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Explanations of and results from different automated shear zone thickness calculation methods, and a brief investigation of the differences in the measured shearing resistance in the Fallow specimens.

This paper is concerned with a late stage of lymphangiogenesis in the trunk of the zebrafish embryo. At 48 hours post-fertilisation (HPF), a pool of parachordal lymphangioblasts (PLs) lies in the horizontal myoseptum. Between 48 and 168 HPF, the PLs spread from the horizontal myoseptum to form the thoracic duct, dorsal longitudinal lymphatic vessel, and parachordal lymphatic vessel. This paper deals with the potential of vascular endothelial growth factor C (VEGFC) to guide the differentiation of PLs into the mature lymphatic endothelial cells that form the vessels. We built a mathematical model to describe the biochemical interactions between VEGFC, collagen I, and matrix metalloproteinase 2 (MMP2). We also carried out a linear stability analysis of the model and computer simulations of VEGFC patterning. The results suggest that VEGFC can form Turing patterns due to its relations with MMP2 and collagen I, but the zebrafish embryo needs a separate control mechanism to create the right physiological conditions. Furthermore, this control mechanism must ensure that the VEGFC patterns are useful for lymphangiogenesis: stationary, steep gradients, and reasonably fast forming. Generally, the combination of a patterning species, a matrix protein, and a remodelling species is a new patterning mechanism.

The remarkable complexity of soil and its importance to a wide range of ecosystem services presents major challenges to the modeling of soil processes. Although major progress in soil models has occurred in the last decades, models of soil processes remain disjointed between disciplines or ecosystem services, with considerable uncertainty remaining in the quality of predictions and several challenges that remain yet to be addressed. First, there is a need to improve exchange of knowledge and experience among the different disciplines in soil science and to reach out to other Earth science communities. Second, the community needs to develop a new generation of soil models based on a systemic approach comprising relevant physical, chemical, and biological processes to address critical knowledge gaps in our understanding of soil processes and their interactions. Overcoming these challenges will facilitate exchanges between soil modeling and climate, plant, and social science modeling communities. It will allow us to contribute to preserve and improve our assessment of ecosystem services and advance our understanding of climate-change feedback mechanisms, among others, thereby facilitating and strengthening communication among scientific disciplines and society. We review the role of modeling soil processes in quantifying key soil processes that shape ecosystem services, with a focus on provisioning and regulating services. We then identify key challenges in modeling soil processes, including the systematic incorporation of heterogeneity and uncertainty, the integration of data and models, and strategies for effective integration of knowledge on physical, chemical, and biological soil processes. We discuss how the soil modeling community could best interface with modern modeling activities in other disciplines, such as climate, ecology, and plant research, and how to weave novel observation and measurement techniques into soil models. We propose the establishment of an international soil modeling consortium to coherently advance soil modeling activities and foster communication with other Earth science disciplines. Such a consortium should promote soil modeling platforms and data repository for model development, calibration and intercomparison essential for addressing contemporary challenges.

The importance of root hairs in uptake of sparingly-soluble nutrients is understood qualitatively, but not quantitatively, and this limits efforts to breed plants tolerant of nutrient-deficient soils. We develop a mathematical model of nutrient uptake by root hairs allowing for hair geometry and the details of nutrient transport to hairs through soil, including diffusion within and between soil particles. Compared with conventional ‘single porosity’ models, this ‘dual porosity’ model predicts greater root uptake because more nutrient is available by slow release from within the soil particles. Also the effect of soil moisture is less important with the dual porosity model because the effective volume available for diffusion within the soil is larger, and the predicted effects of hair length and density are different. Consistent with experimental observations, increases in hair length give greater increases in uptake than increases in hair density per unit main root length. The effect of hair density is less in dry soil because the minimum concentration in solution in the hair zone for net influx is reached more rapidly. The effect of hair length is much less sensitive to soil moisture. Implications for manipulation of root architecture through plant breeding are discussed.

Plants change their shapes, depending on their environment, for example, plant height increases with increasing population density. We examined the density-dependent plasticity in shoot morphology of herbs by analysing a mathematical model which identifies a number of key factors that influence shoot morphology, namely (i) solar radiation captured by leaves; (ii) shading from neighbouring plants; and (iii) utilisation efficiency of resource by leaves, stems and veins. An optimisation theory was used to obtain optimal shoot morphology in relation to maximal light capture by leaves, under trade-offs of resource partition among organs. We first evaluated the solar radiation flux per unit leaf area per day for different shoot forms. Our model predicts that the optimal internodal length of the stem that brings about the maximal light capture by leaves increases with plant population density, and this is consistent with experimental data. Moreover, our simple model can also be extended to explain the morphological plasticity in other herbs (i.e. stemless plants) that are different from our model plants with a stem. These findings illustrate how optimisation theory can be used for the analysis of plasticity in shoot morphology of plants in response to environmental changes, as well as the analysis of diversity in morphology.

