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Magn Reson Med. Author manuscript; available in PMC 2010 April 1.

Published in final edited form as:

PMCID: PMC2711212

NIHMSID: NIHMS120262

Magnetic Resonance Imaging and Spectroscopy Section, National Institute on Aging, National Institutes on Health, Baltimore, MD 21224

Address correspondence to: Richard G. Spencer, Magnetic Resonance Imaging and Spectroscopy Section, GRC 4D-06, National Institute on Aging, National Institutes on Health, Baltimore, MD 21224, Email: vog.hin.xileh@recneps, 410-558-8226, FAX: 410-558-8318

The publisher's final edited version of this article is available free at Magn Reson Med

See other articles in PMC that cite the published article.

MR techniques are sensitive to the initial phases of osteoarthritis, characterized by disruption of collagen and loss of proteoglycan (PG), but are of limited specificity. Here, water compartments in normal and trypsin-degraded bovine nasal cartilage were identified using a non-negative least squares multiexponential analysis of T_{2} relaxation. Three components were detected: T_{2,1} = 2.3 ms, T_{2,2} = 25.2 ms, and T_{2,3} = 96.3 ms, with fractions w_{1} = 6.2%, w_{2} = 14.5%, and w_{3} = 79.3%, respectively. Trypsinization resulted in increased (p<0.01) values of T_{2,2} = 64.2 ms and T_{2,3} = 149.4 ms, supporting their assignment to water compartments that are bound and loosely associated with PG, respectively. The T_{2} of the rapidly-relaxing component was not altered by digestion, supporting assignment to relatively immobile collagen-bound water. Relaxation data were simulated for a range of TE, number of echoes, and SNR to guide selection of acquisition parameters and assess the accuracy and precision of experimental results. Based on this, the expected experimental accuracy of measured T_{2}’s and associated weights was within 2% and 4% respectively, with precision within 1% and 3%. These results demonstrate the potential of multiexponential T_{2} analysis to increase the specificity of MR characterization of cartilage.

Articular cartilage, the connective tissue responsible for transmitting load across joints, is comprised primarily of water and extracellular matrix. The two main matrix constituents are type II collagen, which is configured as a three-dimensional fibrous network resulting in tissue tensile strength, and proteoglycan (PG), a large hydrophilic macromolecule which provides swelling pressure and compressive resistance. By wet weight, cartilage is 65–80% water, 15–25% collagen, and 3–10% PG (1).

A great amount of effort has been focused on non-invasive early detection of osteoarthritis (OA), a highly prevalent and potentially debilitating disease which results in progressive degradation and loss of articular cartilage. The initial phases of OA are characterized by disruption of collagen and loss of PG. MR is increasingly used as a noninvasive diagnostic modality sensitive to early OA, and establishing correlations between MRI parameters such as T_{1}, T_{2}, T_{1ρ}, magnetization transfer and apparent diffusion coefficient and cartilage biochemical properties has been an active area of investigation (2–4). Although attempts have been made to associate average values of such MR parameters to specific tissue characteristics, results have been inconclusive (3).

T_{2} has consistently been reported to increase in cartilage with pathomimetic enzymatic degradation. T_{2} is influenced by the mobility of water molecules associated with various effective tissue compartments, defined by cartilage macromolecular components, which would be expected to exhibit a distinct T_{2} (5).

Multiexponential T_{2} analysis has previously been applied to identify and characterize multiple water compartments in normal and pathologic tissue (6–13). As with previous work in other tissues, we expect this approach to lead to insight into water compartmentation in normal and degraded cartilage. We further conjecture that the ability of multiexponential analysis to detect degradation-induced changes in water compartmentation would support its use as a potential diagnostic approach in early OA.

Therefore, the focus of this work was to investigate water compartmentation in cartilage using multiexponential analysis of T_{2} relaxation data. Analysis was performed using non-negative least squares (NNLS), making no *a priori* assumptions about the number of relaxation components present. We hypothesized that decomposition of relaxation data into multiple components would permit observation of water compartment shifts in response to pathomimetic enzymatic degradation.

T_{2} is known to be influenced by tissue anisotropy, such as that found in articular cartilage. In addition, T_{2} varies with depth in articular cartilage (14,15). Therefore, we used bovine nasal cartilage (BNC), which is largely isotropic and homogeneous, for our investigations (16). We performed enzymatic degradation with trypsin, which cleaves the core and link proteins releasing PG fragments (1).

