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Next-generation prognosis framework for pediatric spinal deformities using bio-informed deep learning networks

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Abstract

Predicting pediatric spinal deformity (PSD) from X-ray images collected on the patient’s initial visit is a challenging task. This work builds on our previous method and provides a novel bio-informed framework based on a mechanistic machine learning technique with dynamic patient-specific parameters to predict PSD. We provide a geometry-based bone growth model that can be utilized in a range of applications to enhance the bio-informed mechanistic machine learning framework. The proposed technique is utilized to examine and predict spine curvature in PSD cases such as adolescent idiopathic scoliosis. The best fit of a segmented 3D volumetric geometry of the human spine acquired from 2D X-ray images is employed. Using an active contour model based on gradient vector flow snakes, the anteroposterior and lateral views of the X-ray images are segmented to derive the 2D contours surrounding each vertebra. Using minimal user input, the snake parameters are calibrated and automatically computed over the dataset, resulting in fast image segmentation and data collection. The 2D segmented outlines of each vertebra are transformed into a 3D image segmentation result. The Iterative Closest Point mesh registration technique is then used to establish a mesh morphing approach and creates a 3D atlas spine model. Using the comprehensive 3D volumetric model, one can automatically extract spinal geometry data as inputs to the mechanistic machine learning network. Moreover, the proposed bio-informed deep learning network with the modified bone growth model achieves competitive or even superior performance against other state-of-the-art learning-based methods.Please check and confirm if the author names and initials are correct for “Yongjie Jessica Zhang” and “Wing Kam Liu”.We confirm they are correct.

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References

  1. Wiggins GC, Shaffrey CI, Abel MF, Menezes AH (2003) Pediatric spinal deformities. Neurosurg Focus 14(1):1–14

    Article  Google Scholar 

  2. Konieczny MR, Senyurt H, Krauspe R (2013) Epidemiology of adolescent idiopathic scoliosis. J Child Orthop 7(1):3–9

    Article  Google Scholar 

  3. Girdler S, Cho B, Mikhail CM, Cheung ZB, Maza N, Cho SK-W (2020) Emerging techniques in diagnostic imaging for idiopathic scoliosis in children and adolescents: A review of the literature. World Neurosurg 136:128–135

    Article  Google Scholar 

  4. Knez D, Nahle IS, Vrtovec T, Parent S, Kadoury S (2019) Computer-assisted pedicle screw trajectory planning using CT-inferred bone density: a demonstration against surgical outcomes. Med Phys 46(8):3543–3554

    Article  Google Scholar 

  5. Mischler D, Windolf M, Gueorguiev B, Nijs S, Varga P (2020) Computational optimisation of screw orientations for improved locking plate fixation of proximal humerus fractures. J Orthop Transl 25:96–104

    Google Scholar 

  6. Goerres J, Uneri A, De Silva T, Ketcha M, Reaungamornrat S, Jacobson M, Vogt S, Kleinszig G, Osgood G, Wolinsky J et al (2017) Spinal pedicle screw planning using deformable atlas registration. Phys Med Biol 62(7):2871

    Article  Google Scholar 

  7. Müller F, Roner S, Liebmann F, Spirig JM, Fürnstahl P, Farshad M (2020) Augmented reality navigation for spinal pedicle screw instrumentation using intraoperative 3D imaging. Spine J 20(4):621–628

    Article  Google Scholar 

  8. Liebmann F, Roner S, von Atzigen M, Wanivenhaus F, Neuhaus C, Spirig J, Scaramuzza D, Sutter R, Snedeker J, Farshad M et al (2020) Registration made easy–standalone orthopedic navigation with hololens. arXiv preprint arXiv:2001.06209

  9. Sarkalkan N, Weinans H, Zadpoor AA (2014) Statistical shape and appearance models of bones. Bone 60:129–140

    Article  Google Scholar 

  10. Campbell JQ, Petrella AJ (2015) An automated method for landmark identification and finite-element modeling of the lumbar spine. IEEE Trans Biomed Eng 62(11):2709–2716

