To investigate the use of a three-pool relaxation model to measure myelin, myelinated-axon, and mixed water-pool fractions in white matter (WM) during myelination.
To investigate the use of a three-pool relaxation model to measure myelin, myelinated-axon, and mixed water-pool fractions in white matter (WM) during myelination.
MRI at 1.9 Tesla, and conventional spin-echo imaging were used to acquire T1 and T2 relaxation data in 15 normal children ranging in age from 3 months to 13 years 4 months. Three equations with three unknowns were solved to calculate three water-pool fractions for each child in a frontal association-fiber area and a frontal-parietal projection-fiber area. The temporal trend of the fractions was compared with a theoretical three-pool myelination model.
The myelin level in the projection-fiber area rose earlier than in the association-fiber area following the standard caudal-to-rostral trend. The temporal trend of the three-pool fractions followed that predicted by the theoretical myelination model in both brain areas. The myelinated-axon and mixed pool sizes were significantly different in the two WM areas following early myelination, although their myelin pools were similar. T1 values correlated more highly with the myelinated-axon and mixed pool fractions than with the myelin pool fraction.
The three-pool relaxation model provides measurements of water-pool fractions in WM that follow values predicted during myelination. J. Magn. Reson. Imaging 2003;17:1–10. © 2002 Wiley-Liss, Inc.
MR IMAGES of the human brain provide excellent contrast between gray matter (GM) and white matter (WM) with a limiting spatial resolution of about 1 mL. Two mechanisms are largely responsible for this high contrast: spin-lattice (T1) and spin-spin (T2) relaxation. The microscopic environment of water in tissue determines the T1 and T2 relaxation times. A single volume element (voxel) in WM contains several distinct microscopic water environments (or water pools) separated by membranes that limit the exchange of water between them. For instance, in the genu of the corpus callosum (CC) there are approximately 100,000 individual axons with differing diameters and myelin content within a volume of 1 mm3 (1). Interspersed among these axons are oligodendrocytes, other glial cells, extracellular fluid, and vessels. The interplay of within-pool relaxation times and between-pool exchange times of water for these biological pools determines the net relaxation properties in WM (2, 3).
T1- and T2-weighted images are commonly used for visual assessments of WM (4, 5), while MRI relaxometry (calculating T1 and T2) is more appropriate when quantitative measures are desired (6). Several investigators have acquired age-related baseline values for T1 and T2 in normal children and adults (6–9). MRI relaxometry has also been used to study myelin disorders, hypomyelination (7, 10), and demyelination (11). While clinical MRI studies and MRI relaxometry are useful in myelin assessments, neither directly indicate myelin levels. Though changes in myelin content are known to produce changes in MR relaxation properties in WM (3, 4, 6, 12), changes in net relaxation properties of WM arise from multiple water pools, not just myelin, confounding accurate assessment of myelin by relaxometry. Various experimental methods to analyze myelin content in humans and animals have been proposed (13–22); however, there is no clear consensus for its use in clinical studies. The three-pool model is based on MRI methods that are readily accessible for use in a clinical setting.
During myelination in the central nervous system, a multilayer myelin sheath is formed around many axons by oligodendrocytes (23, 24) (Fig. 1). The myelin sheath acts as a water diffusion-restricting barrier, forming three major distinct water pools: a myelin pool (contained within dark bands, such as around A8), a myelinated axon pool (surrounded by dark bands), and a mixed pool (all others). Prior to myelination only a mixed pool is present in WM. The mixed pool is made up of water in intracellular (unmyelinated axons and glia) and extracellular (interstitial and intravascular) subcompartments. Each of the three pools is assumed to have distinct T1 and T2 relaxation times, with the myelin pool having the shortest and the mixed pool having the longest relaxation time (Table 1).
