Computational Model of Interstitial Transport in the Spinal Cord using Diffusion Tensor Imaging

Abstract

Local drug delivery methods, including convection-enhanced delivery (CED), are being used to increase distribution in selected regions of nervous tissue. There is a need for 3D models that predict spatial drug distribution within these tissues. A methodology was developed to process magnetic resonance microscopy (MRM) and diffusion tensor imaging (DTI) scans, segment gray and white matter regions, assign tissue transport properties, and model the interstitial transport of macromolecules. Fiber tract orientation was derived from DTI data and used to assign directional dependence of hydraulic conductivity, K, and tracer diffusivity, D _t , transport tensors. Porous media solutions for interstitial fluid pressure, velocity, and albumin distribution were solved using a finite volume method. To test this DTI-based methodology, a rat spinal cord transport model was developed to simulate CED into the dorsal white matter column. Predicted distribution results correspond well with small volume (∼1 μl) trends found experimentally, although albumin loss was greater at larger infusion volumes (>2 μl). Simulations were similar to those using fixed transport properties due to the bulk alignment of white matter fibers along the cord axis. These findings help to validate the DTI-based methodology which can be applied to modeling regions where fiber tract organization is more complex, e.g., the brain.

INTRODUCTION

With the development of promising therapeutic agents for chronic pain, spinal injury, and other neurodegenerative diseases, local drug delivery methods12,28,30,46,49 are increasingly being considered as a solution to overcoming transport barriers encountered by macromolecular, slow-diffusing drugs (e.g., nerve growth factor with an apparent diffusion coefficient in brain tissue of ∼2.8×10⁻⁷ cm²/s53). One such local drug delivery method is convection enhanced delivery (CED), an investigational surgical technique designed to bypass the blood-brain and blood-spinal cord barrier.12,30,31,46 By this method, a small diameter cannula is inserted directly in the tissue and the drug infusate is pumped out of the tip of the cannula and transported through the surrounding interstitial (extracellular) space by convection and diffusion. Previous infusion studies have shown CED into specific regions of the brain, spinal cord, and peripheral nerves to be reproducible and clinically safe.12,14,29 ^– 32,59 Macromolecular distributions were shown to be homogeneous and over larger volumes than could be attained by diffusion alone. Rational design of such regional therapy will require new tools to evaluate drug transport issues specific to nervous tissue physiology. In addition to anatomical boundary concerns, macromolecular tracer distribution can be significantly influenced by the site of delivery and flow of extracellular fluid along white matter tracts.1,12,31

Previous models of local interstitial transport have been developed for nervous tissue using porous media assumptions. Morrison et al. successfully modeled high-flow microinfusion of 180 kDa macromolecules at rates of 0.5–6.0 μl/min in homogenous brain tissue, e.g., gray matter.40 Kalyanasundaram et al. simulated the controlled release of drug from an implant in rabbit brain.24 Diffusion, convection (due to vasogenic edema), and metabolism and clearance of interleukin-2 were modeled using a two-dimensional finite element method (FEM) model formulated using a single image slice of the brain. While capturing characteristics of controlled release, the model does not capture out of plane boundary effects and white matter transport anisotropy. Porous media transport models have also been developed to study macromolecular transport through other biological tissues including solid tumors, blood vessels and cartilage.9,18,20,27,60

Within white matter regions, transport by diffusion and convection has been shown to be anisotropic.12,31,47,48 Studies show transport to be dependent on the orientation of the white matter tracts such that preferential transport occurs within the interstitial space parallel to axonal fibers. Analysis methods to correctly predict such transport behavior are required. Recently, we have presented a model for predicting the local tissue distribution of an injected macromolecule in the rat spinal cord following CED.51 We modeled the tissue as an anisotropic porous media assuming fixed, homogeneous transport properties, with preferential transport in the z-direction which was chosen to be parallel to the axis of the spinal cord. A 3D finite element model was created from a single high resolution magnetic resonance microscopy (MRM) scan of excised, fixed tissue. Use of fixed tissues allowed for the long scans necessary to obtain high resolution microstructural data. Finite element methods were used to calculate the in vivo interstitial fluid flow and transport of the macromolecular tracer albumin. (Albumin is an ideal macromolecule on which to base tracer studies because of its low binding and low reactivity.) Predicted transport was found to be sensitive to the hydraulic conductivity (or permeability) tensor, K, which is a measure of the conductance of fluid through the interstitium. Simulations were found to be weakly dependent on the absolute values of the hydraulic conductivity components, but sensitive to the inhomogeneity (K _gm vs. K _wm) and anisotropy (K _⊥ vs. K _|| ) of conductivity.50,52,59 Also, predicted distributions corresponded well with small volume distribution trends found in in vivo studies,59 although albumin loss across the pial membrane was greater at larger infusion volumes (>440 μl).

