a finite element model for mechanical deformation of single tomato suspension cells

7/18/2019 A Finite Element Model for Mechanical Deformation of Single Tomato Suspension Cells

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A nite element model for mechanical deformationof single tomato suspension cells

E. Dintwa a , P. Jancsók a ,b , H.K. Mebatsion a , B. Verlinden a, P. Verboven a , C.X. Wang b , C.R. Thomas b , E.Tijskens a , H. Ramon a , B. Nicolaï a ,⇑

a Flanders Centre of Postharvest Technology/BIOSYST-MeBioS, Katholieke Universiteit Leuven, Kasteelpark Arenberg 30, B-3001 Leuven, Belgiumb School of Chemical Engineering, The University of Birmingham, Edgbaston, Birmingham B15 2TT, UK

a r t i c l e i n f o

Article history:Received 4 May 2010Received in revised form 20 October 2010Accepted 23 October 2010Available online 2 November 2010

Keywords:Finite elementModelSingle cell compressionTomato cellMicromanipulationTextureFruit

a b s t r a c t

A nite element model was developed to simulate compression experiments on single tomato cells fromsuspension cultures. The cell was modelled as a thin-walled liquid-lled sphere with a permeable wallallowing ow of uid out in response to internal turgor increases due to the compression. The permeabil-ity of the cell wall/plasma lemma was considered to be constant throughout compression. The contactbetween cell andcompression probe was modelled using a soft contact boundary condition. Thecytoplastwas represented as an internal pressure acting on the plasma lemma and cell wall. Assuming linearelastic constitutive behaviour for the cell wall, and using previously determined cell wall material para-meters, the model was found to be remarkably capable of reproducing the force–deformation behaviourof a single cell in compression, as well as its deformed shape, even for large strains. The model might beused as a building block to construct more comprehensive tissue deformation models.

2010 Elsevier Ltd. All rights reserved.

1. Introduction

Fruit and vegetables are an important component of the humandiet and consumers usually expect such produce to be of premiumquality. Texture is a major quality attribute and inuencesconsumer acceptance, shelf-life, resistance, and transportability(Seymour et al., 2002 ). The texture of plant produce and its suscep-tibility to damage are determined by its mechanical properties.There is an enormous body of the literature relating to this butuntil recently a continuum approach has usually been adopted,in which it is assumed that the properties of the material are inde-pendent of the spatial scale of observation ( Ghysels et al., 2009 ). Assuch, the classical techniques for determination of the basicmechanical properties of these materials have been essentiallythe same as those applied for standard engineering materials. Suchapproaches, while they admittedly have been applied with somesuccess, have always been weak in legitimacy. The major concernis that, unlike traditional engineering materials, plant tissue has ahighly complex hierarchical structure. For example, a fruit such asa tomato consists of a complex conglomerate of different tissues(e.g. skin, cortex, core, seeds, etc.), and each tissue has many micro-scopic constituents such as cells, the middle lamella and interstitialspaces. The macroscopic mechanical properties of the fruit depend

on the properties of these constituents, their arrangement andtheir interactions. Mathematical continuum models ignore thisstructure. A newview among practitioners is that the most realistichope for deriving generic, robust and durable constitutive relationsfor the mechanical behaviour of this type of materials lies with amulti-scale approach to the problem ( Ghysels et al., 2009 ). Thecontention is that only by a thorough understanding of the physicsat all the spatial scales within an individual fruit can we be able toderive, in a quantitative way, the critical inuences by the differentproperties at various levels. The ultimate objective is to derive con-stitutive relations that fully determine the mechanical behaviourof the whole fruit by their microscopic properties. This will requirestudies of tissues and cells at microscopic scales.

Studies of the mechanical properties of fruits and vegetables atmicroscopic scales and beyond have understandably beenconstrained by the lack of technology to conduct mechanicalexperiments at such size scales. Recent developments in microma-nipulation techniques ( Mashmoushy et al., 1998; Shiu et al., 1999;Blewett et al., 2000; Thomas et al., 2000; Wang et al., 2005 ),however, have allowed accurate measurements to be performedat micro-scales, hence opening up new possibilities in this area.As arguably the most important primary unit of vegetative tissue,the plant cell represents an obvious starting point for the studyof the mechanical behaviour of fruit tissue that takes into accountthe multi-scale structure of the material. Working on single cells,whether isolated from tissue or taken from suspension cultures,

0260-8774/$ - see front matter 2010 Elsevier Ltd. All rights reserved.doi: 10.1016/j.jfoodeng.2010.10.023

⇑ Corresponding author.E-mail address: [email protected] (B. Nicolaï).

