Hierarchical particle optimization for cortical shape correspondence in temporal lobe resection

Abstract

Background: Anterior temporal lobe resection is an effective treatment for temporal lobe epilepsy. The post-surgical structural changes could influence the follow-up treatment. Capturing post-surgical changes necessitates a well-established cortical shape correspondence between pre- and post-surgical surfaces. Yet, most cortical surface registration methods are designed for normal neuroanatomy. Surgical changes can introduce wide ranging artifacts in correspondence, for which conventional surface registration methods may not work as intended.

Methods: In this paper, we propose a novel particle method for one-to-one dense shape correspondence between pre- and post-surgical surfaces with temporal lobe resection. The proposed method can handle partial structural abnormality involving non-rigid changes. Unlike existing particle methods using implicit particle adjacency, we consider explicit particle adjacency to establish a smooth correspondence. Moreover, we propose hierarchical optimization of particles rather than full optimization of all particles at once to avoid trappings of locally optimal particle update.

Results: We evaluate the proposed method on 25 pairs of T1-MRI with pre- and post-simulated resection on the anterior temporal lobe and 25 pairs of patients with actual resection. We show improved accuracy over several cortical regions in terms of ROI boundary Hausdorff distance with 4.29 mm and Dice similarity coefficients with average value 0.841, compared to existing surface registration methods on simulated data. In 25 patients with actual resection of the anterior temporal lobe, our method shows an improved shape correspondence in qualitative and quantitative evaluation on parcellation-off ratio with average value 0.061 and cortical thickness changes. We also show better smoothness of the correspondence without self-intersection, compared with point-wise matching methods which show various degrees of self-intersection.

Conclusion: The proposed method establishes a promising one-to-one dense shape correspondence for temporal lobe resection. The resulting correspondence is smooth without self-intersection. The proposed hierarchical optimization strategy could accelerate optimization and improve the optimization accuracy. According to the results on the paired surfaces with temporal lobe resection, the proposed method outperforms the compared methods and is more reliable to capture cortical thickness changes.

Keywords: Hierarchical particle optimization; Shape correspondence; Temporal lobe resection.

Fig. 1.

The spherical parametrization with and w/o temporal lobe resection. Circled region shows the distortion in spherical methods after spherical parametrization. Due to the distortion, spherical methods may not fully address cortical correspondence problem in temporal lobe resection.

Fig. 2.

Example shape for post-surgical surface. It defines the number of particles and their distribution. The number of particles is 163842 (∣P_pre∣ = ∣P_post∣ = 163842). They are ordered and located uniformly on the surface. Each of them has 5-6 neighborhood points, which is shown in the zoomed in figure. Initially, the locations of P_pre and P_post are not matched.

Fig. 3.

Schematic overview of particle movement on the triangular mesh. (a)-(b) two cases show that particles move in the direction of g on the triangular mesh. u^(m−1) indicates the initial location of a particle on the pre-surgical surface Ω_pre at iteration m. In (a), particle u^(m−1) is first moved to x′ which is on edge, and then to $p_3^{\prime }$ in the direction of g. In (b), particle u^(m−1) is moved to x′ which is on a vertex. (c) Particle stops moving when the length of the trajectory is equal to α∥g∥. The complete particle movement procedure is summarized in Algorithm 1.

Fig. 4.

Particle locations during hierarchical optimization. (a) initial locations of particles P_pre (blue) on Ω_pre. (b) particles P_pre (blue) on Ω_pre are updated using Algorithm 1 to be matched to P_post on Ω_post (red in (f) and (g)). (c) after the optimization, new neighborhood particles (orange) are added by an icosahedral subdivision scheme, in which they are placed at midpoints of edges formed by the original particles. Then, we repeat the optimization and subdivision procedure in (d-e). The hierarchical particle division repeats up to 7 levels of the icosahedral subdivision (∣P_pre∣ = ∣P_post∣ = 163,842) for a dense shape correspondence.

Fig. 5.

Parcellation map of full optimization and hierarchical optimization on clinical data. The white circles represent regions which are mismatched after correspondence establishment. (a) the parcellation map on pre-surgical cortical surface. (b) the transferred parcellation map on post-surgical surface using full optimization without hierarchical update. (c) the transferred parcellation map on post-surgical surface using hierarchical optimization. After optimization, we expect the post-surgical parcellation is similar to pre-surgical one in (a). The circled regions are mismatched with those in (a), which means an unpromising correspondence. Compared with full optimization, the hierarchical optimization could lead to better correspondence while reducing a chance of trappings in local minima.

