Age estimation in children and young adolescents for forensic purposes using fourth cervical vertebra (C4)

Abstract

The aim of this study was to evaluate the applicability of using the growth of the body of C4 vertebra for the estimation of age in children and young adolescents. We used the fact that the proportions between the radiologic projections of the posterior and anterior sides of the C4 vertebral body, which forms a trapezoidal shape, differ with age: in younger individuals, the posterior side is higher, whereas in older individuals, the projections of the sides of the vertebral body form a rectangular shape with the two sides equal or with the anterior side slightly higher. Cephalograms of 444 Italian subjects (214 female and 230 male individuals) aged between 5 and 15 years and with no obvious development abnormalities were analyzed. The projections of the anterior side (a) and of the posterior side (b) of each C4 body were measured, and their ratio (Vba), as a value of the C4 body development, was used for age estimation. Distribution of the Vba suggested that it does not change after 13 years in female and 14 years in male subjects. Consequently, we restricted our analysis of the Vba growing model until 14 years in both sexes. We used a Bayesian calibration method to estimate chronological age as function of Vba as a predicting variable. The intra- and inter-observer agreement was satisfactory, using intra-class correlation coefficient of Vba on 30 randomly selected cephalograms. The mean absolute errors were 1.34 years (standard deviation 0.95) and 1.01 years (standard deviation 0.71), and the mean inter-quartile ranges of the calibrating distribution were 2.32 years (standard deviation 0.25) in male and 1.72 years (standard deviation 0.39) in female individuals, respectively. The slopes of the regression of the estimated age error to chronological age were 0.02 in male and 0.06 in female individuals, where both values did not result significantly different from 0 (p > 0.12). In conclusion, although our Bayesian calibration method might not really outperform the classical regression models in the precision of its estimates, it appears to be more robust, to greatly reduce the typical bias inherent in the regression model approach, and to have the ability to incorporate multiple predictors.

Appendix

The first step in Bayesian analysis is to choose a probability model for the observed data.

We choose the probability model for the observed data of the form p(x _i |t _i , θ), i = 1, … n; with the unknown vector of parameters θ. Vector θ is supported by parameter space Θ ⊆ ℝ^m and considers a vector of m random variables, the joint prior distribution of which is h(θ).

• 1. We assume that observations are independent but not necessarily identically distributed with the probability model for observed data of the form p(x _i |t _i , θ), i = 1, … n; with unknown vector of parameters θ.

• 2. Given age u and θ, the new observation y is independent of the observed data and follows the same probability model.

With these assumptions, given observations t and x, the posterior distribution for θ may be written as follows:

$$ h\left(\boldsymbol{\theta}\ \Big|\ \boldsymbol{t},\ \boldsymbol{x}\right)=\frac{h\left(\boldsymbol{\theta} \right)p\left(\boldsymbol{x}\ \Big|\ \boldsymbol{t},\ \boldsymbol{\theta} \right)}{{\displaystyle {\int}_{\Theta}}h\left(\boldsymbol{\theta} \right)p\left(\boldsymbol{x}\ \Big|\ \boldsymbol{t},\ \boldsymbol{\theta} \right)d\boldsymbol{\theta}}=\frac{h\left(\boldsymbol{\theta} \right){\displaystyle {\prod}_{i=1}^n}p\left({\boldsymbol{x}}_{\boldsymbol{i}}\ \Big|\ {t}_i,\ \boldsymbol{\theta} \right)}{{\displaystyle {\int}_{\Theta}}h\left(\boldsymbol{\theta} \right){\displaystyle {\prod}_{i=1}^n}p\left({\boldsymbol{x}}_{\boldsymbol{i}}\ \Big|\ {t}_i,\ \boldsymbol{\theta} \right)d\boldsymbol{\theta}}. $$

Lastly, the calibrating distribution may be written as follows:

$$ f\left(u\ \Big|\ y,\ \boldsymbol{t},\boldsymbol{x}\right)=\frac{p(u)\phi \left(y\ \Big|\ u,\ \boldsymbol{t},\boldsymbol{x}\right)}{{\displaystyle {\int}_0^{+\infty }}p(u)\phi \left(y\ \Big|\ u,\ \boldsymbol{t},\boldsymbol{x}\right)du} $$

(1)

where ϕ(y | u, t, x) is the predictive distribution:

$$ \phi \left(y\ \Big|\ u,\ \boldsymbol{t},\boldsymbol{x}\right)={\displaystyle \underset{\Theta}{\int }p\left(y\ \Big|\ u,\ \boldsymbol{\theta} \right)\ h\left(\boldsymbol{\theta}\ \Big|\ \boldsymbol{t},\ \boldsymbol{x}\right)d\boldsymbol{\theta}} $$

(2)

and p(u) is the prior distribution of age.

In our Bayesian calibration approach, the probability model for a typical observation, (x | t, θ), is assumed to be asymmetric Laplace distribution (ALD) (μ, σ, τ), with location parameter μ = μ(t) and scale parameter σ > 0 and a skewness parameter 0 < τ < 1

$$ p\left(x\ \Big|\ t,\ \boldsymbol{\theta}, \tau \right)=\frac{\tau \left(1-\tau \right)}{\sigma } \exp \left\{-{\rho}_{\tau}\left(\frac{x-\mu (t)}{\sigma}\right)\right\} $$

where

$$ {\rho}_{\tau }(z) = \frac{\left|z\right|+\left(2\tau -1\right)z}{2} $$

and θ = (α, β, σ) is the vector of parameters.

A restricted cubic splines were used to model μ = μ(t) allowing nonlinearity in the relation between age and C4 [18].

For each value of parameter τ, we obtain the distribution of the τth quantile of the posterior distribution of age u, conditioned to the value of age predictor y, the vectors of ages t = (t ₁, …, t _n ) and predictor values x = (x ₁, …, x _n ). The calibrating distribution is obtained for τ = 0.5, when the probability model for the observed data, p(x | t, θ), reduces to a Laplace distribution:

$$ p\left(x\ \Big|\ t,\ \boldsymbol{\theta} \right)=\frac{1}{4\sigma } \exp \left\{-\frac{\left|x-\mu (t)\right|}{2\sigma}\right\}. $$

Thus, the calibrating distribution is the distribution of the median of the conditioned posterior distribution for age u. Since no prior information is available on the model parameters, we chose an uninformative prior distribution h(θ) = 1/σ for the model parameters.

Although conjugate prior distribution is not available for ALD, we approximate the predictive distribution, ϕ(y | u, t, x), by the sample average:

$$ \frac{1}{M}{\displaystyle \sum_{m=1}^M}p\left(y\ \Big|\ u\ {\boldsymbol{\theta}}_m\right) $$

where θ _m , m = 1,…, M = 500 are posterior draws from h(θ | t, x) according to the Markov Chain Monte Carlo (MCMC) method described in [46] and implemented in R [48].