A specific objective of our study was to determine the optimum acquisition parameters (TE and number of echoes) and minimum SNR required to resolve multiple T_{2} components. This analysis was performed through NNLS analysis of simulated noisy relaxation data. Results of these simulations allowed us to define experimental acquisition parameters optimized for accuracy and precision in defining T_{2} components, as well as to specify the error in component T_{2}’s and weights as a function of acquisition parameters and SNR.

BNC disks (diameter = 8 mm) were excised from the nasal septum of a skeletally mature cow (Green Village Packing, Green Village, NJ). Samples were moistened with DPBS and stored at 4 °C until testing. Within a week of harvest, each sample was removed individually, blotted dry to remove excess surface water, wrapped in Teflon tape to reduce moisture loss, and placed in a 10 mm NMR tube for MR studies. After the initial MR measurements, samples were removed from the NMR tube, unwrapped and again stored at 4 °C. After scanning, samples were incubated in DPBS with 1mg/ml of trypsin (Sigma-Aldrich, St. Louis, MO) for 24 hours at 37 °C and then washed with DPBS. MR measurements were then repeated after this period of enzymatic digestion.

Data were acquired with a 9.4T Bruker DMX NMR spectrometer (Bruker Biospin, GmbH, Rheinstetten, Germany) at 4 °C in order to minimize degradation of the sample while in the magnet. Relaxation data were measured using a spectroscopic CPMG pulse sequence. Both pre- and post-enzymatic treatment relaxation decays were acquired with parameters TE/TR = 0.6ms/5s and 1024 echoes. In order to achieve desired levels of accuracy and precision, we implemented signal averaging with NEX = 128 in order to achieve SNR in the range of 6000–10000.

We briefly describe the NNLS method used for multiexponential T_{2} analysis (17). Multiple T_{2} component relaxation decay can be represented in discrete form with the following linear system of equations describing measurement of *M-1* relaxation components over N echoes (18):

$${y}_{n}={A}_{nm}{S}_{m},$$

[1]

where *y _{n}* is an array containing the

$$\sum _{n=1}^{N}{\mid \sum _{m=1}^{M}{A}_{nm}{S}_{m}-{y}_{n}\mid}^{2}.$$

[2]

In order to minimize the influence of noise on the fit as well as produce a continuous T_{2} distribution a minimum energy constraint can be imposed so that the following function is minimized:

$$\sum _{n=1}^{N}{\mid \sum _{m=1}^{M}{A}_{nm}{S}_{m}-{y}_{n}\mid}^{2}+\mu {\mid \sum _{m=1}^{M}{S}_{m}\mid}^{2},$$

[3]

where *μ* is referred to as a regularizer term. As *μ* is increased, the permitted misfit increases while reducing the influence of noise on the fit. Eq. [3] is recast in terms of *χ*^{2} by appropriate division by the standard deviation of *y _{n}*,

The T_{2} analysis was implemented in MATLAB (MathWorks, Natick, MA). The first moment and associated fractions for each identified component of the T_{2} distribution were obtained.

Means ± standard deviation (SD) for control and trypsin degraded samples are reported. The differences in the mean T_{2} values between control and trypsin degraded samples were tested for statistical significance using a paired t test, with p < 0.05 indicating statistical significance.

As will be discussed, three T_{2} components were consistently detected by NNLS in control BNC samples. Using the average values for these component T_{2}’s and weights as input parameters, data were simulated using the following expression:

$$y(n\xb7TE)=B+{y}_{0}\sum _{m=1}^{M}{w}_{m}\xb7{e}^{-(n\xb7TE)/{T}_{2,m}}+\epsilon (0,\sigma ),$$

[4]

where *y*(*n* · *TE*) represents the amplitude of the *n*^{th} echo measured at *n* · *TE*, *B* represents a baseline offset, *y _{0}* represents overall signal amplitude,

Simulations were performed for a range of echo spacings TE (0.4 ms to 1.6 ms), number of echoes N (256, 512, or 1024) and SNR, defined as *y _{0}*/

The results in Figure 1 indicate the optimum TE for a given number of echoes for reliable T_{2} relaxation measurements using NNLS in BNC. For all N shown, the relationship between SNR and TE is roughly convex, indicating an optimum echo time for a given N. For example, if N = 256, the required SNR for admissibility is 2600, achieved using a TE = 1.2 ms. Optimum echo times over the range of N simulated were between 0.8 ms and 1.2 ms. In addition, simulations generally indicated an SNR requirement for admissibility that decreased with increasing N.