Fig. 6.

Intermediate transferred maps during the hierarchical optimization on clinical data. (a) pre-surgical surface with parcellation map. (b)-(h) post-surgical surfaces with the transferred parcellation maps in each level. After optimization, we expect the post-surgical parcellation is similar to pre-surgical one in (a). With the increase of levels, the correspondence becomes more accurate. In the highest level (level 7), the established correspondence shows the best performance in (h). The circled region with arrows shows the correspondence improvement in consecutive level. We observe improvement in circled region with the increase of levels.

Fig. 7.

The parcellation transfer in the simulated data. The white circles represent regions which are mismatched after correspondence establishment. After optimization, we expect the post-surgical parcellation is similar to pre-surgical one in (a), which means a better correspondence. The circled regions are mismatched with those in (a), which means an unpromising correspondence. (b-e) parcellation map with the spherical methods. They provide smooth correspondences (no self-intersection (S-I)). However, they are sensitive to spherical mapping distortion. Therefore, they could not achieve an accurate correspondence. (f-h) parcellation map with the point-wise matching methods. (f) and (g) offer good parcellation transfer in boundary alignment. However, their established correspondence could not avoid S-I which causes the small isolated area in (f-h). The self-intersection will be discussed in Section 3.4.3. Our method offers a smooth correspondence (no S-I) with small boundary alignment errors, which means our correspondence is more accurate and smooth.

Fig. 8.

The parcellation transfer in the clinical data. The white circles represent regions which are mismatched after correspondence establishment. After optimization, we expect the post-surgical parcellation is similar to pre-surgical one in (a), which means better a correspondence. The circled regions are mismatched with those in (a), which means an unpromising correspondence. We have similar observations with Fig. 7. The spherical methods in (b-e) show worse correspondences with more mismatched regions in parcellation. Point cloud methods show better correspondences with self-intersection which causes the small isolated area in (f-h).

Fig. 8.

The parcellation transfer in the clinical data. The white circles represent regions which are mismatched after correspondence establishment. After optimization, we expect the post-surgical parcellation is similar to pre-surgical one in (a), which means better a correspondence. The circled regions are mismatched with those in (a), which means an unpromising correspondence. We have similar observations with Fig. 7. The spherical methods in (b-e) show worse correspondences with more mismatched regions in parcellation. Point cloud methods show better correspondences with self-intersection which causes the small isolated area in (f-h).

Fig. 9.

ROI-wise boundary distances in the simulated data. The asterisk (*) in (a) indicates the proposed method outperforms the baseline methods after multi-comparison correction over ROIs by FDR at q = 0.05. Our method has no HD larger than 6mm, while other methods have 3-19 ROIs. This observation indicates that our method has accurate transferred parcellation and establish better correspondence.

Fig. 10.

Self-intersection of ICP, CPD and SW in the simulated and clinical data. The first row is the simulated data and the second row is the clinical data. The scattered pink shows the regions which have self-intersection (S-I). We only found S-I in point-wise matching methods. No scattered regions are found in our method and spherical methods.

Fig. 11.

ROI-wise cortical thickness difference between pre- and post-surgical surfaces in clinical data. The asterisk (*) indicates the proposed method has significant difference with the baseline methods. In some of ROIs, spherical methods have cortical changes larger than 0.6mm. In most of ROIs, most of the methods have thickness changes less than 0.4 mm. Combined with Fig. 12, the larger difference might be caused by the registration errors. Our method could be more reliable in capturing the actual cortical thickness changes after surgery.

Fig. 12.

The parcellation transfer of IstCg in the clinical data. In (b-e) and (h), the transferred parcellation is noticeably off. Point cloud methods (f-g) have small isolated clusters in their shape correspondence. Since our method offers better transferred parcellation, combined with Fig. 11, our method could be more reliable in capturing the actual cortical thickness changes after surgery.

Fig. 13.

Portion of false positives in the non-resected regions of the clinical data across 25 subjects with different threshold. Almost no normal region (0.3% out of a total of the non-resected surface area) is classified as resected after threshold t = 7. Therefore, we set t to 7 mm to estimate the resected region for all subjects in this work.