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Enhanced head-skull shape learning using statistical modeling and topological features

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Abstract

Skull prediction from the head is a challenging issue toward a cost-effective therapeutic solution for facial disorders. This issue was initially studied in our previous work using full head-to-skull relationship learning. However, the head-skull thickness topology is locally shaped, especially in the face region. Thus, the objective of the present study was to enhance our head-to-skull prediction problem by using local topological features for training and predicting. Head and skull feature points were sampled on 329 head and skull models from computed tomography (CT) images. These feature points were classified into the back and facial topologies. Head-to-skull relations were trained using the partial least square regression (PLSR) models separately in the two topologies. A hyperparameter tuning process was also conducted for selecting optimal parameters for each training model. Thus, a new skull could be generated so that its shape was statistically fitted with the target head. Mean errors of the predicted skulls using the topology-based learning method were better than those using the non-topology-based learning method. After tenfold cross-validation, the mean error was enhanced 36.96% for the skull shapes and 14.17% for the skull models. Mean error in the facial skull region was especially improved with 4.98%. The mean errors were also improved 11.71% and 25.74% in the muscle attachment regions and the back skull regions respectively. Moreover, using the enhanced learning strategy, the errors (mean ± SD) for the best and worst prediction cases are from 1.1994 ± 1.1225 mm (median: 0.9036, coefficient of multiple determination (R2): 0.997274) to 3.6972 ± 2.4118 mm (median: 3.9089, R2: 0.999614) and from 2.0172 ± 2.0454 mm (median: 1.2999, R2: 0.995959) to 4.0227 ± 2.6098 mm (median: 3.9998, R2: 0.998577) for the predicted skull shapes and the predicted skull models respectively. This present study showed that more detailed information on the head-skull shape leads to a better accuracy level for the skull prediction from the head. In particular, local topological features on the back and face regions of interest should be considered toward a better learning strategy for the head-to-skull prediction problem. In perspective, this enhanced learning strategy was used to update our developed clinical decision support system for facial disorders. Furthermore, a new class of learning methods, called geometric deep learning will be studied.

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Acknowledgements

The authors would like to thank the Métropole Européenne de Lille (MEL) and the I-SITE ULNE (Université Lille Nord Europe) for funding.

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Authors and Affiliations

  1. Ho Chi Minh City University of Technology and Education, Ho Chi Minh City, Vietnam

    Tan-Nhu Nguyen & Vi-Do Tran

  2. Thu Dau Mot University, Binh Duong, Vietnam

    Ho-Quang Nguyen

  3. Université de technologie de Compiègne, CNRS, Biomechanics and Bioengineering, Centre de Recherche Royallieu, CS 60 319- 60 203, Compiègne Cedex, France

    Duc-Phong Nguyen

  4. Univ. Lille, CNRS, Centrale Lille, UMR 9013 - LaMcube - Laboratoire de Mécanique, Multiphysique, Multiéchelle, 59655 Villeneuve d’Ascq Cedex, F-59000, Lille, France

    Tien-Tuan Dao

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Appendices

Appendix 1

Mathematical formulations of distances, thicknesses, and sampling rays of the back and facial topologies

