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Aims: Recent laboratory studies revealed that root hairs may alter soil physical behaviour, influencing soil porosity and water retention on the small scale. However, the results are not consistent, and it is not known if structural changes at the small-scale have impacts at larger scales. Therefore, we evaluated the potential effects of root hairs on soil hydro-mechanical properties in the field using rhizosphere-scale physical measurements. Methods: Changes in soil water retention properties as well as mechanical and hydraulic characteristics were monitored in both silt loam and sandy loam soils. Measurements were taken from plant establishment to harvesting in field trials, comparing three barley genotypes representing distinct phenotypic categories in relation to root hair length. Soil hardness and elasticity were measured using a 3-mm-diameter spherical indenter, while water sorptivity and repellency were measured using a miniaturized infiltrometer with a 0.4-mm tip radius. Results: Over the growing season, plants induced changes in the soil water retention properties, with the plant available water increasing by 21%. Both soil hardness (P = 0.031) and elasticity (P = 0.048) decreased significantly in the presence of root hairs in silt loam soil, by 50% and 36%, respectively. Root hairs also led to significantly smaller water repellency (P = 0.007) in sandy loam soil vegetated with the hairy genotype (-49%) compared to the hairless mutant. Conclusions: Breeding of cash crops for improved soil conditions could be achieved by selecting root phenotypes that ameliorate soil physical properties and therefore contribute to increased soil health. [ABSTRACT FROM AUTHOR]

Global food production depends on the application of phosphorus (P) fertilisers, usually sourced from rock phosphate, a non-renewable resource. Optimising P use to ensure sustainable P application is necessary to supply food worldwide and to protect the environment from P runoff. However, standard models used to guide P application on fields are limited due to assumptions that fail to consider the short-term dynamics of P in the soil solution. This study combined time-resolved microdialysis sampling with 4D spatial information from X-ray computed tomography to inform an image-based model for assessing P-soil-plant interactions over the start of a growing season. The time-resolved microdialysis measurements revealed that P released from the granules is rapidly removed from the soil solution in the short-term. We demonstrate that the standard equilibrium models typically used to characterise P transport in soil are not representative of the experimental system on the time scales considered. Instead, an Absorption-Diffusion model, where a single sink term accounts for all the processes removing P from the soil solution was required to correctly characterise experimental observations. Our study provides the basis for a model which could be adapted to predict within-season fertilisation scenarios in different soil conditions, and provides a conceptual description of plant/crop yield response to P fertilisation. [Display omitted] • Phosphorus fertiliser dynamics and inefficiency sources are poorly understood. • We investigated spatiotemporal dynamics of P in soil and plant uptake. • We combined microdialysis, X-ray Computed Tomography and image-based modelling. • Explaining rapid P adsorption in P-poor soil required a novel modelling approach. • New models predict P application is effective in a narrow window of P-soil content. [ABSTRACT FROM AUTHOR]

Xylem-limited bacterial pathogens cause some of the most destructive plant diseases. Though imposed measures to control these pathogens are generally ineffective, even among susceptible taxa, some hosts can limit bacterial loads and symptom expression. Mechanisms by which this resistance is achieved are poorly understood. In particular, it is still unknown how differences in vascular structure may influence biofilm growth and spread within a host. To address this, we developed a novel theoretical framework to describe biofilm behaviour within xylem vessels, adopting a polymer-based modelling approach. We then parameterised the model to investigate the relevance of xylem vessel diameters on Xylella fastidiosa resistance among olive cultivars. The functionality of all vessels was severely reduced under infection, with hydraulic flow reductions of 2–3 orders of magnitude. However, results suggest wider vessels act as biofilm incubators; allowing biofilms to develop over a long time while still transporting them through the vasculature. By contrast, thinner vessels become blocked much earlier, limiting biofilm spread. Using experimental data on vessel diameter distributions, we were able to determine that a mechanism of resistance in the olive cultivar Leccino is a relatively low abundance of the widest vessels, limiting X. fastidiosa spread. [ABSTRACT FROM AUTHOR]

Monitoring nutrients in the soil can provide valuable information for understanding their spatiotemporal variability and informing precise soil management. Here, we describe an autonomous in situ analyzer for the real-time monitoring of nitrate in soil. The analyzer can sample soil nitrate using either microdialysis or ultrafiltration probes placed within the soil and quantify soil nitrate using droplet microfluidics and colorimetric measurement. Compared with traditional manual sampling and lab analysis, the analyzer features low reagent consumption (96 μL per measurement), low maintenance requirement (monthly), and high measurement frequency (2 or 4 measurements per day), providing nondrifting lab-quality data with errors of less than 10% using a microdialysis probe and 2-3% for ultrafiltration. The analyzer was deployed at both the campus garden and forest for different periods of time, being able to capture changes in free nitrate levels in response to manual perturbation by the addition of nitrate standard solutions and natural perturbation by rainfall events.

The soil region influenced by plant roots, i.e., the rhizosphere, is one of the most complex biological habitats on Earth and significantly impacts global carbon flow and transformation. Understanding the structure and function of the rhizosphere is critically important for maintaining sustainable plant ecosystem services, designing engineered ecosystems for long-term soil carbon storage, and mitigating the effects of climate change. However, studying the biological and ecological processes and interactions in the rhizosphere requires advanced integrated technologies capable of decoding such a complex system at different scales. Here, we review how emerging approaches in sensing, imaging, and computational modeling can advance our understanding of the complex rhizosphere system. Particularly, we provide our perspectives and discuss future directions in developing in situ rhizosphere sensing technologies that could potentially correlate local-scale interactions to ecosystem scale impacts. We first review integrated multimodal imaging techniques for tracking inorganic elements and organic carbon flow at nano- to microscale in the rhizosphere, followed by a discussion on the use of synthetic soil and plant habitats that bridge laboratory-to-field studies on the rhizosphere processes. We then describe applications of genetically encoded biosensors in monitoring nutrient and chemical exchanges in the rhizosphere, and the novel nanotechnology-mediated delivery approaches for introducing biosensors into the root tissues. Next, we review the recent progress and express our vision on field-deployable sensing technologies such as planar optodes for quantifying the distribution of chemical and analyte gradients in the rhizosphere under field conditions. Moreover, we provide perspectives on the challenges of linking complex rhizosphere interactions to ecosystem sensing for detecting biological traits across scales, which arguably requires using the best-available model predictions including the model-experiment and image-based modeling approaches. Experimental platforms relevant to field conditions like SMART (Sensors at Mesoscales with Advanced Remote Telemetry) soils testbed, coupled with ecosystem sensing and predictive models, can be effective tools to explore coupled ecosystem behavior and responses to environmental perturbations. Finally, we envision that with the advent of novel high-resolution imaging capabilities at nano- to macroscale, and remote biosensing technologies, combined with advanced computational models, future studies will lead to detection and upscaling of rhizosphere processes toward ecosystem and global predictions.