Bulk T1 and T2 relaxation within a voxel of WM is a function of individual pool relaxation rates as well as rate of water exchange (mixing) among the pools (2, 3, 12). The structure of the myelin sheath and geometrical relationship of the three water pools in WM (3, 23, 24) determines the multipool mixing time (see Discussion). Two well known mathematical models of NMR relaxation in multicompartment biological environments, fast exchange and slow exchange, provide the basis for estimating the three water-pool fractions in WM using bulk T1 and T2 relaxation measures (Appendix A). The fast-exchange model applies to multicompartment biological systems wherein the water mixing time among compartments is short compared to relaxation times, leading to monoexponential relaxation. Several investigators have applied the fast-exchange model to T1 relaxation measurements in biological systems (25–27). The fast-exchange model was assumed to be appropriate for modeling T1 relaxation in WM, since T1 relaxation is monoexponential in WM (3, 12). The slow-exchange model applies to biological systems wherein the water mixing time among compartments is long compared to relaxation times. In vivo T2 values are approximately an order of magnitude shorter than in vivo T1 values, and T2 relaxation is usually modeled as a slow exchange. The slow-exchange model is the basis for the multiecho T2 studies that have been used to estimate myelin, cytoplasmic, and extracellular pools in WM (20–22). The slow-exchange model describes T2 relaxation using multiple independent exponential equations, each with a distinct T2 value. The slow-exchange model was assumed to be appropriate for modeling T2 relaxation in WM, consistent with multiexponential T2 modeling by other groups.
This work presents methods for estimating pool fractions and preliminary data using a three-pool model (see Methods) to determine pool fractions in WM in a group of young, normal children. An elementary mathematical model of myelination in WM was developed to compare age-related changes in measured pool fractions with those predicted in the 15 children using the three-pool model.
WM was assessed in 15 normal volunteers ranging in age from 3 months to 13 years 4 months. All imaging was done with the approval of the Institutional Review Board at the University of Texas Health Science Center at San Antonio. MR images for this 15-subject control group were acquired previously as part of a research project studying myelin development in the 18q- syndrome (7), in children haploinsufficient for the myelin basic protein (MBP) gene. Subjects under the age of 5 years were accompanied by a nurse and were sedated, if necessary, using chloral hydrate to ensure minimal movement. Children under 5 years old are expected to show the greatest signal changes in WM (6, 7), and data were available on five volunteers in this age range. Data from three of the youngest children from the 18q- study were also used in the development of the model.
All images were acquired on a large-bore 1.9 Tesla clinical MRI system (Elscint, Haifa, Israel). Three spin-echo images were acquired: a T1-weighted image (TE/TR = 20/500 msec) and dual-echo proton density (PD)-weighted/T2-weighted images (TE 1/TE 2/TR = 20/80/3400 msec), all with NEX = 1. First-order flow compensation was applied prior to the second echo. Presaturation of a slab inferior to the scan volume was used to minimize arterial inflow artifacts for all images. Twenty-two axial images were acquired in a 256 × 256 array (1-mm pixel spacing) with 5-mm slice thickness and a 1-mm gap (132-mm span). Acquisition time was ∼ 20 minutes, mostly due to the long TR (3400 msec) in the dual-echo acquisition. The long TR was used for PD weighting in younger subjects. Subjects were immobilized to minimize movement between scanning sequences. T1 values were calculated using a two-point ratio method (28, 29). This method has been used successfully for many years (7, 9, 10, 30), and the estimated T1 values for frontal WM in adults are consistent with recently reported values (Table 2).
|Reference||Field strength (Tesla)|
|Fischer et al. (12)a||533 msecb||629 msecb||698 msecb|
|Koenig et al. (3)c||510 msecb||602 msecb|
|Kingsley et al. (8)d||636 msec|
|Whittall et al. (14)d||718 msec|
|3-pool modeld||714 msecb||793 msec|
Weight factors for the three-pool model (Eq. ) were determined using manual ROI analysis. ROIs were defined in two distinctive brain areas: a frontopolar WM area (FWM) with many association fiber tracts, and a frontal-parietal WM area (MWM) with many motor and somatosensory projection fiber tracts. These areas were selected because the MWM area is known to myelinate earlier than the FWM area following the well-documented caudal-to-rostral timing for myelination (4, 31–33). Additionally, since these two areas are transected by different classes of fiber tracts, a difference in water-pool makeup was suspected, and the three-pool model was used to investigate this difference. The FWM ROI was defined in an axial section image lateral to and forward of the genu of the CC. Defined in this manner, the FWM ROI contains fiber tracts to the superior and middle frontal gyri. The MWM ROI was defined in the first axial slice superior to the lateral ventricles and approximately centered about the central sulcus along the rostral-to-caudal direction. Fiber tracts to Brodmann areas 1, 2, 4, and 6 were assumed to transect this region.