    Article  Google Scholar 

  11. Andrew J, DivyaVarshini M, Barjo P, Tigga I (2020) Spine magnetic resonance image segmentation using deep learning techniques. In: 6th International conference on advanced computing and communication systems (ICACCS), IEEE, pp 945–950

  12. Li R, Niu K, Wu D, Vander Poorten E (2020) A framework of real-time freehand ultrasound reconstruction based on deep learning for spine surgery. In: 10th Conference on new technologies for computer and robot assisted surgery, 28 September 2020 to 30 September, Barcelona, Spain

  13. Baum T, Bauer JS, Klinder T, Dobritz M, Rummeny EJ, Noël PB, Lorenz C (2014) Automatic detection of osteoporotic vertebral fractures in routine thoracic and abdominal MDCT. Eur Radiol 24(4):872–880 (Springer)

    Article  Google Scholar 

  14. Tajdari F, Golgouneh A, Ghaffari A, Khodayari A, Kamali A, Hosseinkhani N (2021) Simultaneous intelligent anticipation and control of follower vehicle observing exiting lane changer. IEEE Trans Veh Technol 70(9):8567–8577

    Article  Google Scholar 

  15. Tajdari F, Ghaffari A, Khodayari A, Kamali A, Zhilakzadeh N, Ebrahimi N (2019) Fuzzy control of anticipation and evaluation behaviour in real traffic flow. In: 2019 7th International conference on robotics and mechatronics (ICRoM), IEEE, p 248–253

  16. Tajdari F, Toulkani NE, Nourimand M (2020) Intelligent architecture for car-following behaviour observing lane-changer: Modeling and control. In: 2020 10th International conference on computer and knowledge engineering (ICCKE), IEEE, p 579–584

  17. Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J Comput Phys 378:686–707 (Elsevier)

    Article  MathSciNet  MATH  Google Scholar 

  18. Tajdari M, Pawar A, Li H, Tajdari F, Maqsood A, Cleary E, Saha S, Zhang YJ, Sarwark JF, Liu WK (2021) Image-based modelling for adolescent idiopathic scoliosis: mechanistic machine learning analysis and prediction. Comput Methods Appl Mech Eng. 374:113590

  19. Roberts M, Pacheco E, Mohankumar R, Cootes T, Adams J (2010) Detection of vertebral fractures in DXA VFA images using statistical models of appearance and a semi-automatic segmentation. Osteoporos Int 21(12):2037–2046

  20. Al Arif SMR, Gundry M, Knapp K, Slabaugh G (2016) Improving an active shape model with random classification forest for segmentation of cervical vertebrae. In: International workshop on computational methods and clinical applications for spine imaging, Springer, p 3–15

  21. Cootes TF (2017) Fully automatic localisation of vertebrae in CT images using random forest regression voting. In: Computational methods and clinical applications for spine imaging: 4th international workshop and challenge, CSI 2016, held in conjunction with MICCAI 2016, Athens, Greece, October 17, 2016, Springer, Revised Selected Papers, vol 10182, p 51

  22. Bromiley P, Adams J, Cootes T (2015) Localisation of vertebrae on DXA images using constrained local models with random forest regression voting. In: Recent advances in computational methods and clinical applications for spine imaging, Springer, p 159–171

  23. Roberts MG, Cootes TF, Adams JE (2012) Automatic location of vertebrae on DXA images using random forest regression. In: International conference on medical image computing and computer-assisted intervention, Springer, p 361–368

  24. Cootes TF, Taylor CJ, Cooper DH, Graham J (1995) Active shape models-their training and application. Comput Vis Image Underst 61(1):38–59

  25. Lamecker H, Wenckebach TH, Hege H-C (2006) Atlas-based 3D-shape reconstruction from X-ray images. In: 18th International conference on pattern recognition (ICPR’06), IEEE, vol 1, p 371–374