In the current study, we build upon this previous model and present a novel application of diffusion-tensor imaging (DTI) technology for non-invasive quantification of interstitial transport properties. DTI employs a pulsed-gradient spin echo to measure the effective tensor of water diffusion in the tissue, D _e , which is sensitive to the structure of the underlying tissues. DTI captures the multi-dimensional assessment of diffusion data caused by restricted motion of water due to interactions with tissue components, e.g., cell membranes, myelin, and macromolecules. Measured D _e in white matter is largely anisotropic5,19 and the direction of maximum diffusivity is proposed to be parallel to the local fiber orientation. In previous studies, D _e has been used to provide in vivo directional patterns for predicting fiber direction field mapping and fiber-tract trajectories.7,8,15,38,45 In the current study, we use D _e to assign directionality to the interstitial transport tensors of white matter: macromolecule tracer diffusivity, D _t , and hydraulic conductivity, K. It should be noted that D _e measures the volume-averaged water diffusivity, whereas D _t and K are extracellular transport properties. However, a strong relation between the eigenvectors (e.g., direction of maximum transport) of D _e and D _t and K tensors is inferred from the fact that extracellular and intracellular transport processes are related through boundary conditions imposed by the tissue microstructure. Such “cross-property” relations have been investigated previously by Tuchs et al. 58 By using an effective medium approach, Tuchs et al. estimated the statistical moments of the microstructure from D _e , and then derived the electrical conductivity tensor from the statistical moments. Through this derivation, they showed that D _e and a broad class of transport tensors, including the hydraulic conductivity, share the same eigenvectors. By using this result, we equate the eigenvectors of D _e to the eigenvectors of K and D _t . Transport tensor eigenvalues are estimated from previous albumin distribution studies.47,51,55 Previous models utilizing directional data encoded in DTI have been developed to predict sucrose diffusion in the brain36 and the electric field induced by an applied magnet field.37 In experimental studies, diffusion-weighted imaging (DWI) has been used to monitor transient changes in water diffusion following CED in tumors and brain tissues.34,35 In these studies, DWI was used to quantify the extent of cytotoxic tissue response, but not quantify the diffusion rate or direction. Applying DTI data to determine directionality of convection-dominated transport has not been previously studied.

In addition to incorporating DTI data, we also use multi-slice MRM data to build a fully 3D spinal cord model, as compared with our previous spinal cord model that used a single slice that was cloned (extruded along the axis). We used a computational fluid dynamic approach to predict tracer distribution volume and residence time for CED in the white matter dorsal column of the rat spinal cord. By focusing on the spinal cord, this study validates the DTI-based modeling method using a simple system of tissue anisotropy for which there is well-characterized bulk alignment of white matter fibers along the axis of the spinal cord. This selection allows for comparison with our previously developed computational model51 and experimental studies by Oldfield and coworkers.31,59 A flowchart of our modeling method is provided in Fig. 1. All together, the current study presents a DTI-based methodology that can be applied to building advanced, fully 3D transport models. The developed methods may be applied to regions with more complex anatomical boundaries and fiber organization, i.e. the brain.

FIGURE 1.

Flowchart of the modeling process used to predict interstitial distribution of a macromolecular tracer in the spinal cord. The 3D spinal cord model was constructed using MR and DTI data from excised tissues. The eigenvectors, v _i, derived from the DTI measured effective water diffusion tensor, D _e , were combined with eigenvalues (K _||, K _⊥, D _{t ||}, and D _{t ⊥}) estimated from previous experiments50 to assemble the hydraulic permeability and tracer diffusivity tensors, K and D _t . The validation process compares distribution with in vivo CED studies conducted previously.57

METHODS AND MODEL

Tissue Preparation

Excised, fixed rat spinal cord was used in MR scans. Rat surgery was conducted in accordance with the NIH guidelines on the use of animals in research and the regulations of the Animal Care and Use Committee of the University of Florida. Deep anesthesia was induced with pentobarbital anesthesia (80–120 mg/kg IP). The rat was then transcardial exsanguinated using 4% saline, 1 cc heparin and 1 cc sodium nitrite. This was followed by perfusion with 4% paraformaldehyde in phosphate buffered saline (PBS). The animal was stored in a refrigerator for 24 h before the spinal cord was removed and stored in the fixative solution of 4% paraformaldehyde in PBS. A 2.5 cm length of spinal cord, centered at vertebral levels L1-T13, was excised.

For MR measurements, the fixed spinal cord was first removed from the fixative solution then soaked in PBS for ∼12 h (overnight). Prior to MR imaging, the spinal cord was removed from the first PBS solution and placed in a 5 mm tube with fresh PBS. Air bubbles were removed from the solution by loading the spinal cord in the tube while submerging both the cord and tube in PBS solution.

