Published online 29 October 2007
Published in Soil Sci Soc Am J 71:1807-1812 (2007)
DOI: 10.2136/sssaj2006.0113
© 2007 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
SOIL PHYSICS
ModelingThree-Dimensional Microstructure in Heterogeneous Media
J. M. Blaira,
R. E. Falconera,
A. C. Milneb,
I.M. Younga and
J. W. Crawforda,*
a SIMBIOS Center, Univ. of Abertay, Dundee, DD1 1HG UK
b School of Computing and Creative Technologies, Univ. of Abertay, Dundee DD1 1HG UK
* Corresponding author (j.crawford{at}simbios.ac.uk).
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ABSTRACT
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While it is now possible to image the three-dimensional structure of soil using high-resolution tomography, none of the techniques can simultaneously image the distribution of the resident soil microbes. This means that it is not possible to visualize soil microbes in their habitat. Consequently, the impact of soil structure on microbially mediated processes cannot be reliably modeled. Biological thin sections offer the opportunity to simultaneously image microbes in structure but are necessarily restricted to two dimensions. Therefore a methodology is required to simulate three-dimensional structures from two-dimensional thin sections of soil that is extendable to simulate the spatial distribution of a range of soil components. We developed a model that is capable of using data gathered from two-dimensional sections to predict the three-dimensional structure of soil. An object-oriented approach to modeling was used to allow the individual representation of each structure voxel. This allows the model to encapsulate both data, presented here, and the subsequent addition of components such as microbial distribution and related diffusion–respiration processes together in a three-dimensional lattice of voxels. The model was validated using data derived from three-dimensional x-ray tomography images of soil structure, and using two-dimensional sections through that data set to predict three-dimensional structure. A range of metrics was used to compare the modeled and imaged three-dimensional structures. The comparison shows that the metrics for the modeled structures agree with those derived from the three-dimensional images for higher porosities, but that systematic differences occur for the lowest porosity soils (<11%). This is due to problems relating to the prediction of rare events such as the presence of large connected pores in low-porosity samples.
Abbreviations: CT, computed tomography
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INTRODUCTION
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Soils are arguably the most complex biomaterial on the planet. Soil is characterized by spatial heterogeneity on all scales that impact on the nature and evolution of the physical and biological properties of the system (Crawford et al., 2005). At the pore scale, the physical heterogeneity promotes the coexistence of air and water, and by affecting the relative proportion of each is a major determinant of microbial activity and function (Crawford, 1994; Nunan et al., 2002; Renault et al., 2002; Vogel et al., 2002). Ultimately, it is this microstructure that governs the flow of resources through the pore space of the system and creates a high diversity of microenvironments (Young and Crawford, 2004; Zhang et al., 2005). Recently, it has been suggested that the role of microbial activity in the genesis of this structure and its impact on the resultant physical properties of soil results in a feedback between physical and biological processes that leads to self-organization (Young and Crawford, 2004; Feeney et al., 2006). Given the importance of the interaction between microbial and physical processes in soil, it is important that any conceptual framework investigating self-organization should include a pore-scale structure model that can incorporate the distribution of microbes.
Recent progress in computed tomography (CT) technology offers the possibility of imaging the three-dimensional structure of soil, nondestructively and at resolutions of relevance to studying the interaction between physical and biological processes in soil (Nunan et al., 2006). Unfortunately, current-generation scanners and image processing software cannot resolve the microbial cells, and indeed the challenges presented preclude this for some time in the future. Currently, the only way to simultaneously image and quantify the distribution of microbes and soil structure is through the use of biological thin sections (Nunan et al., 2003), but this is obviously restricted to two dimensions. To make progress, therefore, a methodology to extrapolate from two-dimensional spatial data to three dimensions is required.
