Eigenvector Analysis of Spinal Deformities in Idiopathic Scoliosis

denote a data matrix

$\begin{aligned} \text {X} = \begin{bmatrix} {\mathbf {x}_{1}}&{\mathbf {x}_{2}}&\cdots&{\mathbf {x}_{n}} \end{bmatrix}, \end{aligned}$

(1)

where

$\begin{aligned} {{\mathbf {x}}_{i}} = \begin{bmatrix} x_1&x_2&\cdots&x_p \end{bmatrix}^T, \end{aligned}$

(2)

i.e. $\text {X}$ contains

measurements of

variables. Compute the covariance matrix $\mathbf {C}_{X}$ as

$\begin{aligned} {\mathbf {C}_{X}} \approx {\frac{1}{n-1}}(\text {X}-\bar{\text {X}})(\text {X}-\bar{\text {X}})^T, \end{aligned}$

(3)

Define similarly a data matrix $\text {Y}$ .

For PCA, a linear transform $\text {W}$ is estimated such that the variance of the components of $\text {Z} = \text {W}^T \text {X}$ is maximized under the constraint that the components $\mathbf {w}_i$ of $\text {W}$ are orthogonal, i.e. the components of $\text {Z}$ are uncorrelated and $\mathbf {C}_{Z} = \text {W}^T \mathbf {C}_{X} \text {W}$ is diagonal. In CCA, two linear transforms, $\text {W}_X$ and $\text {W}_Y$ , are estimated such that the correlation $\rho _i$ between the reduced variables (canonical variates) of $\text {W}_{X,i}^T \text {X}$ and $\text {W}_{Y,i}^T \text {Y}$ , have been maximized and that the different components of $\text {W}_{X,i}^T \text {X}$ and $\text {W}_{Y,i}^T \text {Y}$ are uncorrelated with respect to each other. Note that for CCA, the data matrices $\text {X}$ and $\text {Y}$ are not required to have the same number of variables, therefore the number of canonical variates will correspond to the smallest number of variables provided by either $\text {X}$ or $\text {Y}$ . Estimating the linear transforms $\text {W}$ in PCA, and $\text {W}_X$ and $\text {W}_Y$ in CCA are done solving an eigenvector problem, hence, the term eigenspine.

An interesting aspect of CCA is its relation with mutual information (MI). As shown by [9], the mutual information between $\text {X}$ and $\text {Y}$ can be estimated as the sum of the mutual information of the reduced variables, given that their statistical dependence is limited to correlation. For normally distributed variables, this relation is given as

$\begin{aligned} \text {MI}(\text {X},\text {Y}) = \frac{1}{2}\sum _i{\log _{2} \left( \frac{1}{(1-\rho _{i}^{2})}\right) }. \end{aligned}$

(4)

This follows from considering a continuous random variable $\mathbf {x}$ with the differential entropy defined as

$\begin{aligned} h(\mathbf {x}) = -\int \limits _{\mathbb {R}^N}{p(\mathbf {x}) \log _2 \left( p(\mathbf {x})\right) \; d\mathbf {x}}, \end{aligned}$

(5)

where $p(\mathbf {x})$ is the probability density function of $\mathbf {x}$ . Consider similarly a continuous random variable $\mathbf {y}$ , then it can be shown that

$\begin{aligned} \text{ MI }(\mathbf {x},\mathbf {y}) = h(\mathbf {x}) + h(\mathbf {y}) - h(\mathbf {x},\mathbf {y}) = \int \limits _{\mathbb {R}^M}{\int \limits _{\mathbb {R}^N}{p(\mathbf {x},\mathbf {y}) \log _2 \left( \frac{p(\mathbf {x},\mathbf {y})}{p(\mathbf {x})p(\mathbf {y})} \right) }d\mathbf {x}d\mathbf {y}}. \end{aligned}$

(6)

Further, consider a Gaussian distributed variable $\mathbf {z}$ , for which the differential entropy is given as

$\begin{aligned} h(\mathbf {z}) = \frac{1}{2}\log _2 \left( (2\pi e)^N |\mathbf {C}|\right) , \end{aligned}$

(7)

where $\mathbf {C}$ is the covariance matrix of $\mathbf {z}$ . In the case of two

