Virtual Fruit tissue Generation Based on Cell Growth Modeling

The fact that production of pome fruits is both season and location dependent calls for preservation methods to maintain the quality of the produce after harvest. The principal method to preserve pome fruit is postharvest storage in cool rooms under Controlled Atmosphere (CA) or Ultra Low Oxygen (ULO) conditions. These methods are based on empirical trials to determine optimum storage conditions for a specific cultivar in terms of temperature, oxygen and carbon dioxide concentration and relative humidity. Improved understanding of the underlying fruit physiology in relation to the gas and water exchange processes during storage will assist in improving postharvest quality and reducing the occurrence of storage disorders. Both experimental and modeling approaches have been followed to investigate the relationship between the gas concentration, gas diffusion, respiration and physiological disorders in apple and pear. Experimental approaches are, however, costly, tiresome and time consuming. Moreover, it is difficult to investigate the time course of the physiological disorders experimentally because of the lack of non-destructive techniques to measure the internal gas concentration in the fruit. Alternatively, mathematical models can be used to study gas and water exchange. They are usually based on the continuum hypothesis where the fruit is considered as a material with transport properties that are independent of the spatial scale. However, unlike the traditional engineering materials fruit tissue has a complex fine structure. The cellular architecture is believed to determine to a large extent the biophysical processes in the fruit. The continuum hypothesis does not hold in this case and a multiscale approach is required in which the model parameters of the model that operates at the macroscale the scale of interest are obtained from simulations with a microscale model that incorporates the actual microstructure of the fruit. For the latter, microscale geometric models of the fruit are required. Pome fruit tissue microscale geometry generators exist today but are based on digitized 2D or 3D images of the cellular architecture. Therefore, although these algorithms generate representative geometries of the tissues, they require experimental input in terms of microscopic images. These approaches need complex image acquisition procedures and expensive infrastructures such as synchrotron radiation sources. Also, they do not allow to parameterize the microscale geometry to, for example, investigate the effect of cell size or shape on the transport properties in a systematic way. The main objective of this dissertation was to develop virtual microscale fruit tissue generators (algorithms) that generate statistically and spatially equivalent virtual tissue microstructures resolving the cell symplast, cell wall and intercellular air spaces in both 2D and 3D and interface them to finite element and/or finite volume codes. To achieve this, we have developed virtual tissue generators that are based on cell growth modeling bytaking into account cell biomechanics. The generators are initiated from a random Voronoi tessellation and growth biomechanics is applied to the tessellation which results in a virtual tissue that has equivalent geometrical properties as that of real tissues obtained from microscopic or synchrotron microtomography images. In a further extension we have also developed a cell division algorithm which is based on cell biomechanics and that is capable of mimicking both symmetric cell division and asymmetric cell division with different degree of anisotropic growth. The cell division algorithm can be used instead of the Voronoi tessellation as an input for the expansive growth models. Initial tessellations obtained from the cell division algorithm will have more realistic representation of the cells than the Voronoi tessellations. The geometric models can be used to carry out in silico simulations to determine transport properties to be used in multiscale framework of gas and moisture exchange studies in pome fruits. This approach helps to include more geometrical details and fewer assumptions than the classical continuum modeling approach, while requiring less computer time compared to solving governing model equations at the resolution of the microscale.Acknowledgements i Abstract iii Beknopte samenvatting vii Table of contents xi Abbreviations and symbols xv 1 General introduction 1 1.1 Introduction 1 1.2 Modeling approaches of transport phenomena in fruit tissues 5 1.2.1 The need for modeling transport phenomena in fruit tissues 5 1.2.2 Continuum modeling of transport phenomena in fruit tissues 8 1.2.3 Importance of microstructure in modeling transport phenomena in fruit tissues 10 1.3 Objectives 11 1.4 Thesis outline 12 Bibliography 13 2 Literature review 21 2.1 Introduction 21 2.2 Relevance of geometry of fruit tissue at different scales 22 25 2.3 Fruit tissue structure 26 2.3.1 Plant tissue types 26 2.3.2 Meristematic versus permanent tissues 26 2.3.3 Pome fruit tissue types 28 2.4 Microscale fruit tissue models based on imaging 32 2.5 Microscale fruit tissue modeling using tesselations 34 2.6 Microscale fruit tissue geometries based on tissue growth models 37 2.7 Conclusions 43 Bibliography 44 3 2D virtual fruit tissue generation based on cell growth modeling 53 3.1 Introduction 53 3.2 Material and methods 54 3.2.1 Sample preparation, image acquisition and processing 54 3.2.2 The growth model 55 3.2.2.1 Governing equations 55 3.2.2.2 Model parameters 58 3.2.2.3 Algorithm 59 3.2.3 Sensitivity analysis 61 3.2.4 Statistical analysis 62 3.3 Results and discussion 63 3.3.1 Parenchyma tissue 63 3.3.2 Complex tissue structures 69 3.3.3 Applications 71 3.4 Conclusions 72 Bibliography 73 4 3D virtual fruit tissue generation based on cell growth modeling 77 4.1 Introduction 77 4.2 Materials and methods 79 4.2.1 The growth model 79 4.2.1.1 Governing equations 79 4.2.1.2 Model parameters 83 4.2.1.3 Algorithm 83 4.2.1.4 Pore generation 86 4.2.2 Sample preparation, image acquisition and processing 86 4.2.3 Sensitivity analysis 87 4.2.4 Statistical analysis 87 4.3 Results 88 4.3.1 Cell size and shape 89 4.3.2 Pore generation 90 4.3.2.1 Schizogenous origin pore generation 90 4.3.2.2 Lysigenous origin pore generation 91 4.3.3 Calibration of the model using image data 92 4.3.4 Complex tissue structures 96 4.3.4.1 The skin 96 4.3.4.2 Stone cells 98 4.4 Discussion 99 4.5 Conclusions 103 Bibliography 103 5 2D cell division algorithm based on ellipse fitting 109 5.1 Introduction 109 5.2 Materials and methods 112 5.2.1 Cell growth algorithm 112 5.2.2 Cell division algorithm 114 5.2.3 Experimental data 115 5.2.4 Geometric and topological properties 116 5.2.4.1 Topology 116 5.2.4.2 Cell shape 116 5.2.4.3 Cell size 118 5.2.4.4 Statistical comparison 118 5.3 Results 118 5.3.1 Topology distribution 120 5.3.2 Cell size distribution 122 5.3.3 Cell shape distribution 123 5.3.3.1 Aspect ratio distribution 124 5.3.3.2 Internal angle distribution 126 5.3.4 Comparison of real and virtual topological and geometrical properties 128 5.4 Discussions 131 5.5 Conclusions 135 Bibliography 136 6 3D cell division algorithm based on ellipsoid fitting 141 6.1 Introduction 141 6.2 Materials and methods 144 6.2.1 Cell growth algorithm 144 6.2.2 Cell division algorithm 146 6.2.3 Geometric and topological properties 148 6.2.4 Statistical comparison 149 6.3 Results 149 6.3.1 Topology distribution 151 6.3.2 Cell size distribution 153 6.3.3 Cell shape distribution 154 6.4 Discussion 156 6.5 Conclusions 159 Bibliography 160 7 General conclusion and future directions 165 7.1 General conclusions 165 7.2 Future directions 168 Bibliography 169 List of publications 173nrpages: 197status: publishe