Journal of Food Engineering 103 (2011) 265–272

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is admittedly a reductionist approach, but it should provide asound basis for the subsequent introduction of complexity. Onepossibility to study the cellular mechanics of single cells is theuse of micromanipulation to perform compression tests betweentwo plates on single cells ( Mashmoushy et al., 1998; Shiu et al.,1999; Thomas et al., 2000; Wang et al., 2005 ). Data (applied forces,displacements, bursting forces) acquired from these single cellcompression experiments can be used with cell models to deter-mine the intrinsic properties of the cell wall. The micromanipula-tion technique has been applied in measurements involving avariety of types of single cells such as mammalian cells ( Zhanget al., 1991 ), yeast cells ( Mashmoushy et al., 1998; Smith et al.,2000a,b,c; Stenson et al., 2009 ), bacterial cells ( Shiu et al., 1999 )as well as tomato fruit and suspension-cultured cells ( Blewettet al., 2000; Wang et al., 2004, 2006 ).

Historically, early attempts to model the mechanical behaviourof vegetative tissue have predominantly been in the form of simplemechanistic theoretical models geared at simulating the behaviourof the tissue to external loads ( Nilsson et al., 1958; Pitt, 1982; Pittand Chen, 1983; McLaughlin and Pitt, 1984; Gao and Pitt, 1991;Gates et al., 1986 ). Alternative approaches to modelling cellular tis-sue involved the use of the nite element method (FEM) ( Akyurtet al., 1972; Pitt and Davis, 1984 ). In general these models werelacking in quantitative accuracy or even validation. However, qual-itatively, they were highlyuseful in studyingvarious features of thetissue. Driven by technological advances in measuring mechanicalbehaviour of single cells, more complex models have appeared inthe literature. Wu andPitts (1999) developed a FE model of a singleapple parenchyma cell compressed between two parallel plates.They used intricate serial micro-sectioning and image analysis toacquire a realistically shaped 3-D FE model of the cell froma seriesof parallel slices of sample tissue. The cell was modelled as a thinshell with surface pressure inside to simulate turgor pressure. Fluidmovement out of the cell and volume changes during compressionwere notaccounted for. Smithet al.(1998) developed a comprehen-sive FE model of the compression of an inated yeast cell, with anexplicit objective to use it to assess the uniqueness or otherwiseof the various crucial parameters involved in the model. The cellwas modelled as a thin-walled liquid-lled sphere with permeablewalls that allowexpulsion of uid out in responseto internal turgorincreases. Volume loss during compression was accounted for. Thecell wall was modelled as an isotropic elastic–plastic material.Using data derived from micromanipulation experiments, Smithet al. (2000a,b,c) appliedthe single cell compression modelof Smithet al. (1998) to extract a variety of mechanical properties of theyeast cell walls. Blewett et al. (2000) and Wang et al. (2006) per-formed single cell compression micromanipulation measurementson, respectively, suspension-cultured tomato cells and tomato fruitcells. In the absence of a FE model of these types of cells, they usedan analytical model of the cell derived from the theory of Yang and

Feng (1970) as adapted by Lardner and Pujara (1978, 1980) to ex-tract the mechanical properties of the cells ( Wang et al., 2004,2006 ). The analytical model, which assumes a permeable cell withlinear elastic constitutive behaviour, yielded very good ts toexperimental force deformation data after parameter optimisationfor fractional deformations up to 20%. The curve ts were achievedwith two free parameters: the initial stretch ratio and the Young’smodulus (i.e. both were estimated at the same time, choosing thebest least squares t to the data). Also, in Wang et al. (2004) thecellwall hydraulic conductivity was assumed to be small enough thatthe effects of water loss could be neglected for small deformations,whilst in Wang et al. (2006) it was claimed that the compressionswere so rapid that there was insufcient time for signicant waterloss to occur.

Aninherentdrawback ofanalytical modelssuch as theoneused byWang et al. (2004, 2006) is the difculty of including more complex

features such as non-linear constitutive behaviour of the cell andaccounting for large strain deformation of the cell wall. Also, it is dif-cult tousean analyticalmodel for asinglecellto predict themechan-ical behaviour ofplant tissues composedof many individualcells.Thelatter is of particular importance, as a better understanding of theforce–deformation behaviour of fruit and vegetable tissue mightguide to betterpostharvest handlingpractices or inspire thedevelop-ment of novel techniques to measure their texture properties.