We previously developed a methodology that simulates the three-dimensional architecture of porous rocks based on images derived from two-dimensional thin sections (Wu et al., 2006). In that study, the method was tested by comparing the permeability, simulated Hg intrusion data, and the topology of the pore space as determined by the Euler number (Vogel and Roth, 2001). The results for the rock samples were encouraging, but soil presents distinctive challenges for the methodology. Among the important differences with soil is the occurrence of coherent structures, e.g., pores, on a scale comparable to the sample, and the lack of a characteristic length scale for the inherent heterogeneities; therefore, structural metrics depend on the scale of measurement (Bartoli et al., 2005). Many probabilistic methods for modeling complex spatial structures, including our own, may fail when the structures have these properties simply because, by definition, large coherent structures are rare events and are notoriously difficult to predict when data is limiting (Kolar, 2004). For these reasons, the model of Wu et al. (2006) must be separately validated for soil.
The metrics used for the validation must relate to the functional consequences of any differences between modeled and real structures. Only then can the significance of any difference be meaningfully interpreted. Where we are interested in the interaction between physical and biological processes in soil, the metrics should relate to how the model captures the features relevant to defining the microenvironment of soil microbes. These features include those that describe the transport of O2, the distribution of moisture, and the available habitat for the microbes as well as the metrics associated with the scaling properties of the structure.
There are a number of metrics available for the characterization of soil physical structure. Vogel and Kretzschmar (1996) digitally reconstructed the pore space from sequential thin sections of soil and skeletonized the results to derive the Euler–Poincare characteristic as an index of the three-dimensional connectivity of the pore space. Vogel et al. (2002) used the same reconstruction technology to identify and describe the pore geometry using pore-volume density, pore-surface density, and the Euler number of the pore space. They showed, using a simulation of gas diffusion through the reconstructed pore space, that these metrics could be related to the effective diffusion coefficient of the structure. Using x-ray CT images of soil, Perret et al. (1999) characterized the pore network using metrics relating to tortuosity and connectivity, and showed that these could be related to convective fluid flow in soil.
The suites of metrics that have been applied to soil to date have not been designed specifically to characterize the interaction between physical and biological processes. Where microbes are imbedded in the soil matrix, acting as sinks for diffusive and convective substrate, metrics that characterize the solid matrix and pore space simultaneously are required together with those that characterize diffusive and convective fluxes. In addition, we need to quantify the scaling properties of the structures in soil in recognition of the fact that sample size must be factored into our interpretation of the images of structure.
The aim of this study was to use CT images of soil structure to validate a method for simulating the complex heterogeneity of the soil habitat as it impacts on the interaction between physical and biological processes. We propose the most relevant suite of metrics for the purpose. The model presented provides a theoretical framework to investigate the evolutionary ecology of the soil–microbe system.
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MATERIALS AND METHODS
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Computed Tomography Imaged Soil Structures
We have used images collected from the study reported in Feeney et al. (2006). In that study, a set of pot experiments was designed to investigate the potential for self-organization in the soil–microbe complex (Young and Crawford, 2004). Pots were filled with sandy loam soil (59% sand, 19% silt, and 7% clay, pH 5.6) and were either planted with a single Lolium perenne L. individual or left unplanted. The pots were incubated for 30 d and the soil was destructively sampled at eight time intervals during that period. Soil was sampled from the pots that did not contain plants (control), and from pots that contained plants at two locations: the bulk soil that did not contain roots and rhizosphere soil close to the plant roots. Samples comprised aggregates of approximately 2 mm in diameter and five of these aggregates from each of the treatments at 0 and 30 d were imaged using the Advanced Photon Source within the Argonne National Laboratory (Argonne, IL) in the GeoSoilEnviro Centre for Advanced Radiation Studies (Station 13-BM-D). The resolution of the images was 4.4 µm and a thresholding algorithm was applied to convert the gray-scale images into binary images, with black corresponding to the soil matrix and white corresponding to the pore space. To avoid artifacts arising from the complex boundary shapes of the aggregates, a central portion of the images was extracted that corresponded to a cube of soil with sides equal to 1.1 mm. In this study, we have used the CT images corresponding to the control soil at zero days (Sample 1), the rhizosphere soil at zero days (Sample 2), the bulk soil at 30 d (Sample 3), and the rhizosphere soil at 30 d (Sample 4). These samples were chosen because they span the full range of porosities observed in the experiment. Full details of the soils, experiments, and imaging protocols can be found in Feeney et al. (2006).