Several strategies were used to minimize partial volume averaging. ROIs were traced well within tissue boundaries and within a single slice, while ensuring that slices above and below had the same WM tissue. The 1-mm gap between slices also served to minimize interslice effects. This ROI placement strategy avoided unwanted signals from the CSF, as reported by others (13, 16). ROIs were placed in the left and right hemispheres, and the average values were calculated. ROIs defined in this manner included data from >100 voxels, which helped reduce random noise effects. Mathcad (MathSoft, Inc., Cambridge, MA) was used to calculate fmy, fma, and fmx for each ROI from Eq. .
The three-pool model uses three equations that independently constrain the three water-pool fractions in WM (Appendix A). The three unknowns (pool fractions) are calculated from the three constraining equations (Eqs. [A3], [A4], and [A7]) using the following matrix equation:
where fmy, fma, and fmx are the pool fractions for the myelin, myelinated-axon, and mixed pools, respectively. Values for the w's in Eq.  can be determined from measurements in the spin-echo images and using T1 and T2 relaxation times designated for the myelin, myelinated axon, and mixed water pools (Table 1). Once the w's are calculated, Eq.  is solved for fmy, fma, and fmx. During model development, three of the required relaxation times (T1mx, T2mx, and T1my) were assigned values based on theoretical considerations and preliminary measurements. The remaining three relaxation times (T2my, T1ma, and T2ma) were adjusted to calibrate the model so that estimated myelin pool fractions at ages near birth and in the oldest children were consistent with published values. The calibration procedure was performed using ROI data from the FWM area.
Shortly after birth the FWM is minimally myelinated (5, 33), and is considered a mixed pool only with bulk relaxation times equal to the mixed pool relaxation times. The presence at an early age of small amounts of compacted myelin can shorten bulk relaxation times, leading to an underestimate of mixed pool times. In an attempt to account for this, T1mx and T2mx were assigned values of slightly longer times (10%) than the average bulk T1 and T2 values measured in FWM in the two youngest subjects studied (3 months normal and 5 months 18q- syndrome). The reported values for T1my ranged from 200 msec for in vitro measurement of human WM tissue (3) (at lower field strength) to 463 msec (15), taken from bovine optic nerve. We assigned a value of 350 msec to T1my based on preliminary testing in a small group of 18q- children.
Seed values for T2my, T1ma, and T2ma were estimated from reported values in Table 1. The three seeded relaxation times were iteratively adjusted so that the myelin fraction calculated using Eq.  approached the reported adult estimate (∼15–20%) (3, 13, 14) for the two children near age 13 years and ∼0% for the children near age zero (5, 6, 33). The early-age group included the youngest child from the normal group and the two young (<1 year of age) 18q- children (7, 9), because their myelin level is likely to be near zero. Each of the three relaxation times was individually adjusted to achieve a best fit, and this was repeated several times to determine the overall best fit. Each repetition was seeded with best-fit results of the previous round. Finally, all six relaxation parameters were adjusted slightly (±10%) to determine whether fitting could be improved, but no changes were indicated. This calibration procedure implicitly assumed that fmx ∼ 1 and fma ∼ 0 in the youngest children. However, no constraints or assumptions were made concerning fmx or fma in the oldest children. The majority of the control group (12 children) was not used in this calibration. The relaxation times determined by calibration (Table 1, row 1) were used for all subsequent studies.