The aim of this work was to develop a nite element model tosimulate the compression of a single suspension-cultured tomatocell, using data from Wang et al. (2004) . This model could serveas a basic building block for more complex models for tissue defor-mation under mechanical loading.

2. Materials and methods

2.1. Cell cultures, physical measurements and compression data

Experimental compression data described by Blewett et al.(2000) and Wang et al. (2004) were used. In summary, single toma-to ( Lycopersicon esculentum L.vf36) cells were obtained from a sus-

pension culture derived from a root radicle callus. The suspensionculture resulted in nearly synchronous growth, and hence the cellsproduced were of similar age. The cells were suspended in 0.03 Mmannitol at pH 5, the same osmolality and pH as the medium atharvest. During compression testing, the cells were held at2 7 ± 1 C. The micromanipulation experiments involved the indi-vidual cells being compressed between the at endof an optic breprobe and a glass surface whilst measuring the force–deformationcurve and video-imaging the cell compression. The probe speedwas 23 l m s 1 . Data on the probe displacement and the compres-sion force from one such experiment were analysed here.

The initial thickness of the cell wall (i.e. the thickness of cell wallwhen the cell is at zero turgor) was determined as the mean cellwall thickness of several cells from the culture using freeze-frac-

ture scanning electron microscopy. The mean initial turgor pressureof the cells was determined earlier by Wang et al. (2004) from di-rect osmotic pressure measurements (the difference between themeasured external osmolality and the internal osmolality mea-sured by freezing point depression). The initial stretch ratio of thecell before compression was not measured but the diameter of the cell before compression was known.

2.2. Image acquisition, analysis and digitization

In order to allow comparison of the shape of the deformed cellaccording tothe FEmodelto that of thereal cell, thecelldeformationduring the compression experiment was obtained from the video.Still images from several stages of the compression were then fur-

therprocessedusingMATLAB (Mathworks,Natick, MA,USA)imageanalysis programs, digitized versions of the contours of thedeformed cell were obtained. Fig. 1 gives a summary of the proce-dure used to obtain these digitized images. First, a series of stillimages was obtained from the movie of the experiment ( Fig. 1 a),the boundary of the cell was then carefully marked manually bymeans of a mouse pointer on the images ( Fig. 1 b). The cell imagewas then further enhanced to allow easy boundary detection bythe program ( Fig. 1 c). The digitized deformed cell contour is asshown in Fig. 1 d.

2.3. Finite element model

2.3.1. Model denition

Fig. 2 gives a schematic description of the cell compression pro-cess. An uninated (zero turgor pressure) cell of a specied outer

266 E. Dintwa et al. / Journal of Food Engineering 103 (2011) 265–272



radius r 0 and initial wall thickness h0 is rst inated by raising itsinternal pressure to a prescribed initial turgor pressure P 0 and initialouter radius r i. Because the uninated cell is assumed to be in equi-librium with the surrounding medium, it has the same osmoticpressure. The inated condition is the state of the cell at the start

of the experimental compression. The ination is then followedby compression under a at rigid (probe) surface at compressionvelocity v . The initial stretch ratio ks of the cell is dened as r i=r 0 .The hydraulic conductivity L p of the wall is considered to be con-stant throughout. The ow of uid from the cell was not explicitlymodelled in this case, rather it was represented as an internal pres-sure change. Volume loss during the compression of a cell can bedescribed as follows ( Kedem and Katchalsky, 1958; Smith et al.,1998 ):

dV dz ¼

L p

v S ðD P D p Þ ð1 Þ

where V is the volume of the cell, z is the vertical displacement of the compression surface, S is the surface area of the cell available

for ow (i.e. excludes the contact areas between cell and probe,and cell and glass surface), P is the hydrostatic pressure while p is

the osmotic pressure. In the equation, DP refers to the hydrostaticpressure difference between the inside and outside of the cell andDp refers to the osmotic pressure difference. In the model, the out-side pressure is held constant throughout (at a reference value of 0 Pa) and the compression is assumed to be fast enough to neglectbiological and physiological processes of the cell that might affectthe wall properties. Furthermore, it was assumed that water owsare small both during ination and compression, so that the con-centration of the cellularuid can be considered constant and at os-motic equilibrium with the medium. Thus, Dp in Eq. (1) vanishesand the equation simplies to

dV dz

¼ L p

v SP ð2 Þ

In principle it is possible to determine the osmotic pressure aswell as the diameter of the uninated cell based on visual observa-tion of cell volume changes in solutions of different osmolality butthese data were not available.