$${{\varvec{V}}}_{i}^{BS}\left[M\times 3\right]\equiv \left[\begin{array}{c}{{\varvec{v}}}_{i1}^{BS}\\ \vdots \\ {{\varvec{v}}}_{iM}^{BS}\end{array}\right]\equiv \left[\begin{array}{ccc}{x}_{i1}^{BS}& {y}_{i1}^{BS}& {z}_{i1}^{BS}\\ \vdots & \vdots & \vdots \\ {x}_{iM}^{BS}& {y}_{iM}^{BS}& {z}_{iM}^{BS}\end{array}\right],i=1, 2, \dots , {N}_{\mathrm{subjects}}$$
(13)
$${{\varvec{V}}}_{i}^{FS}\left[N\times 3\right]\equiv \left[\begin{array}{c}{{\varvec{v}}}_{i1}^{FS}\\ \vdots \\ {{\varvec{v}}}_{iN}^{FS}\end{array}\right]\equiv \left[\begin{array}{ccc}{x}_{i1}^{FS}& {y}_{i1}^{FS}& {z}_{i1}^{FS}\\ \vdots & \vdots & \vdots \\ {x}_{iN}^{FS}& {y}_{iN}^{FS}& {z}_{iN}^{FS}\end{array}\right],i=1, 2, \dots , {N}_{\mathrm{subjects}}$$
(14)
$${{\varvec{V}}}_{i}^{BH}\left[M\times 3\right]\equiv \left[\begin{array}{c}{{\varvec{v}}}_{i1}^{BH}\\ \vdots \\ {{\varvec{v}}}_{iM}^{BH}\end{array}\right]\equiv \left[\begin{array}{ccc}{x}_{i1}^{BH}& {y}_{i1}^{BH}& {z}_{i1}^{BH}\\ \vdots & \vdots & \vdots \\ {x}_{iM}^{BH}& {y}_{iM}^{BH}& {z}_{iM}^{BH}\end{array}\right],i=1, 2, \dots , {N}_{\mathrm{subjects}}$$
(15)
$${{\varvec{V}}}_{i}^{FH}\left[N\times 3\right]\equiv \left[\begin{array}{c}{{\varvec{v}}}_{i1}^{FH}\\ \vdots \\ {{\varvec{v}}}_{iN}^{FH}\end{array}\right]\equiv \left[\begin{array}{ccc}{x}_{i1}^{FH}& {y}_{i1}^{FH}& {z}_{i1}^{FH}\\ \vdots & \vdots & \vdots \\ {x}_{iN}^{FH}& {y}_{iN}^{FH}& {z}_{iN}^{FH}\end{array}\right],i=1, 2, \dots , {N}_{\mathrm{subjects}}$$
(16)
$${{\varvec{V}}}^{BSV}\left[M\times 3\right]\equiv \left[\begin{array}{c}{{\varvec{v}}}_{1}^{BSV}\\ \vdots \\ {{\varvec{v}}}_{M}^{BSV}\end{array}\right]\equiv \left[\begin{array}{ccc}{x}_{1}^{BSV}& {y}_{1}^{BSV}& {z}_{1}^{BSV}\\ \vdots & \vdots & \vdots \\ {x}_{M}^{BSV}& {y}_{M}^{BSV}& {z}_{M}^{BSV}\end{array}\right]$$
(17)
$${{\varvec{V}}}^{FSV}\left[N\times 3\right]\equiv \left[\begin{array}{c}{{\varvec{v}}}_{1}^{FSV}\\ \vdots \\ {{\varvec{v}}}_{N}^{FSV}\end{array}\right]\equiv \left[\begin{array}{ccc}{x}_{1}^{FSV}& {y}_{1}^{FSV}& {z}_{1}^{FSV}\\ \vdots & \vdots & \vdots \\ {x}_{N}^{FSV}& {y}_{N}^{FSV}& {z}_{N}^{FSV}\end{array}\right]$$
(18)
$${{\varvec{d}}}_{i}^{B}\left[M\times 1\right]\equiv \left[\begin{array}{c}{d}_{i1}^{B}\\ \vdots \\ {d}_{iM}^{B}\end{array}\right]\equiv \left[\begin{array}{c}\Vert {{\varvec{v}}}_{i1}^{BH}-{{\varvec{v}}}^{{\varvec{S}}{\varvec{C}}}\Vert \\ \vdots \\ \Vert {{\varvec{v}}}_{iM}^{BH}-{{\varvec{v}}}^{{\varvec{S}}{\varvec{C}}}\Vert \end{array}\right],i=1, 2, \dots , {N}_{\mathrm{subjects}}$$
(19)
$${{\varvec{d}}}_{i}^{F}\left[N\times 1\right]\equiv \left[\begin{array}{c}{d}_{i1}^{F}\\ \vdots \\ {d}_{iN}^{F}\end{array}\right]\equiv \left[\begin{array}{c}\Vert {{\varvec{v}}}_{i1}^{FH}-{{\varvec{v}}}^{{\varvec{S}}{\varvec{C}}}\Vert \\ \vdots \\ \Vert {{\varvec{v}}}_{iN}^{FH}-{{\varvec{v}}}^{{\varvec{S}}{\varvec{C}}}\Vert \end{array}\right],i=1, 2, \dots , {N}_{\mathrm{subjects}}$$
(20)
$${{\varvec{t}}}_{i}^{B}\left[M\times 1\right]\equiv \left[\begin{array}{c}{t}_{i1}^{B}\\ \vdots \\ {t}_{iM}^{B}\end{array}\right]\equiv \left[\begin{array}{c}\Vert {{\varvec{v}}}_{i1}^{BH}-{{\varvec{v}}}_{i1}^{BS}\Vert \\ \vdots \\ \Vert {{\varvec{v}}}_{iM}^{BH}-{{\varvec{v}}}_{iM}^{BS}\Vert \end{array}\right],i=1, 2, \dots , {N}_{\mathrm{subjects}}$$
(21)
$${{\varvec{t}}}_{i}^{F}\left[N\times 1\right]\equiv \left[\begin{array}{c}{t}_{i1}^{F}\\ \vdots \\ {t}_{iN}^{F}\end{array}\right]\equiv \left[\begin{array}{c}\Vert {{\varvec{v}}}_{i1}^{FH}-{{\varvec{v}}}_{i1}^{FS}\Vert \\ \vdots \\ \Vert {{\varvec{v}}}_{iN}^{FH}-{{\varvec{v}}}_{iN}^{FS}\Vert \end{array}\right],i=1, 2, \dots , {N}_{\mathrm{subjects}}$$
(22)
$${{\varvec{I}}}^{B}[M\times 3]\equiv \left[\begin{array}{c}{{\varvec{i}}}_{1}^{B}\\ \vdots \\ {{\varvec{i}}}_{M}^{B}\end{array}\right]\equiv \left[\begin{array}{c}\frac{{{\varvec{v}}}_{1}^{BSV}-{{\varvec{v}}}^{SC}}{\Vert {{\varvec{v}}}_{1}^{BSV}-{{\varvec{v}}}^{SC}\Vert }\\ \vdots \\ \frac{{{\varvec{v}}}_{M}^{BSV}-{{\varvec{v}}}^{SC}}{\Vert {{\varvec{v}}}_{M}^{BSV}-{{\varvec{v}}}^{SC}\Vert }\end{array}\right]$$
(23)
$${{\varvec{I}}}^{F}[N\times 3]\equiv \left[\begin{array}{c}{{\varvec{i}}}_{1}^{F}\\ \vdots \\ {{\varvec{i}}}_{N}^{F}\end{array}\right]\equiv \left[\begin{array}{c}\frac{{{\varvec{v}}}_{1}^{FSV}-{{\varvec{v}}}^{SC}}{\Vert {{\varvec{v}}}_{1}^{FSV}-{{\varvec{v}}}^{SC}\Vert }\\ \vdots \\ \frac{{{\varvec{v}}}_{N}^{FSV}-{{\varvec{v}}}^{SC}}{\Vert {{\varvec{v}}}_{N}^{FSV}-{{\varvec{v}}}^{SC}\Vert }\end{array}\right]$$
(24)

