An efficient subset of morphological measures for articular cartilage in the healthy and diseased human knee

Authors


  • **

    The members of the A9001140 Investigators are listed in the appendix.

Abstract

The relationship between three-dimensional, MRI-based morphologic measurements commonly taken of knee cartilage was examined to determine whether a subset of variables fully reflects differences observed in cartilage in cross-sectional and longitudinal studies. The benefits of a subset of measures include increased statistical power due to reduced multiple comparisons, improved understanding of relationships between the morphologic measures of articular knee cartilage, and greater efficiency in reporting results. One hundred fifty-two women (77 healthy and 75 with knee osteoarthritis) had coronal 3-T MR images of the knee acquired at baseline and at 24 months. Measures of femorotibial cartilage morphology (surface area, thickness, volume, etc.) were determined in the medial and lateral tibia and femur. Cartilage thickness (mean cartilage thickness over the total area of the [subchondral] bone), total subchondral bone area, and percentage of denuded area of the subchondral bone were found to explain over 90% of the cross-sectional and longitudinal variation observed in other measures of cartilage morphology commonly reported in knee osteoarthritis. Hence, these three measures of cartilage morphology explain nearly all variation in a larger set of common cartilage morphology measures both cross-sectionally and longitudinally, both in healthy and in osteoarthritic knees. These variables hence define an efficient subset for describing structural status and change in osteoarthritic cartilage. Magn Reson Med 63:680–690, 2010. © 2010 Wiley-Liss, Inc.

MRI has been instrumental in exploring the development, maintenance, functional adaptation, and degeneration of articular cartilage as it has made it possible to extract the geometric dimensions of the tissue in vivo (1–3). In the study of osteoarthritis (OA) and other joint diseases, morphologic measures of cartilage are frequently obtained to assess disease status or progression. Metrics of cartilage morphology (volume, thickness, and others) based on MRI have been shown to be reproducible in single (1, 3) and multicenter studies (4) and hold promise for evaluating the treatment response of structure/disease-modifying drugs. Several studies have reported the rate and sensitivity to change of cartilage morphology measures in participants with OA (5–11) and healthy persons (12–15) using 1.5T (1, 3, 16) or 3T MRI (17–19). Some of these studies have compared measures of change based on MRI to joint-space narrowing from radiographs (7, 10, 11, 17, 20).

An international group of experts has recommended definitions and nomenclature for MRI-based measures of cartilage (21), and these involve, among others:

  • VC: Volume of the cartilage

  • tAB: Total area of the (subchondral) bone

  • AC: Area of the cartilage surface

  • cAB: tAB covered by the AC

  • dABp: Percentage of tAB not covered by the AC = 100 × (1 – cAB/tAB)

  • ThCtAB.Me: Mean cartilage thickness over the tAB

  • ThCcAB.Me: Mean cartilage thickness over the cAB

  • VCtAB: Volume normalized to the tAB

These measurements can be taken on several knee surfaces, e.g., medial and lateral tibial plates (MT, LT), medial and lateral weight-bearing femoral plates (cMF, cLF), and specific regions of these plates, thus creating a large (if not overwhelming) set of measurements available for examination and statistical testing in a given study. As many of these morphologic measures are strongly related, some may be redundant or contain minimal additional information.

The general goal of this analysis was to identify an efficient subset of core measures that comprises a comprehensive description of cartilage morphology and its longitudinal changes in healthy and diseased cartilage. A subset of measures will be considered efficient if it maximizes the information present in all measures in a given (minimal) number of measurements. Information is equivalent to the observed variation in the measures in the study sample. This exercise could be accomplished through the use of various statistical methods, e.g., principal components; however, maintaining the original measures as endpoints in clinical studies is highly desirable, so the search for a subset was undertaken with this constraint in mind. In practice, particularly if a number of measures are highly correlated, several equally valid efficient subsets may exist. The choice of the efficient subset will therefore also be based on the subjective interpretation of the measures selected.

The first specific objective of this study was to examine the relationship between commonly reported measures of knee cartilage in a relatively large cross-sectional and longitudinal study of participants with and without knee OA. The second specific objective was to identify an efficient subset of cartilage morphology measures that successfully explains most of the variation observed cross-sectionally and longitudinally in other measures not included in the efficient subset.