The compressed sphere is modelled as axisymmetric around theaxis that lies along the line of motion of the compression probe ( z -axis). In addition, the problem symmetry in the horizontal planethrough the centroid of the cell means that only half of the model

needs to be analysed. The cell wall is modelled as isotropic andpurely elastic. Large strains are considered in the model whilethe tangential/shear forces on the wall surface are neglected. Thestructural mechanics module of the commercial package, COMSOL Multiphysics 3.2 (COMSOL Group, Stockholm, Sweden) was usedfor modelling.

2.3.2. Contact modelIn the model, the compression surface is not explicitly modelled

and meshed. Instead a ‘soft contact’ procedure is implemented(Pennec et al., 2007 ). In this procedure the contact is modelled witha force that increases rapidly as the rigid object approaches theedge of the mesh. The contact is represented by a pressure of thetype

pcontact ¼ Ae ðB: g Þ ð3 Þ

Fig. 1. Steps in the acquisition of digitized contours of the deformed cells. First, a series of still images was obtained from the movie of the experiment (a), the boundary of each cell was then carefully marked on the images (b). The cell image was then further enhanced to allow easy boundary detection by the image analysis program (c). Thenal digitized contour was eventually acquired using the MATLAB image analysis programs (d).

cell

bar

Inflated cell

2r i

r 0

v

Fig. 2. Schematic view of the cell compression. An uninated (zero turgor pressure)cell with outer radius r 0 is rst inated to a radius r i by raising its internal pressureto a prescribed initial turgor pressure. Theination is then followed by compressionunder a at rigid (probe) surface at compression velocity v .

E. Dintwa et al. / Journal of Food Engineering 103 (2011) 265–272 267



where g is the gap between the contacting surfaces (the at bar andthe surface of the cell). The constants A and B have no physicalmeaning and should be selected carefully. The value of A shouldbe of the same order as the expected contact pressure. The constantB must be large enough not to cause any signicant forces overphysically important gap distances; however, excessively large val-ues would increase the nonlinearity of the problem, which might bedetrimental to the convergence. The soft contact procedure wasimplemented directly as a boundary expression to the boundarywhere contact is expected. For these models, after examining the lo-cal maxima of the pressures on the cell wall for several combina-tions of the two parameters, a value of 9 MPa was used for A andB was set to 5 10 7 m 1 .

2.3.3. Material propertiesThe cell wall was modelled as isotropic and elastic (Hookean)

material with the following material properties: cell wall densityq w ¼ 1000 kgm 3 (by assumption); hydraulic conductivity L p =4.64 10 13 m (sPa) 1 (Maggio and Joly, 1995 ) Young’s modulusE = 2.3 10 9 Pa ( Wang et al., 2004 ); Poisson’s ratio m¼ 0 :4 (Wanget al., 2004 ). The liquid inside the cell was assumed to be

incompressible and with similar properties to water (densityq f ¼ 1000 kg m 3 ; bulk modulus K f = 2 10 9 Pa).

2.3.4. Constraints and loadsDuring the ination phase of the model, a pressure load is ap-

plied to the internal boundary equal to the prescribed initial turgorpressure. The initial turgor was adopted from Wang et al. (2004) as0.363 MPa. During the compression stage, the soft contact condi-tions are applied to the external boundary. The rigid probe surfaceis lowered in a series of downward displacement steps of 0.2 l m.The active area and pressure are recalculated at every step andthe volume loss is accounted for through a boundary expressionon the internal wall boundary according to Eq. (2). Careful consid-eration should be given to the fact that not only the volume loss

depends on the pressure resulting from the direct FE solution, thispressure in turn is also inuenced by the calculated volume loss.So, instead of determining the volume loss in a single step, onemust do this iteratively ( Smith et al., 1998 ). Hence, an iteration cy-cle was implemented in this model with a set minimum differencebetween two subsequent values for the pressure as a stop criterion.

2.3.5. ImplementationThe nite element mesh for the FE model was produced using

quadratic Lagrange elements ( Fig. 3 ). The model contained a totalof 795 elements. Fig. 3 a shows a portion of the mesh from thetop of the cell model. In order to determine the adequateness of the mesh density, preliminary analysis results were compared tothe results obtained with a ner mesh ( Fig. 3 b). The results were

identical, and therefore the mesh density was judged to beadequate.