Water-pool fractions (fmy, fma, and fmx) for each of the 15 normal children are plotted as a function of age in Figures 2 (FWM) and 3 (MWM). These figures clearly show a rapid early change in pool fractions in both brain areas. Although there were only three children under age 2, a similar pattern was seen in three 18q- children in this age range. The myelin pool fractions in MWM were larger than those in FWM (mean value of 0.140 vs. 0.112) for ages under 90 months (paired t-test, N = 7, P = 0.015). Most of this difference was due to the three youngest children (age < 2 years). This result is consistent with the expected earlier myelination in the MWM area (4, 31–33). However, the myelin pool fractions in the two WM regions were not significantly different for ages over 90 months (N = 8, P > 0.5), where they reached similar plateau levels (Figs. 2 and 3). The water-pool fractions for both the myelinated axon and mixed pools were clearly different in the MWM and FWM areas in the plateau age range (paired t-test, N = 8, P < 0.001). The myelinated axon pool fraction was largest in the FWM area (mean of 0.51 vs. 0.43), while the mixed pool fraction was smallest (0.29 vs. 0.37).
Statistical tests of difference in T1 between the FWM and MWM areas were performed for age >90 months, where the T1 values were more consistent. Mean T1 values (1.9T) were found to be statistically different (paired t-test, N = 8, P < 0.001). The magnitude of this difference was relatively small, with the T1 of MWM (852 msec) being approximately 6% larger than the T1 for FWM (804 msec). The relationship between T1 values and estimated three-pool fractions in the 15 children was studied using correlation analysis. The highest correlation with T1 was seen for the myelinated-axon (correlation coefficient ρ = –0.98) and mixed pools (ρ = +0.98), as compared to the myelin pool (ρ = –0.87).
An elementary mathematical model of myelination was formulated to assess the temporal trend of individual pool fractions estimated by the three-pool WM model (Appendix B). It was based on a three-pool concept whereby myelin segregates water fractions into myelin, myelinated axons, and mixed pools (Fig. 4). The three-pool myelination model estimates the rate and onset of myelination, and makes no assumptions about bulk MRI measurements, pool relaxation times, or the fast- and slow-exchange properties intrinsic to the three-pool WM model.
The myelination model was fitted to the three-pool fractions estimated in the 15 normal children, and results are illustrated by the smooth curves in Figs. 2 and 3. The fit of the myelination model for the FWM and MWM areas was very good, with an overall root mean square (RMS) error of ∼0.025 for each area. The general trend of the estimated water-pool fractions followed the characteristic curves of the myelination model, and the model adapted well to the differences in the two WM areas. The fit quality of the myelination model for ages <5 years was hard to evaluate because of the small sample size. However, the general trend appears appropriate, with a shorter time constant for MWM (∼7 months) compared with FWM (∼10 months), and an earlier onset of myelination (Δ = 0.5 months for MWM; Δ = 1.5 months for FWM). Absolute interpretation of these model values should be made with caution, but they are consistent with the fact that WM areas nearer to the brainstem myelinate earlier (4, 31–33).
The three-pool WM model predicted the expected earlier myelination in a projection fiber area (MWM) than in an association fiber area (FWM). Several additional findings were seen in the plateau region following early-stage myelination. The myelin water-pool fractions for the two WM areas, although clearly different early on, reached similar levels (∼0.18) in the plateau region (Figs. 2 and 3). The myelinated-axon pool fraction was smaller in the MWM area (approximately 83% of the fraction in the FWM area) (Figs. 2 and 3). This might relate to a similar pattern seen in the density of axons (axons per mm2) in corresponding regions of the CC, where the axonal density in the body of the CC has been estimated at ∼86% of the density in the genu/rostrum (1). Although axon density appears to be lower in the MWM area, the predicted myelin-to-myelinated axon water-pool ratio was larger in the MWM (0.4) area than in the FWM area (0.3). This could be due to the large myelinated axons from the primary motor area that form the cortical-spinal (pyramidal) motor tracts. This latter finding was not obvious from the small difference in T1 relaxation times for these two regions.