The following steps were taken to achieve the above objectives.

  • 1The mathematical relationships between morphologic measurements under consideration were examined.
  • 2The hypothesis that volume (VC) can be accurately predicted (both cross-sectionally and longitudinally, i.e., changes in VC) by a simple function of surface areas (tAB, AC, or cAB) and thickness (ThCcAB.Me or ThCtAB.Me) was examined.
  • 3Regression models were constructed to determine whether simple additive models limited to surface area and thickness measures could explain most of the variation in volume, again both cross-sectionally and longitudinally (i.e., change in volume).

The overarching goal of these objectives was to identify an efficient subset of morphologic measures based on known (mathematical) and empirically constructed biologic and physical relationships that maximizes the information provided while minimizing the number of variables in the subset. The potential benefits of identifying a subset of measures include increased consistency and efficiency of reporting results, increased statistical power due to reduction in multiple comparisons made on a larger number of measures, and a better understanding of the relationships between the morphologic measures of articular knee cartilage.

MATERIALS AND METHODS

The study initially included 180 women, of which 152 (age 56.7 ± 8.6 years) completed the baseline and month 24 visits. Inclusion and exclusion criteria have been described previously (4, 17, 22, 23). In brief, the inclusion criteria for the OA participants were mild to moderate radiographic OA (Kellgren-Lawrence grade [KLG] 2 or 3) in the medial femorotibial compartment in conventional weight-bearing extended anterior-posterior radiographs, frequent symptoms in the target knee, and a body mass index ≥30 kg/m2. Healthy control participants had to show a complete absence of symptoms, no sign of radiographic knee OA in the anterior-posterior radiographs (KLG0), and a body mass index ≤28 kg/m2. Additionally, Lyon Schuss radiographs (24, 25) were obtained for all participants. Both radiographic techniques were used to refine the two groups, such that the healthy group (n = 77) was KLG0 in both radiographs, while the OA group (n = 75) had KLG >0 in the anterior-posterior or Lyon Schuss radiographs. The study was conducted in compliance with the ethical principles derived from the Declaration of Helsinki and in compliance with the local institutional review board, informed consent regulations, and the International Conference on Harmonization Good Clinical Practices Guidelines.

MRI at 3.0T was performed at seven clinical sites, with the test-retest precision and stability of the measurements in this multicenter study having been reported previously (4, 22). Validated (26, 27) double-oblique coronal MRI acquisitions were obtained at baseline and 24 months, using water-excitation spoiled gradient echo sequences at a 1.0mm × 0.31mm × 0.31mm resolution (4, 22). Segmentation of the femorotibial cartilages (medial tibial plate, lateral tibial plate, weight-bearing medial femoral condyle, and weight-bearing lateral femoral condyle) was performed using custom software (Chondrometrics GmbH, Ainring, Germany) (4, 22, 23); see Fig. 1. All morphologic measures mentioned previously were computed, and change (between baseline and follow-up) was measured in the four femorotibial cartilage plates. In order to determine the mean cartilage thickness (ThCtAB.Me and ThCcAB.Me), the mean of two distance transformations, one from the tAB to the AC and one from the AC to the tAB, was considered (28).

Figure 1.

Image showing the MRI data used in the study, regions of interest in the knee, and the cartilage morphology metrics investigated. a: Double-oblique coronal SPGR (spoiled gradient recalled) sequence with 1mm slice thickness. b: Same image showing regions of interest and the VC (volume of cartilage) of the femorotibial cartilage plates: medial tibial plate (blue); cMF (yellow) = medial weight-bearing (central) femur; lateral tibial plate (green); cLF (red) = lateral weight-bearing (central) femur. c: Same image showing the segmentation of the tAB (green = total area of subchondral bone) and of the AC (magenta = area of cartilage surface). d: Detail of (c) as indicated by the rectangle in (c). The tAB covered by AC is termed cAB and the tAB not covered by AC, the dAB (denuded area of subchondral bone). Cartilage thickness (ThC) can be computed over the cAB (ThCcAB) or over the entire tAB (ThCtAB), with the dAB counting as 0mm cartilage thickness.