MRM and DTI of Fixed Excised Spinal Cord

Use of excised, fixed tissue in MR scans allowed for the long scan times necessary to obtain high resolution microstructural information. Equivalent high resolution scans of the rat spinal cord would be difficult to obtain in vivo due to respiration movement, reduced magnet strength, and coil placement. High-resolution three-dimensional MRM and multiple-slice diffusion-weighted images (DWI) of the fixed excised spinal cord were measured in a 5 mm tube in a 14.1 Tesla magnet at 600 MHz at approximately 20°C. (When comparing with normal body temperature, eigenvalues of the effective diffusivity of water, D _e , are temperature dependent. However, we are interested in only the eigenvectors of D _e which should not be affected by temperature). The excised cord images were centered at vertebral levels L1-T13. The 3D MRM was measured in 2 h and 44 min using a gradient-echo pulse sequence with a repetition time (TR) of 150 ms, echo time (TE) of 10 ms and 4 averages (NA). The 3D image had a field-of-view (FOV) of 20×5.0×5.0 mm³ in a data matrix of 512×128×128, which resulted in a resolution of 39.1×39.1×39.1 μm³ per pixel. The multiple-slice DWI was measured in ∼11 h using a spin-echo pulse sequence with TR=1400 ms, TE=25 ms. The diffusion-weighting gradient pulses were 1.5 ms and separated by 17.5 ms. A total of 40 slices, with a thickness of 0.3 mm, were measured with an orientation transverse to the long-axis of the cord. These slices were imaged with a FOV=4.3×4.3 mm² in a matrix = 72×72, which resulted in resolution = 59.7×59.7×300 μm³. Each image slice was measured with two diffusion weightings: 0, 1250 s/mm². Images without diffusion-weighting (0 s/mm²) were measured with NA=28. The images with a diffusion-weighting of 1250 s/mm² were measured in 46 gradient-directions specified by the tessellations of an icosahedron on the hemisphere,17,57 with NA=7. As a point of reference, the signal-to-noise ratio (SNR) of the low-diffusion-weighted images is ∼160 in gray matter and ∼120 in white matter and the SNR of the high-diffusion-weighted images is ∼70 in gray matter and ∼40 in white matter. The distribution of the signal averages between diffusion weightings were selected to optimize the measured signal-to-noise ratio21,22 and the angular resolution of the measurements was extended to allow data analysis with diffusion tensors up to and including rank 842 ^– 44 so that more complex structures could be visualized.

Image Processing

Using custom software, written in the Interactive Data Language (Research Systems, Inc., Boulder, CO), images were generated from the raw data by Fourier transformation. Then the diffusion-weighted images were interpolated (bilinear interpolation, with nearest neighbor sampling) by a factor of two in each dimension. After initial image processing, the multiple-slice DTI data was fit to a rank-2 tensor model of diffusion using multiple-linear regression.6 In this calculation, all imaging and diffusion-weighting gradients were included in the calculation of the matrix of diffusion weighting, so that the effect of gradient cross-term were included in the calculation of the diffusion tensor. This resulted in an accurate accounting for the effect of all gradients and provided a precise rank-2 tensor of diffusion, D _e , at each image voxel from which eigenvalues and associated eigenvectors were derived. The diffusion tensor, average diffusivity, fractional anisotropy, and orientation images, from a representative slice of the three-dimensional image data set, are shown in Fig. 2.

FIGURE 2.

Diffusion tensor image of excised, fixed rat spinal cord from a volume image data set (72×72×40 pixels over 4.3×4.3×12 mm FOV obtained on a 14 T imaging magnet). In part A, the top row contains the tensor elements, D _xx , D _xy and D _xz , images in gray scale. The second row contains the the tensor elements, D _yy and D _yz , images in gray scale. The display intensity of the off-diagonal element images (D _xy , D _xz and D _yz ) have been increased by a factor of 5 to make these images more visible. The third row contains the image without diffusion weighting (S ₀), calculated from a fitting routine, along with the tensor element image D _zz . In part B from left to right, the first is the average diffusivity (AD) image, then the fractional anisotropy (FA) image (normalized RMS deviation of eigenvalues from a unit matrix), and finally an orientation-map image representing the direction of the maximum eigenvectors (EV), colored according to the eigenvector direction and scaled by multiplying the eigenvector direction cosine with the fractional anisotropy. In this last image, red represents the eigenvector direction perpendicular to the image plane, blue represents up-and-down and green represents left-to-right. Such color orientation-maps are more difficult to interpret in regions of complex fiber structure, such as the brain, where it is not easy to align the sample in the magnet so that the laboratory coordinate frame is coincident with the principal axes of the diffusion tensor.

Mesh Model

Stacks of MRM images were used as input. Gray-white matter boundaries were segmented semi-automatically by tissue density, and smooth surface contours were generated. From these contours, triangulated surface meshes of tissue structures were generated (Amira v. 2.3, TGS San Diego, CA) and converted to a parametric representation, non-uniform rational B-spline surfaces (NURBS) (Geomagic Studio, Research Triangle Park, NC). Compared with direct import of a triangulated surface mesh, this approach decreases computation and expedites importing and manipulating processes. Following surface import into meshing software (Gambit, Fluent, Lebanon, NH), gray and white matter volumes were formed and the volumes were meshed. In addition, a spherical infusion site corresponding to the outer diameter of a 33 gauge needle (radius = 0.102 mm) was incorporated into the white matter dorsal column. The epicenter of the site was 0.146 mm lateral to the dorsal median fissure located along the midsagittal plane, 0.401 mm posterior to the gray commissure (thin gray matter segment along midsagittal plane), and 0.737 mm anterior to the external surface of the spinal cord. The final mesh utilized 4-node tetrahedral elements (∼600,000 elements), Fig. 3B. The final mesh volume was ∼11.5 mm in length approximately covering vertebral regions L1-T13 of the spinal cord, Fig. 3A.

FIGURE 3.

(A) MRM-derived NURB surface geometry of rat spinal cord without nerve roots, length ∼11.5 mm; (B) transverse view through the tetrahedral spinal cord mesh (∼600,000 elements).