A 3D Fruit Tissue Growth Algorithm Based on Cell Biomechanics

A 3D fruit tissue growth algorithm is presented based on the biomechanics of plant cells in tissues. The algorithm is able to generate realistic virtual fruit tissues. It was used to produce cell architectures of pome fruits with intercellular air spaces. The cell size and shape differences in pear cortex tissue and apple cortex tissue are obtained by implementing different maximum resting length of the cell walls and different cell wall stiffness (e.g., by including a degree of anisotropy) in the model. In addition to cell size and shape, the difference in the size of the intercellular air spaces and their connectivity is recognized in our model by implementing different pore formation mechanisms. The arrangement of different tissue layers which are observed from the skin to the cortex (epidermis, hypodermis and cortex) and particular features such as stone cells were also accounted for by our model. The algorithm was shown to produce cell architectures that are very similar to measured tissue structures of the pear and cortex tissue with intercellular air spaces. The resulting geometric models can be used in finite element simulations to study exchange processes within the fruit tissue and between the tissue and the surrounding environment. The geometric models can also be used to study coupled phenomena of moisture migration and tissue shrinkage in a multiscale approach. These approaches were demonstrated to be useful using our 2D version of the algorithm. The 3D version of the algorithm avoids many of the limitations of the 2D algorithm such as lack of intercellular air space connectivity in the 2D geometric models and hence, will help to better understand the exchange mechanisms.status: accepte