Coefficients of determination, R2, and standard linear regression models were used to assess statistical relationships between the various measures. Analysis of variance of regression models where VC was the dependent variable was used for variance decomposition of VC, with the goal of estimating how much variability in VC can be explained by the predictors included in the model. The analysis of variance methods used here provide type I sums of squares, i.e., the order of inclusion in the model is important unless the model variables are completely uncorrelated (which was not the case). The type I sums of squares (in percentage of total sums of squares) were used to determine the relative importance of a given predictor in the model in explaining the variation in VC, once the previous measures were included in the model.

In summary, the identification of an efficient subset was based more on a simple sequential approach relying on known physical and biologic relationships, rather than undirected statistical techniques that form basis functions empirically. The approach taken to finding this efficient subset was to first look at the known mathematical definitions connecting different measures and then examine empirically those relationships where physical or biologic descriptions of the relationships were not known or defined.

RESULTS

Tests for differences between healthy and OA groups in examined morphology relationships were carried out and found not to be significant after adjusting for multiple comparisons. Only four of a total of 64 tests were significant, with smallest P value >0.02; hence for efficiency, results reported here are based on analyses including all subjects (also see Fig. 2).

Figure 2.

Scatterplot of tAB vs ThCtAB.Me at baseline for each of the cartilage plates. \ represents OA participants; + represents healthy participants.

Theoretical Results

Several morphologic measures listed above are mathematically related. By definition, cAB ≤ tAB and cAB = tAB unless dABp = 100 × (1 − cAB/tAB) >0. Hence, three pairs of measurements, (tAB and dABp), (cAB and tAB), or (cAB and dABp), provide the same mathematical information and qualify as an efficient subset with respect to the measures cAB, tAB, and dABp. Also, by definition ThCtAB.Me = ThCcAB.Me × (cAB/tAB) = ThCcAB.Me × (100 – dABp)/100. Again, three measures, ThCtAB.Me, ThCcAB.Me, and dABp, have a direct mathematical relationship; hence, any of the three pairs (ThCtAB.Me and ThCcAB.Me), (ThCcAB.Me and dABp), or (ThCtAB.Me and dABp), qualify as an efficient subset.

Since there is considerable overlap between the two sets of pairs, the five measures with known relationships may be combined into a single group of three measures, without loss of information: (ThC*AB.aMe, *AB, dABp), with * representing either t or c, i.e., *AB could be either tAB or cAB and ThC*AB.aMe could be either ThCtAB.Me or ThCcAB.Me. Since the four triplets provide the same amount of information, any of the triplets provide a reasonable basis for forming an efficient subset.

Empiric Relationships

Since all pairs in the proposed triplets (see above) are mathematically related except for the pairs *AB and ThC*AB.aMe, the relationships between tAB and ThCtAB.Me or cAB and ThCcAB.Me were explored empirically. The coefficient of determination for the relationship between any pairing of bone surface area (tAB or cAB) and average cartilage thickness (ThCtAB.Me or ThCcAB.Me) was less than 25% over all plates both cross-sectionally (Fig. 2) and longitudinally, so the two measures had little overlap in information.

The next step of the empiric exploration was the examination of AC, which is likely to have its highest correlation with cAB as both ignore denuded areas. All plates displayed an R2 >0.91 for the relationship at baseline; hence, over 90% of variation in AC can be explained by cAB, and hence also by a function of tAB and dABp. For longitudinal studies, R2 between change in AC and change in cAB ranged from 58% to 85% across the cartilage plates.

Functional Relationship Between Cartilage Volume, Surface Area, and Thickness

In principle, VC should be related to surface areas and thickness but may be influenced by joint curvature. The thickness measurement, ThCcAB.Me, is the average cartilage thickness over the bone surface area covered with cartilage (cAB). Hence, in theory, if the bone surface area is flat, VC should simply equate to the product ThCcAB.Me × cAB. To assess the effect of joint curvature, the cartilage was assumed to have a shape of a longitudinal section of a cylinder. Under these conditions, the product ThCcAB.Me × AvgAC, where AvgAC = (cAB + AC)/2, should equal VC. These calculated cartilage volumes, cVC(AvgAC) = AvgAC × ThCcAB.Me and cVC(cAB) = cAB × ThCcAB.Me, are compared to the observed VC to assess which measure more accurately reflects observed VC.