Interstitial Transport Theory

Nervous tissue can be modeled as a porous media within which there are two distinct regions of interest: fluid and solid. The fluid region corresponds to the unbound interstitial fluid surrounding cells, vasculature, and white matter fibers. The solid phase is made up of the extracellular matrix, cells, nerve fibers, and blood vessels. As in our previous model,51 interstitial transport analysis adapted rigid-pore transport models of Baxter and Jain9 and Morrison et al. 40 These studies make a rigid pore assumption that is valid for low endogenous flows and low rates of interstitial infusion where elastic expansion effects are not large.14,46 A rigid pore model is also a reasonable approximation provided that consolidation is relatively complete and transport parameters empirically describe the flow patterns associated with tissue deformation/swelling for a specific flow rate. (We calculated the characteristic time $t^* = a^2 /(H_{\rm A} K)$ for a poroelastic material, where a represents the characteristic distance and H _A the aggregate modulus (H _A=E(1−v)/[(1−2v)(1+v)]) (Holmes, Lai et al. 1985). Using mechanical parameters from previous poroelastic models of brain tissue,4,23,39,56 we calculated t ^* to range between 0.2 s to 13 min for a=1.0 mm. Thus, there is some uncertainty in the time to reach steady-state conditions. Implications of our rigid pore assumption are discussed in the discussion.) In addition, local sources and sinks of fluid can be neglected in the infusion models because tissues of the CNS lack an active lymphatic system,13 are characterized by low rates of fluid transfer across the capillary walls at the pressures encountered during interstitial infusion at a moderate flow rate,40 and have negligibly low rates of water formation by metabolism.²⁴ Transport in a rigid porous medium is described by the continuity relation and Darcy’s law,10,41

$$\nabla \cdot {\bf v} = 0$$

(1)

$${\bf v} = - {\bf K} \cdot \nabla p$$

(2)

where v is the tissue averaged interstitial fluid velocity, K is the hydraulic conductivity tensor, and p is the interstitial fluid pressure. K is dependent on the pore geometry (extracellular matrix) and fluid viscosity and is a measure of the conductance of the material to fluid flow.

FIGURE 4.

(A) Transverse view of the maximum eigenvectors of D _e (unit eigenvector corresponding to the largest eigenvalue) for a fixed rat spinal cord suspended in a fluid medium; (B) Axial and transverse views. All vectors have the same magnitude. Red coloring corresponds to the magnitude of the z-component. DTI scans used a spin-echo pulse sequence, 59.7×59.7×300 μm³ resolution.

Transport of a non-binding macromolecule is governed by convection and diffusion,

$$\phi \frac{{\partial c}}{{\partial t}} + \nabla \cdot ({\bf v}c) = \nabla \cdot (\phi {\bf D}_t \cdot \nabla c)$$

(3)

where c is the concentration volume averaged with respect to tissue volume, t is time, φ is porosity, and D _t is the macromolecule diffusivity tensor in the porous medium (a volume averaged term). For low infusion rates, dispersion may be neglected if little pore-level mixing and flow-induced enhancement of tracer spread is expected. Concentration was solved in terms of normalized units, $\tilde c = c/(c_i \phi )$ where c _i is the infusate concentration.

Interstitial Transport Tensors

In this study, gray matter hydraulic conductivity and diffusion tensors were set to be isotropic. Computational nodes in the white matter were assigned spatially varying, anisotropic transport tensor values which account for changes in fiber directions. A flow chart of the steps used to assign transport tensor values is given in Fig. 1. The directions of maximum interstitial transport correspond to the directions of maximum effective water diffusivity, Fig. 4. Hydraulic conductivity, K, and D _t tensors were assumed to share the same eigenvectors, v _i , which were calculated voxel by voxel from the DTI derived data, D _e , of the rat spinal cord tissue. The conductivity components assigned to each node in the computational model were determined using the relation,

$${\bf K} = {\bf V}\left[ {\begin{array}{*{20}c}{K_ \bot } & 0 & 0 \\0 & {K_ \bot } & 0 \\0 & 0 & {K_\parallel } \\\end{array}} \right]{\bf V}^{\rm T} \quad {\rm where}\;{\bf V} = [{\bf v}_1 {\bf v}_2 {\bf v}_3 ]$$

(4)

The hydraulic conductivity eigenvalues, K _⊥ and K _|| , were assumed to be transversely isotropic along the fiber tracts and were initially assigned based on curve fits to in vivo albumin distribution studies in the rat spinal cord.51 v ₃ is the unit eigenvector corresponding to the largest eigenvalue of D _e . Since there are only two unique eigenvalues, v ₁ and v ₂ are arbitrary unit eigenvectors in the plane orthogonal to v ₃. Simulations using this DTI-based K were compared with additional simulations assuming homogeneous tissue with a fixed K tensor and bulk alignment of white matter fibers,

$${\bf K}_{{\rm fixed}} = \left[ {\begin{array}{*{20}c}{K_ \bot } & 0 & 0 \\0 & {K_ \bot } & 0 \\0 & 0 & {K_\parallel } \\\end{array}} \right]$$

(5)

where the z-direction was set to be parallel to the axis of the cord (cranio-caudal axis).

The same relations as Eqs. (4) and (5) with different eigenvalues were used to assign the interstitial diffusion tensor of the macromolecular tracer, D _t and D _{t_fixed}. Tracer diffusivity eigenvalues, D _t⊥ and D _t||, were estimated by scaling measures of extracellular TMA+ diffusivity taken parallel and transverse to the white matter alignment previously measured by Sykova et al. 54 with extracellular albumin diffusivity measured by Tao and Nicholson in cortical brain slices,55 Table 1.