-dimensional variables, then (6) becomes

$\begin{aligned} \text{ MI }(\mathbf {x},\mathbf {y}) = \frac{1}{2}\log _2 \left( \frac{|\mathbf {C}_{xx}| |\mathbf {C}_{yy}|}{\mathbf {C}}\right) , \end{aligned}$

(8)

where

$\begin{aligned} \mathbf {C} = \left[ \begin{matrix}{{\mathbf {C}}_{xx}} &{} {\mathbf {{{C}}}_{xy}}\\ {\mathbf {{{C}}}_{yx}} &{}{\mathbf {{{C}}}_{yy}}\end{matrix}\right] . \end{aligned}$

(9)

For two one-dimensional Gaussian distributed variables, (8) reduces to

$\begin{aligned} \text{ MI }(x,y) = \frac{1}{2}\log _2 \left( \frac{\sigma _x^2 \sigma _y^2}{\sigma _x^2 \sigma _y^2 - \sigma _{xy}^2}\right) = \frac{1}{2}\log _2 \left( \frac{1}{1-\rho _{xy}^2}\right) , \end{aligned}$

(10)

where $\sigma _x^2$ and $\sigma _y^2$ are the variances of

and

, $\sigma _{xy}^2$ is the covariance of

and

and $\rho _{xy}$ is the correlation between

and

. Given that information is additive, for statistically independent variables, and that the canonical variates are uncorrelated, i.e. $\text {W}_{X,i}^T \text {X}$ and $\text {W}_{Y,i}^T \text {Y}$ , hence, the mutual information between $\mathbf {X}$ and $\mathbf {Y}$ is the sum of the mutual information between the variates.

Note that using the $\log$ -function with the base

provides an MI measure defined in bits. This measure will be employed in the subsequent analysis for quantifying the dependence between different measures.

3 Experiments

To demonstrate the use of the data analysis scheme, measurements of the position and the orientation of the vertebrae for a number of patients were analyzed to determine which of these measures that have the strongest linear dependence.

3.1 Image Data

Image data from 22 patients (19 female and three male) were retrospectively gathered and extracted from the local picture archiving and communications system. The only criteria for inclusion was that the patient suffered from idiopathic scoliosis and that the CT data had a resolution higher than $1\times 1\times 1$ mm $$ ^3 $$

. The data sets depicted all lumbar and thoracic vertebrae, i.e. 17 vertebrae per patient. The requirement on the resolution was needed in order to be able to distinguish adjacent vertebrae in the subsequently applied method for obtaining the position and rotation of each vertebra. The patients had an average age of $16.0\pm 3.1$ years at the time of their respective examinations and an average Cobb angle of $60.4^{\circ } \pm 9.6$ (standing position). Most patients were classified has having a scoliosis of Lenke type 3C or 4C.

The images were captured as a part of the standard routine for pre-operational planning and they were anonymized before being exported by clinical staff. Note that for patients of similar age as included in this retrospective study, it is often questionable whether a CT scan is appropriate or not, due to the exposure to radiation. However, at the local hospital there is a protocol in place for acquiring low-dose CT examinations with maintained image quality, targeted towards examinations of the spine. With the use of this protocol, the radiation dose is approximately $$0.4$$

mSv. More on this can be found in [8].

3.2 Curvature Measures

Each data set was processed with the method presented in [5], which is based on the following steps; extraction of the spinal canal centerline, disc detection, vertebra centerpoint estimation and vertebra rotation estimation. A graphical overview of the method is provided in Fig. 1. In [5], the method was shown to have a variability, when compared with manual measurements, that is on par with the inter-observer variability for measuring the axial vertebral rotation. This was supported by Bland-Altman plots and high values of the intraclass correlation coefficient, thus, showing that the method can be used as a replacement for manual measurements.

The method estimates, for each vertebra, the position $$ [x,y,z] $$

and the rotation matrix $\text {R}$ , from which the rotation angles $[\theta _{X},\theta _{Y},\theta _{Z}]$ can be derived. The rotation angles were computed as the Euler angles (using a fixed world frame) of the rotation matrix $\text {R}$ . Note the order of the rotational angles, $\text {R} = \text {R}_Z(\theta _{Z}) \text {R}_Y(\theta _{Y}) \text {R}_X(\theta _{X})$ . $\theta _{Z}$ corresponds to axial vertebral rotation, $\theta _{Y}$ to frontal rotation and $\theta _{X}$

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