Transport in porous media plays an essential role for many physical, engineering, biological and environmental processes. Novel synchrotron imaging techniques and image-based models have enabled more robust quantification of geometric structures that influence transport through the pore space. However, image-based modelling is computationally expensive, and end users often require, while conducting imaging campaign, fast and agile bulk-scale effective parameter estimates that account for the pore-scale details. In this manuscript we enhance a pre-existing image-based model solver known as OpenImpala to estimate bulk-scale effective transport parameters. In particular, the boundary conditions and equations in OpenImpala were modified in order to estimate the effective diffusivity in an imaged system/geometry via a formal multi-scale homogenisation expansion. Estimates of effective pore space diffusivity were generated for a range of elementary volume sizes to estimate when the effective diffusivity values begin to converge to a single value. Results from OpenImpala were validated against a commercial finite element method package COMSOL Multiphysics (abbreviated as COMSOL). Results showed that the effective diffusivity values determined with OpenImpala were similar to those estimated by COMSOL. Tests on larger domains comparing a full image-based model to a homogenised (geometrically uniform) domain that used the effective diffusivity parameters showed differences below 2 % error, thus verifying the accuracy of the effective diffusivity estimates. Finally, we compared OpenImpala's parallel computing speeds to COMSOL. OpenImpala consistently ran simulations within fractions of minutes, which was two orders of magnitude faster than COMSOL providing identical supercomputing specifications. In conclusion, we demonstrated OpenImpala's utility as part of an on-site tomography processing pipeline allowing for fast and agile assessment of porous media processes and to guide imaging campaigns while they are happening at synchrotron beamlines., Supplementary Information: The online version contains supplementary material available at 10.1007/s11242-023-01993-7., Competing Interests: Conflicts of interestThe authors have no relevant financial or non-financial interests to disclose., (© The Author(s) 2023, corrected publication 2023.)

Nitrogen fertilization is vital for productive agriculture and efficient land use. However, globally, approximately 50% of the nitrogen applied is lost to the environment, causing inefficiencies, pollution, and greenhouse gas emissions. Rainfall and its effect on soil moisture are the major components controlling nitrogen losses in agriculture. Thus, changing rainfall patterns could accelerate nitrogen inefficiencies. We used a mechanistic modeling platform to determine how precipitation-optimal nitrogen fertilization timings and resulting crop nitrogen uptake have changed historically (1950-2020) and how they are predicted to change under the RCP8.5 climate scenario (2021-2069) in the South East of England. We found that historically, neither precipitation-optimal fertilization timings nor resulting plant uptake changed significantly. However, there were large year-to-year variations in both. In the 2030s, where it is projected to get wetter, precipitation-optimal fertilization timings are predicted to be later in the season and the resulting plant uptake noticeably lower. After 2040, the precipitation-optimal uptakes are projected to increase with earlier precipitation-optimal timings closer to historical values, corresponding to the projected mean daily rainfall rates decreasing to the historical values in these growing seasons. It seems that the interannual variation in precipitation-optimal uptake is projected to increase. Ultimately, projected changes in precipitation patterns will affect nitrogen uptake and precipitation-optimal fertilization timings. We argue that the use of bespoke fertilization timings in each year can help recuperate the reduced N uptake due to changing precipitation., Competing Interests: The authors declare no competing financial interest., (© 2022 The Authors. Published by American Chemical Society.)

Phosphorus (P) is essential for plant growth. Arbuscular mycorrhizal fungi (AMF) aid its uptake by acquiring P from sources distant from roots in return for carbon. Little is known about how AMF colonise soil pore-space, and models of AMF-enhanced P-uptake are poorly validated. We used synchrotron X-ray computed tomography to visualize mycorrhizas in soil and synchrotron X-ray fluorescence/X-ray absorption near edge structure (XRF/XANES) elemental mapping for P, sulphur (S) and aluminium (Al) in combination with modelling. We found that AMF inoculation had a suppressive effect on colonisation by other soil fungi and identified differences in structure and growth rate between hyphae of AMF and nonmycorrhizal fungi. Our results showed that AMF co-locate with areas of high P and low Al, and preferentially associate with organic-type P species over Al-rich inorganic P. We discovered that AMF avoid Al-rich areas as a source of P. Sulphur-rich regions were found to be correlated with higher hyphal density and an increased organic-associated P-pool, whilst oxidized S-species were found close to AMF hyphae. Increased S oxidation close to AMF suggested the observed changes were microbiome-related. Our experimentally-validated model led to an estimate of P-uptake by AMF hyphae that is an order of magnitude lower than rates previously estimated - a result with significant implications for the modelling of plant-soil-AMF interactions., (© 2022 The Authors New Phytologist © 2022 New Phytologist Foundation.)

Vegetation enhances soil shearing resistance through water uptake and root reinforcement. Analytical models for soils reinforced with roots rely on input parameters that are difficult to measure, leading to widely varying predictions of behaviour. The opaque heterogeneous nature of rooted soils results in complex soil-root interaction mechanisms that cannot easily be quantified. The authors measured, for the first time, the shear resistance and deformations of fallow, willow-rooted and gorse-rooted soils during direct shear using X-ray computed tomography and digital volume correlation. Both species caused an increase in shear zone thickness, both initially and as shear progressed. Shear zone thickness peaked at up to 35 mm, often close to the thickest roots and towards the centre of the column. Root extension during shear was 10-30% less than the tri-linear root profile assumed in a Waldron-type model, owing to root curvature. Root analogues used to explore the root-soil interface behaviour suggested that root lateral branches play an important role in anchoring the roots. The Waldron-type model was modified to incorporate non-uniform shear zone thickness and growth, and accurately predicted the observed, up to sevenfold, increase in shear resistance of root-reinforced soil., (© 2022 The Authors.)