TABLE 1. Baseline simulation parameters.

Computational Model

Equations (1)–(3) for interstitial fluid flow, pressure, and tracer transport were solved for infusion into the white matter dorsal column, Fig. 3B. A weakly coupled problem was assumed, and transient concentration was solved with a fixed velocity field. Equations were solved with a computational fluid dynamics package (FLUENT, Fluent, Lebanon, NH) which allowed for flexibility in unstructured and large mesh geometries. Conservation of momentum for a fluid was simplified to Darcy’s law (Eq. (2)) by defining a momentum source term (−K ⁻¹ v) and reducing the viscosity term. (Convective acceleration was assumed negligible. Simulations removing the convective acceleration term, (v·∇)v, show negligible change in pressure and velocity profiles.) The albumin transport equation (Eq. (3)) was solved using a user-defined flux term accounting for diffusional anisotropy. Governing Eqs. (1)–(3) were discretized with a control-volume based technique. By using a segregated numerical method, discretized equations were linearized implicitly and solved sequentially using a point implicit (Gauss-Seidel) linear equation solver and the algebraic multigrid method. The SIMPLE (Semi-Implicit Method for Pressure-Linked Equations,2) pressure-velocity coupling method was chosen, and convergence criterion was set (1E-3 to 1E-5). Temporal terms were calculated with an implicit time integration scheme using fixed time steps. Initial conditions for albumin transport assume no initial tracer in the tissue, $\tilde c = 0$.

Simulation results were shown to be independent of the mesh. Negligible changes in interstitial velocity, pore pressure, and tracer distribution were found after increasing the mesh element number to ∼1.15 million. Simulations were conducted on a PC with Pentium IV 3.2 GHz processor and 1.0 GB RAM. Steady state velocity and pressure simulations took ∼30 min and transient tracer transport simulations (up to 2 μl) took ∼2 h.

Boundary Conditions

A constant infusion rate, 0.1 μl/min, was applied along the embedded spherical boundary (constant pressure boundary condition was iteratively applied until the desired flow rate was obtained, to establish the correct velocity distribution at the infusion site, see explanation in51). A zero fluid pressure condition, p=0, was applied along cut ends and along the external pial membrane surface (a fibrous membrane adherent to the external surface of the spinal cord). Concentration of albumin at the infusion site boundary was constant, $\tilde c = 1$. Albumin transport at the cut ends and across the pial membrane into the surrounding cerebrospinal fluid (CSF) was assumed to be dominated by convection with negligible loss due to diffusion $({\bf D}_{\rm t} \cdot \nabla \tilde c) \cdot {\bf n} = 0$ where n is the unit normal vector to the surface.

The pial membrane is not thought to hinder transport of fluid,11,25 but its role as a potential barrier to macromolecular transport is less certain. Boundary conditions with fluid flux (p=0) but various levels of albumin hindrance were conducted. The convection term in Eq. (3) has the general form,$\nabla \cdot {\bf v}c$. After integrating the convection term over a quadrilateral element and applying Gauss theorem, it turns out that for the two-dimensional case,

$$\int\limits_{\rlap{\hbox{--}} V} {\nabla \cdot \rho {\bf v}cd\rlap{\hbox{--}} V} = \oint\limits_S {(\rho {\bf v}c) \cdot d\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over S} } = \int\limits_{NS} {(\rho {\bf v}c) \cdot d\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over S} }\\ \quad+ \int\limits_{SS} {(\rho {\bf v}c) \cdot d\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over S} } + \int\limits_{ES} {(\rho {\bf v}c) \cdot d\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over S} } + \int\limits_{BS} {(\rho {\bf v}c) \cdot d\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over S} }$$

(6)

where ρ is fluid density, $\rlap{--} V$ is the volume of the element, $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over S}$ is the surface of the element, and NS, SS, ES, and BS are the four faces of the quadrilateral element. To introduce a partial barrier and limit the concentration flux across the boundary BS, we multiplied the last term in Eq. (6) with a partition coefficient, m (m=0 corresponds to total barrier and m=1 corresponds to no barrier).

Parameter Analysis

As in our previous model,51 the baseline value of gray matter hydraulic conductivity, K _gm, was taken to be substantially lower than that of white matter, Table 1. The ratio of the transverse components, K _⊥/K _gm, was 100. This value is consistent with previous observations that show gray matter spread to be approximately diffusion-limited. In convection dominated transport, white matter anisotropy arises from large values of K _|| /K _⊥, the ratio of the hydraulic conductivity parallel to the fiber tracts to the conductivity in the transverse direction. We have previously estimated K _|| /K _⊥ to be ∼20 when using a fixed K.51 In the current study, DTI-based K values with different K _|| /K _⊥ were used {14, 16, 20, 22, 24, 30 and 40}, Table 2.

TABLE 2. Simulation parameters for hydraulic conductivity parameter analysis.

Simulations were compared with experimental distribution measures including the volume of distribution, the percentage recovery of solute, and axial distribution length at each of the observed infusate volumes.⁵⁹ Axial length of the fixed spinal cord sample limited analysis to infusion volumes less than 2 μl. Axial distribution lengths were calculated using a threshold of ∼15% of the maximum concentration. Distributed volume was calculated as the integrated albumin concentration over the cord volume. The recovery ratio was calculated by dividing the distributed volume by the total albumin infused.

