Statistical model for diffusion attenuated MR signal

Growing interest in diffusion MRI stems from numerous clinical and research applications (for example, see recent reviews in special issues of NMR in Biomedicine (1,2)). Most applications rely on a Stejskal-Tanner (3) pulsed gradient spin echo (PGSE) experiment and an assumption that the diffusion-attenuated MR signal can be expressed as a monoexponential function given by (3): S=S0exp(−b ADC). [1] Here ADC is the apparent diffusion coefficient for tissue-water or other MR active species under consideration; the so-called b-value depends on the shape of the diffusion-sensitizing gradient pulse waveform G(t) (4): b=γ2∫0tdt′[∫0t′dt″ G(t″)]2, [2] where γ is the magnetogyric (late gyromagnetic) ratio of the nuclide under consideration. For example (3), for the case of bipolar rectangular gradient pulses with amplitude G, duration δ and interval between pulse centers Δ: b=(γGδ)2(Δ−13δ). [3] However, numerous studies of the diffusion of water and/or other metabolites in brain tissue and other biological systems have documented a non-monoexponential behavior of the MR signal S as a function of the b-value at fixed diffusion times (e.g., see Refs. 5–17). Most authors report that their data can be fit well by a biexponential function with two different diffusion coefficients (large/fast and small/slow) and suggest ascribing the two exponential components to two physical compartments (extra-and intracellular) in a tissue. However, as noted by Le Bihan and van Zijl (18), the origin of fast- and slow-diffusion pools is “still mysterious.” Moreover, Sehy et al. (19) have directly observed biexponential diffusion MR signal behavior within the intracellular space of a single cell, the frog oocyte. Generally speaking, Eqs. [1] and [3] describe the diffusion-attenuated MR signal in a PGSE experiment only for unrestricted diffusion in homogeneous media. However, in most in vivo experiments each imaging or spectroscopic voxel contains numerous cells with different cell types, sizes, geometries, orientations, membrane permeabilities, and presumably different T2 and T1 relaxation time constants. Hence, the expectation that such a system can be described in terms of a simple unrestricted diffusion (monoexponential) model Eq. [1] is not reasonable. Practically any of the above-mentioned issues can lead to a deviation from monoexponential behavior—a point clearly illustrated in a recent article by Chin et al. (20). These authors carried out numerical simulations for a realistic geometrical structure of a rat spinal cord and found that under a variety of conditions the signal behavior is not monoexponential and can be generally described in terms of biexponential diffusion attenuation. Theoretical modeling of diffusion in multicompartment systems with cylindrical geometry by van der Weerd et al. (21) also demonstrated that “multi-exponential analysis of diffusion signal behavior cannot be freely related to the geometrical parameters of the system.” Moreover, for short diffusion times this laboratory has shown that biexponential signal behavior is predicted theoretically even for a single homogeneous compartment (22). A number of theoretical models based on the complex structure of biological objects have also been developed to describe non-monoexponential PGSE MR signal behavior (10,16, 23–27,32). While in some cases a biexponential behavior of an MR signal can obviously be related to the tissue compartmentalization or geometrical structure, in most cases it simply reflects the fact that a biexponential function describes signal behavior better than does a monoexponential function without a simple obvious relationship between the biexponential model’s parameters and the physical parameters of the system under investigation. It would seem reasonable to develop a well-defined physical model that recognizes the presence of variable length scales for restrictions and hindrances to water diffusion in biological systems. Such systems, which by their very nature represent highly complicated structures, present an architectural arrangement characterized by an extraordinary range of length scales. The MR diffusion experiment is sensitive to a subset of the full range of possible length scales, dependent on experimental parameters and the practical constraints of the measurement (SNR, etc.). The MR relevant range of length scales for restrictions and hindrances to water diffusion should be reflected in any model of MR diffusion data from biological systems. The critical issue is how best to do this. Below we propose a statistical method/model to reflect in a well-defined manner the consequences of a distribution of length scales for restrictions and hindrances to water diffusion (i.e., by a resulting diffusion coefficient probability distribution characterized by a mean value and a distribution width). This general phenomenological model can describe a rather large number of experimental results related to the structure of PGSE diffusion-attenuated MR signal in biological systems. We apply this model to describe MR data in a human brain and demonstrate good agreement with data obtained from practically all regions of brain for a b-value interval from zero to 2 ms/μm2.