SIAM News Blog

Image-based Modeling of Plant-soil Interactions

By Tiina Roose and Siul Ruiz

Despite its status as a generally underappreciated resource, soil is the largest terrestrial carbon pool in the world — bigger than plant and atmospheric pools combined. Its sustainable management is therefore essential for climate change mitigation, as humans must store increasingly more carbon in the soil and reduce carbon release from intensive agricultural practices. All human food is plant-based (including meat, since livestock eat grass and grains), and plant production depends on high-quality soil. Soil provides plants with 13 essential minerals and serves as a vessel that delivers water, carbon dioxide, and sunlight.

The interaction and integration of belowground and aboveground soil processes are ripe for mathematical modeling because they allow researchers to predict the way in which different soil management practices help to optimize crop production and environmental mitigation. However, the opacity of soil obscures these interactions. Recent efforts have started to overcome this obstacle by adopting X-ray imaging techniques. But X-ray computed tomography (XCT) requires another layer of mathematics and computation in order to turn images into models with predictive power. Ultimately, such models might potentially illuminate the amount of water and fertilizer that plants consume and investigate the way in which fertilizer timing and other factors can minimize the environmental impacts of commercial food production.

As the process of observing and modeling plant-soil interactions becomes more technologically intensive, it likewise becomes more challenging to highlight the different areas wherein pure and applied mathematics can make progress. Here, we discuss the pipeline of image-based modeling for plant-soil interactions and highlight specific places where mathematicians can make transformative contributions by accelerating the modeling processes. 

Image-based modeling begins with the acquisition of images. Standard XCT obtains approximately 3,000 X-ray adsorption projections while rotating the sample between the X-ray beam and detector; it then uses a computational filtered backprojection algorithm (i.e., Fourier slice theorem coupled with a Ram-Lak filter) to reconstruct the three-dimensional (3D) image of the sample. Obtaining 3,000 projections is fine for samples of composite materials, but doing so becomes problematic when imaging a live sample because the X-ray dose is likely to influence the biological behavior and properties. This consideration is critical when dealing with four-dimensional imaging (three spatial dimensions plus time). Scientists hence seek to minimize the number of projections that are required to detect the features of interest while also accounting for a priori knowledge about feature geometry — i.e., plant roots are cylindrical objects in a branching structure that is embedded in a soil matrix of granular material. We believe that an approach like compressed sensing—as done by Terence Tao—could potentially work in this scenario. Such a rigorous approach requires fundamental mathematical footing, however, so we encourage more pure and applied mathematicians to get involved in sampling and image reconstruction. 

Additionally, the reconstruction methods for X-ray diffraction maps that investigate 3D stress fields within soil are currently rudimentary. No standard computational methods exist because the nuances of powder diffraction imaging require a step change in the mathematical treatment of projected spectra and their reconstruction into a 3D image. Present-day methods are underpinned by the assumption of a two-dimensional (2D) plane stress field, but this postulation is unlikely to be a valid for highly dynamic plant-soil interaction problems. 3D diffraction reconstruction methods are therefore necessary.

Figure 1. Sample image and model for root hair phosphorus uptake. \(C_l\) is the nutrient’s concentration in the soil pore water, \(C_a\) is the amount of nutrient that is attached to the soil particle surfaces, \(k_d\) is the rate of nutrient desorption from the soil particle surfaces into the soil pore water, \(k_a\) is the rate of nutrient adsorption from the soil pore water to the soil particle surfaces, \(D\) is the nutrient diffusion coefficient in water, \(F_m\) and \(K_m\) are Michaelis-Menten constants that describes nutrient uptake by root surfaces, and \(C_\textrm{min}\) is the nutrient pore water concentration when root nutrient uptake ceases. Figure adapted from [3].

After obtaining the reconstructed images, an image-based modeler segments out the features of interest. XCT images are essentially grayscale 3D maps of voxels that demonstrates the density distribution within the sample; the denser material absorbs more X-rays and is often illustrated with light gray voxels, while darker voxels depict the less dense material. Therefore, detecting the boundaries between plant roots and soil water, plant roots and soil particles, and so forth—all of which can be similar in density—is the resulting challenge. Although the human eye can relatively easily differentiate between a cylindrical root and soil granular material that consists of soil minerals, organic matter, water, and clay particles, automation of this detection for consistent application to large numbers of images is notoriously difficult. Most plant root segmentation packages effectively step through 2D slices of the XCT stack and label portions of the image as “soil” and “roots” based on the grayscale density thresholding. But this technique rarely works automatically on all images and XCT settings, often requires major human intervention, and generally does not account for a priori geometry knowledge on the computational software side. While the approach is sometimes categorized as a form of artificial intelligence (AI), it relies heavily on the training of the AI tool and the skillset of the trainer (especially for 3D datasets); despite efforts over the last decade, no robust tools have yet emerged. We hope that mathematicians with new ideas on 3D image segmentation will make a transformative contribution to this field in the near future.

