Investigation of influence of packing density of axons on diffusivity
Diffusion-weighted MRI is sensitive to biological tissue architecture on a micrometer scale. Determining whether it is possible to infer the specific mechanisms that underlie changes in DW-MRI could lead to new diffusion contrasts specific to particular brain white-matter degeneration processes. We have developed a renormalization group method in order to explore the effects of a large range of microparameters on apparent diffusion and have applied it to different kinds of biological tissue tessellations. Our approach takes the influence of disorder into the consideration and it allows quantitative investigation of the sensitivity of apparent diffusion to the variations of the dominant set of microparameters.
The signal in diffusion-weighted (DW) magnetic-resonance imaging (MRI) experiments is exquisitely sensitive to water molecular dynamics in the local geometrical and physiological environment. Determining whether it is possible to infer the specific mechanisms that underlie changes in the DW-MRI signal is an intense area of investigation and could lead to new modelling approaches for generating DW-MRI contrasts that are specific to particular brain white matter degeneration processes. However, the complexity of the diffusion behaviour due to compartmentalization, exchange, restrictions and anisotropy imposed by cellular microstructure, hinders the establishment of relations between dynamics and structure in a quantitative manner.
In recent years, increasing efforts have been made to interpret diffusion in brain tissue using geometrical models [1–3]. In our work, the renormalization group (RG) method [4, 5] of an enhanced Basser-Sen (BS) model  was performed taking into account possible different packing densities of white matter axons. Various tessellations were modelled using symmetry properties of the Wigner-Seitz (WS) [6–9] cell with a random probability of occupation.
Diffusion in brain white matter was modelled in a cross-sectional raster filled with infinitely long, parallel-aligned cylinders representing myelinated axons immersed in an extracellular matrix (Fig.1a). Following the BS model, we characterize axons by Dm (myelin-sheath diffusion), Da (axon diffusion) and corresponding proton densities, cm and ca, which have outer, Rm, and inner, Ra, radius. Such a compartment is immersed in an extra-cellular space with diffusion De, proton density Ce and linear size L. The properties of all components are color coded as in Fig.1b. If we suppose that white matter structure can be tessellated by WS cells with different symmetries, for example, square (sq, Fig.1a) and hexagonal (hex, Fig.1a) then it is possible to calculate the diffusive properties using the BS model . The two types of WS cells can be randomly distributed on the lattice with a probability p that WS cell is empty and (1 – p) that WS is occupied with fibers. In Fig.2a,b, a black WS cell indicates fiber occupation and the overlaid yellow lattice is pristine one. Vertices of this dual lattice are assigned to the random WS cells.
Random occupation of the WS cells is the opposite of the ordered spatial distribution of fibers described in the BS model . On the square and hexagonal lattices with randomly occupied cells, it is possible to outline a RG unit [4, 5]. All non-degenerative configurations of black and white WS cells for such unit are presented in Fig.3a for square tessellation and in Fig.3b for hexagonal tessellation. In Fig.4a,b the process of scale renormalization is depicted for the specific distribution of WS cells. Mathematically, such RG process can be described by a system of nonlinear equations (Eq.1). In the Table 1, equations for probability rnk and their degeneracy numbers for classes of renormalization groups (Fig.3a,b) are presented. These equations comprise Eq.1a describing the RG process for an extra-cellular region. Eqs.(1c,d) were derived according to the rules given in the last column of Table1 and probability density (Eq.1b, where δ(x) is a Dirac delta-function) [4–9].
We input the microscopic parameters, X, taken from Table 2 into the BS model to estimate lower (L superscript in notation, black square or hexagon in Fig.3a,b) and upper (U superscript in notation, white square or hexagon in Fig.3a,b) bounds of transverse diffusivity (t subscript in notation). The packing density of axons was taken into account. Then we were solving Eq.1 for different values of the extra-cellular volume fraction p. During the RG process, effective diffusion approaches the stable point DUt, n→∞ = DtL, n→∞ = Dtn→∞ which depends on p and tissue tessellation type (Fig.5, red and black colours represent square and hexagonal packing). The first iteration step i gives the classical results from the BS model. It is clear that as in classical models, as in enhanced models, the packing density of axons influences the effective transversal diffusivity. In Fig.5 for both packing density cases we observe large linear fluctuations at the beginning of iterations (I) which approach stable point (III) via regime of nonlinearity (III).
In Fig.6 the sensitivity S (Eq.3) of ADCeff (Eq.2) to the various X microparameters changes in the biologically relevant limit p ~ 0.2 are presented. For comparison, we give the classical BS results. We denote X1 as extracellular volume fraction, X2 extracellular diffusion, X3 myelin sheath diffusion, X4 axon diffusion, X5 mean axon size, X6 mean myelin sheath size, X7 extracellular proton density, X8 myelin sheath proton density, X9 axon proton density.
DISCUSSION AND CONCLUSIONS
Using RG method, we have calculated the sensitivity of ADCeff to the different microparameters variations and packing densities of axons. We found that ADCeff exhibits it’s the strongest sensitivity to the extra-cellular volume fraction and diffusivity for all types of axon packing density. These findings suggest a possible mechanism to explain ADCeff changes during neurodegenerative disease progression. The ADCeff demonstrates more nonlinear behaviour vs p changes due to blocking effects which are absent in an ordered model of the brain white matter.
The method illustrated in the article can be applied to other tessellations in two (2D) and three (3D) Eucleadean dimensions. The results for the 2D square and 3D cubic tessellations are presented below (Fig.1Aa). The development for both these cases follows the procedure outlined in section "Methods". Hence specificity of results is mentioned along with points of differences between lattices. Because of additional dimension, 3D case is characterized with a large number of configurations and it is not possible to illustrate all of them as in Fig.3. Fig.1Ab is a plot of convergence of diffusion coefficient for different spatial dimensions. We observe significant difference between 2D and 3D for p ~ 0.35. It can be explained different percolation thresholds of clusters in 2D and 3D. Lower concentration of components generates less cross-links and approaches to 1D curled and branched chain. However, the probability of path of connecting opposite sides of WS cell is high enough leading to the obvious differences in diffusivity in 2D and 3D. ■