The Model
The method used to simulate the microstructure of the soil is described fully in Wu et al. (2006) and will be described again here. The method uses data in the form of two-dimensional digitized images of soil sections to predict the structure of a three-dimensional simulated volume of soil. The model has already been validated for a range of rock types in Wu et al. (2006) using morphological metrics. In this study, we tested the applicability of the method to soil, where we have both two- and three-dimensional images of the structure, and where we use metrics relevant to key soil processes.
The basis of the model is to use two-dimensional data to estimate the probability that a particular voxel (three-dimensional pixel element) is in a particular state, conditioned on the states of voxels in a local three-dimensional neighborhood. In principle, the state of a voxel could include whether it is pore or solid, and whether a microbe or cluster of microbes was present. Here, we focus on modeling structure alone, since the CT images do not capture microbes, and therefore the state of a voxel is defined to be either pore or solid. The method comprises a series of steps beginning with the derivation of the equivalent two-dimensional conditional probabilities for each of the orthogonal sections. These probabilities are calculated by successively applying the six-cell neighborhood shown in Fig. 1a
to each pair of "target" pixels (i,j and i,j+1) of the image of the two-dimensional section. For each combination of states of the target pixels, the number of instances of each of the observed combinations of states of the neighborhood cells is determined. The probability of realizing a specific state of the target pixels, given the state of the neighborhood pixels, is estimated by dividing the number of occurrences of the given neighborhood state that coincide with the specific target cell state, and divide that by the total number in instances of the given neighborhood state. This process is repeated for each of the three orthogonal images.
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Fig. 1. Neighborhoods for simulating structure: (a) six-cell neighborhood; (b) 15-cell neighborhood. Gray scale shows target cells used in algorithm.
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The simulation of the three-dimensional structure is initiated by first specifying the states of the cells in the bottom layer of the structure. This is done using the conditional probabilities derived for the horizontal section, as described in Wu et al. (2004). The states of the pixels along one edge of the layer are estimated by first randomly choosing the state of a corner pixel such that the probability that the pixel is a pore is given by the porosity. The state of the cell next to the corner cell is determined using the above method, applied to a two-cell neighborhood. Given the state of the corner cell, the state of the next cell is determined using the associated two-cell conditional probability. This is repeated until all pixels along the edge have been determined. The first pixel of the next row is determined using the two-cell conditional probability. The states of all other pixels in that row are determined using the six-cell neighborhood described above. This process is repeated until the states of all pixels in the first layer are determined.
Each further layer of the three-dimensional lattice is obtained by combining the six-cell neighborhoods from each orthogonal section, resulting in a localized neighborhood where the state of the target voxels depends on 15 neighborhood states, as shown in Fig. 1b. Each orthogonal section offers an estimate of the state of one of the target voxels. Thus, a means of combining these states to achieve the state of the voxel on the three-dimensional lattice is required. Since the orthogonal slices are not spatially referenced, there is no formal method for doing this. Wu et al. (2006) developed a weighting scheme for combining the states to achieve the voxel state on the three-dimensional lattice. The important element of this weighting scheme is an optimization step that determines the weighting factor by iteration until the porosity of the simulated three-dimensional structure, 
sim, is sufficiently close to the average porosity of the two-dimensional orthogonal sections,
2D, such that
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This algorithm is used to estimate the states of the voxels in all remaining layers, using modifications of the neighborhood to deal with voxels on the boundary in direct analogy with the description of the simulation of the first layer provided above, and detailed in Wu et al. (2006). Finally, the three-dimensional lattice of voxels is exposed to a smoothing algorithm that removes isolated voxels.
