This contribution is part of a project concerning the creation  of an artificial dataset  comprising  3D  head  scans  of craniosynostosis patients for a deep-learning-basedclassification. To conform to real data, both head and neck are required in the 3D scans. However, during patient recording, the neck is often covered by medical staff. Simply pasting an arbitrary neck leaves large gaps in the 3D mesh. We therefore use a publicly available statistical shape model (SSM) for neck reconstruction.   However, mostSSMs of   the   head are constructed    using    healthy   subjects,   so    the    full    head reconstruction loses the craniosynostosis-specific head shape. We  propose  a method  to  recover  the  neck while  keeping  the pathological   head   shape   intact. We propose   a   Laplace-Beltrami-based refinement step to deform the posterior mean shape  of  the  full  head  model  towards  the  pathological  head. The  artificial  neck  is  created  using  the  publicly  available Liverpool-York-Model.  We  apply  our  method  to  construct artificial  necks  for  head  scans  of  50  scaphocephaly  patients. Our  method  reduces  mean  vertex  correspondence  error  by approximately  1.3  mm  compared  to  the  ordinary  posterior mean  shape,  preserves  the  pathological  head  shape,  and creates  a  continuous  transition  between  neck  and  head.  The presented  method  showed  good  results  for  reconstructing  a plausible neck to craniosynostosis patients. Easily generalized it might also be applicable to other pathological shapes