Table 1 shows the relationship between the observed volume (VC), as measured by numerical integration of all segmented voxels and the calculated volume (cVC) at baseline for the two different models of surface area, cVC(cAB), cVC(AvgAC). While R2 was high (>98%) for all cVCs versus VC, the intraclass correlation coefficient was slightly lower for cVC(cAB) than for cVC(AvgAC) for all plates. The systematic bias as indicated by the mean difference was considerably smaller in the femoral plates using cVC(AvgAC) (−0.9%, −1.2%) compared to the bias using cVC(cAB) (5.5%, 7.7%). The flatter tibial plates showed more comparable biases between the two measures, with biases of (−1.5%, −2.2%) for cVC(AvgAC) and (1.3%, 3.3%) for cVC(cAB). Scatterplots (Figs. 3 and 4) of the relationship between VC and cVC(AvgAC) at baseline and for change over time, respectively, show the strong relationship between these two measures and the overlap in values between OA and healthy participants. AvgAC appeared to be the best choice of a surface area measure to approximate VC and will be used in this role in the remainder of the report.

Table 1. Summary Statistics for Association Between Observed Volume (VC) and Estimates of Volume Based on Surface Area (AC, cAB, or AvgAC) and ThCcAB.Me
Surface areaComparison at baselineComparison of change at 24 mo
Platemean.diffSlopeR2ICCmean.diffSlopeR2ICC
  1. ICC, intraclass correlation coefficient; cLF, weight-bearing lateral femoral plate; cMF, weight-bearing medial femoral plate; mean.diff, mean difference between observed and estimated volume computed from product of cartilage surface area and cartilage thickness.

AvgACcLF13.4290.9780.9970.996−0.1580.9710.9910.995
AvgACLT43.4360.9600.9970.991−1.6100.8920.9580.975
AvgACcMF8.8470.9830.9980.9980.8921.0050.9780.989
AvgACMT28.1490.9640.9980.9950.0860.9590.9750.987
cABcLF−60.3881.0780.9940.952−2.0421.0640.9630.977
cABLT−62.5601.0210.9960.9830.7320.9960.9800.990
cABcMF−76.5951.0800.9920.926−0.3021.0580.9710.983
cABMT−22.3030.9930.9950.9950.4300.9790.9670.983
Figure 3.

Scatterplot of VC vs cVC(AvgAC) at baseline for each of the cartilage plates. \ represents OA participants; + represents healthy participants.

Figure 4.

Scatterplot of change in VC vs change in cVC(AvgAC) for each of the cartilage plates. \ represents OA participants; + represents healthy participants.

When examining change over time in cVC versus VC (Table 1), cVC(AvgAC) and cVC(cAB) were very comparable. More important, the R2 for cVC(AvgAC) and VC were very high (>97%) across all plates both cross-sectionally and longitudinally. The goal here was not to replace VC with cVC, but given the observed strong relationship, cVC can be used as a starting point to decompose the variation observed in VC in terms of variables that make up cVC.

The strength of the relationship between cVC and VC provides useful insights into the relationship between VCtAB and average cartilage thickness, ThCtAB.Me. Substituting cVC(AvgAC) for VC, VCtAB ≈ cVC(AvgAC)/tAB = ThCtAB.Me × (1 + (AC – cAB)/2cAB). cAB and AC are very similar, so AC – cAB is relatively small compared to cAB; in fact, (AC – cAB)/2cAB ≈ 0.06 ± 0.03. The close association between VCtAB and ThCtAB.Me shown mathematically by approximation to cVC(AvgAC) can also be shown empirically. The R2 for VCtAB and ThCtAB.Me was >98.6% and the percentage difference between the two measures was <9% across all plates for baseline (Fig. 5), while for change over time R2 was >95.6% and percentage difference relative to baseline <1% across all plates.

Figure 5.