  26. Fotsin TJT, Vázquez C, Cresson T, De Guise J (2019) Shape, pose and density statistical model for 3D reconstruction of articulated structures from X-ray images. In: 2019 41st Annual international conference of the IEEE engineering in medicine and biology society (EMBC), IEEE, p 2748–2751

  27. Ehlke M, Ramm H, Lamecker H, Hege H-C, Zachow S (2013) Fast generation of virtual X-ray images for reconstruction of 3D anatomy. IEEE Trans Visual Comput Graph 19(12):2673–2682

  28. Kim H, Lee K, Lee D, Baek N (2019) 3D reconstruction of leg bones from X-ray images using CNN-based feature analysis. In: International conference on information and communication technology convergence (ICTC), IEEE, p 669–672

  29. Reyneke CJF, Lüthi M, Burdin V, Douglas TS, Vetter T, Mutsvangwa TE (2018) Review of 2-D/3-D reconstruction using statistical shape and intensity models and X-ray image synthesis: toward a unified framework. IEEE Rev Biomed Eng 12:269–286

  30. Deschênes S, Charron G, Beaudoin G, Labelle H, Dubois J, Miron M-C, Parent S (2010) Diagnostic imaging of spinal deformities: reducing patients radiation dose with a new slot-scanning X-ray imager. Spine 35(9):989–994

  31. Le Bras A, Laporte S, Mitton D, De Guise J, Skalli W (2003) Three-dimensional (3D) detailed reconstruction of human vertebrae from low-dose digital stereoradiography. Eur J Orthop Surg Traumatol 13(2):57–62

  32. Dubousset J, Charpak G, Dorion I, Skalli W, Lavaste F, Deguise J, Kalifa G, Ferey S (2005) A new 2D and 3D imaging approach to musculoskeletal physiology and pathology with low-dose radiation and the standing position: the EOS system. Bulletin de l’Academie Nationale de Medecine 189(2):287–97

    Article  Google Scholar 

  33. Bashkuev M, Reitmaier S, Schmidt H (2018) Effect of disc degeneration on the mechanical behavior of the human lumbar spine: a probabilistic finite element study. Spine J 18(10):1910–1920

  34. Bashkuev M, Reitmaier S, Schmidt H (2020) Relationship between intervertebral disc and facet joint degeneration: a probabilistic finite element model study. J Biomech 102

  35. Bah MT, Nair PB, Browne M (2009) Mesh morphing for finite element analysis of implant positioning in cementless total hip replacements. Med Eng Phys 31(10):1235–1243

  36. Pasha S, Shah S, Newton P, Group HS et al (2021) Machine learning predicts the 3D outcomes of adolescent idiopathic scoliosis surgery using patient–surgeon specific parameters. Spine 46(9):579–587

    Article  Google Scholar 

  37. Peng L, Lan L, Xiu P, Zhang G, Hu B, Yang X, Song Y, Yang X, Gu Y, Yang R et al (2020) Prediction of proximal junctional kyphosis after posterior scoliosis surgery with machine learning in the Lenke 5 adolescent idiopathic scoliosis patient. Front Bioeng Biotechnol 8:1–10

  38. Liang R, Yip J, To K-TM, Fan Y (2021) Machine learning approaches to predict scoliosis. In: International conference on applied human factors and ergonomics, Springer, p 116–121

  39. Cho J-S, Cho Y-S, Moon S-B, Kim M-J, Lee HD, Lee SY, Ji Y-H, Park Y-S, Han C-S, Jang S-H (2018) Scoliosis screening through a machine learning based gait analysis test. Int J Precis Eng Manuf 19(12):1861–1872

  40. Saha S, Gan Z, Cheng L, Gao J, Kafka OL, Xie X, Li H, Tajdari M, Kim HA, Liu WK (2021) Hierarchical deep learning neural network (HIDENN): an artificial intelligence (AI) framework for computational science and engineering. Comput Methods Appl Mech Eng 373