The change in T1 in the FWM area, from ∼2800 msec following birth to ∼800 msec by age 13 years, had the highest correlation (near unity) with changes in the mixed and myelinated-axon water fractions. The mixed-pool fraction in the FWM area dropped from ∼1 to ∼0.3, and the myelinated-axon pool fraction rose from ∼0 to ∼0.5, while the myelin pool fraction rose from ∼0 to ∼0.2 (Fig. 2). This finding suggests that the shift of water from the mixed pool (longest T1) to the myelinated axon pool (intermediate T1) during myelination was the largest contributor to the observed large change in T1 during myelination.
The temporal trend of estimated three-pool fractions from 15 different children followed the trend of the elementary myelination model surprisingly well, and is consistent with nonquantitative reports of myelination from large-group studies (31, 33). The consistent temporal trend seen in these different children suggests that myelination proceeds in a preprogrammed manner for the two WM areas studied.
It should be emphasized that the three-pool relaxation model was formulated for use in subcortical WM areas, and might not be appropriate where microanatomy differs significantly. For example, in nerves where axons are tightly bound, such as the optic nerve, the three-pool model may need to be modified to predict pool fractions correctly. The relaxation times for the myelin and myelinated-axon pools designated for the three-pool model should be adequate for most WM areas since these pools are similarly structured throughout the central nervous system.
The three-pool model requires three equations each independently constraining the three-pool fractions. For this study in normal young children, the modeling was done using a T1-constraining equation (Eq. [A4]), a T2-constraining equation (Eq. [A7]), and a constraint that the three-pool fractions must sum to unity (Eq. [A3]). This approach was based on the acquisition protocol used in the previously acquired 18q- syndrome control subject images. However, the three-pool model does not explicitly require a T1 constraint. For instance, from a three-echo spin-echo acquisition, two independent T2-constraining equations can be derived from separate pairs of the three images using Eq. [A7], and a third equation from Eq. [A3]. A key point is that T2 relaxation, following a slow-exchange model, can provide more than one constraining equation, while T1 relaxation, following a fast-exchange model, can only provide one.
Many assumptions were used in formulating the three-pool relaxation model for WM. As stated earlier, the bulk T1 relaxation for WM was assumed to follow the fast-exchange model, and the bulk T2 relaxation was assumed to follow the slow-exchange model. Since the shortest T1 value in the model is T1my = 350 msec, and the longest T2 value in the model is T2mx = 130 msec, our fast/slow-exchange assumption is that the three-pool mixing time is between these two values. We based this assumption on numerous reports (3, 12, 27, 34) that indicated a fast exchange for T1 relaxation and a slow exchange for T2 relaxation in WM tissues. No attempt was made to accommodate for differences between the ideal fast/slow exchange and the marginal fast/slow exchange that occur in WM. However, preliminary kinetic modeling of the three pools demonstrated that monoexponential T1 relaxation can be achieved simultaneously with multiexponential T2 relaxation with T1 and T2 relaxation curves similar to those observed in this study. Further evaluation of three-pool kinetic modeling is planned in animals, in which in vitro measurements can be used to test predictions of the kinetic model.
A basic assumption for the three-pool model is that each pool has distinct T1 and T2 relaxation times. Since each pool is a heterogeneous microenvironment, containing subpools, it is not obvious that pool relaxation could result in single T1 and T2 values. To have a single relaxation time for a pool with multiple subrelaxation environments, its mixing time must be short compared to subpool relaxation times, i.e., the fast-exchange model must apply within the pool. Also, since the first signal recorded is at 20 msec, subpools with significantly shorter T2 values will not directly contribute to the signals.
Calculations of within-pool mixing times indicate that the fast-exchange model is reasonable within each pool of the three-pool WM model. For myelinated axons, the distance a water molecule must diffuse to sample all relaxation environments (rmix) is the order of the unmyelinated axon diameter (∼1 μm). The apparent diffusion coefficient (ADCic) for intracellular water was estimated as 0.06 × 10–3 mm2/s, using data from Pfeuffer et al (35). Using this ADCic as the axonal water ADC and the Einstein self-diffusion equation, the mixing time or time to sample all relaxation environments (tmix) for the myelinated axon pool was estimated as ∼3 msec. This mixing time is very short compared with the T1 and T2 relaxation times reported for a variety of isolated cells (34), and supports the fast-exchange model for T1 and T2 within the myelinated axon pool.