Eisenmann, Urs

Freudlsperger, Christian

Hagen, Niclas

Kühle, Reinald

Nahm, Werner

Ringwald, Friedemann

Schaufelberger, Matthias

Wachter, Andreas

Weichel, Frederic

English

KITopen

  Matthias Schaufelberger*, Reinald Kühle, Frederic Weichel, Andreas Wachter, Niclas Hagen, Friedemann Ringwald, Urs Eisenmann, Christian Freudlsperger and Werner Nahm   Laplace-Beltrami Refined Shape Regression Applied to Neck Reconstruction for Craniosynostosis Patients Combining posterior shape models with a Laplace-Beltrami based approach for shape reconstructionAbstract: This contribution is part of a project concerning the creation of an artificial dataset comprising 3D head scans of craniosynostosis patients for a deep-learning-based classification. To conform to real data, both head and neck are required in the 3D scans. However, during patient recording, the neck is often covered by medical staff. Simply pasting an arbitrary neck leaves large gaps in the 3D mesh. We therefore use a publicly available statistical shape model (SSM) for neck reconstruction. However, most SSMs of the head are constructed using healthy subjects, so the full head reconstruction loses the craniosynostosis-specific head shape. We propose a method to recover the neck while keeping the pathological head shape intact. We propose a Laplace-Beltrami-based refinement step to deform the posterior mean shape of the full head model towards the pathological head. The artificial neck is created using the publicly available Liverpool-York-Model. We apply our method to construct artificial necks for head scans of 50 scaphocephaly patients. Our method reduces mean vertex correspondence error by approximately 1.3 mm compared to the ordinary posterior mean shape, preserves the pathological head shape, and creates a continuous transition between neck and head. The presented method showed good results for reconstructing a plausible neck to craniosynostosis patients. Easily generalized it might also be applicable to other pathological shapes. Keywords: Statistical Shape Modeling, Craniosynostosis, Scaphocephaly, Laplace-Beltrami, Shape Reconstruction, Posterior Shape Model https://doi.org/10.1515/cdbme-2021-2049 1 Introduction 1.1 Clinical Motivation Craniosynostosis manifests in characteristic head deformities and can lead to elevated intracranial pressure resulting in limited brain growth. Early diagnosis is crucial for surgical planning and clinical treatment. Although radiation-free classification approaches by means of machine learning have been proposed [1], computed tomography (CT) remains the gold standard for classification. It has been shown that head deformities of craniosynostosis can be captured using 3D photogrammetric scans [2]. Robust classifiers using 3D surface scans or 2D image data derived from surface scans could eliminate the need of CT thus avoiding ionizing radiation for infants. However, 3D surface scans of craniosynostosis patients are sparse, so we plan to construct a large dataset craniosynostosis patients using a Statistical Shape Model (SSM). Our long-term goal is to create 3D and 2D image data for deep-learning-based classifiers. The synthetic image data needs to contain both the characteristic head deformity of craniosynostosis as well as a full head including face and neck to conform to real data.  1.2 Shape Reconstruction  Posterior shape modeling is one of the most suitable methods for shape reconstruction. It allows incorporating a-priori ______ *Corresponding author: Matthias Schaufelberger: Karlsruhe Institute of Technology, Institute of Biomedical Engineering, Karlsruhe, publications@ibt.kit.edu  Reinald Kühle, Frederic Weichel, Christian Freudlsperger: Clinic of Oral and Maxillofacial Surgery, Heidelberg University Hospital, Heidelberg, Germany Niclas Hagen, Friedemann Ringwald, Urs Eisenmann: Institute of Medical Informatics, Heidelberg University Hospital, Heidelberg, Germany Andreas Wachter, Werner Nahm: Karlsruhe Institute of Technology, Institute of Biomedical Engineering, Karlsruhe, Germany DE GRUYTER Current Directions in Biomedical Engineering 2021;7(2): 1        1-1  4 Open Access. © 2021 The Author(s), published by De Gruyter.  This work is licensed under the Creative Commons Attribution 4.0 International License. 1919 9Laplace-Beltrami Refined Shape Regression Applied to Neck Reconstruction for Craniosynostosis Patients knowledge about the shape, modeled as a probabilistic function based on real data [3]. Consequently, the shape reconstruction is restricted to shape attributes provided by the shape model, i.e., to the training data which was used to construct the SSM. If an SSM is constructed from subject with normal head shape, the posterior mean shape will also be a normally-shaped head. To the best of our knowledge, there are no publicly available head models constructed from craniosynostosis patients to date, so we have to use a publicly available SSM of the healthy head with the necessity to recover the pathological head shape. In this contribution, we propose a Laplace-Beltrami-based deformation step to obtain a full head with neck and the craniosynostosis-specific head deformity. We propose a two-step-approach using a posterior shape model [3] and a modified version of a Laplace-Beltrami-based As-Rigid-As-Possible morphing [4]. 