  41. Zhang YJ (2016) Geometric modeling and mesh generation from scanned images. Chapman and Hall/CRC, New York

  42. Kass M, Witkin A, Terzopoulos D (1988) Snakes: active contour models. Int J Comput Vis 1(4):321–331

  43. Tajdari F, Roncoli C, Papageorgiou M (2020) Feedback-based ramp metering and lane-changing control with connected and automated vehicles. IEEE Trans Intell Transport Syst 1–13

  44. Tajdari F, Ebrahimi Toulkani N (2021) Implementation and intelligent gain tuning feedback-based optimal torque control of a rotary parallel robot. J Vib Control 1–18

  45. MATLAB Convhull Function (2020) The MathWorks. Natick, MA, USA

  46. Tajdari M, Tajdari F, Pawar A, Zhang J, Liu W (2021) 2D to 3D volumetric reconstruction of human spine for diagnosis and prognosis of spinal deformities. In: Conference: 16th US national congress on computational mechanics

  47. Pawar A, Zhang Y, Jia Y, Wei X, Rabczuk T, Chan CL, Anitescu C (2016) Adaptive FEM-based nonrigid image registration using truncated hierarchical B-splines. Comput Math Appl 72(8):2028–2040

  48. Pawar A, Zhang YJ, Anitescu C, Jia Y, Rabczuk T (2018) DTHB3D-Reg: dynamic truncated hierarchical B-spline based 3D nonrigid image registration. Commun Comput Phys 23(3):877–898

    Article  MathSciNet  MATH  Google Scholar 

  49. Pawar A, Zhang YJ, Anitescu C, Rabczuk T (2019) Joint image segmentation and registration based on a dynamic level set approach using truncated hierarchical B-splines. Comput Math Appl 78:3250–3267

  50. Amberg B, Romdhani S, Vetter T (2007) Optimal step nonrigid ICP algorithms for surface registration. In: IEEE Conference on computer vision and pattern recognition, p 1–8

  51. Tajdari F, Kwa F, Versteegh C, Huysmans T, Song Y. Dynamic 3D mesh reconstruction based on nonrigid iterative closest-farthest points registration. In: International design engineering technical conferences and computers and information in engineering conference, vol 2022, p 1–9

  52. Tajdari F, Eijck C, Kwa F, Versteegh C, Huysmans T, Song Y (2022) Optimal position of cameras design in a 4D foot scanner. In: International design engineering technical conferences and computers and information in engineering conference, vol 2022, p 1–9

  53. Cheung JPY, Cheung PWH, Samartzis D, Luk KD-K (2018) Curve progression in adolescent idiopathic scoliosis does not match skeletal growth. Clin Orthop Relat Res 476(2):429

  54. Stokes I (2002) Mechanical effects on skeletal growth. J Musculoskelet Neuron Interact 2(3):277–280

  55. MATLAB Deep Learning Toolbox (2018) The MathWorks. Natick, MA, USA

  56. Agarap AF (2018) Deep learning using rectified linear units (RELU). arXiv preprint arXiv:1803.08375

  57. Stone M (1974) Cross-validatory choice and assessment of statistical predictions. J R Stat Soc Ser B (Methodol) 36(2):111–133

  58. Tajdari F, Toulkani NE, Zhilakzadeh N (2020) Intelligent optimal feed-back torque control of a 6dof surgical rotary robot. In: 2020 11th Power electronics, drive systems, and technologies conference (PEDSTC), IEEE, p 1–6

  59. Yang Y, Yuan T, Huysmans T, Elkhuizen WS, Tajdari F, Song Y (2021) Posture-invariant three dimensional human hand statistical shape model. J Comput Inf Sci Eng 21(3)

  60. Tajdari F, Tajdari M, Rezaei A (2021) Discrete time delay feedback control of stewart platform with intelligent optimizer weight tuner. In: 2021 IEEE international conference on robotics and automation (ICRA), IEEE, p 12701–12707