Functional Metrics Used to Quantify the Structure
To validate the model, comparisons between the simulated and CT-imaged structures were made using a number of metrics. These are: porosity (); the surface-connected porosity (sc); the percentage of surface-connected pore space (
= sc/); the frequency distribution of distances of matrix voxels to the nearest surface-connected pore; the pore size distribution; and the fractal dimension. Each of these metrics was chosen because it can be related to important physical and biological processes in soil.
Porosity is one of the most important metrics affecting physical transport rates. The porosity of the simulated structures is important in identifying the reliability of the model. Although the model is calibrated using the average porosity of the orthogonal two-dimensional sections, this does not guarantee that the porosity values corresponding to the simulated three-dimensional structures will correspond to those of the CT-imaged structures.
The surface-connected porosity of a structure is the fraction of the soil volume comprising pore space that is connected to the boundary. This is a useful measure, as it is the connected pore pathways that affect the transport of resources into the volume. The percentage of pore space that is surface connected () quantifies the partitioning of the pore space between isolated and boundary-connected pores. The computation of both these metrics requires that we define what constitutes a connection between adjacent voxels, and there are three ways this can be done. We denote the first of these neighborhoods as the "faces" neighborhood, and this assumes that a voxel is connected to another voxel only if they meet at any face. The second connection neighborhood is denoted as the "edges" neighborhood, and this assumes that a voxel is connected to another voxel if it is in the faces neighborhood or if they meet along any edge. The final connection neighborhood is denoted the "vertices" neighborhood, and this assumes a voxel is connected to another if it is in its edges neighborhood or if the voxels meet at any vertex. The sensitivity of the results to the choice of connection neighborhoods was investigated.
The distribution of solid voxels around surface-connected pores is a further metric used to quantify structures in this study. These distributions approximate the O2 distributions in soil if each matrix voxel has the same potential respiration rate (Rappoldt and Verhagen, 1999). The frequency distribution of voxel distances to the nearest connected pore was calculated for each structure sample, both CT and simulated, allowing comparisons to be made between the two sets of data.
The final metric we considered is the fractal dimension of the pore space, which is a measure of its clustering properties (Pachepsky et al., 2000). The method used here uses the box counting method to estimate the fractal (strictly the Hausdorff) dimension (Baveye et al., 1998). A log–log plot of the number of boxes required to cover the pattern against box size gives a straight line whose slope is the fractal dimension. While the fractal dimension gives an estimate of the clustering of the space, it does not give a measure of the density of the pores. Therefore, structures with many small pores could have the same fractal dimension as structures with a few larger pores. The intercept of the plotted line described above with the vertical axis gives a measure (called the lacunarity) of the density of the pore space, and a pattern with the same fractal dimension but with smaller pores would have a larger intercept (Mandelbrot, 1977). Therefore, both the slope and the height of the plot characterize the distribution.
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RESULTS
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Porosity
These porosity data (Table 1
) show that the modeled structures capture the variation of porosity found across the CT-imaged sets. Statistical t-tests were performed on the porosity results and three of the four simulated structures were not significantly different from the CT structures (P > 0.05). Although not significant, there is a general trend, however, to underestimate the porosity of the simulated structure compared with the CT-imaged data sets. Furthermore, Sample 1 (lowest porosity structure) has a P value of 0.026 (P < 0.05); therefore, we reject the null hypothesis that there is no difference between the data sets and conclude that the low-porosity simulated structure is significantly different from the CT-imaged sample. Figure 2
shows a three-dimensional visualization of the CT-imaged and simulated structures for Sample 1.
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Table 1. Averaged porosity metrics obtained for computed tomography imaged (CT) and simulated structures; standard error shown in parentheses.
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Fig. 2. Comparison of (a) computed-tomography-imaged and (b) simulated structure for one of the replicates of Sample 1.
