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Hernandez,´ Frangi, and Sapiro

Table 5.6: Repeatability and agreement study within observers and methods, respectively. ObsI and ObsII stand for each observer, and MB indicates the model-based technique. The table shows

µ ± SD of the difference of the measurements in mm. The standard deviation has been corrected for repeated measurements in the agreement study [33]

 

Neck (mm)

Width (mm)

Depth (mm)

 

 

 

 

ObsI

−0.07 ± 1.09

0.94 ± 1.87

−0.65 ± 2.41

ObsII

−0.51 ± 0.86

−0.34 ± 1.35

0.18 ± 1.50

ObsI vs ObsII

−0.03 ± 1.22

0.34 ± 1.91

−0.70 ± 2.45

ObsI vs MB

−0.47 ± 1.05

0.23 ± 1.86

−0.69 ± 2.12

ObsII vs MB

−0.44 ± 0.91

−0.11 ± 1.43

0.00 ± 1.55

5.4.2.2 Bland–Altman Study

The Bland–Altman plot is a statistical method of comparison of two clinical measurement techniques. The agreement between the two techniques can be quantified using the standard deviation of the differences between observations made on the same subjects. Bland–Altman graphs show the distribution of the differences by plotting the mean against the differences of paired measurements. The information of the Bland–Altman graphs can be summarized by providing the bias (µ) and standard deviation (SD) of the differences of the measurements. The limits of agreement, defined as µ ± 1.96 · SD, provide an interval within which the 95% of the differences between measurements are expected to lie. When repeated measurements from two techniques are available, a corrected standard deviation is computed [33]. A very similar analysis to the limits of agreement approach can be applied to quantify the repeatability of a method from replicated measurements obtained from the same measurement technique.

The results of the Bland-Altman study are shown in Table 5.6. Figure 5.10 shows the Bland-Altman graphs.

5.5 Discussion

Classic GAC approaches were unsatisfactory for segmenting the cerebral vasculature from CTA and more sophisticated speed functions introducing statistical

neck

neck

 

6

 

 

 

 

 

 

 

 

 

4

 

 

 

 

 

 

 

 

[mm]

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limits of agreement

 

 

 

 

 

 

 

 

 

difference

 

 

 

 

 

 

 

 

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bias

 

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mean [mm]

 

 

 

 

 

 

 

 

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width

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

limits of agreement

 

 

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bias

difference

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depth

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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limits of agreement

 

 

 

 

 

 

 

 

 

 

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difference

 

 

 

 

 

 

 

bias

 

 

 

 

 

 

 

 

 

 

 

0

 

 

 

 

 

 

 

 

 

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−4

 

 

 

 

 

 

 

 

 

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mean [mm]

 

 

 

 

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[mm]

 

 

 

 

 

 

 

limits of agreement

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mean [mm]

 

 

 

 

 

 

 

 

 

(b)

 

 

 

 

 

6

 

 

 

 

 

width

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

4

 

 

 

 

 

 

limits of agreement

 

 

 

 

 

 

 

 

 

 

 

[mm]

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difference

−2

 

 

 

 

 

 

 

 

bias

 

 

 

 

 

 

 

 

 

 

 

0

 

 

 

 

 

 

 

 

 

 

 

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6

 

 

 

 

 

depth

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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limits of agreement

 

 

 

 

 

 

 

 

 

 

 

[mm]

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difference

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mean [mm]

 

 

 

(f)

Figure 5.10: In the first column, Bland–Altman graphs comparing the two manual measurements of each observer. Symbols and stand for ObsI and ObsII, respectively. In the second column, Bland–Altman graphs comparing the intersession measurement for manual and computerized methods. Symbol stands for ObsI vs ObsII study, for ObsI vs MB and for ObsII vs MB. In all plots the horizontal axis of the plot indicates the average and the vertical axis indicates the difference between measurements. The bias and the limits of agreement are indicated.

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(a)

(b)

(c)

(d)

Figure 5.11: Cross-section with the probability density function estimated from the Gaussian model. Brighter areas correspond to higher probabilities. (a) Original gray level image. (b–d) Probability density functions for vessel, background and bone, respectively.

information from the image were required to improve them. Most of the approaches found in the literature use a Gaussian model for the intensities of each region. Figure 5.11 shows an example of the tissue probability density functions modelled by Gaussian distributions. Compared to Fig. 5.6, it can be appreciated that the probability of vessel is higher in the transition between bone and background. The probability of bone in the interior of the aneurysm is also higher. Background tissue inside the bone has high probability for vessel tissue. The introduction of these features in the region-based term makes the model less robust and very sensitive to the parameter settings of the algorithm, which have to be tuned for each patient to compensate the effect of the misclassification.