  61. Tajdari F, Toulkani NE, Zhilakzadeh N (2020) Semi-real evaluation, and adaptive control of a 6dof surgical robot. In: 2020 11th Power electronics, drive systems, and technologies conference (PEDSTC), IEEE, p 1–6

  62. Tajdari F, Roncoli C (2021) Adaptive traffic control at motorway bottlenecks with time-varying fundamental diagram. IFAC-PapersOnLine 54(2):271–277

    Article  Google Scholar 

  63. Tajdari F, Roncoli C (2022) Online set-point estimation for feedback-based traffic control applications. arXiv preprint arXiv:2207.13467

  64. Tajdari F, Huysmans T, Yang Y, Song Y (2022) Feature preserving non-rigid iterative weighted closest point and semi-curvature registration. IEEE Trans Image Process 31:1841–1856

  65. Dekker M (1986) Mathematical programming. CRC, May 4

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Acknowledgements

We would like to thank the Division of Orthopaedic Surgery and Sports Medicine at the Ann and Robert H. Lurie Children’s Hospital of Chicago for their collaboration on this project, which was made possible by a philanthropic grant. W. K. Liu would like to acknowledge the support of NSF CMMI-1762035. M. Tajdari would like to thank Madeleine Handwork for her assistance in data processing. A. Pawar and Y. J. Zhang were supported in part by the NSF grant CMMI-1953323. Moreover, F. Tajdari is partly supported by the Dutch NWO Next UPPS under Grant 15470 - Integrated Design Methodology for Ultra Personalised Products and Services Project.

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Author notes

  1. Mahsa Tajdari and Farzam Tajdari contributed equally to this work.

Authors and Affiliations

  1. Department of Mechanical Engineering, Northwestern University, Evanston, 60208, IL, USA

    Mahsa Tajdari & Wing Kam Liu

  2. Faculty of Industrial Design Engineering, Technical University of Delft, Mekelweg 5, Delft, 2628 CD, Netherlands

    Farzam Tajdari, Toon Huysmans & Yu Song

  3. Department of Computer Science, Virginia Tech, Blacksburg, VA, 24061, USA

    Pouyan Shirzadian

  4. School of Mechanical Engineering, Purdue University, West Lafayette, IN, 47907, USA

    Aishwarya Pawar

  5. Stanford Cardiovascular Institute and Department of Radiology, Stanford University, Stanford, CA, 94305, USA

    Mirwais Wardak

  6. Theoretical and Applied Mechanics, Northwestern University, Evanston, IL, 60208, USA

    Sourav Saha & Chanwook Park

  7. Department of Mechanical Engineering, Carnegie Mellon University, Pittsburgh, PA, 15213, USA

    Yongjie Jessica Zhang

  8. Department of Orthopaedic Surgery, Northwestern University Feinberg School of Medicine, Chicago, IL, 60611, USA

    John F. Sarwark

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Appendices

Appendix A: Image segmentation using the Snakes method

Image segmentation using the Snakes method [42] is a fast and efficient technique to detect important object contours from images. The limitation of this method is that it requires initialization of the contour to be done manually. The initial contour determines the accuracy of the segmentation method. The framework involves a variational formulation in which the total energy functional consists of three main terms, namely the image energy term which attracts the contour to the salient features in the image, internal spline energy term which introduces smoothness and regularity in the evolving contour and external constrain term which aligns the contour near local minima. \(v(s)=(x(s),y(s))\) is the parametric representation of a snake where parameter \(s \in [0,1]\). As s changes smoothly, a closed contour on a plane is traced. The total energy functional proposed in [42] considers both image and constraint forces is given as

$$\begin{aligned} E(\mathbf{v} (s))= & {} \int _0^1 \! E_{\text {int}}(\mathbf{v} (s)) + E_{\text {con}}(\mathbf{v} (s)) \nonumber \\&\quad + E_{\text {img}}(\mathbf{v} (s)) \, \mathrm {d}s, \end{aligned}$$
(A1)