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Surface-Connected Porosity
For three of the four samples, for each of the faces, edges, and vertices neighborhood schemes used to calculate the surface-connected porosity, the t-test results indicate no significant difference between the simulated and CT samples (P > 0.05). Statistical analysis of Sample 3, however, infers that the simulated and CT samples are significantly different for the faces and edges neighborhoods (P = 0.03 and 0.01, respectively) but not for the vertices neighborhood (P = 0.402). Furthermore, the sc metric calculated for the lower porosity simulated structures (Samples 1 and 2) vary more from the values for the corresponding CT-imaged structures.
Percentage of Surface-Connected Pore Space
These results are consistent with the surface-connected porosity results. For three of the four samples, for each of the faces, edges, and vertices neighborhood schemes used to calculate the percentage of surface-connected pore space, the t-test results indicate no significant difference between the simulated and CT samples (P > 0.05). Statistical analysis of Sample 3, however, infers that the simulated and CT samples are significantly different for the faces neighborhood (P = 0.03) but not for the edges and vertices neighborhoods (P = 0.06 and 0.15, respectively). Again, there is a general trend for the simulated percentage of surface-connected pore space to be underestimated. As expected, the connection neighborhoods that offer each voxel a larger number of possible connected neighbors result in higher values for the sc and metrics.
Frequency Distributions
For each CT-imaged sample, the frequency distribution of the distance to the nearest connected pore for each voxel was calculated. The frequency distributions obtained from the simulated structures follow similar qualitative trends to the CT-imaged structures. It was found that high-porosity structures (Samples 3 and 4) resulted in maximum distances to the nearest connected pore of approximately half the maximum distances found in lower porosity structures (Samples 1 and 2). It can be seen that there are a greater number of voxels with distances of <135 µm to a connected pore in the high-porosity samples than in the low-porosity samples (Fig. 3
). This is a consequence of the higher surface-connected porosities of these structures.
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Fig. 3. Frequency distributions of voxel distances to nearest connected pore for Samples 1 and 4 (S1 and S4, respectively) for computed-tomography-imaged data (CT) and simulated data (Sim).
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The agreement between the distributions for the simulated and CT structures is better for short distances than it is for larger distances; this is evident from Fig. 3, as the simulated and CT-derived distributions overlap for small distances but diverge for larger ones. There is no general trend toward over- or underestimation as porosity increases. Larger distance measures will be sensitive to the simulated distribution of smaller pores and there is evidence from the next set of results that the simulation is weakest in the prediction of the small-scale pore space.
Fractal Dimension and Pore Size Distributions
Two further measures of the soil structure were calculated to aid in analysis of the structures: the fractal dimension and the pore size distribution. Both of these measures give an indication of the clustering properties of the pore space. The fractal dimensions obtained for the simulated and CT-imaged structures are not significantly different (P > 0.05), as can be seen from Table 2
. Figure 4
illustrates the results of the box counting method for CT-imaged and simulated structures with a low and high porosity. The slope of the lines characterizing the fractal dimension for CT-imaged and simulated Sample 1 are 2.69 and 2.84, respectively. The fractal dimension for the CT-imaged and simulated Sample 4 are 2.79 and 2.80, respectively.
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Table 2. Comparison of averaged fractal dimension for computed tomography imaged (CT) and simulated structures; standard error shown in parentheses.
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Fig. 4. Example of fractal dimension for Samples 1 and 4 (S1 and S4, respectively) for computed-tomography-imaged data (CT) and simulated data (Sim).
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The pore size distribution for the simulated and CT-imaged structures is shown in Fig. 5
. The distribution for the simulated structure for the higher porosity case (Sample 4) is closer to that of the corresponding CT-imaged structure than in the lower porosity case (Sample 1).
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Fig. 5. Example of pore size distributions for Samples 1 and 4 (S1 and S4, respectively) for computed-tomography-imaged data (CT) and simulated data (Sim).
