscale model are listed in table 1. These parameters were taken from our previous studies and from different literature resources.
2.1.5.Computation of apparent diffusivity and mechanical properties
The microscale model was used for in silico analysis of water loss and deformation of cells in apple fruit tissue, and for computing the apparent properties of the apple tissue as a whole. The water capacity and dry mass density were combined into an apparentdiffusion coefficient Dapp (m2 s-1) of the tissue from the apparentwater conductivity Kapp(kg m-1Pa-1 s-1) obtained from the microscale simulationsusing equation (19).
/ (19)The water capacity cψ(kgkgdm-1 Pa-1) was calculated from the steady state calculated water content and water potential of the microscale tissues for RHvalues of 98.5% and 97.5% and temperature of 25 °C using microscalemodel simulations. The apparentwater conductivity was obtained from the calculated flux for a specified gradient in water potential across the tissue sample:
/ (20)with J (kg m-2 s-1) the total flux through the fruit tissue, Δψ(Pa) the assigned water potential difference between the two opposite sides and h(m) the thickness of the simulated tissue.
The apparent mechanical properties of the tissue were calculated from the microscale model by assuming that the change in displacement (deformation) of tissue is due to a change in turgor pressure. This is logical, since turgor pressure generates a stress that leads to the expansion of cellwall[39]. The reversephenomenon, shrinkage of cells, takes place as a result of turgor loss during dehydration. Thus, in the current study, weassumed that the history of turgor pressure and the resulting expansion or shrinkage of the cellwall networkcould determine the mechanical properties of the tissue in general. The current approach isplausible from the common type of mechanical tests where external forcesare applied to determine tissue properties, because the stresses and the resulting shrinkage can be obtained from the natural dehydration processes.Based on this approach, the average changes in turgor and the corresponding displacementof the tissuein the direction of the applied gradient (see section 2.1.4)were calculated from microscale model simulations. Afterwards, mechanical properties were estimated by fitting these data points into the Yeoh hyperelastic potential with three parameters[2], (see equation(21)).
/ (21)2.2.Macroscale model of water transport coupled with deformation
A macroscopic phenomenological approach was used to model water transport in fruit tissue, where water moves in the tissue as a consequence of a gradient in water potential [40]:
/ (22)Materials undergoing large deformations are best described by nonlinear elasticity theory according to [41], instead of by linear elasticity theory. The stresses in the fruit tissue were calculated from the following mechanical equilibrium, assuming no body and surface forces [42]:
/ (23)The model equation (23) is based on the theory of compressible hyperelasticity with the decoupled representation of the Helmholtz free energy function with the internal variables[41]. The way in which the second Piola-Kirchhoff stress tensor (S) and the deformation gradient tensor (G) are defined is clearly explained in [2].The coupling between water transport and the mechanical response (large deformations) is based on [2].