Simulation results using for the minimum SNR required for admissible experiments over various combinations of TE and number of echoes (N). SNR values are admissible on and above each curve. In general, the SNR requirement is relaxed with increasing N. **...**

Figure 2 shows the accuracy of T_{2} and associated weight determination for TE = 1.2 ms and N = 512. These acquisition parameters are those used experimentally, since only data from even numbered echoes were analyzed. As seen, accuracy increases with increasing SNR. An SNR of 8000, comparable to experimental values, yielded accuracy in T_{2}’s and associated weights within 2% and 4% respectively. At low SNR, the algorithm underestimated the relaxation time of each component. The fractional weight w_{2} was underestimated while w_{1} and w_{3} were overestimated at low SNR. Since weights must sum to 100%, this result is not surprising. As also seen in Fig. 2, at low SNR the percent errors in T_{2} and w were substantially greater magnitude for component 2 than for those of the other components.

Simulation results for the percent error in component relaxation times and fractions as a function of SNR for TE = 1.2 ms and N = 512. Percent error is calculated as fitted value minus input value divided by input value. This figure shows how accuracy **...**

Figure 3 indicates the precision of T_{2} and associated weight determinations, again for TE = 1.2 ms and N = 512. Parameter precision was defined by their coefficient of variation calculated by dividing the standard deviation over the trials with different noise realizations by the simulation input value for each parameter. For SNR = 8000, the coefficients of variation for T_{2}’s and their associated weights were within 1% and 3%, respectively. T_{2,1} and w_{1} exhibited the largest CV, primarily due to the small denominators in the corresponding calculations. However, T_{2,2} and w_{2} exhibited the largest SD’s in absolute terms, consistent with Fig. 2.

Results for monoexponential and NNLS analysis of the experimental decay curves are presented in Table 1. Monoexponential analysis of data from control samples yielded T_{2} = 84 ± 3 ms, while three distinct compartments were consistently detected with NNLS. These components were: T_{2,1} = 2.3 ± 0.6 ms, T_{2,2} = 25.2 ± 6.2 ms, and T_{2,3} = 96.3 ± 4.7 ms, with associated fractions: w_{1} = 6.2 ± 2.5%, w_{2} = 14.5 ± 3.9%, and w_{3} = 79.3 ± 6.1%, respectively. As expected, trypsin degradation resulted in a significant increase in the monoexponential T_{2}, to 134.5 ± 5.2 ms. T_{2,1} was not affected by degradation, while T_{2,2} and T_{2,3} increased with degradation, to 64.2 ± 19.3 ms and 149.4 ± 11.8 ms, respectively. The corresponding weight fractions were 3.1± 1.3%, 10.9 ± 10.5%, and 86.0 ± 11.2%. It is important to note that these fractions represent relative changes in water compartment size, as the net increase or decrease of water in the sample was not detected. From this analysis it is therefore impossible to determine whether the absolute size of a particular water compartment changed with degradation, and direct comparison of fractions pre- and post-degradation are not valid unless one assumes that total tissue water content is unchanged. However, assessment of relative shifts of water protons between compartments is independent of this assumption. Figure 4 shows typical T_{2} distribution results for control and trypsin degraded BNC.

Typical T_{2} distributions obtained using NNLS analysis of relaxation data acquired from a BNC sample before and after trypsin degradation. Trypsin degradation resulted in a negligible change in T_{2,1} but a lengthening of T_{2,2} and T_{2,3}.