where \(E_{\text {int}}(\mathbf{v} (s))\), \(E_{\text {img}}(\mathbf{v} (s))\) and \(E{\text {con}}(\mathbf{v} (s))\) are the energy functionals associated with internal spline energy, image force and external constraint energy, respectively. The \(E_{\text {int}}(\mathbf{v} (s))\) term is given as

$$\begin{aligned} \begin{aligned} E_{\text {int}}(\mathbf{v} (s)) = \alpha \left|\mathbf{v} '(s) \right|^2 + \beta \left|\mathbf{v} ''(s) \right|^2, \end{aligned} \end{aligned}$$
(A2)

where \(\alpha\) and \(\beta\) are weights associated with the first- and second-order regularization terms which are elastic length and stiffness of the contour. \(E_{\text {img}}(\mathbf{v} (s))\) is defined as

$$\begin{aligned} \begin{aligned}&E_{\text {img}}(\mathbf{v} (s)) = {}\, w_{\text {line}} E_{\text {line}}(\mathbf{v} (s)) + w_{\text {edge}}E_{\text {edge}}(\mathbf{v} (s)) \\&\quad + w_{\text {term}}E_{\text {term}}(\mathbf{v} (s)), \end{aligned} \end{aligned}$$
(A3)

where \(w_{\text {line}}\), \(w_{\text {edge}}\) and \(w_{\text {term}}\) are the weighting coefficients associated with the energy functionals \(E_{\text {line}} = I(x,y)\), \(E_{\text {edge}} = -|\bigtriangledown I(x,y)|^2\) and \(E_{\text {term}} = \frac{C_{yy}C_{x}^2 - 2C_{xy}C_xC_y + C_{xx}C_{y}^2}{(C_x^2 + C_y^2)^{\frac{2}{3}}}\), I(x, y) is the image intensity, \(C(x,y)=G_\sigma (x,y)*I(x,y)\) and \(G_\sigma\) is a Gaussian of standard deviation \(\sigma\) [42].

Appendix B: Point cloud registration

This section introduces the employed non-rigid ICP approach, where a concise description of the approach is given based on the conventional ICP algorithm [50].

1.1 B.1 The approach

In the registration process of the non-rigid ICP, the source surface \(\mathcal {S} = (\mathcal {V}, \mathcal {E})\), consisting of n vertices in \(\mathcal {V}\) and m edges in \(\mathcal {E}\), is registered to the target surface \(\mathcal {T}\) step by step. Fig. 17 illustrates a step of the registration process. In the figure, the meshes are assumed to be triangular meshes, and the vertices are labeled by numbers. In this step, first, the correspondences between vertices \(v_i\) in the source surface \(\mathcal {S}\) (green) and vertices \(u_i\) in the target surface \(\mathcal {T}\) (red) are established.

In the use of a conventional ICP method, given a point on \(\mathcal {S}\), the closest point on \(\mathcal {T}\) is considered as its corresponding point [59]. Then \(v_i\) is transformed by locally affine transformations (\(X_i\)) towards the target surface \(\mathcal {T}\) (red). The transformed source surface is \(\mathcal {S}(X)\) (blue). This procedure iterates till an optimal stable state [60,61,62,63] is obtained.

Fig. 17
figure 17

Match the source surface to the target surface [64]

Full size image

1.2 B.2 3D mesh registration

Here, based on the established correspondences (\(v_i, u_i\)), a cost function consisting of different terms is defined and then minimized with guaranteed stability, convergence, and robustness [50]. In the following, we introduce each term in the cost function first, and then we describe the optimization process based on the cost function.