Because of the natural repetitive nature of myelin, rmix is of the order of the spacing between adjacent layers (∼160 Å). Assuming that radial-directed diffusion is the limiting factor, that water diffuses equally inwardly and outwardly between adjacent subpools in the myelin sheath, and that a conservatively small estimate of membrane permeability (Pm = 2 × 10–4 cm/second) is used (3, 36), tmix is estimated as ∼3 msec. Reported T1 relaxation times of myelin are quite long compared with this tmix, and reported T2 relaxation times range from 10–20 msec (Table 1). These data are supportive of the fast-exchange model for T1 and T2 relaxation in the myelin pool.
The value of rmix is not as clearly defined for the mixed pool. Subpools include both cellular (axons and glia) and extracellular (interstitial and intravascular) water, and each is assumed to have its own characteristic T1 and T2 relaxation times. Self-diffusion of water is much more rapid in the extracellular fluid than within cells (ADCec = 45 × ADCic) (35); therefore, the mixing time for the mixed pool is approximately the time for water to exchange between cells and the surrounding extracellular space. A well known physiology textbook (37) reported that red blood cells exchange internal and external water at a rate of ∼100 times per second, indicating that tmix ∼ 10 msec. Pfeuffer et al (35) reported that tmix ∼ 50 msec for several different perfused cancer-cell models (cell diameters 6–8 μm). We calculated that tmix ∼ 11 msec for 1-μm-diameter axons using cylindrical geometry and Pm = 2.33 × 10–3 cm/second (derived from the Pfeuffer et al (35) data for cancer cells). These data suggest that tmix is in the range of 10–50 msec for the mixed pool in WM, but is likely near the low end of this range because of the large fraction of axonal water in the mixed pool. The shortest T1 and T2 relaxation times estimated for a subpool of the mixed pool are assumed to be approximately the same as in the myelinated axon pool (Table 1), and these times are longer than tmix, indicating that the fast-exchange model is appropriate within the mixed pool.
An additional assumption for the three-pool model was that T1 and T2 relaxation times should remain constant. This is particularly important if the three-pool model is to be used to study pool fractions in children 0–5 years old. It seems reasonable to assume that the relaxation times within an axon would be unaffected by myelination since the myelin sheath is an external structure. Increases in axon diameter up to about 3 μm (large for WM) should still follow the fast-exchange model (tmix ∼ 27 msec) for both T1 and T2. The relaxation time within compact myelin should be similar regardless of the number of layers, since each layer is nearly identical (24). The relaxation environment in the mixed pool might vary due to its major reconfiguration during myelination. However, there are offsetting effects that moderate changes in the mixed pool's relaxation environment. The major factor influencing change in the mixed pool is the balance between its intracellular (axons and glia) and extracellular (interstitial and intravascular) subfractions. Myelin formation leads to a loss of axonal intracellular water from the mixed pool as myelinated axons are formed. However, the extracellular fraction is also reduced due to the capture of extracellular water between myelin layers and the displacement of extracellular water by the myelin. The loss of axonal intracellular water from the mixed pool is proportional to the axon volume, while the loss of extracellular water is proportional to the volume of the myelin sheath. Also, there is brain growth during this time, which potentially can displace more extracellular water. We assumed that the subpool fractions of the mixed pool in subcortical WM were constant when the three-pool model was formulated.
The period of time between myelin formation and compaction is not well defined, but the lamellae in the central myelin tend to compact earlier than in the peripheral myelin (23) (Fig. 1). Further investigation is needed to determine the effect of this on the three-pool model. The present assumption is that only a small fraction of myelin sheaths will be noncompact at any time, and that the three pools are formed as compaction occurs.