Phosphorus is an essential nutrient for crops. Precise spatiotemporal application of P fertilizer can improve plant P acquisition and reduce run-off losses of P. Optimizing application would benefit from understanding the dynamics of P release from a fertilizer pellet into bulk soil, which requires space- and time-resolved measurements of P concentration in soil solutions. In this study, we combined microdialysis and X-ray computed tomography to investigate P transport in soil. Microdialysis probes enabled repeated solute sampling from one location with minimal physical disturbance, and their small dimensions permitted spatially resolved monitoring. We observed a rapid initial release of P from the source, producing high dissolved P concentrations within the first 24 h, followed by a decrease in dissolved P over time compatible with adsorption onto soil particles. Soils with greater bulk density (i.e., reduced soil porosity) impeded the P pulse movement, which resulted in a less homogeneous distribution of total P in the soil column at the end of the experiment. The model fit to the data showed that the observed phenomena can be explained by diffusion and adsorption. The results showed that compared with conventional measurement techniques (e.g., suction cups), microdialysis measurements present a less invasive alternative. The time-resolved measurements ultimately highlighted rapid P dynamics that require more attention for improving P use efficiency., Competing Interests: The authors declare that there is no conflicts of interest., (© 2020 The Authors. Soil Science Society of America Journal published by Wiley Periodicals LLC on behalf of Soil Science Society of America.)

Aims: We sought to develop a novel experimental system which enabled application of iodinated contrast media to in vivo plant roots intact in soil and was compatible with time-resolved synchrotron X-ray computed tomography imaging. The system was developed to overcome issues of low contrast to noise within X-ray computed tomography images of plant roots and soil environments, the latter of which can complicate image processing and result in the loss of anatomical information., Methods: To demonstrate the efficacy of the system we employ the novel use of both synchrotron X-ray computed tomography and synchrotron X-ray fluorescence mapping to capture the translocation of the contrast media through root vasculature into the leaves., Results: With the application of contrast media we identify fluid flow in root vasculature and visualise anatomical features, which are otherwise often only observable in ex vivo microscopy, including: the xylem, metaxylem, pith, fibres in aerenchyma and leaf venation. We are also able to observe interactions between aerenchyma cross sectional area and solute transport in the root vasculature with depth., Conclusions: Our novel system was capable of successfully delivering sufficient contrast media into root and leaf tissues such that anatomical features could be visualised and internal fluid transport observed. We propose that our system could be used in future to study internal plant transport mechanisms and parameterise models for fluid flow in plants., Supplementary Information: The online version contains supplementary material available at 10.1007/s11104-020-04784-x., Competing Interests: Competing interestsWe have no competing interests., (© The Author(s) 2020.)

Root citrate exudation is thought to be important for phosphate solubilization. Previous research has concluded that cluster-like roots benefit most from this exudation in terms of increased phosphate uptake, suggesting that root structure plays an important role in citrate-enhanced uptake (additional phosphate uptake due to citrate exudation). Time-resolved computed tomography images of wheat root systems were used as the geometry for 3D citrate-phosphate solubilization models. Citrate-enhanced uptake was correlated with morphological measures of the root systems to determine which had the most benefit. A large variation of citrate-enhanced uptake over 11 root structures was observed. Root surface area dominated absolute phosphate uptake, but did not explain citrate-enhanced uptake. Number of exuding root tips correlated well with citrate-enhanced uptake. Root tips in close proximity could collectively exude high amounts of citrate, resulting in a delayed spike in citrate-enhanced uptake. Root system architecture plays an important role in citrate-enhanced uptake. Singular morphological measurements of the root systems cannot entirely explain variations in citrate-enhanced uptake. Root systems with many tips would benefit greatly from citrate exudation. Quantifying citrate-enhanced uptake experimentally is difficult as variations in root surface area would overwhelm citrate benefits., (© 2020 The Authors. New Phytologist © 2020 New Phytologist Trust.)

The subtle relationship between vascular network structure and mass transport is vital to predict and improve the efficacy of anticancer treatments. Here, mathematical homogenisation is used to derive a new multiscale continuum model of blood and chemotherapy transport in the vasculature and interstitium of a vascular tumour. This framework enables information at a range of vascular hierarchies to be fed into an effective description on the length scale of the tumour. The model behaviour is explored through a demonstrative case study of a simplified representation of a dorsal skinfold chamber, to examine the role of vascular network architecture in influencing fluid and drug perfusion on the length scale of the chamber. A single parameter, P, is identified that relates tumour-scale fluid perfusion to the permeability and density of the capillary bed. By fixing the topological and physiological properties of the arteriole and venule networks, an optimal value for P is identified, which maximises tumour fluid transport and is thus hypothesised to benefit chemotherapy delivery. We calculate the values for P for eight explicit network structures; in each case, vascular intervention by either decreasing the permeability or increasing the density of the capillary network would increase fluid perfusion through the cancerous tissue. Chemotherapeutic strategies are compared and indicate that single injection is consistently more successful compared with constant perfusion, and the model predicts optimal timing of a second dose. These results highlight the potential of computational modelling to elucidate the link between vascular architecture and fluid, drug distribution in tumours., (© 2020 The Authors. International Journal for Numerical Methods in Biomedical Engineering published by John Wiley & Sons Ltd.)

Rhizosphere soil has distinct physical and chemical properties from bulk soil. However, besides root-induced physical changes, chemical changes have not been extensively measured in situ on the pore scale. In this study, we couple structural information, previously obtained using synchrotron X-ray computed tomography (XCT), with synchrotron X-ray fluorescence microscopy (XRF) and X-ray absorption near-edge structure (XANES) to unravel chemical changes induced by plant roots. Our results suggest that iron (Fe) and sulfur (S) increase notably in the direct vicinity of the root via solubilization and microbial activity. XANES further shows that Fe is slightly reduced, S is increasingly transformed into sulfate (SO 4 2- ) and phosphorus (P) is increasingly adsorbed to humic substances in this enrichment zone. In addition, the ferrihydrite fraction decreases drastically, suggesting the preferential dissolution and the formation of more stable Fe oxides. Additionally, the increased transformation of organic S to sulfate indicates that the microbial activity in this zone is increased. These changes in soil chemistry correspond to the soil compaction zone as previously measured via XCT. The fact that these changes are colocated near the root and the compaction zone suggests that decreased permeability as a result of soil structural changes acts as a barrier creating a zone with increased rhizosphere chemical interactions via surface-mediated processes, microbial activity and acidification., (© 2019 The Authors. New Phytologist © 2019 New Phytologist Trust.)