Prediction of water loss and viscoelastic deformation of apple tissue using a multiscale model

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 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.status: publishe

A novel plant cell division algorithm based on ellipse/ellipsoid fitting

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Microscale modeling of water transport in fruit tissue

A model was developed to describe water transport in fruit tissue, taking into account the microstructural architecture of the cell assemblies in the tissue, which leads to a better understanding of the underlying phenomena causing water loss. Pear (Pyrus communis L. cv. Conference) was chosen as a model system. The fruit tissue architecture was generated by means of a cell growth model. The transport of water in the intercellular space, the cell wall network and cytoplasm was predicted using transport laws using the chemical potential as the driving force for water exchange between different microstructural compartments. The model equations were solved on the pear cortex tissue geometry (referred here after as geometry) using the finite element method. The different water transport properties of the microstructural components were obtained experimentally or from literature. The effective water conductivity of pear cortex tissue was calculated based on the microscale simulations. The values corresponded well with measured values of tissue water transport parameters. The model helped to explain the relative importance of the different microstructural features (intercellular space, cell wall, membrane and cytoplasm) for water transport. The cell membrane was shown to have the largest effect on the apparent macroscopic water conductivity.status: publishe

Virtual Fruit Tissue Generation Using Cell Growth Modelling

Fruit tissues are very heterogeneous at the microscale and the cellular architecture determines to a large extent the behaviour and development of the fruit and their behaviour during postharvest storage. The cellular architecture is established during the growth of the fruit after fertilization. Understanding the development and the changes of the microstructure of fruits would be an important step to help explain and optimize fruit production and postharvest storage. Pome fruit tissue generators exist today but are based on digitized 2-D or 3-D images of the cellular architecture, which require experimental input in terms of microscopic images. Furthermore, the algorithms today do not provide insight in the reasons why a certain tissue structure develops. To close this knowledge gap, a cell growth-based algorithm is being developed using the biomechanics of plant cells in tissues to help explain the typical differences in cellular architecture found between different fruit species and cultivars. The cell is considered as a closed thin walled structure, maintained in tension by turgor pressure. The cell walls of adjacent cells are modeled as parallel and linearly elastic elements which obey Hooke's law. A Voronoi tessellation is used to generate the initial topology of the cells. Cell expansion is then resulted from turgor pressure acting on the yielding cell wall material. To find the sequence positions of each vertex and thus the shape of the layer with time, a system of differential equations for the positions and velocities of each vertex are established and solved using a forward Euler method. The model is implemented in Matlab and is used to generate realistic fruit tissue structures composed of cells of random shapes and sizes, cell walls and intercellular spaces. Comparison is made with fruit tissue micrographs at different development stages. The virtual tissues can be applied to study tissue mechanics and exchange processes of important metabolites.status: publishe

Virtual Fruit Tissue Generation Based on Cell Growth Modelling

Abstract: A cell-growth-based algorithm is presented based on the biomechanics of plant cells in tissues to help explain the typical differences in cellular architecture found between dif- ferent pome fruit species, cultivars and tissues. The cell was considered as a closed thin-walled structure, maintained in tension by turgor pressure. The cell walls of adjacent cells were modelled as parallel and linearly elastic elements, which obeyed Hooke’s law. A Voronoi tessellation was used to generate the initial topology of the cells. Cell expansion then resulted from turgor pressure acting on the yielding cell wall material. To find the sequence positions of each vertex of the cell walls, and thus, the shape of the cells with time, a system of differential equations for the positions and velocities of each vertex were established and solved using a Runge–Kutta fourth and fifth order (ODE45) method. The model was used to generate realistic 2D fruit tissue structures composed of cells of random shapes and sizes, cell walls and intercellular spaces. Comparison was made with fruit tissue micrographs. The virtual tissues can be used for numerical simulation of heat and mass transfer phenomena or mechanical deformation during controlled atmosphere storage of fresh pome fruit.Acknowledgements: Financial support by the Flanders Fund for Scientific Research (project FWO G.0603.08), K.U. Leuven (project OT 08/023) and the EC (project InsideFood FP7-226783) is gratefully acknowledged.status: publishe