RESULTS

DTI

Large pulsed field gradients caused a diffusion weighting that was used to infer microscopic structural organization. High resolution DTI scans clearly delineated nerve roots and other small structures in the fixed, excised rat spinal cord, Fig. 2. Anisotropic diffusion coefficients of D _e are presented in Fig. 2; structural anisotropy is portrayed in the six unique images D _xx , D _yy , D _zz , D _xy , D _xz , and D _yz . The average diffusivity ([D _xx +D _yy +D _zz ]/3) of D _e is also shown. An orientation map image representing the direction of maximum eigenvectors is colored according to the eigenvector direction and scaled by multiplying the eigenvector direction cosine with the fractional anisotropy (normalized RMS deviation of eigenvalues from a unit matrix). Red represents the eigenvector direction perpendicular to the image plane. The concentrated red coloring in the white matter regions of the map highlights the strong bulk alignment along the axis of the cord in white matter regions. Some anisotropy in gray matter regions is also evident with left-to-right orientation (blue) across the cord in the gray commissure and anterior-posterior orientation (green) in the gray matter dorsal horn. However, gray matter tissues are not as anisotropic as white matter. Fractional anisotropy was greater in the white matter than in gray matter. Thus, our assumption that gray matter is isotropic compared to white matter generally holds. Average D _e values over selected regions of interest, Fig. 5, are presented in Table 3.

FIGURE 5.

Transverse view of fixed rat spinal cord suspended in a fluid medium (image without diffusion weighting at spinal cord level ∼L3). Anatomical regions are dorsal funiculus (DF), substantia gelatinosa (SG), dorsal horn (DG), and gray commissure (GC).

TABLE 3. Tabulated D _e data for the fixed rat spinal cord.

Interstitial Flow Analysis

The steady state solution for interstitial fluid flow was determined for 0.1 μl/min infusion into the spinal cord. Velocity and pressure profiles within the white matter dorsal column were solved using the DTI-based K tensor data set. The established flow and pressure fields for the K _|| /K _⊥=16 case are presented in Figs. 6 and 7. Predicted interstitial fluid pressure exhibited an approximately exponential decrease near the site of infusion, and decayed to the prescribed pressure boundary conditions near the boundary. Interstitial pressure decreased most rapidly in the transverse plane from the injection site to the pial membrane (horizontal line, Fig. 6). Pressure decay along the axial direction was more gradual. Interstitial velocity was predicted to decrease in an exponential manner within a given tissue region (i.e, white or gray matter region), Fig. 7B. Also, interstitial fluid flow was preferentially channeled along the axis of the cord with larger velocity components in the z-direction. In the transverse plane, the discontinuity in the pressure and velocity slope seen at the tissue boundary is due to the change in hydraulic conductivity in gray matter. The increased tissue resistance of gray matter limited flow by increasing the pressure gradient required to enter the tissue. As a result, interstitial velocities were reduced in gray matter. Transverse flow was skewed slightly towards the external pial boundary due to the lower CSF pressure boundary condition. Infused fluid escaped into the adjacent CSF through the pial membrane boundary, Fig. 7A.

FIGURE 6.

Transverse and axial profiles of pressure from the point of infusion in the white matter dorsal column (0.1 μl/min CED). Horizontal and vertical lines match lines indicated in the reference figure. The axial line is in the z-direction (out of the plane of the page). The gray-white matter interface corresponds to the vertical line only. K _|| /K _⊥=16.

FIGURE 7.

(A) Pathlines colored by interstitial pressure (dyne/cm²) predicted in the rat spinal cord model for 0.1 μl/min CED. (B) Velocity magnitude along transverse and axial lines intersecting the infusion site. Horizontal and vertical lines match lines indicated in the reference figure, Fig. 6. In Fig. 6, the axial line is out of the plane of the page. The gray-white matter interface corresponds to the vertical line only. K _|| /K _⊥=16.

Albumin Transport Analysis

Albumin transport solutions utilizing the DTI-based K are shown in Figs. 8 and 9. Preferential spread was along the direction of the structured white matter tracts. Distribution characteristics of Fig. 10 match predicted distribution patterns in the y-z plane, Fig. 8, with preferential transport along the axis of the cord, and limited penetration into adjacent gray matter regions.31,59 Albumin tissue distribution increased with continued infusion with axial spread matching that of the interstitial fluid velocity, Fig. 9B. There was also some loss across the pial membrane (outer) boundary, even at low infusion volumes. Predicted distributions also show flat concentration profiles with a rapid concentration drop at the advancing front (axial line in Fig. 9A). This profile reflects convection-dominated transport with modest dispersional or diffusional spread. The calculated Peclet number was as high as ∼1300 near the infusion site for the 0.1 μl/min infusion rate, and decreased proportionally with the velocity magnitude, Fig. 11. Convection-dominated regions extend significant distances from the point of infusion.

FIGURE 8.

Predicted albumin distribution through various planes of the spinal cord after 1.0 and 2.0 μl CED infusion (0.1 μl/min, K _|| /K _⊥=16, D _{t ||}/D _{t ⊥}=1.7).

FIGURE 9.