This thesis deals with the mathematical modelling of nutrient uptake by plant roots. It starts with the Nye-Tinker-Barber model for nutrient uptake by a single bare cylindrical root. The model is treated using matched asymptotic expansion and an analytic formula for the rate of nutrient uptake is derived for the first time. The basic model is then extended to include root hairs and mycorrhizae, which have been found experimentally to be very important for the uptake of immobile nutrients. Again, analytic expressions for nutrient uptake are derived. The simplicity and clarity of the analytical formulae for the solution of the single root models allows the extension of these models to more realistic branched roots. These models clearly show that the `volume averaging of branching structure' technique commonly used to extend the Nye-Tinker-Barber with experiments can lead to large errors. The same models also indicate that in the absence of large-scale water movement, due to rainfall, fertiliser fails to penetrate into the soil. This motivates us to build a model for water movement and uptake by branched root structures. This model considers the simultaneous flow of water in the soil, uptake by the roots, and flow within the root branching network to the stems of the plant. The water uptake model shows that the water saturation can develop pseudo-steady-state wet and dry zones in the rooting region of the soil. The dry zone is shown to stop the movement of nutrient from the top of the soil to the groundwater. Finally we present a model for the simultaneous movement and uptake of both nutrients and water. This is discussed as a new tool for interpreting available experimental results and designing future experiments. The parallels between evolution and mathematical optimisation are also discussed.

Root hairs are thought to play an important role in mediating nutrient uptake by plants. We develop a mathematical model for the nutrient transport and uptake in the root hair zone of a single root in the soil. Nutrients are assumed to diffuse both in the soil fluid phase and within the soil particles. Nutrients can also be bound to the soil particle surfaces by reversible reactions. Using homogenization techniques we derive a macroscopic dual porosity model for nutrient diffusion and reaction in the soil which includes the effect of all root hair surfaces.

The rhizosphere is a dynamic region where multiple interacting processes in the roots and surrounding soil take place, with dimensions set by the distance to which the zone of root influence spreads into the soil. Its complexity is such that some form of mathematical modelling is essential for understanding which of the various processes operating are important, and a minimal model of the rhizosphere must provide information on (a) the spatiotemporal concentration changes of mobile solutes in the root-influenced soil, and (b) the cumulative uptake of solutes per unit length of root over time. Both are unique for a given set of parameters and initial conditions and hence the model is fully deterministic. 'Up-scaling' to uptake by whole plants by integrating individual fluxes requires a measure of the growth and senescence of the root system. Root architecture models are increasingly successful in providing this. The spatio-temporal scales of the rhizosphere and roots are sufficiently different that they can be treated separately, and this greatly simplifies modelling. The minimal model has been successfully applied to the more-mobile nutrients in soil, such as nitrate or potassium, but much less successfully to less-soluble nutrients such as phosphorus, because other, undescribed processes become important. These include transfers from unavailable forms, heterogeneity of resource distribution, root competition, water redistribution and adaptive processes. Incorporating such processes into models can disrupt independent scaling. In general, scaling from the scale of the individual root to that of the whole plant does not pose insuperable problems. Paradoxically, the major challenge in introducing more complexity is that experimental corroboration of the model is required at the individual root scale.

This review will outline a number of illustrative mathematical models describing the growth of avascular tumours. The aim of the review is to provide a relatively comprehensive list of existing models in this area and discuss several representative models in greater detail. In the latter part of the review, some possible future avenues of mathematical modelling of avascular tumour development are outlined together with a list of key questions.

Cluster roots are thought to play an important role in mediating nutrient uptake by plants. In this paper we develop a mathematical model for the transport and uptake of phosphate by a single root. Phosphate is assumed to diffuse in the soil fluid phase and can also solubilised due to citrate exudation. Using multiple scale homogenisation techniques we derive an effective model that accounts for the cumulative effect of citrate exudation and phosphate uptake by cluster roots whilst still retaining all the necessary information about the microscale geometry and effects.