The growth of rice in submerged soils depends on its ability to form continuous gas channels-aerenchyma-through which oxygen (O 2 ) diffuses from the shoots to aerate the roots. Less well understood is the extent to which aerenchyma permits venting of respiratory carbon dioxide (CO 2 ) in the opposite direction. Large, potentially toxic concentrations of dissolved CO 2 develop in submerged rice soils. We show using X-ray computed tomography and image-based mathematical modelling that CO 2 venting through rice roots is far greater than thought hitherto. We found rates of venting equivalent to a third of the daily CO 2 fixation in photosynthesis. Without this venting through the roots, the concentrations of CO 2 and associated bicarbonate (HCO 3 - ) in root cells would have been well above levels known to be toxic to roots. Removal of CO 2 and hence carbonic acid (H 2 CO 3 ) from the soil was sufficient to increase the pH in the rhizosphere close to the roots by 0.7 units, which is sufficient to solubilize or immobilize various nutrients and toxicants. A sensitivity analysis of the model showed that such changes are expected for a wide range of plant and soil conditions., (© 2019 The Authors. Plant, Cell & Environment published by John Wiley & Sons Ltd.)

This investigation establishes a system of gold nanoparticles that show good colloidal stability as an X-ray computed tomography (XCT) contrast agent under soil conditions. Gold nanoparticles offer numerous beneficial traits for experiments in biology including: comparatively minimal phytotoxicity, X-ray attenuation of the material and the capacity for functionalization. However, soil salinity, acidity and surface charges can induce aggregation and destabilize gold nanoparticles, hence in biomedical applications polymer coatings are commonly applied to gold nanoparticles to enhance stability in the in vivo environment. Here we first demonstrate non-coated nanoparticles aggregate in soil-water solutions. We then show coating with a polyethylene glycol (PEG) layer prevents this aggregation. To demonstrate this, PEG-coated nanoparticles were drawn through flow columns containing soil and were shown to be stable; this is in contrast with control experiments using silica and alumina-packed columns. We further determined that a suspension of coated gold nanoparticles which fully saturated soil maintained stability over at least 5 days. Finally, we used time resolved XCT imaging and image based models to approximate nanoparticle diffusion as similar to that of other typical plant nutrients diffusing in water. Together, these results establish the PEGylated gold nanoparticles as potential contrast agents for XCT imaging in soil., Competing Interests: We have no competing interests., (© 2019 The Authors.)

In this paper, we use multiple scale homogenisation to derive a set of averaged macroscale equations that describe the movement of nutrients in partially saturated soil that contains growing potato tubers. The soil is modelled as a poroelastic material, which is deformed by the growth of the tubers, where the growth of each tuber is dependent on the uptake of nutrients via a sink term within the soil representing root nutrient uptake. Special attention is paid to the reduction in void space, resulting change in local water content and the impact on nutrient diffusion within the soil as the tubers increase in size. To validate the multiple scale homogenisation procedure, we compare the system of homogenised equations to the original set of equations and find that the solutions between the two models differ by [Formula: see text]. However, we find that the computation time between the two sets of equations differs by several orders of magnitude. This is due to the combined effects of the complex three-dimensional geometry and the implementation of a moving boundary condition to capture tuber growth.

Lung lymphatics maintain fluid homoeostasis by providing a drainage system that returns fluid, cells and metabolites to the circulatory system. The 3D structure of the human pulmonary lymphatic network is essential to lung function, but it is poorly characterised. Image-based 3D mathematical modelling of pulmonary lymphatic microfluidics has been limited by the lack of accurate and representative image geometries. This is due to the microstructural similarity of the lymphatics to the blood vessel network, the lack of lymphatic-specific biomarkers, the technical limitations associated with image resolution in 3D, and sectioning artefacts present in 2D techniques. We present a method that combines lymphatic specific (D240 antibody) immunohistochemistry (IHC), optimised high-resolution X-ray microfocus computed tomography (μCT) and finite-element mathematical modelling to assess the function of human peripheral lung tissue. The initial results identify lymphatic heterogeneity within and between lung tissue. Lymphatic vessel volume fraction and fractal dimension significantly decreases away from the lung pleural surface (p < 0.001, n = 25 and p < 0.01, n = 20, respectively). Microfluidic modelling successfully shows that in lung tissue the fluid derived from the blood vessels drains through the interstitium into the lymphatic vessel network and this drainage is different in the subpleural space compared to the intralobular space. When comparing lung tissue from health and disease, human pulmonary lymphatics were significantly different across five morphometric measures used in this study (p ≤ 0.0001). This proof of principle study establishes a new engineering technology and workflow for further studies of pulmonary lymphatics and demonstrates for the first time the combination of correlative μCT and IHC to enable 3D mathematical modelling of human lung microfluidics at micrometre resolution.

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.

Soil adjacent to roots has distinct structural and physical properties from bulk soil, affecting water and solute acquisition by plants. Detailed knowledge on how root activity and traits such as root hairs affect the three-dimensional pore structure at a fine scale is scarce and often contradictory. Roots of hairless barley (Hordeum vulgare L. cv Optic) mutant (NRH) and its wildtype (WT) parent were grown in tubes of sieved (<250 μm) sandy loam soil under two different water regimes. The tubes were scanned by synchrotron-based X-ray computed tomography to visualise pore structure at the soil-root interface. Pore volume fraction and pore size distribution were analysed vs distance within 1 mm of the root surface. Less dense packing of particles at the root surface was hypothesised to cause the observed increased pore volume fraction immediately next to the epidermis. The pore size distribution was narrower due to a decreased fraction of larger pores. There were no statistically significant differences in pore structure between genotypes or moisture conditions. A model is proposed that describes the variation in porosity near roots taking into account soil compaction and the surface effect at the root surface., (© 2018 The Authors. New Phytologist © 2018 New Phytologist Trust.)

We develop a model with which to study the poorly understood mechanisms of uranium (U) uptake by plants. The model is based on equations for transport and reaction of U and acids and bases in the rhizosphere around cylindrical plant roots. It allows for the speciation of U with hydroxyl, carbonate, and organic ligands in the soil solution; the nature and kinetics of sorption reactions with the soil solid; and the effects of root-induced changes in rhizosphere pH. A sensitivity analysis showed the importance of soil sorption and speciation parameters as influenced by pH and CO 2 pressure; and of root geometry and root-induced acid-base changes linked to the form of nitrogen taken up by the root. The root absorbing coefficient for U, relating influx to the concentration of U species in solution at the root surface, was also important. Simplified empirical models of U uptake by different plant species and soil types need to account for these effects.