(A) Predicted albumin distribution along various lines originating at the injection site during 1.0 μl CED infusion, t=10 min. Horizontal and vertical lines match lines indicated in the reference figure, Fig. 6. In Fig. 6, the axial line is out of the plane of the page. The gray-white matter interface corresponds to the vertical line only. (K _|| /K _⊥=16, D _{t ||}/D _{t ⊥}=1.7). (B) Velocity of the concentration front along the axial line using a 15% concentration threshold. vc+ is along the positive z-direction, vc-along the negative z-direction, and vi is the interstitial velocity magnitude along the positive z-direction.

FIGURE 10.

Axial spread following CED of macromolecular tracer (¹⁴C-albumin) into the rat spinal cord (57).

FIGURE 11.

Peclet number contours on x- y- and z-planes. Pe=vl/D. v is velocity magnitude; l is typical length (=1 mm); and D is typical diffusivity coefficient (=1.67×10⁻⁷ cm²/s).

The axial distribution, distributed volume and the percentage recovery of albumin tracer were found to be comparable with the previously developed model and experimental measures, Fig. 12. A discrete parameter analysis of K _|| /K _⊥ over a range of values resulted in a best fit with Wood et al.’s distribution data for K _|| /K _⊥=16 (minimum chi squared value, χ ²=5.7) which is close to K _|| /K _⊥=20 estimated previously using an idealized rat spinal cord geometry and a fixed K tensor.51 Distribution results comparing fixed K with the DTI-based K (K _|| /K _⊥=16) were similar due to the bulk alignment of white matter tissue along the axis of the cord. Simulations using DTI-based transport properties predicted slightly larger tissue distributions. However, both approaches underestimate the distributed volume and percentage recovery (percentage of infused drug remaining in tissue) for larger infusions.

FIGURE 12.

Comparison of simulated albumin distribution results (varying K _|| /K _⊥ and boundary conditions) with experimental data measured by Wood et al. ⁵⁷ Error bars represent ±1 standard deviation. ‘k#’ corresponds to the ratio, K _|| /K _⊥=#. ‘K const’ corresponds to the simulation with fixed K tensor (Eq. (5), K _|| /K _⊥=16). ‘k16 (m=0.5)’ corresponds to the simulation with K _|| /K _⊥=16 and a barrier coefficient of m=0.5 acting on albumin flux at the pial boundary. (A) Axial (cranio-caudal) distribution; (B) distributed volume; (C) percentage recovery (percentage infused albumin remaining in tissue).

FIGURE 13.

Transverse albumin concentration profile for varying boundary conditions. The concentration profile is along the horizontal line indicated in Fig. 6. m is the partition coefficient (m=0 for a complete albumin barrier and m=1 for no barrier effect). There is no barrier to fluid flow at the pial membrane. (1 μl CED infusion, K _|| /K _⊥=16).

One option to increase the percentage recovery is to increase the K _|| /K _⊥ ratio. An effect of increasing this ratio is amplification of noise in the eigenvector field. Increasing K _|| /K _⊥ ratio to values up to 100 resulted in distribution patterns that were regular at the advancing front and that were symmetric about the point of infusion. These simulation results show relatively smooth contours which indicates that such effects are secondary. Corresponding increases in percentage recovery were modest; there was still significant flux of albumin into the adjacent CSF.

Introduction of a partial barrier to albumin at the pial membrane surface resulted in an accumulation of albumin at the membrane boundary. Increasing the barrier (m=0.2) resulted in concentrations reaching 5–6 times that of the infusion concentration without a barrier (m=1), Fig. 13. Partial barrier effects were localized to regions adjacent to the pial membrane. Concentration profiles along vertical and axial lines were little changed, Fig. 12A. Percentage recovery was seen to increase slightly compared with the free boundary flux case.

DISCUSSION

Advantages of using MRI and DTI data are the full three-dimensional capability, excellent soft tissue contrast, good spatial resolution (<1 mm), and potentially noninvasive nature. In this study, use of high resolution 3D MRM and DTI allowed for micron scale resolution of an excised rat spinal cord. DTI-derived directional data was incorporated into a 3D computational model that predicts in vivo transport within nervous tissue. Interstitial fluid flow and macromolecular transport in white and gray matter regions were solved. This model accounts for the next step in modeling transport along white matter tracts by providing a methodology for assigning directions of highest interstitial fluid flow and drug transport that change with position. We also provide a process for incorporating realistic 3D anatomical boundaries.

The spinal cord is an ideal location for validation studies because of its well-characterized bulk alignment of white matter fibers along its axis. Overall, albumin distribution predicted using the DTI-based K tensor approach was similar to our previous model that assumes K _fixed and idealized geometry.51 Thus, the DTI-based K is able to capture the bulk alignment of white matter in the spinal cord. Computational prediction was also compared with previously published experimental measures of CED in the dorsal column of the rat spinal cord by Oldfield and coworkers.31,59 Both models predict albumin distribution to follow observed trends with preferential transport along the axis of the cord and limited distribution in gray matter for small volume infusions. Model simulations using the DTI-based K estimate a K _|| /K _⊥∼16 that is was similar to that predicted using our previous model (K _|| /K _⊥∼2051). This ratio is an order of magnitude greater than the corresponding tracer diffusivity ratio. This may be explained by the fact that hydraulic conductivity in the direction parallel to channels increases by a higher power of the channel radius than does the corresponding diffusional transport.16 Small expansion of the extracellular space between white matter fibers such as would occur with moderate edema would explain the disproportionate increase in K _|| /K _⊥.