Scatterplot of ThCtAB.Me vs VCtAB at baseline for each of the cartilage plates. \ represents OA participants; + represents healthy participants.

Decomposition of VC Variability

As it has been shown that VC can be accurately described by the product of AvgAC and ThCcAB.Me, the question is how much of the variability can be attributed to each, particularly with regard to longitudinal change over time. Also, given that tAB is part of a triplet being suggested as a basis for an efficient subset and tAB and AvgAC are highly correlated, another question is whether tAB is an acceptable substitute for AvgAC.

Regression models for each plate were constructed to assess which measures, if any, explain the observed variation in VC. The initial regression models for examining baseline measures of the different plates were defined as VC = ThCcAB.Me + AvgAC + ThCcAB.Me × AvgAC. The variance decomposition of VC observed across all subjects at baseline for the four plates is shown in Table 2; results for different subgroups (healthy reference group, OA participants) were very similar. The results for three models are shown (Table 2): The results for model 1, which includes ThCcAB.Me first, showed that ThCcAB.Me explains about 67–75% of the variability in VC at baseline and that AvgAC explains nearly all the rest, with only about 1% explained by the cross-product or unexplained (residuals). When AvgAC was included in the model first, model 2, the variance decomposition of VC was split more evenly between AvgAC and ThCcAB.Me. This indicates that AvgAC does not explain as much of the variation in VC on its own as ThCcAB.Me; hence, one could consider ThCcAB.Me as the primary variable, although both are important. The final model described in Table 2 included tAB and dABp. The results show that once tAB is included, AvgAC, AvgAC × ThCcAB.Me, and residuals explain only 1-2% of the variation in VC.

Table 2. Variance Decomposition of VC at Baseline for All Subjects in Specified Plate
 cLFLTcMFMT
  1. Results are presented as percentage of total variability. Decomposition is based on type I sums of squares; hence, the percentage of variation explained by a variable is conditional on which variables are already in the model. The order in which variables are listed in the table is the order of inclusion for the different regression models.

Terms in model 1    
 ThCcAB.Me67.472.066.376.2
 AvgAC31.727.032.623.1
 ThCcAB.Me:AvgAC0.60.70.90.5
 Residuals0.30.30.20.2
Terms in model 2    
 AvgAC52.952.052.666.4
 ThCcAB.Me46.147.046.332.9
 AvgAC:ThCcAB.Me0.70.670.90.5
 Residuals0.30.320.20.2
Terms in model 3    
 ThCcAB.Me67.472.066.376.2
 tAB31.527.032.022.8
 dABp0.10.10.40.1
 AvgAC0.20.00.40.2
 ThCcAB.Me:AvgAC0.70.50.70.5
 Residuals0.20.30.20.2

The same decomposition was performed for the longitudinal data. The rate of change for VC as a product of AvgAC and ThCcAB.Me can be written as d(VC)/dt ≈ d(cVC)/dt = ThCcAB.Me(t) × d(AvgAC)/dt + AvgAC(t) × d(ThCcAB.Me)/dt. The discrete version measuring change over time has the form ΔVC ≈ ΔcVC = ΔThCcAB.Me × Mean(AvgAC) + ΔAvgAC × Mean(ThCcAB.Me), where Δ represents the change in measure between baseline and month 24, and Mean(.) is the average of month 24 and baseline visits.

The regression model included all linear and cross-product terms of this equation; hence, the model was ΔVC = ΔThCcAB.Me + ΔAvgAC + Mean(ThCcAB.Me) + Mean(AvgAC) + ΔThCcAB.Me × Mean(AvgAC) + ΔAvgAC × Mean(ThCcAB.Me).

The results shown in Table 3 indicate that the simple additive model including change in ThCcAB.Me and change in AvgAC explained over 95% of the variation in change in VC in each of the plates when all subjects were included. If change in ThCcAB.Me was included in the model first (model 1), the equivalent of including only change inThCcAB.Me in the model, it explained 74-90% of the variability of VC change, depending on the specific cartilage plate, while including only AvgAC (Model 2) explained only 31–47% of variation. Change in AvgAC explained 8-21% of variation in VC change, once change in ThCcAB.Me was included in the model (model 1; Table 3).

