A Fractional-Order Model for T2 Relaxation in Normal and Degraded Cartilage

Introduction: MRI is increasingly used as a sensitive, noninvasive diagnostic modality for OA, but suffers from lack of specificity for matrix degradation. The initial phases of OA are characterized by disruption and loss of collagen and proteoglycan, the two major matrix components of cartilage. While changes in monoexponential relaxation times (e.g. T1, T2, and T1rho) have been observed to accompany cartilage degradation (1,2), these parameters exhibit limited sensitivity to cartilage pathology and limited specificity for particular cartilage matrix components. Non-negative least squares (NNLS) fits have previously been used to quantify and characterize underlying relaxation components in cartilage (3,4) in an attempt to improve sensitivity and specificity, but require high SNR. Recently, we derived a fractional-order relaxation model through imposing a memory kernel in the integral form of the Bloch equations. This leads to a stretched-exponential (Str-Exp) function for transverse magnetization decay (5). This physically-motivated model captures important features of the relaxation through the fractional-order parameter α, which reflects matrix microstructure. We previously observed a correspondence between α and the concentration of pure cartilage matrix components in solution, suggesting its sensitivity to matrix composition. Here, we extend this analysis by applying Str-Exp fits to normal and enzymatically degraded cartilage, showing the sensitivity of the fractional-order parameter α to loss of matrix proteoglycan. We also compare the T2 distributions derived from the fractional-order model with T2 distributions derived using NNLS, which makes no a prior assumption regarding the number of underlying relaxation components; these comparisons demonstrate the ability of the fractional-order approach to capture features of the relaxation distribution through adjustment of a single parameter, α. In addition, we find that our StrExp analysis is more stable in the presence of noise as compared to NNLS analysis, supporting its potential for clinical MRI applications.