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DISCUSSION
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The results presented show that the model produces simulated structures that capture the variation in structure found across the real CT-imaged structures. The analysis performed on the CT-imaged data samples and the simulated structures show a distinct split between high-porosity (>11%) and low-porosity (<11%) treatments.
The simulations performed well for the high-porosity samples (>11%); indeed, for these samples there is excellent agreement between the simulated and CT structures, as quantified by the metrics used in this study. There is, however, a tendency for the , sc, and metrics to be underestimated, although this was not significant.
For extremely low-porosity values, 
6%, the structures were not so good at replicating the bulk porosity. Furthermore, unless the vertices neighborhood was used in the calculation of the sc and metrics, the simulated structures were not similar to the CT structures for 8.1%. Furthermore, for 8.1, the simulated pore size distributions varied greatly from the CT pore size distributions. The most likely reason for these discrepancies is the inability of the method to simulate rare events, particularly the appearance of large-scale correlated structure in low-porosity media.
Our analysis showed that, in general, the higher porosity structures had more surface-connected pathways than lower porosity sets (Table 1). Furthermore, the high-porosity sets show greater numbers of soil voxels concentrated around surface-connected pores than in lower porosity structures, with the maximum distance to a connected pore in high-porosity structure sets being approximately half the maximum distance found in lower porosity sets (Fig. 3). As this distribution relates to the distribution of O2 in the soil matrix, the correspondence between the distributions from the simulated and CT structures, particularly for smaller distances, suggests that the model should be able to capture the functionally important characteristics of high-porosity structures.
The use of a detection neighborhood for computing the various porosity measures was investigated and, as expected, the larger neighborhoods resulted in an increase in the sc and measures, resulting in convergence of agreement between the simulated and CT structures. This outcome is due to the larger neighborhoods providing more possible connections between voxels, therefore compensating for the general underestimation of the metrics. It can be noted from Table 1 that the different detection neighborhoods used for computing the sc metric on CT-imaged and simulated structures showed little change between the faces and edges algorithms and the largest effect was between the edges and vertices algorithms for both simulated and CT structures.
The fractal dimension measurements show good agreement between CT and simulated structures for all porosities. The gradients and heights of the lines of the log–log plots show close agreement between CT-imaged and simulated structures; therefore, the model can replicate the clustering properties of the CT-imaged structure. As the porosity increases, so does the fractal dimension. Overall, there is a general trend across all metrics for higher porosity simulations to show better correlation with the CT-imaged structures than lower porosity sets.
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CONCLUSIONS
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The work presented here has used CT-imaged soil structures to validate a method of simulating the three-dimensional soil habitat. It has been shown that the model adequately captures the pore architecture of structures where the porosity is >11%. Below this value, the simulated structures capture the main porosity values but not the detailed characteristics of porosity, such as connectivity and clustering properties as characterized by the fractal dimension. The reason for this failure is due to the inability of the model to simulate the rare events, such as large pores in low-porosity structures. The fact that we have focused here on small samples at high resolution means that the porosities are substantially smaller than normally found in soil at larger scales. Our results suggest that the model would work well at these scales and porosities.
Nevertheless, the model has been found to simulate the general trends of the soil microstructure found across different porosity classes. The simulated structures provide the three-dimensional-habitat space in which a theoretical framework can be built to study the evolutionary ecology of the soil–microbe system.
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ACKNOWLEDGMENTS
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We gratefully acknowledge Dr. Debbie Feeney, Dr. Naoise Nunan, Prof. Karl Ritz, Dr. Mark Rivers, and Dr. Kejian Wu for helpful discussions and technical support during this project. We also thank the Biotechnology and Biological Sciences Research Council (BBSRC) for the provision of a Strategic Studentship to J.M. Blair.
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NOTES
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All rights reserved. No part of this periodical may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Permission for printing and for reprinting the material contained herein has been obtained by the publisher.
Received for publication March 13, 2006.
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TOP
ABSTRACT
INTRODUCTION