Table 3. Variance Decomposition of Change in VC at 24 Months for All Subjects in Specified Plate
 cLFLTcMFMT
  1. Results are presented as percentage of total variability. Decomposition is based on type I sums of squares; hence, the percentage of variation explained by a variable is conditional on which variables are already in the model. The order in which variables are listed in the table is the order of inclusion for the different regression models.

Terms in model 1    
 ΔThCcAB.Me89.977.074.576.4
 ΔAvgAC8.419.320.620.0
 Remaining terms0.80.53.01.3
 Residuals0.93.21.92.3
Terms in model 2    
 ΔAvgAC31.545.335.146.9
 ΔThCcAB.Me66.851.060.049.4
 Remaining terms0.80.53.01.3
 Residuals0.93.21.92.3

Given that tAB and dABp are related to AvgAC, these two variables were included in the model to explore whether these terms successfully explained all variation currently explained by AvgAC. Results shown in Table 4 suggest that after change in tAB and change in dABp were included in the model, the percentage of variation explained by change in AvgAC dropped to less than 3% in all plates, regardless of which subset of participants was examined.

Table 4. Decomposition of Variability in Change at 24 Months for VC Using an Explanatory Model That Includes Change in tAB and Change in Percentage dAB for Various Subsets of Subjects in Specified Plate
 cLFLTcMFMT
  1. Results are presented as percentage of total variability. Decomposition is based on type I sums of squares; hence, the percentage of variation explained by a variable is conditional on which variables are already in the model. The order in which variables are listed in the table is the order of inclusion for the different regression.

All subjects    
 ΔtAB15.743.29.027.6
 ΔThCcAB.Me81.450.078.161.3
 ΔdABp0.64.77.66.4
 ΔAvgAC0.70.21.51.2
 Remaining terms0.80.52.11.3
 Residuals0.91.41.72.3
Healthy group    
 ΔtAB23.926.412.49.7
 ΔThCcAB.Me74.070.882.984.5
 ΔAvgAC0.60.70.31.4
 Remaining terms0.90.41.91.5
 Residuals0.61.72.52.9
OA group    
 ΔtAB11.152.07.635.1
 ΔThCcAB.Me85.439.578.053.6
 ΔdABp1.16.78.97.2
 ΔAvgAC0.80.12.61.0
 Remaining terms0.70.81.71.2
 Residuals0.90.91.22.0

The variation in change in VC was then examined for the healthy reference group and OA participants separately. In the healthy reference subjects, there were no subjects with changes in denuded area, so change in dABp was not included in the model. Results for this group indicated that the percentage of variation explained by change in AvgAC was <1.4% in all plates once change in tAB was included in the model. The amount explained by change in tAB was around 25% laterally, while it was only around 10% in medial cartilage plates. In the OA participants, the percentage of variation in longitudinal change in VC explained by change in tAB was different than in healthy reference group: 52% and 35% of the variation in change in VC was explained by change in tAB in the lateral and medial tibial compartments, respectively, compared to 11% and 7% in the femoral compartments. The percentage of change in VC explained by changes in denuded area (ΔdABp) ranged from 1.1% to 9% (Table 4).

The sums of squares (Table 4) are dependent on the order of variables included in the model. Table 5 shows the variance decomposition for change in VC for the OA group as presented in Table 4, but with change in dABp included in the model before change in ThCcAB.Me. Percentage of variation explained by dABp was similar in both models for the lateral plates, but the variation explained by dABp was higher in the medial plates, with 37% of variation in longitudinal change in VC explained by change in dABp in the medial femoral plate (medial femoral condyle). The variation in change in VC explained by change in ThCcAB.Me changed inversely with variation explained by change in dABp.

Table 5. Variance Decomposition of Change in VC for OA Subjects With No Change With dABp Included in Model Before ThCcAB.Me
ThCcAB.MeOA groupcLFLTcMFMT
  1. Compare to results with comparable participant group in Table 4, where dABp is included in the model after ThCcAB.Me.