The model analysis was performed using a non-linear paramet-ric solver with a direct linear equation system solver and largedeformation. The span of the load stepping parameter was chosento provide similar deformation as the corresponding experiments.

2.4. Simulation

The model was used to simulate data for a single cell from theexperiment reported by Wang et al. (2004) . Hence the cell modelwas developed with the dimensions and material properties as

determined (or estimated) in that work. The FE model analysiswas then performed and the results were compared to the exper-imental data. Comparisons were made for force–deformation dataas well as cell deformed shapes at different stages of the compres-sion. The main properties of the cell from that particular experi-ment were as follows: cell diameter before compression = 62 l m;initial turgor pressure = 0.363 MPa; initial thickness of the cellwall = 126 ± 16 nm. The other constants were as given in Section2.3.3. The speed of the cell compression was 23 l m s 1 .

3. Results

Fig. 4 shows the deformed shape of the cell after a simulatedcompression with a at rigid bar to a prescribed maximum vertical

Fig. 3. A zoom view of the mesh of the cell model from the top end. The mesh in (a) was used in the formulation while the ner mesh in (b) was used to assess theadequateness of the mesh density.

Fig. 4. Typical deformed shape plot of the cell after compression. Cell initialdiameter = 62 l m. A surface plot of the von Mises stresses is also included (greyscale). The von Mises stresses are highest at theedge of the contact surface (approx.

0.5 GPa) and lowest at the centre of the contact surface (approx. 13 MPa).




displacement. The undeformed (but inated) shape of the cell is in-cluded for comparison. The von Mises stresses are calculated fromthe stress components and may be used to predict whether the(ductile) material may yield ( Barber, 2002 ). Here the von Misesstresses appear to reach a maximumat the edge of the contact sur-face with the at rigid manipulator, suggesting that cell burstingwould occur at this edge (assuming the cell wall material isductile).

3.1. Validation of the FE model

3.1.1. Force–deformation curvesFig. 5 shows a comparison of the results of the FE model to those

of experiment. In the gure the external applied force is plotted

against d, the fractional deformation of the cell. The fractionaldeformation is dened as follows:

d ¼ 1 ðr i z Þ

r ið4 Þ

where z refers to thevertical displacement of thecell surface in con-tact with the at object and r i is the initial outer radius of the cell

after ination and before compression. There is a remarkable agree-ment between the FE model and experiment in the lower deforma-tion ranges, up to a fractional deformation of approximately 0.2. Atthis stage a systematic departure clearly occurs with the real cellprole appearing to atten and follow a more straightened proleuntil the cell bursts.

3.1.2. Deformed shapes of the cellFig. 6 shows a comparison of the deformed shapes of the cell in

the compression experiment (obtained from image analysis anddigitization of real cell images) to those generated with the FEmodel for a series of stages of the compression (indicated by thefractional deformations). In agreement with force–deformationcurve discussed above, the FE model makes a very accurate predic-

tion of the deformed shapes of the cell compression in the lowerdeformation ranges. Fromthe gure, cell shapes of fractional defor-mations up to 0.23 are predicted reasonably accurately by themodel. At higher deformations the model predictions start to de-part systematically from the measured data; the latter show ahigher lateral expansion than the former.

The validation results show that the FE model is capable of accurately predicting the force deformation behaviour of a cellundergoing small to moderate compression and hence would beuseful for determining the mechanical properties of the cell. Thediscrepancy between the model and the actual cell experimentsat larger deformations is understandable, because at that stagethe cell is nearing bursting. A possible interpretation of thisbehaviour would be that, at these rates of compression, the cell

Fig. 5. Comparison of the force–deformationproles determinedfrom the FE modeland those obtained from experiment. The experiment and material properties areaccording to Wang et al. (2004) : r 0 ¼ 62 nm; h0 ¼ 126 nm; qw ¼ 1000 kg m 3 ;L p ¼ 4 :64 10 13 m ðs Pa Þ 1 ; E ¼ 2 :3 10 9 Pa; P 0 ¼ 3 :63 10 5 Pa; m¼ 0 :4 andv ¼ 23 l m s 1 .