Each voxel contains a collection of many axons, often with different diameters and with myelin content varying by axon diameter (23, 38). This variation in diameters could lead to a variation in mean water mixing times between the axons and their exterior. From a study of conduction times in myelinated callosal fiber diameters, the mean diameter of myelinated axons was reported to be ∼1.0 μm, with almost none below 0.25 μm (39). While most WM axons fall within the range of 0.75–1.25 μm, the impact of different diameters on water exchange times and fast/slow-exchange assumptions is not well understood. Future research will attempt to study this in animals.
The nodal region of myelinated CNS axons provides a more direct route for exchange of water between the myelinated axon and the mixed pool (23). We do not account for this in our model, but since nodal spacing is approximately ∼100× the diameter of myelinated axons (23) and ∼1 μm wide, this mixing route was assumed to account for less than 1% of myelinated axon water diffusion.
While the three-pool model was not tested in WM disease studies, it should be useful for patient studies in which model assumptions are valid. This should be the case in the early stages of demyelinating diseases, during which serial analyses could be helpful in monitoring the change in water distribution among the three pools. The T1-relaxation parameters will need to be altered to account for differences in field strength, but the basic model assumptions should remain valid for field strengths near 1.9 Tesla. The three-pool model should be very useful for studying developmental hypomyelination. This is based on data from the 18q- syndrome study, which indicated that children are haploinsufficient for the MBP gene (7). Preliminary modeling indicates that the myelin pool fraction in 18q- children is less than one-half the value seen in normal children.
Thanks to Carmen L. Contreras-Sesvold and Amanda Pasquali for help with data analysis.
The three-pool WM model is derived from two well-known approaches for modeling T1 and T2 relaxation in a multicompartment setting: the fast- and slow-exchange models. The combination of these models and signal measurements from spin-echo MR images led to the three-pool WM model for estimating water fractions from each pool.
If water molecules excited in one pool have an equal probability of relaxing in any pool, then a single net relaxation rate is seen for the pools. This happens if between-pool water mixing times are short compared to within-pool relaxation times. This is usually referred to as ideal “fast exchange,” or “fast mixing,” and the net relaxation rate is a weighted average of each pool's relaxation rate. Zimmerman and Brittin (2) showed that the net relaxation rate could be modeled mathematically for this case as follows:
where R = net relaxation rate (this can be either 1/T1 or 1/T2), Ri = relaxation rate for pool “i,” and fi = fraction of water molecules in pool “i.”
If water molecules excited in one pool relax before diffusing into another pool, then their relaxation rate is solely determined by that pool. This can occur if the relaxation times are short compared with the time it takes water molecules to diffuse between pools. This is usually referred to as ideal “slow exchange” or “slow mixing,” and the net relaxation signal has distinct components for each pool. Zimmerman and Brittin (2) showed that the net relaxation signal for this case can be modeled as follows:
where S(t) = magnitude of the net relaxation signal as a function of time, Si(0) = magnitude of the signal for pool “i” at time = 0, and Ti = relaxation time for pool “i” (this can be either T1 or T2).
Three basis equations are needed for the three-pool model of WM. The first equation states that the net MR water signal is the sum of signals from just three pools (myelin, myelinated axon, and mixed).
where fmy, fma, and fmx = fraction of signal from the myelin, myelinated axon, and mixed pools, respectively.
The second basis equation follows from modeling T1 relaxation as “fast exchange” (Eq. [A1]):
where w1my = R1my/R1; w1ma = R1ma/R1; w1mx = R1mx/R1; R1 = measured net relaxation rate (1/T1) for WM; and R1my, R1ma, and R1mx = relaxation rates for myelin (1/T1my), myelinated-axon (1/T1ma), and mixed (1/T1mx) pools, respectively. Equation [A4] provides the spin-lattice (T1) constraint for the three-pool WM model.