ΔtAB11.152.07.635.1
ΔdABp1.82.836.816.2
ΔThCcAB.Me84.743.450.144.5
ΔAvgAC0.80.12.61.0
Remaining terms0.70.81.71.2
Residuals0.90.91.22.0

DISCUSSION

The overarching goal of this study was to determine whether there exists a subset of MRI-based measures of knee cartilage morphology that contains nearly all the information available in an original (larger) set of measures in cross-sectional and longitudinal studies of OA.

Based on mathematical definitions relating five of these measures, it was possible to derive an efficient subset of three measures. Given the mathematical relationships, there was not a single possible efficient subset, but several. All the subsets had a common thread: one measure of the triplet was a cartilage thickness measure (ThCcAB.Me or ThCtAB.Me), one was a bone surface area measurement (cAB or tAB), and the third was percentage denuded area, dABp.

There were three other measures, VCtAB, AC, and VC, to also be considered for inclusion. The empiric results showed a strong relationship between VCtAB and cartilage thickness (ThCtAB), indicating that VCtAB could be swapped for ThCtAB, but including both would be redundant. It appeared reasonable to consider ThCtAB.Me as the best choice for inclusion in the efficient subset because it is the simpler and more intuitive measure.

For cross-sectional studies, there was little benefit to adding AC to the efficient subset as the information provided by AC was almost fully explained by cAB. The relationship between change in AC and change in the measures in the efficient subset was also fairly high for longitudinal studies, but some variation in change in AC (15-40%) was not explained by measures in the efficient subset. Hence, while not included in the efficient subset here, AC may warrant consideration for inclusion in the efficient subset in longitudinal studies.

It was shown that for cVC, the product of ThCcAB.Me and AvgAC (the average of AC and cAB), provided a highly correlated and minimally biased surrogate of the observed VC. The high correlation between calculated VC and observed VC provides an opportunity to use the calculated VC to help gain a better understanding of the relationship between VC or VCtAB, thickness, and surface area. One instance where evidence of these fundamental mathematical relationships can be used to gain further insight is through the examination of the propagation of errors inherent in the construction of measures such as VCtAB and VC. All morphology measures are derived from definitions of cartilage based on the segmentation of MR images. There is likely to be some variation or “error” in the identification of the segmentation from one image or observer to the next. The relative error can be described by the coefficient of variation for each morphology measure. Given that VC is closely estimated by the product of ThCcAB.Me and AvgAC, the rules for measuring the propagation of error show that the coefficient of variation (VC) ≈ coefficient of variation (cVC) = sqrt(coefficient of variation (ThCcAB.Me)2 + coefficient of variation (AvgAC)2). This formula assumes that AvgAC and ThCcAB.Me are not correlated, but the results indicated that the correlation between ThCcAB.Me and the surface area was minimal. Hence, this paper provides theoretical evidence and a rationale why the relative measurement error for VC is larger than for cartilage thickness or surface area, as recently reported for this study (4). Similar results hold for VCtAB.

Despite the prospect that relative measurement error is greater in the calculation of volume compared to measures of thickness or surface area, a drawback of not including volume in an efficient subset would be scenarios where changes in cartilage thickness and bone surface area are marginally significant, but change in VC may be significant. This situation can be easily monitored, however, and VC brought in at a later point to prove this case. It should be noted that the opposite case, where significant changes in tAB and ThCcAB.Me cause VC to show minimal change, may also be true and even more likely, because significant increases in tAB with aging have recently been reported in longitudinal studies (4, 29–32).

There are several potential limitations in this study. First, while the broader definitions of cartilage and bone characteristics measurements are generally agreed upon, methodological differences in how they are measured may produce different relationships. Given the general agreement in the fundamentals of these measurements (21) and the fundamental principles of measuring the basic geometry of an object, however, it seems unlikely that differences between methodologies would produce substantially different results. Second, while this study examined a number of subjects with no signs of knee OA, as well as subjects identified as having knee OA, the study inclusion criteria limited the severity of disease (KLG2 and KLG3, joint space width >2mm) presented in the study; hence, relationships may differ in subjects with more severe OA than observed in this study or in patients with other diseases. With the two samples (healthy and OA) and the underlying basic principles in measuring standard characteristics of size, however, we feel that this is a strong foundation for arguing the universal conditions indicated by the results. Moreover, the OA disease severity examined here (KLG2 and KLG3) corresponds to that typically chosen in clinical trials, for which these considerations should apply.

