An image‐based finite difference model for simulating restricted diffusion

Abstract

Water diffusion in tissues is generally restricted and often anisotropic. Neural tissue is of particular interest, since it is well known that injury alters diffusion in a characteristic manner. Both Monte Carlo simulations and approximate analytical models have previously been reported in attempts to predict water diffusion behavior in the central nervous system. These methods have relied on axonal models, which assume simple geometries (e.g., ellipsoids, cylinders, and square prisms) and ignore the thickness of the myelin sheath. The current work describes a method for generating models using synthetic images. The computations are based on a 3D finite difference (FD) approximation of the diffusion equation. The method was validated with known analytic solutions for diffusion in a cylindrical pore and in a hexagonal array of cylinders. Therefore, it is envisioned that, by exploiting histologic images of neuronal tissues as input model, current method allows investigating the water diffusion behavior inside biological tissues and potentially as- Diffusion-sensitized MRI provides a means of probing the architectural features of structures that are much smaller than a voxel. In the CNS the degree of diffusion anisotropy is sensitive to local differences in nerve fiber orientation (1-3). Various mechanisms have been proposed to explain changes in the apparent diffusion coefficient (ADC) during pathologic states. For example, in the study of cat brain, cellular swelling in the initial stages of ischemia may account for the decreased ADC since water protons originally in the faster-diffusing extracellular space migrate into the intracellular space (4) In addition, the cellular swelling reduces the extracellular space and, thus, increases the tortuosity of the diffusion paths (5). In the rat spinal cord diffusion was found to be anisotropic (6 -8). Spinal cord injury resulted in reduced diffusion anisotropy (9) which has also been attributed to axonal swelling or loss and inflammation (10,11). Changes in the myelin sheath may also affect the ADC. For example, the demyelination resulting from mechanical trauma (12) and the varying levels of remyelination during recovery (13) may cause changes in the permeability of the barrier between the intracellular and extracellular compartments. Furthermore, regenerating and sprouting axon fibers are typically tortuous, small, and unmyelinated (14). The effect of these tissue parameters on the ADC may potentially be exploited to assess the extent of neural damage and repair in vivo. Reduced diffusion anisotropy in multiple sclerosis lesions have also been attributed to loss of axons and myelin (15,16). Diffusion-weighted imaging may thus potentially provide insight into the pathophysiologic changes in MS lesions if the relationship between morphology and the diffusion-sensitized MR signal are better understood.Previous attempts to predict the effects of morphologic changes have relied on Monte Carlo (MC) simulations (11,17) or approximate analytic expressions (18). Exact solutions...

Biexponential diffusion attenuation in the rat spinal cord: Computer simulations based on anatomic images of axonal architecture

“…The method was previously validated for restricted diffusion bounded by parallel planes and diffusion within a sphere, where the numerical results essentially replicated existing analytical solutions (11), and a detailed description of the method is the subject of another article by some of the authors of the present work. Since the method has been presented in abstract form only, a brief summary is given below.…”

“…Pixels (0.16 ϫ 0.16 m 2 pixel size; 256 ϫ 256 matrix) were then assigned to ECS, ICS (intra-axonal), or myelin ( Fig. 2b and e) using a segmentation algorithm reported previously (11). To decrease computation time and computer memory requirements, the segmented images were resampled to 64 ϫ 64 matrix (0.64 ϫ 0.64 m 2 pixel size; Fig.…”

MR connectomics: Principles and challenges

“…Therefore our ability to evaluate the quality of the tracing just by comparing the shape of reconstructed tracts with our a-priori knowledge is only limited to a crude analysis of a dozen major fiber bundles (Hagmann et al, 2003;Mori et al, 2002). The approach consisting of creating synthetic diffusion models for validation is useful in the development phase to characterize the behavior of a given algorithm, but in our view not adequate to predict the performance in biological tissue (Alexander, 2008;Assaf and Basser, 2005;Hwang et al, 2003;Szafer et al, 1995). Chemical tracing certainly has an important role to play in animal models where some species have been explored relatively extensively-though incompletely and not homogenously (Schmahmann and Pandya, 2006).…”

Indirect measurement of regional axon diameter in excised mouse spinal cord with q-space imaging: Simulation and experimental studies

“…Q-space simulations were performed by numerically solving the diffusion equation using a 3D finite-difference model developed by Hwang et al (Hwang et al, 2003) for a pulsed-gradient spin-echo (PGSE) sequence. This method had previously been validated for restricted diffusion bounded by cylindrical pores and diffusion within hexagonal array of cylinders, where the simulated data agreed well with existing analytical solutions (Hwang et al, 2003).…”

“…A custom built 50 T/m z-gradient coil (Wright et al, 2007) enabled QSI imaging experiments under optimal conditions allowing sub-millisecond diffusion gradient durations in order to fulfill the SGPA and, for the first time, considering that axon diameter is on the order of or less than one micrometer, sub-micrometer displacement resolution. Additionally, using a diffusion simulation program developed previously (Hwang et al, 2003), we investigated the effects of axon shape and diameter distribution, and the presence of ECS and ICS signal on QSI with ellipsoidal and circular axon models and histologic images from the specimens. Finally, with commercial gradient amplitude limitations in mind, we also experimentally investigated the errors in quantifying axonal architecture arising from failure to fulfill the SGPA and from low displacement profile resolution.…”