Experimental distribution studies show an approximately linear relationship between distribution volume and infusion volume.31,59 This suggests that little albumin was lost to the surrounding CSF, which is consistent with the relatively constant percentage recovery of label that was found. Greater loss of albumin (lower percentage recovery) was predicted by both computational modeling approaches (DTI-based K and fixed K) than measured by Wood et al. 59 This discrepancy may be due to a number of factors. In the following paragraphs, we focus on tissue volume, membrane barrier, and DTI effects. However, other differences between simulated conditions and experimental conditions, including interspecies variability, exact site of infusion, and infusion site geometry, may also explain differences of predicted outcome with the Wood et al. measures.

To attain high resolution, we used scans of excised, fixed tissues to create our 3D model. Limitations of this approach are associated with changes in tissue structure between excised and living tissue. We do not expect changes in fiber orientation, i.e, the eigenvectors of D _e . However, fixation may result in some tissue shrinkage and result in the 3D model having a smaller tissue volume than the living tissue in the Wood et al. study. Thus, the tracer has a shorter distance to travel before reaching the outer tissue boundary and lower percentage recovery is predicted. Also in living tissue, expansion of the extracellular space due to infusion (e.g. tissue swelling and/or edema) may account for transient changes that increase the tissue volume. Either of these tissue volume related effects (fixation shrinkage and/or infusion swelling) may contribute to a lower predicted percentage recovery.

Several studies have examined the transport of small molecules across the pial membrane.11,25,26 These studies have not revealed any quantitative support for the membrane acting as a barrier for transport across it, and suggest the unimpeded transport of water. However, transport of macromolecules may be hindered by the membrane, providing a means of increasing tracer retention in the tissue. Simulations implementing a macromolecular barrier show accumulation adjacent to the membrane. Experimental concentration profiles in the spinal cord do not reveal a significant accumulation of macromolecules near the boundary.31,59 Thus, model results do not support a filtration mechanism.

Simulation differences may also be due to factors associated with the DTI data set such as volume averaging at the CSF-spinal cord boundary. If maximum eigenvectors are not aligned with the exterior surface, fluid in the associated voxels will preferentially flow out of the spinal cord. However, small differences in percentage recovery between DTI-based K and K _fixed tensor (where maximum eigenvectors are aligned with the exterior surface) simulations suggest that these factors may not be significant. Noise in the DTI data set, which results in erroneous eigenvectors and orientation assignment, did not appear to be a significant factor. We did not encounter any convergence problems, and increasing K _|| /K _⊥ ratio up to a value of 100 resulted in relatively smooth contours.

In summary, pial membrane effects and DTI data effects do not appear to contribute to simulation differences in percentage recovery. Future study will focus on clarifying tissue volume effects, specifically the extent of tissue shrinkage during fixation and the extent of tissue expansion during infusion. Rather than a rigid media assumption, poroelastic or poroviscoelastic constitutive models may account for regional tissue swelling and edema. Poroelastic and biphasic models have been developed to study coupled fluid-solid interactions at the site of injection.3,4,14,23 By accounting for changing pore fraction, Gillies and coworkers predict pore fraction near the infusion site to increase with flow rate in an agarose gel-tissue substrate.14 Since increasing pore fraction may change drug or tracer distribution results, extension of these results to nervous tissue needs to be confirmed through experimental measure of poroelastic tissue properties, i.e., hydraulic conductivity and tissue modulus. Also, experimental infusion line pressures have been shown to exhibit some time dependent behavior.12 Transient changes in the tissue transport properties over the course of infusion may be accounted for using poroviscoelastic constitutive models. However, whether or not transient changes have a significant impact on final distribution volumes is unclear and needs to be further investigated. Efforts to quantify poroelastic and/or poroviscoelastic properties of nervous tissue have been hindered by the difficulty of measuring induced pressure gradients in vivo and the difficulty of maintaining the intact interstitial space in in vitro samples large enough for perfusion testing. Alternatively, increasing field strength or selecting a tissue region less susceptible to movement artifacts may allow for in vivo DTI and MRM scans. In vivo scans would allow for a direct measurement of transient tissue changes during infusion.

When applying this modeling approach to structures of the brain, loss of agents into the surrounding CSF may be less of an issue if the structure is not adjacent to a the exterior surface or ventricles. Further validation of the DTI-based methodology in more complex fiber structures, e.g., nerve root entry, brainstem, or corpus collosum, is required. Future work will focus on experimental validation of predicted spinal cord tracer distributions in regions of more complex fiber structure. MR scans have been used to measure the distribution of liposomes and Gd-albumin.31,33 MR measurement of the in vivo infusion of albumin bound to gadolinium will provide distribution and tissue geometry data sets. These studies will also quantify changes in tissue volume after fixation of tissues.

With further development and validation, the physiologically-based tracer transport modeling methodology developed in this study may eventually improve CED by providing tools to predict transient, 3D distribution for a given infusion site and flow rate in selected regions of the spinal cord or brain. Also, with the inclusion of reactions such as extracellular protein degradation and cellular binding and uptake,52 these models may eventually be applied directly to specific therapeutic agents (e.g., protein toxins, growth factors, genetic vectors, and radioimmunoconjugates). With the accessibility of in vivo DTI measures, the developed methods may eventually be used for patient-specific therapeutic treatment and computer-aided chemical surgery that is based on an individual’s medical imaging data.