Conventional models of solute transport in soil consider only soil volumes large enough to average over microscale heterogeneities, and it is assumed that microscale variations are unimportant at the macroscale. In this research we test this assumption for cases in which the microscale distribution of solute‐sorbing sites is patchy. We obtain a set of equations at the macroscale that allow for the effect of the microscale distribution with the mathematical technique of homogenization. We combine these equations with an image‐based model that describes the true microscale pore geometry in a real, structured soil measured with X‐ray computed tomography. The resulting models are used to test the microscale averaging assumptions inherent in conventional models. We show that, in general, macroscale diffusion is little affected by microscale variation in the distribution of sorption sites. Therefore, for most purposes the assumption of microscale averaging used in conventional models is justified. The effects of microscale heterogeneity are noticeable only when (i) the rate of sorption is slow compared with diffusion, but still fast enough to affect macroscale transport and (ii) the defined macroscale volume approaches the microscale. We discuss the effects when these conditions are met.

Conventional models of solute transport in soil consider only soil volumes large enough to average over microscale heterogeneities, and it is assumed that microscale variations are unimportant at the macroscale. In this research we test this assumption for cases in which the microscale distribution of solute-sorbing sites is patchy. We obtain a set of equations at the macroscale that allow for the effect of the microscale distribution with the mathematical technique of homogenization. We combine these equations with an image-based model that describes the true microscale pore geometry in a real, structured soil measured with X-ray computed tomography. The resulting models are used to test the microscale averaging assumptions inherent in conventional models. We show that, in general, macroscale diffusion is little affected by microscale variation in the distribution of sorption sites. Therefore, for most purposes the assumption of microscale averaging used in conventional models is justified. The effects of microscale heterogeneity are noticeable only when (i) the rate of sorption is slow compared with diffusion, but still fast enough to affect macroscale transport and (ii) the defined macroscale volume approaches the microscale. We discuss the effects when these conditions are met

The remarkable complexity of soil and its importance to a wide range of ecosystem services presents 133 major challenges to the modeling of soil processes. Although major progress in soil models has 134 occurred in the last decades, models of soil processes remain disjointed between disciplines or 135 ecosystem services, with considerable uncertainty remaining in the quality of predictions and several 136 challenges that remain yet to be addressed. Firstly, there is a need to improve exchange of knowledge 137 and experience amongst the different disciplines in soil science and to reach out to other Earth science 138 communities. Secondly, the community needs to develop a new generation of soil models based on a 139 systemic approach comprising relevant physical, chemical, and biological processes to address critical 140 knowledge gaps in our understanding of soil processes and their interactions. Overcoming these 141 challenges will facilitate exchanges between soil modeling and climate, plant, and social science 142 modeling communities. It will allow us to contribute to preserve and improve our assessment of 143 ecosystem services and advance our understanding of climate-change feedback mechanisms, amongst 144 others, thereby facilitating and strengthening communication among scientific disciplines and society. 145 In this paper we review the role of modeling soil processes in quantifying key soil processes that shape 146 ecosystem services with focus on provisioning and regulating services. We then identify key 147 challenges in modeling soil processes including the systematic incorporation of heterogeneity and 148 uncertainty, the integration of data and models, and strategies for effective integration of knowledge 149 on physical, chemical and biological soil processes. We discuss how the soil modeling community 150 could best interface with modern modeling activities in other disciplines such as climate, ecology, and 151 plant research and how to weave novel observation and measurement techniques into soil models. We 152 propose to establish an international soil modeling consortium to coherently advance soil modeling 153 activities and foster communication with other Earth science disciplines. Such a consortium should 154 promote soil modeling platforms and data repository for model development, calibration and inter-155 comparison essential for addressing contemporary challenges.

Fmiso-PET is a non-invasive modality used for the assessment of tumour hypoxia, and increasingly for planning radiotherapy. However, the availability and contrast properties of Fmiso are not ideal. Recent efforts to compare FDG binding with that of Fmiso, in order to ascertain FDG's potential as a marker of hypoxia, have met with mixed results. The potential reasons for correlated and disparate binding patterns between the two tracers have been postulated, but not formally outlined as yet. We present a model of a key component of the image formation process - tracer pharmacokinetics. This involves a series of coupled PDEs, describing the interplay between concentrations of oxygen, glucose, HIF, Fmiso and FDG. We use this model to assess the general feasibility of FDG as a surrogate marker of hypoxia and find that its utility is dependent on activity of oncogenic pathways.