2 Methods 2.1 Dataset and Preprocessing For this study, we used 3D stereo photographs of scaphocephaly patients, acquired in the Department of Oral and Maxillofacial Surgery of the Heidelberg University Hospital, Heidelberg, Germany in the years 2011 to 2020. All scans were recorded using a VECTRA-360-nine-pod system (Canfield Science, Fairfield, NJ, USA) and annotated with 10 cephalometric landmarks by trained medical staff. We selected 50 patients diagnosed with scaphocephaly. The mean age of the selected patients is 0.55 ± 0.29 years. As the children were held by medical staff during recording, neck and upper torso are either covered by the children's clothes or the hands of medical staff. We therefore aim for reconstructing an artificial, but realistic-looking neck using a SSM. We use the Liverpool York Child Head Model (LYCHM) constructed from children between 2 and 15 years (8.9 ± 3.3 years) [4] with the idea to reconstruct a plausible neck for all our patients.  For correspondence establishment, we extracted the craniofacial part of the LYCHM and performed the LB regularized projection of [4] using the mean shape of the LYCHM. Note that the template shape is arbitrary, if it shares the same point identifiers as the SSM and has the same triangular mesh structure we use for shape reconstruction later. The full observation (cranium and neck) has 𝑝 = 11510 points, the partial observation (cranium only) has 𝑝 = 10620 points.  2.2 Shape regression with Laplace Beltrami regularized refinement When dealing with observations and SSMs general, we need to remove non-shape related components (rotation, translation, and scaling) from the 𝑝	points of any unaligned observation 𝐗* ∈ ℝ-×/ first. This alignment step can be performed using Procrustes Analysis applied to vertex-to-vertex correspondences to compute rotation matrix 𝐑 ∈ ℝ/×/, translation vector 𝐭 ∈ ℝ/, and isotropic scaling 𝑠 between 𝐗* and the mean shape 𝚳 ∈ ℝ-×/ of the SSM. We can therefore obtain the aligned observation 𝐗 using   𝐗 = 𝑠 ⋅ 𝐑 ⋅ 𝐗𝒖 + 𝐭𝑻 .     (1) To align partial observations with 𝑞 points with 𝑞 < 𝑝, we simply select the 𝑞 vertices from the mean shape 𝚳 and use eq. 1 in an analogous manner. Regarding the shape model notation, we will mainly rely on [3]. We convert the block matrix 𝐗 into column vector 𝐱 ∈ℝ/- (and analogous 𝚳 into 𝛍) using   𝐱 = 𝐜𝐨𝐥(𝐗) =	ABCDCECBF⋮EHI, 𝛍 = 𝐜𝐨𝐥(𝐌).  (2) The SSM consists of mean shape 𝛍 ∈ ℝ/- and sample covariance matrix 	𝚺 ∈ ℝ/-×/-. 𝚺 can be decomposed using eigenvalue decomposition 𝚺 = 𝐔𝐃N𝐔O ∈ ℝ/-×/-. It can be reduced to its first 𝑛	eigenvalues in diagonal form 𝐃 ∈ ℝQ×Q and the first 𝑛	principal components 𝐔 ∈ ℝ/-×Q. With the principal component matrix 𝐐 = 𝐔𝐃 ∈ ℝ/-×Q, we can represent a shape of the SSM as a combination of 𝛍, 𝐐, and coefficient vector 𝛂 ∈ ℝQ using  𝐱 = 𝛍 +𝐐 ⋅ 𝛂.    (3) For a given partial observation of 𝑞 points 𝐱T = col(𝐗T) ∈ℝ/X, we usually cannot find any 𝛂 according to eq. 3 to match the observation perfectly. We rather match it in terms of an observation noise 𝛜 ∈ ℝ/X. By selecting the corresponding rows and columns in 𝐐 for the 𝑞 points we obtain	𝐐T ∈ ℝ/X×Q and can model 𝐱T as   𝐱𝒈 = 𝛍𝒈 +𝐐𝒈 ⋅ 𝛂 + 𝛜.     (4) According to [3], the posterior mean shape (PMS)  𝝁\ ∈ ℝ/- of the conditional distribution can be calculated using   𝛍𝒄 = 𝛍 +𝐐^	𝐐𝒈𝑻𝐐𝒈 + 𝝈𝟐𝐈𝒏cd𝟏𝐐𝑻(𝐱𝒈 − 𝛍𝒈).  (5) 192Laplace-Beltrami Refined Shape Regression Applied to Neck Reconstruction for Craniosynostosis Patients 𝜎 models the variance of the observation. For 𝜎 → 0, we demand that the SSM match the observation accurately, which usually leads to an over-emphasis of the noisy components of the SSM. Increasing 𝜎 allows the SSM more flexibility for fitting the observation.  The PMS is a representation of the full shape created from the SSM. Features not present in the training data are not yet represented in the reconstruction. If mapped the partial observation directly onto the reconstructed shape, we would create a gap between the partial observation and the reconstruction.  By converting 𝛍\ = col(𝚳\) and 	𝐱T = col^𝐗Tc back into their respective matrix form (analogous to eq. 2), we can proceed with the LB regularized refinement. We use the LB regularized projection [4] to deform the reconstruction to match the partial observation. 𝐋\ ∈ ℝ-×- denotes the LB Operator of the PMS 𝚳\ ∈ ℝ-×/. We solve the system of linear equations for the refined points 𝐗j ∈ ℝ-×/, which is given by  k𝝀𝐋𝒄𝐒𝒄n 𝐗𝒓 =	 k𝝀𝐋𝒄 ⋅ 𝐌𝒄𝐈𝒈 ⋅ 𝐗𝒈n.   (6) 𝐈T ∈ ℝX×X denotes the identity and 𝐒\ ∈ ℝX×-  denotes a Boolean selection matrix. For each of the observed points 𝑞 of 𝐗T we selected the corresponding vertex from the 𝑝 vertices of 𝐗j which we are solving for. This equation balances retaining the morphology of the PMS 𝐌\ and mapping the partial observation 𝐗T. By changing the regularization parameter 𝜆, we can put more weight to either the correspondences of the morphology or the PMS. It can therefore be described as the "stiffness" of the transformation. For 𝜆 → ∞, the result is the rigidly transformed 𝚳\, for	𝜆 → 0, we map the observation almost directly and create an irregular mesh. For the two parameters 𝜎 and 𝜆 introduced in eq. 5 and eq. 6, we choose 𝜎 = 1 and 𝜆 = 1.  