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The use of non-parametric statistical information provides more accurate segmentations with minimal sensitivity to the selection of the parameters. In the case of Gaussian distributions, the region descriptors are time and front dependent. In contrast, the non-parametric approach presented in this work does not impose these two constraints.

The features used by the kNN rule are computed at a single scale. It would seem that, due to the nature of the object to be segmented, a multiscale approach should provide better results. However, it was observed that results were worse than when using a single scale. This could be explained by the fact that as the number of scales increases, the dimensionality of the feature space also increases. This may deteriorate the performance of the classifier owing to the peaking phenomenon [34]. We are currently working on improvements on the PDF estimation technique using a multiscale approach and dimensionality reduction strategies.

The aneurysms involved in the study had a mean size of 2.81 mm for the neck diameter, 5.40 mm for the width, and 6.44 mm for the depth with standard deviations of 0.84, 2.95 and 3.10 mm, respectively. The ANOVA reported an intraand interobserver standard deviation of less than 1.50 mm in all the cases. Results obtained in the Bland-Altman study with the manual method showed that both observers have a similar performance in independent sessions. The repeatability study shows a bias less than 0.94 mm in all the cases. The standard deviation is larger in the measurements of the aneurysm width and depth than in the neck diameter.

The agreement study indicated a bias less than 0.70 mm in all cases. The standard deviation is larger in the measurement of the aneurysm width and depth than in the neck diameter as happened in the repeatability study. This is logical as minimal variations in the selection of the view angle can yield large variations in the saccular dimensions of the aneurysm when measured on the 2D projection images. These variations are less significative at the neck due to its smaller size and symmetry.

When comparing manual and computerized measurements, it can be observed that the bias is, in the worst case, approximately of the order of a voxel (−0.69 mm). The standard deviations are lower than in the agreement study between observers in all cases. Therefore, the computerized method has a higher agreement with each observer separately than the agreement achieved between the observers themselves.

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In some of the patients, MIP images tended to induce misinterpretation of the overlapping vessels leading to wrong estimates of the neck size. Therefore, particularly in these cases, aneurysm quantification from 3D models may help in a more accurate determination of coil dimensions for the surgical intervention.

5.6 Conclusion

We have presented a method for three dimensional quantification of brain aneurysms for the purpose of surgical planning and the corresponding evaluation study. This study demonstrates the feasibility of using implicit deformable models combined with non-parametric statistical information to quantify aneurysm morphology and to obtain clinically relevant parameters. In summary, the technique presented in this work will contribute to the computerized surgical planning of coiling procedures by allowing more accurate and truly 3D quantification of brain aneurysms.

Acknowledgments

The authors would like to acknowledge Dr. R. Barrena and Dr. G. Hernandez´ from the Aragon Institute of Health Sciences (Zaragoza, Spain) for providing the data, measurements and clinical background. We also would like to acknowledge F. Memoli´ for interesting conversations during this work. MHG is supported by the grant AP2001-1678 from the Spanish Ministry of Education, Culture and Sport. AFF is supported by a Ramon y Cajal Research Fellowship from the Spanish Ministry of Science and Technology (MCyT). GS is supported by ONI28NSF. This research was partially supported by the grants TIC2002-04495- C02 and TIC2000-1635-C04-01 from the MCyT, and G03/185 from ISCIII.

Questions

1.Why are normalized the feature vectors? In the normalization of the sample feature vectors, the mean and standard deviation extracted from the training set are used. Explain why.

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2. Infer the gradient descent flow of the surface S associated with the kNN region-based energy functional

Eregion(t) = in (t) kin(x; t) dx + out (t) kout(x; t) dx

(5.23)

3.Explain the influence of each of the terms involved in the evolution of the surface driven by

S(x; t) = + −→ −→

ζ(kout in) n η(gκ g, n ) n (5.24)

t

4.How could we determine the convergence of the level set algorithm?

5.Explain what could be the advantages of using 3D models for quantification

against 2D MIP images. Regarding the evaluation study presented in this chapter, was it taking real profit of the 3D measurement capability?k −→ −

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