Previous work has demonstrated the utility of multiexponential T_{2} analysis to characterize tissue water compartments in a variety of tissues. In studies of normal muscle, Saab et al. (20) found four T_{2} components, with relaxation times of <5 ms assigned to macromolecule-bound water, ~20 ms and ~40 ms assigned to intracellular water compartments, and >100 ms assigned to extracellular water. Assignment of T_{2} components to intra- and extracellular water compartments was performed in the study of a rat model in which edema was presumed to result in an increase in extracellular water fraction and a decrease in the intracellular water fraction (21). MacKay et al. found that brain white matter exhibited three T_{2} components: a minor component with relaxation time between 10 and 55 ms associated with myelin membranes, an abundant component with T_{2} between 70 and 95 ms associated with cytoplasmic and extracellular spaces, and a slowly-relaxing compartment with T_{2} > 1 s associated with cerebral spinal fluid (22). Multicomponent T_{2} analysis has also demonstrated that the myelin-associated water fraction is reduced or absent in pathologic conditions known to effect myelination, such as multiple sclerosis and possibly schizophrenia (10,11). Our work echoes these previous investigations in the context of cartilage water compartments, where multiexponential analysis indicates changes in relative water pool sizes under pathomimetic intervention.

Multiexponential T_{2} analysis has previously been used to evaluate anisotropy in a variety of tissues, including bovine articular cartilage (23). In that study, also using NNLS analysis, the change in the measured T_{2} distribution as a function of cartilage orientation was taken as an indicator of macromolecular organization and anisotropy. The examples reported qualitatively resemble our results for the intermediate and long T_{2} components in relative size and T_{2} values. Henkelman et al. attribute the appearance of multiple T_{2} components in articular cartilage to distinct tissue layers exhibiting different relaxation times. In the case of BNC, as discussed above, layered organization is not present to any significant degree and the identified T_{2} components are attributable to tissue compartments that are not spatially distinct. Mosher et al. also observed multiexponential T_{2} relaxation in porcine articular cartilage, using using nonlinear least-squares fits to mono-, bi-, and triexponentials (24). These authors found T_{2} relaxation distributions compatible with a three compartment model; as was also the case in Henkelman et al., this was primarily attributed to spatial heterogeneity, with different tissue regions contributing different values of T_{2}. It is important to note that in both of these previous studies, as also for our work, T_{2} determinations were made from non-localized spectroscopic experiments, rendering it impossible to differentiate regional variation in relaxation properties from pools that are co-localized but exhibit different relaxation rates. It was for this reason that we performed our study using BNC, which is known to exhibit much less tissue anisotropy and inhomogeneity as compared to articular cartilage; this permits us to interpret our results in terms of water compartments without the confounding effect of regional differences. We note that the matrix composition of BNC is otherwise very similar to that of articular cartilage, so that our results should correspond at least qualitatively to results obtained in analyses of articular cartilage.

Graham et al. (19) used simulated decay data to determine the accuracy of multicomponent T_{2} analysis in white matter, muscle, and breast. Similarly, our simulations were directed towards evaluating the reliability, accuracy, and precision in fitting multiple T_{2} components in BNC using NNLS. Consistent with the finding of Graham et al., we find that the utility of multiexponential analysis relies upon the appropriate selection of TE and N, in addition to sufficient SNR.

In general, the minimum required SNR decreased with increasing number of echoes for a given TE. This result relies upon our selection of reasonable combinations of TE and N. That is, sampling the relaxation decay extends throughout the full duration of that decay without extending beyond the point of nearly-complete signal decay. This avoids sampling of noise without underlying signal.

Our simulations, based on experimental component T_{2}’s and weights for BNC, also allowed us to predict the error in component T_{2}’s and weights as a function of SNR for experimental TE and N. As expected, the relationship between error and SNR was influenced by the input T_{2} values and weights. For example, when using input parameters from measurements of undigested BNC, w_{2} was underestimated while w_{1} and w_{3} were overestimated. However, when using simulation input values corresponding to degraded tissue, for which the values of T_{2,2} and T_{2,3} were closer together than in control tissue, w_{1} and w_{2} were underestimated while w_{3} was overestimated (data not shown). We also found that the minimum required SNR for admissibility was generally determined by the accuracy criterion, rather than by the precision (data not shown). When using NNLS analysis of relaxation data to categorize cartilage as healthy or degraded, the required minimum SNR could be relaxed when the change in a particular component of interest is expected to be large. In particular, the 10% criterion on accuracy and precision used here for admissibility could be raised to a larger value.

The optimum acquisition parameters and minimum SNR for admissibility were found to be sensitive to the T_{2}’s and weights of each component. In particular, when T_{2} values of two water components are within a factor of roughly 2, the minimum SNR necessary for consistently resolving those components significantly increases. This is consistent with previous findings (18). Given that T_{2} components change with matrix degradation, NNLS evaluation of cartilage must make use of experimental parameters and SNR values that will permit appropriate evaluation at all stages of degradation.