A plant cell division algorithm based on biomechanics and ellipse fitting

Abstract • Background and aim: The importance of cell division models in cellular pattern studies has been acknowledged since the 19th century. Most of the available models to date are limited to symmetric cell division with isotropic growth. Often, the actual growth of the cell wall is either not considered or updated intermittently in separate time scale from the mechanics. Here, we have presented a generic algorithm that accounts for both symmetrically and asymmetrically dividing cells with isotropic and anisotropic growth. The actual growth of cell wall is simulated simultaneously with the mechanics. • Methods: The cell is considered as a closed thin walled structure, maintained in tension by turgor pressure. The cell walls are represented as linear elastic elements which obey Hooke's law. Cell expansion is induced by turgor pressure acting on the yielding cell wall material. A system of differential equations for the positions and velocities of the cell vertices as well as for the actual growth of the cell wall is established. Readiness to divide is determined based on cell size. An ellipse fitting algorithm is used to determine the position and orientation of the dividing wall. The cell vertices, walls and cell connectivity are then updated and the cell expansion resumes. Comparison has been made with experimental data from literature. • Key results: A generic plant cell division algorithm has been successfully implemented. It can handle both symmetrically and asymmetrically dividing cells coupled with isotropic and anisotropic growth modes. The importance of ellipse fitting to produce randomness (biological variability) even in symmetrically dividing cells is highlighted. Unlike previous models, differential equation is formulated for the resting length of the cell wall to simulate actual biological growth and is solved simultaneously with position and velocity of the vertices. • Conclusions: The algorithm presented here can produce different tissues varying in topological and geometrical properties. This capability allows the model to be used in in silico cellular pattern studies of specific cases.status: publishe

In silico study of the role of cell growth factors in photosynthesis using a virtual leaf tissue generator coupled to a microscale photosynthesis gas exchange model

Computational tools that allow in silico analysis of the role of cell growth and division on photosynthesis are scarce. We present a freely available tool that combines a virtual leaf tissue generator and a two-dimensional microscale model of gas transport during C3 photosynthesis. A total of 270 mesophyll geometries were generated with varying degrees of growth anisotropy, growth extent, and extent of schizogenous airspace formation in the palisade mesophyll. The anatomical properties of the virtual leaf tissue and microscopic cross-sections of actual leaf tissue of tomato (Solanum lycopersicum L.) were statistically compared. Model equations for transport of CO2 in the liquid phase of the leaf tissue were discretized over the geometries. The virtual leaf tissue generator produced a leaf anatomy of tomato that was statistically similar to real tomato leaf tissue. The response of photosynthesis to intercellular CO2 predicted by a model that used the virtual leaf tissue geometry compared well with measured values. The results indicate that the light-saturated rate of photosynthesis was influenced by interactive effects of extent and directionality of cell growth and degree of airspace formation through the exposed surface of mesophyll per leaf area. The tool could be used further in investigations of improving photosynthesis and gas exchange in relation to cell growth and leaf anatomy.status: Published onlin

Automatic analysis of the 3-D microstructure of fruit parenchyma tissue using X-ray micro-CT explains differences in aeration

Background: 3D high-resolution X-ray imaging methods have emerged over the last years for visualising the anatomy of tissue samples without substantial sample preparation. Quantitative analysis of cells and intercellular spaces in these images has, however, been difficult and was largely based on manual image processing. We present here an automated procedure for processing high-resolution X-ray images of parenchyma tissues of apple (Malus × domestica Borkh.) and pear (Pyrus communis L.) as a rapid objective method for characterizing 3D plant tissue anatomy at the level of single cells and intercellular spaces. Results: We isolated neighboring cells in 3D images of apple and pear cortex tissues, and constructed a virtual sieve to discard incorrectly segmented cell particles or unseparated clumps of cells. Void networks were stripped down until their essential connectivity features remained. Statistical analysis of structural parameters showed significant differences between genotypes in the void and cell networks that relate to differences in aeration properties of the tissues. Conclusions: A new model for effective oxygen diffusivity of parenchyma tissue is proposed that not only accounts for the tortuosity of interconnected voids, but also for significant diffusion across cells where the void network is not connected. This will significantly aid interpretation and analysis of future tissue aeration studies. The automated image analysis methodology will also support pheno- and genotyping studies where the 3D tissue anatomy plays a role.status: publishe