Once the image is segmented out from the XCT stack, the modeler can run models to learn more about the function of the imaged system. In the most conventional approach, one would import this segmented image geometry into a software package, mesh the geometry, choose the modeling equations for each subdomain, and then utilize a computational approach to solve the equations. These steps generally involve the use of commercial packages like SimpleWare for the meshing component and finite element packages like COMSOL Multiphysics to solve the model. This approach only works well when the study is exploratory and does not involve large numbers of 3D images, namely because it is very time consuming; most of these meshing and finite element packages were designed for simpler engineering geometries rather than heterogeneous and complex image-based modeling [1]. As such, all of these processes must be accelerated as the trend for image-based modeling moves toward higher numbers of images that allow for the assessment of functional heterogeneity. This direction is particularly important for biological investigations where some properties are not directly observable and thus require ensemble averages to infer function. 

Open-access fast approaches, such as OpenFOAM and OpenImpala, circumvent many of the aforementioned steps by running models directly on images. These models—which are based on finite difference, finite volume, and lattice Boltzmann methods—are applied directly to binarized images and require parallel programming for 3D calculations or some sort of graph simplification. High-throughput, image-based modeling is important in certain cases but often comes at with a corresponding time cost because staff must learn and implement these packages. How can one optimize the speed from image acquisition to solution in terms of staff and computational time? If only a small number of simulations is necessary, the optimal choice often involves employing existing packages that are easy to use but run more slowly, as the savings in staff time compensate for the calculation speed. Conversely, when many simulations must run in the same computational pipeline, the investment of staff time to develop computationally fast approaches is often desirable.

Postprocessing and visualization of 3D simulation results is a known challenge, since displaying 3D volume results in 2D formats (such as papers and presentations) is no easy task. While some individual institutions do have 3D augmented reality setups, they are not widely available and are certainly not utilized in academic publications. Use of the software packages in the XCT image analysis/segmentation step to analyze and visualize the simulation results offers one possible alternative. But this course of action requires access to all of these packages, and imaging and image analysis facilities are often separate from modeling and simulation facilities. Furthermore, 3D visualization software comes with costly hardware requirements. Imaging facilities often have custom machines that are high in RAM and GPU power, which is not standard for off-the-shelf computers. 

Animation 1. Illustration of the plant root and root hair imaging assay and imaging results from the synchrotron imaging experiments. Animation created by Sam Keyes and courtesy of Tiina Roose’s group.

We’ll conclude with an example that demonstrates image-based modeling’s contributions to new knowledge. Root hairs are unicellular extensions of root epidermal cells that are seemingly important for the uptake of all essential mineral nutrients, especially phosphorus. Since the hairs appear in a semi-regular grid on the root surface, the first attempt at modeling them used the multiscale homogenization theory [4]. This work resulted in three effective models to describe plant phosphorus uptake, depending on the rate of phosphorus uptake by individual root hairs. However, it was difficult for researchers to determine which model was correct because parameterization of the root hair geometry was always based on experiments that were conducted in Petri dishes rather than actual soil. Assuming that the root hair geometry would look different in soil is not unrealistic, since the presence of soil particles modifies the growth process.

To illustrate this effect (see Figure 1 and Animation 1), we grew plant roots in soil and used synchrotron XCT to image the root hair geometry with respect to soil particle position at a 0.7-micron resolution [3]. By performing this XCT observation and completing an image-based calculation that explicitly accounts for soil pore space and root hair geometry, we determined which homogenized model is most appropriate and quantified the effect of local versus global phosphorus transport from soil particle surfaces to root surfaces. Because these images were obtained in small soil volumes (XCT imaging necessity \(\varnothing\)5 millimeters), we later used matched asymptotic expansions to expand our image-based modeling efforts to include larger volumes of soil; doing so accounts for continuation processes from the image-based domain to the soil continuum domain [2]. 

Tiina Roose delivered an invited presentation on this research at the 2023 SIAM Conference on Mathematical & Computational Issues in the Geosciences, which took place in Bergen, Norway, in June 2023.

[1] Blunt, M.J. (2017). Multiphase flow in permeable media: A pore-scale perspective. Cambridge, U.K.: Cambridge University Press.
[2] Daly, K.R., Keyes, S.D., Masum, S., & Roose, T. (2016). Image-based modelling of nutrient movement in and around the rhizosphere. J. Exp. Bot., 67(4), 1059-1070.
[3] Keyes, S.D., Daly, K.R., Gostling, N.J., Jones, D.L., Talboys, P., Pinzer, B.R., … Roose, T. (2013). High resolution synchrotron imaging of wheat root hairs growing in soil and image based modelling of phosphate uptake. New Phytol., 198(4), 1023-1029.
[4] Leitner, D., Klepsch, S., Ptashnyk, M., Marchant, A., Kirk, G.J.D., Schnepf, A., & Roose, T. (2010). A dynamic model of nutrient uptake by root hairs. New Phytol., 185(3), 792-802.

Tiina Roose holds a Chair of Biological and Environmental Modelling and is deputy head of the School for Research within the School of Engineering at the University of Southampton in the U.K. She holds a Ph.D. in applied mathematics from the University of Oxford and completed postdoctoral appointments at Harvard University and then Oxford, after which she was appointed as a Royal Society University Research Fellow and then a professor at Southampton. 
Siul Ruiz is a newly appointed Royal Society University Research Fellow within the School of Engineering at the University of Southampton. He earned his Ph.D. in soil biophysics at ETH Zürich and is currently working on a wide range of problems that relate to plant-soil interaction modeling and imaging. 
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