Summary We hypothesized that plant exudates could either gel or disperse soil depending on their chemical characteristics. Barley (Hordeum vulgare L. cv. Optic) and maize (Zea mays L. cv. Freya) root exudates were collected using an aerated hydroponic method and compared with chia (Salvia hispanica L.) seed exudate, a commonly used root exudate analogue. Sandy loam soil was passed through a 500‐μm mesh and treated with each exudate at a concentration of 4.6 mg exudate g−1 dry soil. Two sets of soil samples were prepared. One set of treated soil samples was maintained at 4°C to suppress microbial processes. To characterize the effect of decomposition, the second set of samples was incubated at 16°C for 2 weeks at −30 kPa matric potential. Gas chromatography–mass spectrometry (GC–MS) analysis of the exudates showed that barley had the largest organic acid content and chia the largest content of sugars (polysaccharide‐derived or free), and maize was in between barley and chia. Yield stress of amended soil samples was measured by an oscillatory strain sweep test with a cone plate rheometer. When microbial decomposition was suppressed at 4°C, yield stress increased 20‐fold for chia seed exudate and twofold for maize root exudate compared with the control, whereas for barley root exudate decreased to half. The yield stress after 2 weeks of incubation compared with soil with suppressed microbial decomposition increased by 85% for barley root exudate, but for chia and maize it decreased by 87 and 54%, respectively. Barley root exudation might therefore disperse soil and this could facilitate nutrient release. The maize root and chia seed exudates gelled soil, which could create a more stable soil structure around roots or seeds. Highlights Rheological measurements quantified physical behaviour of plant exudates and effect on soil stabilization.Barley root exudates dispersed soil, which could release nutrients and carbon.Maize root and chia seed exudates had a stabilizing effect on soil.Physical engineering of soil in contact with plant roots depends on the nature and origin of exudates.

The parameters in Richards’ equation are usually calculated from experimentally measured values of the soil–water characteristic curve and saturated hydraulic conductivity. The complex pore structures that often occur in porous media complicate such parametrization due to hysteresis between wetting and drying and the effects of tortuosity. Rather than estimate the parameters in Richards’ equation from these indirect measurements, image-based modelling is used to investigate the relationship between the pore structure and the parameters. A threedimensional, X-ray computed tomography image stack of a soil sample with voxel resolution of 6 µm has been used to create a computational mesh. The Cahn–Hilliard–Stokes equations for two-fluid flow, in this case water and air, were applied to this mesh and solved using the finite-element method in COMSOL MULTIPHYSICS. The upscaled parameters in Richards’ equation are then obtained via homogenization. The effect on the soil–water retention curve due to three different contact angles, 0◦, 20◦ and 60◦, was also investigated. The results show that the pore structure affects the properties of the flow on the large scale, and different contact angles can change the parameters for Richards’ equation.

Capillaries in tumours are often severely buckled (in a plane perpendicular to the axis) and / or chaotic in their direction. We develop a model of these phenomena using nonlinear solid mechanics. Our model focusses on the immediate surrounding of a capillary. The vessel and surrounding tissue are modelled as concentric annulii. The growth is dependent on the concentration of a nutrient (oxygen) diffusing from the vessel into the tumour interstitium. The stress is modelled using a multiplicative decomposition of the deformation gradient F=F_e F_g. The stress is determined by substituting the elastic deformation gradient F_e (which gives the deformation gradient from the hypothetical configuration to the current configuration) into a hyperelastic constitutive model as per classical solid mechanics. We use a Blatz-Ko model, parameterised using uniaxial compression experiments. The entire system is in quasi-static equilibrium, with the divergence of the stress tensor equal to zero. We determine the onset of buckling using a linear stability analysis. We then investigate the postbuckling behaviour by introducing higher order perturbations in the deformation and growth before using the Fredholm Alternative to obtain the magnitude of the buckle.Our results demonstrate that the growth-induced stresses are sufficient for the capillary to buckle in the absence of external loading and / or constraints. Planar buckling usually occurs after 2-5 times the cellular proliferation timescale. Buckles with axial variation almost always go unstable after planar buckles. Buckles of fine wavelength are initially preferred by the system, but over time buckles of large wavelength become energetically more favourable. The tumoural hoop stress T_{ThetaTheta} is the most invariant (Eulerian) variable at the time of buckling: it is typically of the order of the tumoural Young's Modulus when this occurs.