In their analysis of multiple T_{2} compartments in peripheral muscle (19), Graham et al. also found three relaxation components and demonstrated a convex relationship between SNR and TE using simulation inputs applicable to muscle. These authors used a different criterion for admissibility, targeted to distinguishing between slow and fast twitch muscle based on the accuracy of the first moment of the entire T_{2} distribution as opposed to the accuracy of the first moment of each individual component T_{2} and weight as in the present work. Using this criterion, Graham et al. found a minimum required SNR in the range of 100 to 400. We used a more restrictive constraint to select minimum SNR since in our application we were interested in identifying differences in water compartments between normal and degraded BNC for which the T_{2} values were unknown.

We used trypsin as a degradative enzyme to model the loss of PG associated with early stages of OA. Our results are consistent with previous work that reported an increase in the monoexponential T_{2} of cartilage as a result of trypsinization. For example, Menezes et al. reported an increase in monoexponential T_{2} from 83 ms to 95 ms with trypsinization in bovine cartilage from the femoral groove (3). These authors also attributed this increase in T_{2} to an increase in the size of the unbound water compartment.

Ghiassi-Nejad et al. (25) characterized the stoichiometry of articular cartilage through use of dehydration and deuteration. They performed a fit of T_{2} decay data to a biexponential function under various conditions. In deuterated samples, rapidly-relaxing tightly bound protons were visible (using direct measurements of the free induction decay signal) as well as a component with T_{2} ranging from 15 to 30 ms, assigned to non-exchangeable protons on mobile PG side-chains. The slowly-relaxing component of non-deuterated samples exhibited a T_{2} of 60 to 80 ms, indicating greater average mobility (25). As noted, the NNLS analysis used in our study makes no *a priori* assumption about the number of water compartments present, allowing for the separate identification of water tightly bound to PG and water loosely associated with PG. Our reported T_{2,2} and T_{2,3} correspond closely to the slowly-relaxing T_{2} component of deuterated and hydrated cartilage samples respectively, as reported by Ghiassi-Nejed et al. These similarities support our assignment of T_{2,2} and T_{2,3} to water tightly bound to PG and water associated loosely with PG, respectively.

We found that enzymatic degradation with trypsin resulted in a significant lengthening of T_{2,2} and T_{2,3} in addition to a small corresponding shift from w_{2} to w_{3}. This shift is consistent with a loss of water tightly bound to PG accompanied by a relative increase in water loosely associated with PG. The lengthening of both relaxation components is consistent with the fact that while trypsin cleaves the core and link protein of PG, it does not ensure the diffusion of PG fragments from the matrix. Water associated with GAG fragments would be expected to demonstrate an increased T_{2} as compared to that of intact PG. The lengthening of both T_{2,2} to T_{2,3} with trypsin suggests that both of these compartments are associated with PG, and further supports their assignment to water tightly bound to PG and water associated loosely with PG. Lattanzio et al. also investigated cartilage before and after trypsin degradation using a biexponential fit of T_{2} relaxation data (5). Consistent with our results, they described a slowly-relaxing T_{2} component, assigned to PG, whose relaxation time increased with degradation. No significant change in water fractions was observed. In addition, this slowly-relaxing T_{2} component accounted for 97% of the signal, similar to the sum of w_{2} and w_{3} in our work, in which these slowly-relaxing pool fractions accounted for 92% and 98.6% of the signal in control and degraded tissue, respectively.

Due to the markedly lower mobility of collagen-bound water in comparison with PG (26–28), we associateT_{2,1} with a collagen-bound water compartment. This is supported by the lack of observed change in T_{2,1} upon degradation with trypsin. Knauss et al. observed biexponential T_{2} relaxation in collagen gel using multiexponential T_{2} analysis, reporting fast and slow components with T_{2}’s of 6.5 ms and 120 ms, respectively (29). They assigned the rapidly-relaxing component to collagen-bound water and the more slowly-relaxation component to bulk water. The short T_{2} of the collagen-bound water fraction is comparable to our value of T_{2,1} = 2.3 ms. Nightingale et al. (30) described four relaxation components in the nucleus pulposus of the intervertebral disc, and used multiple regression to correlate the water fractions of matrix components with magnetization fractions. Collagen correlated strongly with the most rapidly-relaxing component, with T_{2} = 3.1 ms and fractional weight of 3%. These are comparable to our values of T_{2,1} = 2.3 ms and w_{1} = 6.2%, respectively. We note that the nucleus pulposus has a similar matrix composition to BNC (16), so that these results support the assignment of the rapidly-relaxing pool in our studies to being collagen-bound water.