The variance decompositions for VC depended on the order of variables included in the model. The use of type I SSs in the variance decompositions provided the desired evidence for considering which measures, if any, were not contributing significantly (from a practical rather than statistical viewpoint) to explaining variation in VC or change in VC. Results showed that additive models including only ThCcAB.Me and tAB explained over 90% of variation in VC, with ThCcAB.Me being able to explain over 60% on its own cross-sectionally or longitudinally. Given these results, only two of the three measurements, ThCcAB.Me, tAB, and VC, are needed in the efficient subset. Since ThCcAB.Me and tAB were not strongly correlated and ThCcAB.Me and tAB were both strongly correlated with VC, choosing ThCcAB.Me and tAB over VC created an efficient subset with the most unrelated and most sensitive (see comments on error propagation above) measures. The regression models showed that longitudinal changes in VC were explained by changes in thickness and/or bone surface area. Given that cartilage thickness and bone surface area are the elemental components of VC, they also provide a more detailed assessment of changes in cartilage than VC.

When considering those variables that remained in the model, however, the order was somewhat arbitrary. It made sense to include tAB first, as this was the first measure taken during the segmentation process. Including ThCcAB.Me before dABp in the model boosts the strength of ThCcAB.Me as a predictor, because then it accounts for variation in volume change that could have been attributed to dABp. Including dABp in the model before ThCcAB.Me highlighted that much of the volume change (15-30%) related to ThCcAB.Me was associated with increases in denuded area.

While the purpose in generating the cVC was to examine the behavior of VC with respect to cartilage surface area and thickness, the cVC can also be used for quality control purposes. Given the tight relationship between VC and the product of cAB or AC and ThCcAB.Me, if the cVC deviates from observed VC too much, the result may be flagged for further review.

These arguments leave us with the triplets (*AB, ThC*AB.aMe, and dABp) as the preferred efficient subsets, because these sets maximize the information while minimizing the measures included. The possible exception is AC, where the inclusion of AC would provide minimal additional information in longitudinal studies. The triplets are all equivalent with regard to information content, so choosing between them will by definition be based on more subjective measures. The variables tAB, dABp, and ThCtAB.Me are put forth as the efficient subset of measurements for the following reasons:

  • 1The metric dABp provides information about the relationship between cAB and tAB that is not immediately apparent when considering the pair (cAB, tAB) itself.
  • 2The bone surface area, tAB, is a fundamental measure of epiphyseal bone, e.g., we know cAB ≤ tAB, so tAB is an upper bound for cAB.
  • 3The measurement ThCtAB.Me provides a summary of cartilage changes both from thinning and denuding of cartilage, unlike ThCcAB.Me, which provides a summary of thinning only and is less correlated with tAB than VC.

CONCLUSIONS

This study shows that three measures of cartilage morphology, tAB, dABp, and ThCtAB.Me, explain nearly all variation in a larger set of common cartilage morphology measures observed in cross-sectional or longitudinal studies, both in healthy and in osteoarthritic knees. These three parameters hence define an efficient subset for describing structural change in cartilage in OA. Reporting on this efficient subset of knee cartilage morphology measures in studies of knee OA using MRI-based cartilage morphology is encouraged. The benefits include increased power due to reduction in multiple comparisons, a better understanding of relationships between the morphologic measures of articular knee cartilage, and a higher efficiency in reporting results.

Appendix

The A9001140 Investigators: Eric Vignon MD, Steven A. Mazzuca PhD, Kenneth D. Brandt MD, Muriel Piperno MD, H. Cecil Charles PhD, David J. Hunter MD, Christopher Jackson MD, Virginia Byers Kraus MD, PhD, Thomas M. Link MD, Sharmila Majumdar PhD, Pottumarthi V. Prasad PhD, Thomas J. Schnitzer MD, PhD, Austin Vaz MD PhD.

Acknowledgements

This study was supported by Pfizer Inc.

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