Spatially averaged models of root-soil interactions are often used to calculate plant water uptake. Using a combination of X-ray computed tomography (CT) and image-based modelling, we tested the accuracy of this spatial averaging by directly calculating plant water uptake for young wheat plants in two soil types. The root system was imaged using X-ray CT at 2, 4, 6, 8 and 12 d after transplanting. The roots were segmented using semi-automated root tracking for speed and reproducibility. The segmented geometries were converted to a mesh suitable for the numerical solution of Richards' equation. Richards' equation was parameterized using existing pore scale studies of soil hydraulic properties in the rhizosphere of wheat plants. Image-based modelling allows the spatial distribution of water around the root to be visualized and the fluxes into the root to be calculated. By comparing the results obtained through image-based modelling to spatially averaged models, the impact of root architecture and geometry in water uptake was quantified. We observed that the spatially averaged models performed well in comparison to the image-based models with <2% difference in uptake. However, the spatial averaging loses important information regarding the spatial distribution of water near the root system., (© 2017 John Wiley & Sons Ltd.)

The rhizosphere is a zone of fundamental importance for understanding the dynamics of nutrient acquisition by plant roots. The canonical difficulty of experimentally investigating the rhizosphere led long ago to the adoption of mathematical models, the most sophisticated of which now incorporate explicit representations of root hairs and rhizosphere soil. Mathematical upscaling regimes, such as homogenisation, offer the possibility of incorporating into larger-scale models the important mechanistic processes occurring at the rhizosphere scale. However, we lack concrete descriptions of all the features required to fully parameterise models at the rhizosphere scale. By combining synchrotron X-ray computed tomography (SRXCT) and a novel root growth assay, we derive a three-dimensional description of rhizosphere soil structure suitable for use in multi-scale modelling frameworks. We describe an approach to mitigate sub-optimal root hair detection via structural root hair growth modelling. The growth model is explicitly parameterised with SRXCT data and simulates three-dimensional root hair ideotypes in silico, which are suitable for both ideotypic analysis and parameterisation of 3D geometry in mathematical models. The study considers different hypothetical conditions governing root hair interactions with soil matrices, with their respective effects on hair morphology being compared between idealised and image-derived soil/root geometries. The studies in idealised geometries suggest that packing arrangement of soil affects hair tortuosity more than the particle diameter. Results in field-derived soil suggest that hair access to poorly mobile nutrients is particularly sensitive to the physical interaction between the growing hairs and the phase of the soil in which soil water is present (i.e. the hydrated textural phase). The general trends in fluid-coincident hair length with distance from the root, and their dependence on hair/soil interaction mechanisms, are conserved across Cartesian and cylindrical geometries.

In this paper, we provide direct evidence of the importance of root hairs on pore structure development at the root-soil interface during the early stage of crop establishment. This was achieved by use of high-resolution (c. 5 μm) synchrotron radiation computed tomography (SRCT) to visualise both the structure of root hairs and the soil pore structure in plant-soil microcosms. Two contrasting genotypes of barley (Hordeum vulgare), with and without root hairs, were grown for 8 d in microcosms packed with sandy loam soil at 1.2 g cm -3 dry bulk density. Root hairs were visualised within air-filled pore spaces, but not in the fine-textured soil regions. We found that the genotype with root hairs significantly altered the porosity and connectivity of the detectable pore space (> 5 μm) in the rhizosphere, as compared with the no-hair mutants. Both genotypes showed decreasing pore space between 0.8 and 0.1 mm from the root surface. Interestingly the root-hair-bearing genotype had a significantly greater soil pore volume-fraction at the root-soil interface. Effects of pore structure on diffusion and permeability were estimated to be functionally insignificant under saturated conditions when simulated using image-based modelling., (© 2017 The Authors. New Phytologist © 2017 New Phytologist Trust.)

The use of in vivo X-ray microcomputed tomography (μCT) to study plant root systems has become routine, but is often hampered by poor contrast between roots, soil, soil water, and soil organic matter. In clinical radiology, imaging of poorly contrasting regions is frequently aided by the use of radio-opaque contrast media. In this study, we present evidence for the utility of iodinated contrast media (ICM) in the study of plant root systems using μCT. Different dilutions of an ionic and nonionic ICM (Gastrografin 370 and Niopam 300) were perfused into the aerial vasculature of juvenile pea plants via a leaf flap (Pisum sativum). The root systems were imaged via μCT, and a variety of image-processing approaches used to quantify and compare the magnitude of the contrast enhancement between different regions. Though the treatment did not appear to significantly aid extraction of full root system architectures from the surrounding soil, it did allow the xylem and phloem units of seminal roots and the vascular morphology within rhizobial nodules to be clearly visualized. The nonionic, low-osmolality contrast agent Niopam appeared to be well tolerated by the plant, whereas Gastrografin showed evidence of toxicity. In summary, the use of iodine-based contrast media allows usually poorly contrasting root structures to be visualized nondestructively using X-ray μCT. In particular, the vascular structures of roots and rhizobial nodules can be clearly visualized in situ.

The lymphatic system of a vertebrate is important in health and diseases. We propose a novel mathematical model to elucidate the lymphangiogenic processes in zebrafish embryos. Specifically, we are interested in the period when lymphatic endothelial cells (LECs) exit the posterior cardinal vein and migrate to the horizontal myoseptum of a zebrafish embryo. We wonder whether vascular endothelial growth factor C (VEGFC) is a morphogen and a chemotactic factor for these LECs. The model considers the interstitial flow driving convection, the reactive transport of VEGFC, and the changing dynamics of the extracellular matrix in the embryo. Simulations of the model illustrate that VEGFC behaves very differently in diffusion and convection-dominant scenarios. In the former case, it must bind to the matrix to establish a functional morphogen gradient. In the latter case, the opposite is true and the pressure field is the key determinant of what VEGFC may do to the LECs. Degradation of collagen I, a matrix component, by matrix metallopeptidase 2 controls the spatiotemporal dynamics of VEGFC. It controls whether diffusion or convection is dominant in the embryo; it can create channels of abundant VEGFC and scarce collagen I to facilitate lymphangiogenesis; when collagen I is insufficient, VEGFC cannot influence the LECs at all. We predict that VEGFC is a morphogen for the migrating LECs, but it is not a chemotactic factor for them.