in which \({x}_{ij}^{BS}\)(or \({x}_{ij}^{FS}\)), \({y}_{ij}^{BS}\)(or \({y}_{ij}^{FS}\)), and \({z}_{ij}^{BS}\)(or \({z}_{ij}^{BS}\)) are \(x\), \(y\), and \(z\) coordinates of the \({j\mathrm{th}}\) back (or facial) skull feature point on the \({i\mathrm{th}}\) head-skull pair. \({x}_{ij}^{BH}\)(or \({x}_{ij}^{FH}\)), \({y}_{ij}^{BH}\)(or \({y}_{ij}^{FH}\)), and \({z}_{ij}^{BH}\)(or \({z}_{ij}^{FH}\)) are \(x\), \(y\), and \(z\) coordinates of the \({j\mathrm{th}}\) back (or facial) head feature point on the \({i\mathrm{th}}\) head-skull pair. \({x}_{j}^{BSV}\)(or \({x}_{j}^{FSV}\)), \({y}_{j}^{BSV}\)(or \({y}_{j}^{FSV}\)), and \({z}_{j}^{BSV}\)(or \({z}_{j}^{FSV}\)) are \(x\), \(y\), and \(z\) coordinates of the \({j\mathrm{th}}\) back (or facial) sampling vertices. \({d}_{ij}^{B}\) (or \({d}_{ij}^{F}\)) is the Euclidean distance between the \({j\mathrm{th}}\) back (or facial) head feature point of the \({i\mathrm{th}}\) head-skull pair and the centroid of the first skull convex (\({{\varvec{v}}}^{SC}\)). \({t}_{ij}^{B}\) (or \({t}_{ij}^{F}\)) is the Euclidean distance from the \({j\mathrm{th}}\) back (facial) head feature point to the \({j\mathrm{th}}\) back (facial) skull feature point of the \({i\mathrm{th}}\) head-skull pair. \(M\) is the number of back sampling rays. \(N\) is the number of facial sampling rays. \({N}_{\mathrm{subjects}}\) is the number of head-skull pairs (\({N}_{\mathrm{subjects}} = 329\)) in the whole dataset.

