Figure shows a typical

*h(t) *curve computed using the proposed model-independent deconvolution method overlaid on the corresponding measured tissue enhancement curve, C

_{tis}(t), and the estimated tissue enhancement curve computed using

*h(t)*, for one subject in the study. From the plot, it is noted that after a brief time delay, while the contrast agent distributes in the myocardium,

*h(t) *begins with an initial maximum amplitude and then approximates a monotonically decaying function similar to estimates of

*h(t) *used in other model-based deconvolution methods such as 2-compartment modeling or Fermi function modeling. From the central volume principle, the maximum amplitude of

*h(t) *is related to the rate of perfusion in the tissue:

*Flow *=

*max [h(t)]/(dt*SG*_{myo}*) *[

7].

Figure shows one L-curve plot for each of the five subjects in the study. Each of the L-curves shown in Figure was computed for one representative region of the 6-region data in each of the five subjects at rest. These L-curves were typical of the L-curves computed from multiple regions of tissue enhancement in all five subjects. The dark marker at the region of maximum curvature on each L-curve represents the optimal value of λ that provides a near optimal balance of the goodness-of-fit of the estimated tissue enhancement curves and the temporal smoothness of *h(t)*, when six regions of interest per slice were used. The five optimal regional values of λ for the subjects imaged in the study ranged from 0.02 to 0.042. Subsequently, an average regularization weighting parameter of λ = 0.03 was used for the perfusion analysis of the 6-region data for all the subjects in the study.

Perfusion estimates (at rest and stress) computed using the single average λ value varied by ~1% from perfusion estimates computed using the optimal regional λ values for each subject. This small variation in perfusion estimates demonstrates the robustness of the choice of λ in the regularization process. In the iterative minimization process, fewer than 200 iterations were required for most of the CMR datasets to estimate *h(t) *with an L_{2}-norm curve-fit error less than 5%. For the cases in which the L_{2}-norm curve-fit error never reached a steady-state value less than 5%, a maximum of 600 iterations was used.

Figure shows a plot of the optimal λ values computed using L-curve analysis for all five subjects in the study versus the CNR of the measured tissue enhancement data in large uniform regions of enhancement down to enhancement data measured in individual pixels. For each subject there is a nonlinear relationship between the optimal regularization weight parameter and the CNR of the measured enhancement data. And, although the correlation between these two variables is different for each subject, the figure illustrates the range of optimal λ values between subjects. The robustness of perfusion estimates to non-optimal λ values is evaluated in Figure and Figure .

Figure shows a scatter plot of perfusion estimates in 40 single pixel regions of the myocardium (eight pixels in five LV slices; one slice per subject) when using the optimal λ value for each pixel region separately versus using one average λ value for each subject or when using a single average λ value for all 40 pixel enhancement regions. Perfusion estimates using a subject specific λ value or a single average λ value for all the subjects were not significantly different than the perfusion estimates computed using optimal λ values for each pixel region separately (p = 0.54 and p = 0.14, respectively).

Figure shows plots of the mean perfusion estimates (left) and the coefficient of variation (right) of perfusion estimates from all five subjects in the study versus changes in the regularization weight parameter, λ. The average optimal λ value for these perfusion estimates was λ = 0.03. While there may be relatively large changes in perfusion estimates across a broad range of λ values, small deviations from the optimal λ value resulted in insignificant changes in mean perfusion estimates. This demonstrates that model-independent analysis is relatively insensitive to deviations from the optimal regularization parameter. The coefficient of variation of the perfusion estimates remained nearly constant over a large range of λ values.

Regional perfusion estimates

Figure shows a scatter plot of the mean rest and stress myocardial perfusion estimates from 3 T CMR and dynamic

^{13}N-ammonia PET for three coronary artery territories of the LV in all five subjects in the study. At rest and stress, the regional perfusion estimates using 3 T CMR were 1.03 ± 0.76 ml/g/min and 2.97 ± 1.59 ml/g/min, respectively. The corresponding perfusion estimates at rest and stress using dynamic

^{13}N-ammonia PET were 0.80 ± 0.24 ml/g/min and 3.04 ± 1.14 ml/g/min. The combined rest and stress perfusion estimates were not significantly different between CMR and PET (p = 0.42) for coronary artery territorial regions of the myocardium. Similarly, the perfusion estimates were not significantly different between CMR and PET (p = 0.11) in the individual 16-segment regions of the myocardium. Figure shows a plot of the MPR values for CMR and PET imaging in the same three AHA coronary artery territory regions [

32]. The mean MPR for 3 T CMR and PET were 3.2 ± 1.7 and 3.7 ± 0.7, respectively. The best-fit linear regression between 3 T CMR and PET perfusion estimates was y = 0.90x + 0.24 (r = 0.85). Table gives a summary of the aggregate rest and stress perfusion estimates and the MPR estimates using the model-independent analysis method with 3 T CMR and PET. Figure shows a Bland-Altman plot indicating that CMR perfusion estimates had a mean overestimation of 0.12 ml/min/g versus PET perfusion estimates.

| **Table 2**Aggregate rest and stress perfusion estimates and MPR values for all five subjects imaged with 3 T CMR and dynamic ^{13}N-ammonia PET. |

Pixelwise perfusion estimates

Aggregate pixelwise perfusion estimates at rest and stress were comparable to the 6-region perfusion estimates for all five subjects in the study (6-region: 1.0 ± 0.8 ml/min/g and 3.0 ± 1.6 ml/min/g versus pixelwise: 1.1 ± 0.9 ml/min/g and 2.9 ± 2.3 ml/min/g). For this comparison, different λ values were used for the 6-region data and the pixelwise data, according to the CNR of the enhancement data. The CNR of the regional enhancement data in all five subjects was 17.3 ± 8.5 and the CNR of the pixelwise enhancement data was 4.8 ± 2.0. Figure shows 6-region and pixelwise perfusion maps (left and right columns, respectively) at rest and stress (top and bottom rows, respectively) in one LV slice from one representative subject in the study. When the average 6-region λ value was used for the pixelwise C_{tis}(t) data, aggregate pixelwise perfusion estimates from all five subjects were 25% higher than the 6-region perfusion estimates. This overestimation was partially due to using a non-optimal λ value that under-regularized the noisy pixelwise enhancement data.