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

x 10-5

0

0.5

1

1.5

2

2.5

3

3.5

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4.5x 10

-5

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

x 10-5

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2.5

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4.5x 10

-5

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x 10-5

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0.5

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-5

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x 10-5

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1.5

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-5

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x 10-5

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-5

simulation experiment

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

x 10-5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

-5

δ = 0.00δ = 0.085 δ = 0.162

δ = 0.230 δ = 0.394 δ = 0.443

Fig. 6. Comparison of the deformed shapes of the real cells to those of the FE model at different stages of the cell compression. d is the fractional deformation of the cell.r 0 = 62 l m; h0 = 126 nm; qw = 1000kgm 3 ; L p = 4.64 10 13 m (sPa) 1 ; E = 2.3 10 9 Pa; P 0 = 3.63 10 5 Pa; m = 0.4 and v = 23 l m s 1 . Dimensions in the gure are metres.




wall constitutive behaviour is characterised by an elastic–plasticresponse where a plastic yield point might be identied at thepoint of substantial departure from the FE model. The yield pointwould then be followed by a portion of ‘softening’ before the cellnally bursts. However, the validation of this hypothesis wouldrequire tensile tests on isolated cell walls which is currently notfeasible.

3.2. Sensitivity of the FE model to the thickness of the cell wall,Poisson’s ratio, hydraulic conductivity and initial turgor pressure

In order to assess the robustness of the cell model developed tovarious crucial variables and properties of plant cells, the model asdescribed above was analysed for its sensitivity to each of the vari-ables by repeating the analysis at different values of each variablewhile holding the rest and comparing the resultant force–deforma-tion curves. Four parameters were investigated, namely: the initialthickness of the cell wall; the Poisson’s ratio of the cell wall; thehydraulic conductivity of the cell wall and the initial turgor pres-sure of the cell. All the other cell properties and experimental vari-ables were held as described in the validation above. The inuenceof the cell wall elastic modulus has been addressed by others be-fore ( Smith et al., 1998; Wang et al., 2004 ) and hence was notinvestigated.

3.2.1. Sensitivity to cell wall thicknessFig. 7 shows the force–deformation response of the cell model

for three values of the initial cell wall thickness, based on the rangeof values found by Wang et al. (2004) . There is a clear inuence of this variable as a thicker wall results in a stiffer response of the cell.This inuence is to be expected as a thicker cell should providemore structural rigidity to the cell.

3.2.2. Sensitivity to cell wall Poisson’s ratioThe effect of for three different cell wall Poisson’s ratios (0.3, 0.4

and 0.49) was investigated (results not shown). No substantialinuence by the Poisson’s ratio of the cell wall on the overallforce–deformation behaviour of the cell could be observed. A sim-ilar nding was reported by Wang et al. (2004) .

3.2.3. Sensitivity to the initial turgor pressureIn the analytical model of Wang et al. (2004) , a pre-determined

value of the initial turgor pressure was used to avoid the need totreat this as an adjustable parameter. Wang et al. (2004) deter-mined a mean value experimentally. Because individual cellsmay vary, a sensitivity analysis was done using the FE model.Fig. 8 shows a plot of the force–deformation curves of the cellmodel at three different values of initial turgor pressure. Clearly

this has an inuence on the force–deformation behaviour, withhigher turgor pressures resulting in stiffer responses. In additionto the direct effect of pressure, there will be an additional effectthrough consequential changes in the initial stretch ratio of the cellwall. It is consistent that Wang et al. (2004) found that higher ini-tial stretch ratios led to stiffer force–deformation responses. It isclear that changes in initial turgor pressure from 0.2 to 0.5 MPa,do not have a huge effect on the simulated force–deformationcurves.

3.2.4. Sensitivity to the hydraulic conductivity of the cell wallTherange of values of hydraulic conductivity L p for plant cells in

generalis reportedtobein therangeof 2 10 8 to10 5 ms 1 MPa 1

(Wang et al., 2004; Maurel, 1997 ) while for tomato cells the value isreported to be around 10 7 ms 1 MPa 1 (Wang et al., 2004; MaggioandJoly, 1995; Hukinet al., 2002 ). In themodeldescribed hereinthevalue of L p used for the tomato cell is: 4.64 10 7 ms 1 MPa 1 . Toinvestigate the sensitivity of the cell model to L p, the simulation

was repeatedfor a number ofdifferent valuesof L p and comparisonswere made not just for the force–deformation responses of the cell,but also for the volume versus deformation curves. The values of L p

used ranged from0 to0.01 ms 1 MPa 1 (i.e.way outsidethe rangeof notjust tomatocells, but plantcells in general). Forall 4 values of L p,therewas no noticeable effect on the force–deformationcurve of thecell models ( Fig. 9 a). For the volume-deformation curve ( Fig. 9 b),still there was no noticeable difference between the simulated re-sults for realistic values of L p. A signicant effect on the volume of the cell only appeared when a value of 0.01 ms 1 MPa 1 (which is5 orders of magnitude higher than the normal 10 7 ms 1 MPa 1

and 3 orders of magnitude higher than the range for all plants)was used. This extreme value however still did not show any effecton the force–deformation curve.