Appendix 2

Mathematical formulations of predictor and response variables of the topology-based head-to-skull predicting method

$$\left\{{{\varvec{x}}}_{{\varvec{i}}}^{B}=\left({{\varvec{d}}}_{{\varvec{i}}1}^{{\varvec{B}}},\boldsymbol{ }{\dots ,\boldsymbol{ }{\varvec{d}}}_{{\varvec{i}}{\varvec{j}}}^{{\varvec{B}}},\boldsymbol{ }\dots ,\boldsymbol{ }{{\varvec{d}}}_{{\varvec{i}}{\varvec{M}}}^{{\varvec{B}}}\right)\boldsymbol{ }\right|\boldsymbol{ }{\varvec{i}}=1,\boldsymbol{ }2,\boldsymbol{ }\dots , {N}_{\mathrm{training}}\}$$
(25)
$${{\varvec{X}}}^{B}\left[M\times {N}_{\mathrm{training}}\right]\equiv \left[\begin{array}{c}{{\varvec{x}}}_{1}^{B}\\ {{\varvec{x}}}_{2}^{B}\\ \vdots \\ {{\varvec{x}}}_{{N}_{\mathrm{training}}}^{B}\end{array}\right],$$
(26)
$$\left\{{{\varvec{y}}}_{{\varvec{i}}}^{B}=\left({t}_{{\varvec{i}}1}^{{\varvec{B}}},\boldsymbol{ }{\dots , t}_{{\varvec{i}}{\varvec{j}}}^{{\varvec{B}}},\boldsymbol{ }\dots ,\boldsymbol{ }{t}_{{\varvec{i}}{\varvec{M}}}^{{\varvec{B}}}\right)\boldsymbol{ }\right|\boldsymbol{ }{\varvec{i}}=1,\boldsymbol{ }2,\boldsymbol{ }\dots ,\boldsymbol{ }{N}_{\mathrm{training}}\}$$
(27)
$${{\varvec{Y}}}^{B}\left[M\times {N}_{\mathrm{training}}\right]\equiv \left[\begin{array}{c}{{\varvec{y}}}_{1}^{B}\\ {{\varvec{y}}}_{2}^{B}\\ \vdots \\ {{\varvec{y}}}_{{N}_{\mathrm{training}}}^{B}\end{array}\right],$$
(28)
$$\left\{{{\varvec{x}}}_{{\varvec{i}}}^{F}=\left({{\varvec{d}}}_{{\varvec{i}}1}^{F},\boldsymbol{ }{\dots ,\boldsymbol{ }{\varvec{d}}}_{{\varvec{i}}{\varvec{j}}}^{F},\boldsymbol{ }\dots ,\boldsymbol{ }{{\varvec{d}}}_{{\varvec{i}}N}^{F}\right)\boldsymbol{ }\right|\boldsymbol{ }{\varvec{i}}=1,\boldsymbol{ }2,\boldsymbol{ }\dots ,\boldsymbol{ }{N}_{\mathrm{training}}\}$$
(29)
$${{\varvec{X}}}^{F}\left[N\times {N}_{\mathrm{training}}\right]\equiv \left[\begin{array}{c}{{\varvec{x}}}_{1}^{F}\\ {{\varvec{x}}}_{2}^{F}\\ \vdots \\ {{\varvec{x}}}_{{N}_{\mathrm{training}}}^{F}\end{array}\right],$$
(30)
$$\left\{{{\varvec{y}}}_{{\varvec{i}}}^{F}=\left({t}_{{\varvec{i}}1}^{F},\boldsymbol{ }{\dots , t}_{{\varvec{i}}{\varvec{j}}}^{F},\boldsymbol{ }\dots ,\boldsymbol{ }{t}_{{\varvec{i}}N}^{F}\right)\boldsymbol{ }\right|\boldsymbol{ }{\varvec{i}}=1,\boldsymbol{ }2,\boldsymbol{ }\dots ,\boldsymbol{ }{N}_{\mathrm{training}}\}$$
(31)
$${{\varvec{Y}}}^{F}\left[N\times {N}_{\mathrm{training}}\right]\equiv \left[\begin{array}{c}{{\varvec{y}}}_{1}^{F}\\ {{\varvec{y}}}_{2}^{F}\\ \vdots \\ {{\varvec{y}}}_{{N}_{\mathrm{training}}}^{F}\end{array}\right],$$
(32)

in which \({N}_{\mathrm{training}}\) is the number of head-skull pairs in the training dataset.

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Nguyen, TN., Tran, VD., Nguyen, HQ. et al. Enhanced head-skull shape learning using statistical modeling and topological features. Med Biol Eng Comput 60, 559–581 (2022). https://doi.org/10.1007/s11517-021-02483-y

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