We found a good correspondence between the water components identified in our experiments, where the weights associated with water bound to collagen, water tightly bound to PG, and bulk water loosely associated with PG were ~6%, ~14%, and ~80%, respectively, and previous biochemical analysis of BNC, in which fractions of collagen, PG, and water by wet weight were ~10%, ~13% PG, and ~75% respectively (16). Our finding of a smaller value for collagen-associated water as compared to the actual matrix fraction of collagen is consistent with the reportedly smaller binding capacity of collagen as compared to PG per mg (25). In spite of the correspondence of our results with biochemical analysis, we note that several factors can influence the relationship between MR-detected magnetization fractions and macromolecular composition. One such factor is signal visibility based on T_{2}; non-exchangeable protons in collagen have been reported to have a T_{2} of ~30 μs (5), and so would not be detectable with the TE used in our study. In addition, the relationship between magnetization fractions and matrix components may be affected by proton exchange between compartments. For example, Lattanzio et al. used a four-spin component exchange model in their study of articular cartilage (5). A magnetization exchange rate between PG and collagen of 120 s^{−1} was found, which is intermediate to the corresponding nominal relaxation rates in our analysis of 1/T_{2,1} = 438 s^{−1} and 1/T_{2,2} = 39.6 s^{−1}. We would therefore expect that the T_{2}’s and associated fractions measured in our experiment would be influenced by this exchange.

As indicated previously, avoiding the confounding effect of regional variation is an essential element in interpreting multicomponent T_{2} data in terms of water compartmentation. For studies of articular cartilage, including clinical application to osteoarthritis, this would be accomplished through use of imaging experiments and region-of-interest analysis of T_{2} decay in multi-echo experiments. Important differences between spectroscopic and imaging multi-echo sequences are the longer minimum TE’s and lower SNR’s generally available in the latter. Both simulated and experimental data demonstrate that, for realistic values of SNR, a rapidly relaxing water compartment cannot be detected when the minimum TE approaches its T_{2} (data not shown). Therefore, the compartment we have identified as T_{2,1} would not be detectable using a clinical imaging sequence, although T_{2,2} and T_{2,3} would be visible. To assess the adequacy of imaging experiments for accurate description of these more slowly-relaxing components, we have performed simulations with an SNR of 500 and which consider only T_{2,2} and T_{2,3}. The results indicate reliability (as defined in the Methods above) of 100% with accuracy and precision of at least ~12% and ~5% for the derived T_{2,2} values, respectively, with even more favorable results being obtained for T_{2,3}. We also found the accuracy and precision of the estimate of w_{2} to be ~19% and ~6%, respectively, with even more favorable results being obtained for w_{3}. This is clearly adequate for describing the large changes of ~155% and ~55% in T_{2,2} and T_{2,3} demonstrated in this work using trypsin degradation, as well as the possibly much smaller changes expected during the early stages of degenerative cartilage disease.

As noted above, relaxation of the admissibility criteria used in the current study would lead to a decrease in the required SNR for performing acceptable studies. Detection of only the two more slowly-relaxing T_{2} components in a clinical imaging experiment as detailed here is, in effect, an example of this; violations of admissibility in the spectroscopic simulations were primarily attached to the rapidly-relaxing T_{2,1} component.

In conclusion, multiexponential T_{2} analysis detected three distinct water compartments in normal and degraded BNC, assignable to collagen bound water, PG bound water, and water loosely associated with PG. The two slowly-relaxing T_{2} components were significantly affected by trypsinization of cartilage. These results indicate the potential for multiexponential T_{2} analysis to play a diagnostic role in the early detection of OA through monitoring PG depletion. This is the first study in which simulations were used to determine the optimum acquisition parameters for resolving multiple T_{2} components in cartilage. Simulations of decay data over a range of TE, N, and SNR allowed for characterization of the reliability, accuracy and precision of T_{2} NNLS analysis.

This work was supported by the Intramural Research Program of the NIH, National Institute on Aging.

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