In view of the foregoing, it can be concluded that, for tomatocells being compressed under the conditions described here, thehydraulic conductivity does not signicantly affect the force–deformation behaviour and thus is not a crucial parameter. A sim-ilar nding was reported by Wang et al. (2004) .

4. Discussion

Thenite element model developed in this manuscript is equiv-alent to the model of Wang et al. (2004) and incorporates the samephysics. However, unlike the latter which essentially is an analyt-ical model, the main advantage of the nite element model is thatother constitutive equations for the mechanical behaviour of thecell wall material can be incorporated easily. Further, it is suf-

ciently exible to serve as a building block for a more comprehen-sive numerical model of tissue deformation. Such a model may

0.0E+00

1.0E-03

2.0E-03

3.0E-03

4.0E-03

5.0E-03

6.0E-03

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35Fractional deformation

F o r c e

( N )

1.42E-07 m1.26E-07 m1.10E-07 m

Fig. 7. Inuence of initial cell wall thickness to the force–deformation behaviour of

the cell. r 0 = 62 l m; qw = 1000kgm 3 ; L p = 4.64 10 13 m (sPa) 1 ; E = 2.310 9 Pa; P 0 = 3.63 10 5 Pa; m = 0.4 and v = 23 l m s 1 .

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

0.0045

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35Fractional Deformation

F o r c e

( N )

Turgor Pressure = 0.363 M Pa



Fig. 8. Inuence of the initial turgor pressure on the force–deformation of the cellFE model. r 0 = 62 l m; h0 = 126 nm; q w = 1000 kgm 3 ; L p = 4.64 10 13 m (sPa) 1 ;E = 2.3 10 9 Pa; m = 0.4 and v = 23 l m s 1 .




allow to understand how fruit behaves under mechanical loading.This knowledge may guide towards adapting postharvest handlingprocedures to minimise damage or to development novel methodsto measure fruit texture.

The model appears to reproduce remarkably well the experi-mental force–deformation behaviour of the cell from experimentsbased on the parameters as determined by Wang et al. (2004) . Itwas discussed earlier that the mechanical properties as deter-mined by Wang et al. (2004) lack certainty. This is because themethod used to extract the elastic modulus of the cell relied onleast squares curve ttings that were achieved with two freeparameters: the initial stretch ratio and the Young’s modulus. Inaddition, the cell wall hydraulic conductivity was not explicitlymeasured for these specic cells but adopted from the literature.In view of the observations of Smith et al. (1998) regarding thenon-uniqueness of solutions for compression of single cells there

was uncertainty as to the values of the parameters since severalcombinations of parameters could still have produced good ts.With the FE model, the initial stretch ratio does not need to bere-estimated as a tting parameter since it is xed by the modelonce the initial turgor pressure and the cell dimensions are known.This is a major advantage as the (mean) initial turgor pressure andthe cell dimensions are easy to measure accurately ( Mashmoushyet al., 1998; Shiu et al., 1999; Blewett et al., 2000; Thomas et al.,2000; Wang et al., 2005 ), although the initial turgor pressure of aspecic cell under test is not known. This simplication is achiev-able by assuming the osmotic pressure of the cells does not changemarkedly from the uninated state to the compressed state. This isreasonable as the volume changes during this processes are small(<4%). Fromthe studies on the sensitivity of the FE model of the cell

to the hydraulic conductivity, it has emerged that the hydraulicconductivity does not have any noticeable effect of the force–deformation, at least for a practical range of values. Since all theother parameters that could inuence the force–deformationcurves (initial cell wall thickness and the (mean) initial turgorpressure) were determined accurately from experiments, it canbe concluded that the elastic modulus of the cells as determinedby Wang et al. (2004) was correct.