2.3 Evaluation of reconstruction methods We quantitatively evaluated our proposed LB refined morphing approach on 50 scaphocephaly patients using correspondence errors on the head between the reconstruction and the observation. With respect to the neck, a smooth transition from the head to the artificial neck is desirable. We assessed the transition by computing the gap length reduction between the boundary sets of head and neck, compared with the distance when simply mapping the partial observation onto the PSM. Qualitatively, we evaluated the fit in terms of regularity of the mesh on the transition of head and neck.  3 Results Although evaluated on all 50 subjects, for qualitative metrics, we exemplify our results on one subject (S1). Quantitative results were evaluated on all 50 subjects. Figure 1 shows the reconstruction for the PMS and our proposed LB refined approach. While the PSM shows large regions with absolute correspondence errors higher than 2 mm on the face and the cranium, the LB refined approach shows errors between ± 1 mm. Note that we restricted the color scale to ± 2 mm for the sake of comparability, although the maximum absolute correspondence error of the PSM is 9.7 mm. The PMS resembles a shape without scaphocephaly while the proposed LB refined approach shows good accordance with the elongated head of a scaphocephaly patient. Figure 2 shows the region where the left neck transitions into the left head parts. As opposed to a direct mapping, neither the PMS, nor the proposed LB refined approach show a gap.  Figure 3 shows the correspondence errors for the PMS and the LB refined reconstructions for all 50 subjects. The PMS based reconstruction consistently yield higher errors than the proposed LB refined approach. While median correspondence errors for PMSs are approximately 1.9 mm, median Figure 1: Correspondence errors of reconstructions exemplified on  S1. Top row: PMS, bottom row: LB refined. Left: Front view, right: top view. Color scale limited to ±2 mm. Figure 2: Boundary of head and neck exemplified on S1. From left to right: PMS, mapped, and LB refined. 193Laplace-Beltrami Refined Shape Regression Applied to Neck Reconstruction for Craniosynostosis Patients correspondence errors for the LB refined reconstructions are around 0.6 mm. Figure 4 shows that both PMS and the proposed LB refined approach reduce the mean gap distance for all 50 subjects using both methods. The mean and median reduction is at around 0.3 mm. This corresponds to a reduction of more than 10% as the mean of the mean edge length of the PMSs was 2.62 mm. 4 Discussion We proposed a refined shape reconstruction method using an LB-based refinement of a posterior shape model. The proposed LB refined reconstruction method reduces reconstruction errors compared with the PMS while also allowing a smooth transition from the original observation to the reconstructed part. Shape features not present in the PMS could be re-established using the proposed LB refined reconstruction. This includes shape differences and the subjects’ faces most likely introduced by age and ones introduced by head shape. We provided a general description of our method which makes it applicable to a variety of shapes. Our study has several limitations. The first limitation concerns the limited variability of both dataset and application that we provided. Given the nature of craniosynostosis, we only tested our approach on 50 patients younger than 1.5 years. We also applied our method only to neck reconstruction. When generalizing, the results should be interpreted carefully. Secondly, we used a large, homogeneous, and connected region as partial observation which might explain why the LB refined approach was able to fit the observation to the PMS well. Furthermore, the partial observations contain little noise. As the LB refined fit does not consider observation noise, it might be very sensitive to noisy observations or wrong correspondences, especially for isolated and scattered observation points. 5 Conclusion We showed that our proposed LB refined shape reconstruction method shows good results for reconstructing a plausible neck for 50 scaphocephaly patients. As opposed to ordinary posterior shape modeling, our method can be applied even if the SSM has significant differences in its training data population. Our approach should be tested in terms of robustness and its suitability for other shapes. Further applications of our method include the creation of synthetic dataset for machine learning approaches which are dependent on a conform dataset.   Author Statement Research funding: This study is part of the HEiKA research project HEiKA_19-17. Conflict of interest: Authors state no conflict of interest. References [1] G. de Jong, E. Bijlsma, J. Meulstee, M. Wennen, E. van Lindert, T. Maal, R. Aquarius, and H. Delye, “Combining deep learning with 3D stereophotogrammetry for craniosynostosis diagnosis.,” Sci Rep, vol. 10, no. 1, p. 15346, Sep. 2020. [2] J. W. Meulstee, L. M. Verhamme, W. A. Borstlap, F. Van der Heijden, G. A. De Jong, T. Xi, S. J. Bergé, H. Delye, and T. J. J. Maal, “A new method for three-dimensional evaluation of the cranial shape and the automatic identification of craniosynostosis using 3D stereophotogrammetry.,” Int J Oral Maxillofac Surg, vol. 46, no. 7, pp. 819–826, Jul. 2017. [3] T. Albrecht, M. Lüthi, T. Gerig, and T. Vetter, “Posterior shape models.,” Med Image Anal, vol. 17, no. 8, pp. 959–73, Dec. 2013.  [4] H. Dai, N. Pears, W. Smith, and C. Duncan, “A 3D Morphable Model of Craniofacial Shape and Texture Variation,” in 2017 IEEE International Conference on Computer Vision, 2017, pp. 3104–3112 Figure 3: Mean correspondence error on the head for all 50 subjects. Figure 4: Mean gap reduction on the boundary of head and neck for all 50 subjects. 194

Laplace-Beltrami Refined Shape Regression Applied to Neck Reconstruction for Craniosynostosis Patients Combining posterior shape models with a Laplace-Beltrami based approach for shape reconstruction

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Laplace-Beltrami Refined Shape Regression Applied to Neck Reconstruction for Craniosynostosis Patients Combining posterior shape models with a Laplace-Beltrami based approach for shape reconstruction

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