Mechanosensory neurons detect physical events in the local environments of the tissues that they innervate. Studies of mechanosensitivity of neurons or nerve endings in the gut have related their firing to strain, wall tension, or pressure. Digital image correlation (DIC) is a technique from materials engineering that can be adapted to measure the local physical environments of afferent neurons at high resolution. Flat-sheet preparations of guinea pig distal colon were set up with arrays of tissue markers in vitro. Firing of single viscerofugal neurons was identified in extracellular colonic nerve recordings. The locations of viscerofugal nerve cell bodies were inferred by mapping firing responses to focal application of the nicotinic receptor agonist 1,1-dimethyl-4-phenylpiperazinium iodide. Mechanosensory firing was recorded during load-evoked uniaxial or biaxial distensions. Distension caused movement of surface markers which was captured by video imaging. DIC tracked the markers, interpolating the mechanical state of the gut at the location of the viscerofugal nerve cell body. This technique revealed heterogeneous load-evoked strain within preparations. Local strains at viscerofugal nerve cell bodies were usually smaller than global strain measurements and correlated more closely with mechanosensitive firing. Both circumferential and longitudinal strain activated viscerofugal neurons. Simultaneous loading in circumferential and longitudinal axes caused the highest levels of viscerofugal neuron firing. Multiaxial strains, reflecting tissue shearing and changing area, linearly correlated with mechanosensory firing of viscerofugal neurons. Viscerofugal neurons were mechanically sensitive to both local circumferential and local longitudinal gut strain, and appear to lack directionality in their stretch sensitivity., (Copyright © 2016 the American Physiological Society.)

In this study, we developed a spatially explicit model for nutrient uptake by root hairs based on X-ray computed tomography images of the rhizosphere soil structure. This work extends our previous work to larger domains and hence is valid for longer times. Unlike the model used previously, which considered only a small region of soil about the root, we considered an effectively infinite volume of bulk soil about the rhizosphere. We asked the question: At what distance away from root surfaces do the specific structural features of root-hair and soil aggregate morphology not matter because average properties start dominating the nutrient transport? The resulting model was used to capture bulk and rhizosphere soil properties by considering representative volumes of soil far from the root and adjacent to the root, respectively. By increasing the size of the volumes that we considered, the diffusive impedance of the bulk soil and root uptake were seen to converge. We did this for two different values of water content. We found that the size of region for which the nutrient uptake properties converged to a fixed value was dependent on the water saturation. In the fully saturated case, the region of soil we needed to consider was only of radius 1.1mm for poorly soil-mobile species such as phosphate. However, in the case of a partially saturated medium (relative saturation 0.3), we found that a radius of 1.4mm was necessary. This suggests that, in addition to the geometrical properties of the rhizosphere, there is an additional effect of soil moisture properties, which extends further from the root and may relate to other chemical changes in the rhizosphere. The latter were not explicitly included in our model., (© The Author 2016. Published by Oxford University Press on behalf of the Society for Experimental Biology.)

The lymphatic system returns fluid to the bloodstream from the tissues to maintain tissue fluid homeostasis. Lymph nodes distributed throughout the system filter the lymphatic fluid. The afferent and efferent lymph flow conditions of lymph nodes can be measured in experiments; however, it is difficult to measure the flow within the nodes. In this paper, we present an image-based modelling approach to investigating how the internal structure of the node affects the fluid flow pathways within the node. Selective plane illumination microscopy images of murine lymph nodes are used to identify the geometry and structure of the tissue within the node and to determine the permeability of the lymph node interstitium to lymphatic fluid. Experimental data are used to determine boundary conditions and optimise the parameters for the model. The numerical simulations conducted within the model are implemented in COMSOL Multiphysics, a commercial finite element analysis software. The parameter fitting resulted in the estimate that the average permeability for lymph node tissue is of the order of magnitude of [Formula: see text]. Our modelling shows that the flow predominantly takes a direct path between the afferent and efferent lymphatics and that fluid is both filtered and absorbed across the blood vessel boundaries. The amount that is absorbed or extravasated in the model is dependent on the efferent lymphatic lumen fluid pressure.

Background and Aim: Recycled sources of phosphorus (P), such as struvite extracted from wastewater, have potential to substitute for more soluble manufactured fertilisers and help reduce the long-term threat to food security from dwindling finite reserves of phosphate rock (PR). This study aimed to determine whether struvite could be a component of a sustainable P fertiliser management strategy for arable crops., Methods: A combination of laboratory experiments, pot trials and mathematical modelling of the root system examined the P release properties of commercial fertiliser-grade struvite and patterns of P uptake from a low-P sandy soil by two different crop types, in comparison to more soluble inorganic P fertilisers (di-ammonium phosphate (DAP) and triple super phosphate (TSP))., Results: Struvite had greatly enhanced solubility in the presence of organic acid anions; buckwheat, which exudes a high level of organic acids, was more effective at mobilising struvite P than the low level exuder, spring wheat. Struvite granules placed with the seed did not provide the same rate of P supply as placed DAP granules for early growth of spring wheat, but gave equivalent rates of P uptake, yield and apparent fertiliser recovery at harvest, even though only 26 % of struvite granules completely dissolved. Fertiliser mixes containing struvite and DAP applied to spring wheat have potential to provide both optimal early and late season P uptake and improve overall P use efficiency., Conclusions: We conclude that the potential resource savings and potential efficiency benefits of utilising a recycled slow release fertiliser like struvite offers a more sustainable alternative to only using conventional, high solubility, PR-based fertilisers.

This study investigates fluid flow and elastic deformation in tissues that are drained by the primary lymphatic system. A model is formulated based on the Rossi hypothesis that states that the primary lymphatic valves, which are formed by overlapping endothelial cells around the circumferential lining of lymphatic capillaries, open in response to swelling of the surrounding tissue. Tissue deformation and interstitial fluid flow through the tissue are treated using the Biot equations of poroelasticity and, the fluid flux (into the interstitium) across the walls of the blood capillaries, is assumed to be linearly related to the pressure difference across the walls via a constant of proportionality (the vascular permeability). The resulting model is solved in a periodic domain containing one blood capillary and one lymphatic capillary starting from a configuration in which the tissue is undeformed. On imposition of a constant pressure difference between blood and lymphatic capillaries, the solutions are found to settle to a steady state. Given that the magnitude of pressure fluctuations in the lymphatic system is much smaller than this pressure difference between blood and lymph, it is postulated that the resulting steady-state solution gives a good representation of the state of the tissue under physiological conditions. The effects of changes to the Young's modulus of the tissue, the blood-lymphatic pressure difference, vascular permeability and valve dimensions on the steady state are investigated and discussed in terms of their effects on oedema in the context of age- and pregnancy-related changes to the body.