For a non-rigid registration, the distance of the deformed source and the target should be minimized. Thus, a distance term is selected as the first component of the cost function to be minimised as

$$\begin{aligned} E_d&= \sum _{v_i \in \mathcal {V}}^{} w_i \left|X_iv_i-u_i \right|^2, \end{aligned}$$
(B4)

where \(w_i\) is the weight of the distance term and X describes a set of transformations of displaced source vertices \(\mathcal {V}(X)\). The transformation matrix \(X_i\) for each vertex in the source is a \(3 \times 4\) transformation matrix as:

$$\begin{aligned} X_i =\left[ \begin{matrix} r_{xx} &{} r_{xy} &{} r_{xz} &{} d_{x}\\ r_{yx} &{} r_{yy} &{} r_{yx} &{} d_{y}\\ r_{zx} &{} r_{zy} &{} r_{zz} &{} d_{z} \end{matrix} \right] , \end{aligned}$$
(B5)

where r, and d define all afine transformations. The transformation matrix X of all vertices is described in a \(4n \times 3\) matrix as \(X =\left[ X_1 \ldots X_n\right] ^{\text {T}}\).

A canonical form of Eq. (B4) is addressed in Eq. (B6), introduced by swapping the position of transformation matrix, and correspondences \((v_i, u_i)\). The sparse matrix D is formed to facilitate the transformation of the source vertices with the individual transformations contained in X via matrix multiplication, and denoted as \(D = diag(v_1^{\text {T}}, v_2^{\text {T}}, \ldots , v_n^{\text {T}})\). The corresponding points are also arranged as \(U~=~\left[ u_1 \ldots u_n\right] ^{\text {T}}\) and the distance term can be derived as:

$$\begin{aligned} E_d = \left|W\left( DX-U\right) \right|_F^2 \end{aligned}$$
(B6)

where W is a diagonal matrix consisting of weights \(w_i\). To regularise the deformation, an additional stiffness term is employed. Using the Frobenius norm \(\left|. \right|_F\), the stiffness term penalizes difference of the transformations of neighboring vertices, through a weighting matrix \(G = diag(1, 1, 1, \gamma )\). We have

$$\begin{aligned} E_s = \sum _{i,j \in \mathcal {E}}\left|\left( X_i-X_j\right) G \right|_F^2. \end{aligned}$$
(B7)

During the deformation, \(\gamma\) is a parameter to stress differences in the skew and rotational part against the translational part of the deformation. The value of \(\gamma\) can be specified based on data units and the types of deformation [50].

Addressing the function of the stiffness term to penalise differences of transformation matrices of the neighboring vertices, the node-arc incidence matrix M (e.g. Dekker [65]) of the template mesh topology is employed to convert the stiffness term functional into a matrix form. As the matrix is fixed for directed graphs, the construction is one row for each edge of the mesh and one column per vertex. To establish the node-arc incidence matrix of the source topology, the indices (i.e. the subscripts) of edges and vertices are addressed, for any edge of r which is connected to vertices (i, j) , in \(r^{th}\) row of M, and the nonzero entries are \(M_{ri} = -1\) and \(M_{rj} = 1\). Therefore, we formulate the stiffness term as

$$\begin{aligned} E_s = \left|\left( M \otimes G\right) X \right|^{2}_{F}. \end{aligned}$$
(B8)

Briefly, the Amberg’s method accounts for an optimal step with non-rigid ICP approach being capable to employ different regularisations, while they are using a range of lowering stiffness parameter. Thus, the cost function of Eq. (B6) and Eq. (B8) are changed to:

$$\begin{aligned} E(X) = \left|\left[ \begin{matrix} \gamma M \otimes G\\ WD\end{matrix}\right] X-\left[ \begin{matrix} 0\\ WU\end{matrix}\right] \right|^{2}_{F}. \end{aligned}$$
(B9)

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Tajdari, M., Tajdari, F., Shirzadian, P. et al. Next-generation prognosis framework for pediatric spinal deformities using bio-informed deep learning networks. Engineering with Computers 38, 4061–4084 (2022). https://doi.org/10.1007/s00366-022-01742-2

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