In spite of the foregoing, it is worth remarking that the elasticmodulus of the cell as determinedby Wang et al. (2004) and imple-mented in the FE model ( E = 2.3 10 9 Pa) is surprisingly high,especially when placed against the elastic moduli of fruit tissue.Apple fruit tissues, for example, have elastic moduli in the regionof 5 MPa while tomato fruit tissue has its elastic moduli in the re-gion of 0.5 MPa. The modulus of the cell wall used here is thereforeat least 3 orders of magnitude higher than that of tissue made of

similar cells. This is possible since the tissue, as a higher ordercomposite material, has its mechanical properties inuenced bymany other factors, such as the properties of the binding matrix,the structural architecture of the constituent cells, and water owsthrough the structure. A review of the literature on mechanicalproperties of plant cell walls reveals a range of elastic moduli fromas high as 15 GPa for the primary cell walls of wood cell walls,through 0.4 GPa, for cell walls of the shoots of seedlings of whitespruce, to 0.3 GPa, for cell walls of green algae ( Marshall andDumbroff, 1999; Gindl and Gupta, 2002; Wei and Lintilhac,2007 ). Hiller et al. (1996) established a value of 3 GPa for the cellwall of potato tuber parenchyma tissue cells. In a more recent pa-per, Wang et al. (2006) found values of 30–80 MPa for the cell wallof inner pericarp tissue of tomato.

Regarding the prediction of the constitutive behaviour of thecell material, the results of the FE model suggest that an elastic

constitutive relation with large strains is sufcient to explain theforce–deformation behaviour of the cell at fractional deformationof up to 0.2. At higher fractional deformations, it would appear thatelastic–plastic constitutive behaviour with softening and an ulti-mate bursting point would be more appropriate. It is an advantageof the FE model that it might eventually be possible to incorporatesuch elastic–plastic behaviour. The FE model developed here has asimilar level of accuracy as the model of Smith et al. (1998) . How-ever, it is much less computationally intensive. The implementa-tion of the ‘soft contact’ algorithm for this model, rather than thefully meshed at compression bar implemented by the former, sig-nicantly simplies this model without sacricing modellingaccuracy.

5. Conclusions

A nite element model was developed to simulate micromanip-ulation compression testing of single plant cells. It was suitable forsituations with large strains as found in cell compression experi-ments and used a ‘soft contact’ algorithm to reduce computationalintensity. The model was validated using data from a real singlecell compression experiment performed on a suspension culturedcell. Assuming linear elastic constitutive behaviour of the cell wall,the model was found to be capable of reproducing the force–defor-mation behaviour as well as the deformed shapes of the cell up tofractional deformations of about 0.2.

Unlike the analytical model of Wang et al. (2004) , the modelmight be used to simulate single cells of any plant or any similar

cells by adjusting the relevant parameters and experimental vari-ables. Because of its computationally simplicity it is believed that

4.81E-11

4.64E-13

0.00E+00

1.00E-08

4.81E-11

4.64E-13

0.00E+00

1.00E-08

5.756E-14

5.758E-14

5.760E-14

5.762E-14

5.764E-14

5.766E-14

5.768E-14

5.770E-14

5.772E-14

5.774E-14

5.776E-14

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

Fractional deformation

V o

l u m e ( m

³ )

4.81E-11

4.64E-13

0.00E+00

1.00E-08

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

Fractional deformation

D i m e n s

i o n l e s s

F o r c e

4.81E-11

4.64E-13

0.00E+00

1.00E-08

a b

Fig. 9. Inuence of the hydraulic conductivity on the force–deformation (a) and the volume (b) of the cell FE model. r 0 = 62 l m; h0 = 126 nm; qw = 1000 kg m 3 ;E = 2.3 10 9 Pa; P 0 = 3.63 10 5 Pa; m = 0.4 and v = 23 l m s 1 .




this nite element model can serve as a building block for a morecomprehensive numerical model of tissue deformation. Such amodel can be used to improve our understanding of theforce–deformation behaviour of fruit and vegetable tissue undermechanical loading. This might guide to better postharvest han-dling practices for fruit and vegetables or inspire the developmentof novel techniques to measure their texture properties.

Acknowledgements

The authors wish to thank the Research Council of theK.U. Leu-ven (OT 08/023), the Flanders Fund for Scientic Research (ProjectG.0603.08; 3E060094) and the Engineering and Physical SciencesResearch Council, UK for nancial support.

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a finite element model for mechanical deformation of single tomato suspension cells

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