Understanding the dynamics of water distribution in soil is crucial for enhancing our knowledge of managing soil and water resources. The application of X-ray computed tomography (CT) to the plant and soil sciences is now well established. However, few studies have utilized the technique for visualizing water in soil pore spaces. Here this method is utilized to visualize the water in soil in situ and in three-dimensions at successive reductive matric potentials in bulk and rhizosphere soil. The measurements are combined with numerical modelling to determine the unsaturated hydraulic conductivity, providing a complete picture of the hydraulic properties of the soil. The technique was performed on soil cores that were sampled adjacent to established roots (rhizosphere soil) and from soil that had not been influenced by roots (bulk soil). A water release curve was obtained for the different soil types using measurements of their pore geometries derived from CT imaging and verified using conventional methods, such as pressure plates. The water, soil, and air phases from the images were segmented and quantified using image analysis. The water release characteristics obtained for the contrasting soils showed clear differences in hydraulic properties between rhizosphere and bulk soil, especially in clay soil. The data suggest that soils influenced by roots (rhizosphere soil) are less porous due to increased aggregation when compared with bulk soil. The information and insights obtained on the hydraulic properties of rhizosphere and bulk soil will enhance our understanding of rhizosphere biophysics and improve current water uptake models., (© The Author 2015. Published by Oxford University Press on behalf of the Society for Experimental Biology.)

· Root hairs are known to be highly important for uptake of sparingly soluble nutrients, particularly in nutrient deficient soils. Development of increasingly sophisticated mathematical models has allowed uptake characteristics to be quantified. However, modelling has been constrained by a lack of methods for imaging live root hairs growing in real soils. · We developed a plant growth protocol and used Synchrotron Radiation X-ray Tomographic Microscopy (SRXTM) to uncover the three-dimensional (3D) interactions of root hairs in real soil. We developed a model of phosphate uptake by root hairs based directly on the geometry of hairs and associated soil pores as revealed by imaging. · Previous modelling studies found that root hairs dominate phosphate uptake. By contrast, our study suggests that hairs and roots contribute equally. We show that uptake by hairs is more localized than by roots and strongly dependent on root hair and aggregate orientation. · The ability to image hair-soil interactions enables a step change in modelling approaches, allowing a more realistic treatment of processes at the scale of individual root hairs in soil pores., (© 2013 The Authors. New Phytologist © 2013 New Phytologist Trust.)

This study investigates the fluid flow through tissues where lymphatic drainage occurs. Lymphatic drainage requires the use of two valve systems, primary and secondary. Primary valves are located in the initial lymphatics. Overlapping endothelial cells around the circumferential lining of lymphatic capillaries are presumed to act as a unidirectional valve system. Secondary valves are located in the lumen of the collecting lymphatics and act as another unidirectional valve system; these are well studied in contrast to primary valves. We propose a model for the drainage of fluid by the lymphatic system that includes the primary valve system. The analysis in this work incorporates the mechanics of the primary lymphatic valves as well as the fluid flow through the interstitium and that through the walls of the blood capillaries. The model predicts a piecewise linear relation between the drainage flux and the pressure difference between the blood and lymphatic capillaries. The model describes a permeable membrane around a blood capillary, an elastic primary lymphatic valve and the interstitium lying between the two.

Lymphatic capillary drainage of interstitial fluid under both steady-state and inflammatory conditions is important for tissue fluid balance, cancer metastasis, and immunity. Lymphatic drainage function is critically coupled to the fluid mechanical properties of the interstitium, yet this coupling is poorly understood. Here we sought to effectively model the lymphatic-interstitial fluid coupling and ask why the lymphatic capillary network often appears with roughly a hexagonal architecture. We use homogenization method, which allows tissue-scale lymph flow to be integrated with the microstructural details of the lymphatic capillaries, thus gaining insight into the functionality of lymphatic anatomy. We first describe flow in lymphatic capillaries using the Navier-Stokes equations and flow through the interstitium using Darcy's law. We then use multiscale homogenization to derive macroscale equations describing lymphatic drainage, with the mouse tail skin as a basis. We find that the limiting resistance for fluid drainage is that from the interstitium into the capillaries rather than within the capillaries. We also find that between hexagonal, square, and parallel tube configurations of lymphatic capillary networks, the hexagonal structure is the most efficient architecture for coupled interstitial and capillary fluid transport; that is, it clears the most interstitial fluid for a given network density and baseline interstitial fluid pressure. Thus, using homogenization theory, one can assess how vessel microstructure influences the macroscale fluid drainage by the lymphatics and demonstrate why the hexagonal network of dermal lymphatic capillaries is optimal for interstitial tissue fluid clearance., (Copyright © 2011 Elsevier Ltd. All rights reserved.)

Rice (Oryza sativa L.) secretes far smaller amounts of metal-complexing phytosiderophores (PS) than other grasses. But there is increasing evidence that it relies on PS secretion for its zinc (Zn) uptake. After nitrogen, Zn deficiency is the most common nutrient disorder in rice, affecting up to 50% of lowland rice soils globally. We developed a mathematical model of PS secretion from roots and resulting solubilization and uptake of Zn, allowing for root growth, diurnal variation in secretion, decomposition of the PS in the soil, and the transport and interaction of the PS and Zn in the soil. A sensitivity analysis showed that with realistic parameter values for rice in submerged soil, the typically observed rates of PS secretion from rice are sufficient and necessary to explain observed rates of Zn uptake. There is little effect of diurnal variation in secretion on cumulative Zn uptake, irrespective of other model parameter values, indicating that the observed diurnal variation is not causally related to Zn uptake efficiency. Rooting density has a large effect on uptake per unit PS secretion as a result of overlap of the zones of influence of neighbouring roots. The effects of other complications in the rice rhizosphere are discussed., (© 2011 Blackwell Publishing Ltd.)

