Growth modeling of human mandibles using non-Euclidean metrics

Abstract

From a set of 31 three-dimensional computed tomography (CT) scans we model the temporal shape and size of the human mandible for analysis, simulation, and prediction purposes. Each anatomical structure is represented using 14851 semi-landmarks, and mapped into Procrustes tangent space. Exploratory subspace analyses are performed leading to linear models of mandible shape evolution in Procrustes space. The traditional variance analysis results in a one-dimensional growth model. However, working in a non-Euclidean metric results in a multimodal model with uncorrelated modes of biological variation related to independent component analysis. The applied non-Euclidean metric is governed by the correlation structure of the estimated noise in the data. The generative models are compared, and evaluated on the basis of a cross validation study. The new non-Euclidean analysis is completely data driven. It not only gives comparable results w.r.t. previous studies of the mean modeling error, but seems to better correlate to growth, and in addition provides the data analyst with alternative hypothesis of plausible shape evolution; hence aiding in the understanding of cranio-facial growth.

Active Shape Analysis of Mandibular Growth

“…The PLS study thus confirms that growth of the mandible is primarily contained in a one-dimensional linear component in Procrustes tangent space. This is in agreement with the findings in [1,11] and does not conflict with the non-linear growth observed in biological coordinate references systems, [4]. In Figure 5 we show the first three modes of shape variation of the mean shape ±3 standard deviations (std.)…”

“…CCA maximizes the correlation between linear combinations of two multivariate groups of variables, see [5,14,16]. We jointly analyse pairs of landmark variables (x, y), with dispersions Σ 11 and Σ 22 and cross-covariance Σ 12 = Σ 21 T , and find sets of linear combinations (called canonical variates, CVs) of the zero mean original variables that maximize correlation ρ = Corr{a T x, b T y}, under a T Σ 11 a = b T Σ 22 b = 1. Solving the generalized eigenvalue problems…”

Statistical shape models for 3D medical image segmentation: A review

“…Among others, it was applied to shape modeling by Üzümcü et al (2003) and Suinesiaputra et al (2004) for the characterization of myocardial diseases. Another technique to obtain sparse modes of variation is the maximum autocorrelation factor (MAF) analysis as used by Hilger et al (2003). An extensive comparison between PCA, MAF and minimum noise fraction (MNF -another non-Euclidean decomposition) is presented by Larsen and .…”

3-D Volumetric Evaluation of Human Mandibular Growth

“…For example, it has been used tocomparedeviations in mandibles to a defined normal [26], while other studies used longitudinal scans to create a linear growth model [5, 27]. The advantage of using the following method is that we are able to visualize the direction and magnitude of bone growth in 3D by a simple subtraction of two surfaces - consequently providing clinicians and developmental biologists with a minimally laborious procedure that illustrates bone growth from regular dental data.…”