The third basis equation follows from modeling T2 relaxation as “slow exchange” (Eq. [A2]). It is derived using two samples of T2 relaxation. These were from a dual-echo spin-echo image pair, a proton density-weighted (PDW) image, and a T2-weighted (T2W) image. The two signals are represented mathematically as
where SPDW, ST2W = measured signals from the PDW, T2-weighted images; S(0) = signal from all three pools at time zero; TE1 and TE2 = echo times for the PDW, T2-weighted images in msec; and T2my, and T2ma, and T2mx = T2 relaxation times for myelin, myelinated axon, and mixed pools, respectively (in msec).
A single equation can be formed from the ratio of Eqs. [A5] and [A6]. This eliminates the S(0) term, which is dependent on T1, TR, and receiver gain. This ratio equation can be simplified leading to the third basis equation for the three-pool model.
where w2my = S · K2my – K1my; w2ma = S · K2ma – K1ma; and w2mx = S · K2mx – K1mx, and
S = SPDW/ST2W
K1my = e–TE1/T2my K2my = e–TE2/T2my
K1ma = e–TE1/T2ma K2ma = e–TE2/T2ma
K1mx = e–TE1/T2mx K2mx = e–TE2/T2mx
Equation [A7] provides a spin-spin (T2) constraint for the three-pool WM model. The three equations with three unknowns (Eqs. [A3], [A4], and [A7]) can be directly solved for the three unknowns (fmy, fma, and fmx).
Myelination is modeled to begin at t = Δ with all water molecules in the mixed pool (Nmxo). Myelination is assumed to proceed at a constant rate following onset, modeled by a single time constant (TC). The number of water molecules lost from the mixed pool by myelination (Nmxl) is equal to the number of water molecules gained by the other compartments (myelin (my), myelinated axon (ma), and lost water (l)). Only water in the mx, my, and ma pools is measured by MRI, as indicated by the dashed line enclosing them.
A solution of elementary differential equations for this model leads to equations for the number of water molecules in each of the compartments. The number for the mixed pool is
where Nmxo = number of water molecules in the mixed pool at onset of myelination, Nmxl = number of water molecules lost from mixed pool by myelination, t = age in months, Δ = age myelination assumed to start (months), and TC = time constant for the model (months).
The numbers of molecules in each of the other three pools are
where fy, fa, and fl = fractional distribution of water lost from mixed pool into the myelin, myelinated-axon, and lost-water compartments, respectively. This model assumes conservation of water, so
where fmy = fy · fmxl, fma = fa · fmxl, and flost = fl · fmxl are the fractions of water lost from the mixed pool (fmxl) assigned to the myelin, myelinated-axon, and lost-water compartments, respectively.
There is no MR signal from the lost water. The three-pool model, as applied to T1 and T2 relaxation, requires that fmx + fmy + fma = 1, and the myelination model has to be modified to accommodate this. The myelination model states that fmx(t) + fmy(t) + fma(t) = 1 – flost(t). Dividing both sides of this equation by 1 – flost(t) adapts the myelination model such that three modified pool model fractions sum to one. The modified myelination model equations are as follows:
Equations [B10]–[B13] were used in fitting the measured three-pool fractions. Since there was no measure of the lost water fraction it was estimated to be proportional to the myelin pool fraction, flost(t) = kfmy(t) in the model. Several values of k were assigned during fitting, ranging from 0 to 4. The best fit of the data determined by RMS error was found with smaller k-values, and k = 1 was selected for fitting. Also, by assigning a fixed value to k, its contribution to the model-fitting variability was eliminated.
Fitting was done using MathCad Plus 6 for Macintosh (MathSoft, Cambridge MA). The objective function minimized was the total sum-squared error calculated for the three-pool model equations (Eqs. [B10]–[B12]). Seed values for the fit were TC = 20 months, Δ = 0, fmxl = 0.76, fma = 0.5, and fmy = 0.2. Both FWM and MWM three-pool fraction data were easily fit using these seed values. Each seed value was varied to test for sensitivity of the fit. While testing was not exhaustive, only small changes (1–2%) were observed in the model parameters or the estimated fractions, indicating that fitting WM by the myelination model was not sensitive to seed values.