Arbuscular mycorrhizas, associations between plant roots and soil fungi, are ubiquitous among land plants. Arbuscular mycorrhizas can be beneficial for plants by overcoming limitations in nutrient supply. Hyphae, which are long and thin fungal filaments extending from the root surface into the soil, increase the volume of soil accessible for plant nutrient uptake. However, no models so far specifically consider individual hyphae. We developed a mathematical model for nutrient uptake by individual fungal hyphae in order to assess suitable temporal and spatial scales for a new experimental design where fungal uptake parameters are measured on the single hyphal scale. The model was developed based on the conservation of nutrients in an artificial cylindrical soil pore (capillary tube) with adsorbing wall, and analysed based on parameter estimation and non-dimensionalisation. An approximate analytical solution was derived using matched asymptotic expansion. Results show that nutrient influx into a hypha from a small capillary tube is characterized by three phases: Firstly, uptake rapidly decreases as the hypha takes up nutrients, secondly, the depletion zone reaches the capillary wall and thus uptake is sustained by desorption of nutrients from the capillary wall, and finally, uptake goes to zero after nutrients held on the capillary wall have been completely depleted. Simulating different parameter regimes resulted in recommending the use of capillaries filled with hydrogel instead of water in order to design an experiment operating over measurable time scales., (© The Author(s) 2011. This article is published with open access at Springerlink.com)

Direct contact of nanoparticles with the plasma membrane is essential for biomedical applications such as intracellular drug delivery and imaging, but the effect of nanoparticle association on membrane structure and function is largely unknown. Here we employ a sensitive electrophysiological method to assess the stability of protein-free membranes in the presence of silica nanospheres of different size and surface chemistry. It is shown that all the silica nanospheres permeabilize the lipid bilayers already at femtomolar concentrations, below reported cytotoxic values. Surprisingly, it is observed that a proportion of the nanospheres is able to translocate over the pure-lipid bilayer. Confocal fluorescence imaging of fluorescent nanosphere analogues also enables estimation of the particle density at the membrane surface; a significant increase in bilayer permeability is already apparent when less than 1% of the bilayer area is occupied by silica nanospheres. It can be envisaged that higher concentrations of nanoparticles lead to an increased surface coverage and a concomitant decrease in bilayer stability, which may contribute to the plasma membrane damage, inferred from lactate dehydrogenase release, that is regularly observed in nanotoxicity studies with cell cultures. This biophysical approach gives quantitative insight into nanosphere-bilayer interactions and suggests that nanoparticle-lipid interactions alone can compromise the barrier function of the plasma membrane.

The growth of trees and other plants occurs through the interactive combination of photosynthesis and carbon (and other nutrient) assimilation. Photosynthesis enables the production of carbohydrate that can then be used in growing foliage, whereby photosynthesis is enabled. We construct a mathematical model of carbon uptake and storage, which allows the prediction of the growth dynamics of trees. We find that the simplest model allows uncontrolled foliage production through the positive feedback outlined above, but that leaf shading provides an automatic saturation to carbon assimilation, and hence to foliage production. The model explains the necessity for finite leaf area production at outbreak, and it explains why foliage density reaches a constant value during a growing season, while also non-leaf tissue also continues to grow. It also explains why trees will die when their carbon stores are depleted below a certain threshold, because the cost of foliage growth and maintenance exceeds the dynamic supply of carbon by photosynthesis.

In this paper, we set out to illustrate and discuss how mathematical modelling could and should be applied to aid our understanding of plants and, in particular, plant-soil interactions. Our aim is to persuade members of both the biological and mathematical communities of the need to collaborate in developing quantitative mechanistic models. We believe that such models will lead to a more profound understanding of the fundamental science of plants and may help us with managing real-world problems such as food shortages and global warming. We start the paper by reviewing mathematical models that have been developed to describe nutrient and water uptake by a single root. We discuss briefly the mathematical techniques involved in analysing these models and present some of the analytical results of these models. Then, we describe how the information gained from the single-root scale models can be translated to root system and field scales. We discuss the advantages and disadvantages of different mathematical approaches and make a case that mechanistic rather than phenomenological models will in the end be more trustworthy. We also discuss the need for a considerable amount of effort on the fundamental mathematics of upscaling and homogenization methods specialized for branched networks such as roots. Finally, we discuss different future avenues of research and how we believe these should be approached so that in the long term it will be possible to develop a valid, quantitative whole-plant model.

In this paper, we present a model that explains the prepatterning of lymphatic vessel morphology in collagen gels. This model is derived using the theory of two phase rubber material due to Flory and coworkers and it consists of two coupled fourth order partial differential equations describing the evolution of the collagen volume fraction, and the evolution of the proton concentration in a collagen implant; as described in experiments of Boardman and Swartz (Circ. Res. 92, 801-808, 2003). Using linear stability analysis, we find that above a critical level of proton concentration, spatial patterns form due to small perturbations in the initially uniform steady state. Using a long wavelength reduction, we can reduce the two coupled partial differential equations to one fourth order equation that is very similar to the Cahn-Hilliard equation; however, it has more complex nonlinearities and degeneracies. We present the results of numerical simulations and discuss the biological implications of our model.

The distribution of chemotherapeutics in solid tumours is poorly understood and the contribution it makes to treatment failure is unknown. Novel approaches are required to understand how the three-dimensional organisation of cancer cells in solid tumours affects drug availability. Since convective drug transport is limited by increased interstitial pressure in poorly vascularised cancers, the aim of this study was to measure the diffusive hindrance exerted by solid tumour tissue. Multicell layer tumour models comprising DLD1 colon cancer cells were characterised and fluxes were determined for [3H]-vinblastine and [14C]-sucrose. The mathematical models provided the diffusion coefficients for both compounds and predicted higher exposure of cells in the vicinity of vessels. The diffusion of vinblastine was three times slower than that of sucrose. Although slow diffusion delays vinblastine penetration into the avascular regions of tumours, the proliferating cells are generally in the marginal area of tumours. The mathematical model that we have developed enabled accurate quantification of drug pharmacokinetic behaviour, in particular, the diffusivity of vinblastine within solid tissue. This mathematical model may be adapted readily to incorporate the influence of factors mediating pharmacokinetic drug resistance.

In this paper, we investigate the role of arbuscular mycorrhizal fungi in plant phosphorus nutrition. We develop a mathematical model which quantitatively assesses the contribution of external fungal hyphae to plant phosphate uptake. We derive an equation for solute uptake by a growing fungal mycelium which we couple with a model for root uptake. We analyse the model using nondimensionalization and numerical simulations. Simulations predict that removal of phosphate from soil is dominated by hyphal uptake as opposed to root uptake. Model analysis shows that the depletion zones around hyphae overlap within 8 h and that the transfer between fungus and root is a critical step for the behaviour of phosphorus within the mycelial phase. We also show that the volume fraction of mycelium is negligibly small in comparison to other soil phases. This is the first model to quantify the contribution of mycorrhizal fungi to plant phosphate uptake. A full data set for model parametrization and validation is not currently available. Therefore, more complete sets of experimental measurements are necessary to make this model more applicable.