Myelin [3, 4] is a fatty sheath, composed of a bi-layer of lipids, which surround axons in the human brain. The biochemical composition is approximately 70% lipid and 30% protein (myelin basic protein, proteolipid proteins) [ 5]. In the central nervous system (brain, cerebellum, and spinal cord) the myelin sheath are processes from oligodendroglial cells, and in the peripheral nervous system, from Schwann cells. Each oligodendrocyte can myelinate as many as 50 different axons [ 6]. Each oligodendrocyte process wraps around the axon multiple times. The sheath is interrupted, along the axon, by the Nodes of Ranvier, approximately 1 mm apart [ 6], and, therefore, the electrical conduction jumps from one Node of Ranvier to the next at approximately 100 m/s. This jumping (saltatory conduction) results in faster conductivity, relative to an unmyelinated axon, and requires less energy. To maintain the conduction velocity of a myelinated axon 20 microns in diameter, an unmyelinated axon would have to be several millimeters in diameter [ 3]. The thickness of the each myelin bilayer is approximately 0.1 µm [ 7] and each radial line in the myelin sheath is approximately 300 – 400 ˚A apart .
There are many diseases (e.g., multiple sclerosis, adrenoleukodystrophy [ 9, 10], and attention deﬁcit-hyperactivity disorder [ 11]) which affect the myelin, and therefore, the water trapped between the myelin bilayers. To assess the disease progression or drug effect, it is important to analyze the progression of demyelination in vivo. MRI is well suited for this and therefore, will continue to be an important factor in assessment, research and treatment of disease.
In 1978, Vasilescu et al. ((V. Vasilescu et al., “Water compartments in the myelinated nerve. III. Pulsed NMR result,” Cellular and Molecular Life Sciences 34, no. 11 (November 1, 1978): 1443-1444.)) published quantitative results of multi-exponential T2 decay curve analysis of frog sciatic nerve. They found three distinct T2 water compartments based on a “peeling-off ” mathematical procedure applied to multi-echo data. A slowly relaxing compartment, T2 ≈ 300 − 500 ms, attributed to water in the intra-cellular space. An intermediate relaxing component, T2 ≈ 80 ms ascribed to axoplasmic water. And a fast relaxing component, T2 ≈ 20 ms, ascribed to water closely associated with proteins and phospholipids. This paper was the ﬁrst one to describe multi-exponential behavior in myelinated nerve.
A little later, in 1986, Kroeker et al. ((R. M. Kroeker and R. M. Henkelman, “Analysis of biological NMR relaxation data with continuous distributions of relaxation times,” Journal of magnetic resonance 69, no. 2 (1986): 218–235.)) suggested the problem of analyzing biological relaxation times to allow the data determine the appropriate model. They appropriately assumed the data can be described by an integral of weighted exponentials and used the CONTIN curve ﬁtting program (written by Provencher) to solve integral equations for multi-exponential data.
T_2 decay curve fitting is a very non-trivial task in MRI. Typically it is treated as a single exponential decay in most tissues in the body. Unfortunately, that is a bit of an over-simplification. There are at least several tissues that are better represented as at least bi-exponential if not multi-exponential. Not the least of which, white matter in the brain.
An example decay curve typical of MRI data.
Given a problem, there are always multiple solutions. Some easier to implement and some harder to implement. Some are more accurate/robust and some are less accurate/robust. We always hope that the easier solutions are the accurate and robust, but not always.
There are several methods to fit T2 decay curves in MRI, a couple will be treated here:
* Two echoes and assumed mono-exponential data.
* More than two echoes and assumed mono-exponential data.
* More than two echoes and multi-exponential data using NNLS.
* More than two echoes and multi-exponential data using regularized NNLS. (<– probably the best one if you want a quick solution)