A patient-specific multibody kinematic model for representation of the scoliotic spine movement in frontal plane of the human body

Abstract

Multibody models of scoliotic spine have shown great promise in planning scoliosis surgery by providing predictive information concerning the surgery outcome. To provide good predictive information, it is important that the kinematic models underlying the movement of the spine models would be personalized to give good estimates of the spine in different positions, which is lacking in the existing literature. This paper aims to develop a patient-specific multibody kinematic model of the scoliotic spine to represent its movement in frontal plane of the human body. The model is an open-chain mechanism comprising rigid links interconnected with rotary joints. To represent the movement, the mechanism lays on the spine curve and estimates the curve and the location and orientation of vertebrae. To personalize the mechanism for a patient, a minimization problem is defined to give the number of the links and their length by using X-rays of different spine positions. The feasibility and capabilities of our patient-specific model are tested by using the data from preoperative X-rays of five positions of 10 AIS (adolescent idiopathic scoliosis) patients; three of the X-rays are routine in scoliosis standard care. The mechanism is personalized to each patient by using the three routine X-rays, and it is used to estimate all the five positions. Root-mean-square-errors (RMSE) of the curve, location, and orientation are 2e–5 mm, 0.27 mm, and 0.25°, respectively. The small RMSEs imply that our kinematic model is capable of estimating the scoliotic spine positions in the frontal plane and thus of describing the scoliotic spine movement in this plane. Our personalization using X-rays of three spine positions helps to set better values for the kinematic parameters (such as the length of the links) for more accurate estimates of the spine in the frontal plane.

Appendix

A 2D curve ($y=f (z)$) can be approximated by a polygonal chain (a piecewise linear curve or polyline [61]. The chain is obtained by connecting a finite number of points on the curve using line segments. The points ($z,y$) can be defined by segmentation of the $z$-axis, for example, $z_{0}$, $z_{0} +t$, $z_{0} +2t, \ldots ,\mbox{ and }t$ is a real number and positive (Fig. 12a). Alternatively, the points can be specified by parameterization of the curve [62]. The parameterization by equal-length line segments (Fig. 12b) can be given by

$$ ( z_{i} - z_{i-1} ) ^{2} + ( y_{i} - y_{i-1} ) ^{2} = L_{\mathrm{seg}}^{2},\quad i=1,2,\dots ,n, $$

(A.1)

where $L_{\mathrm{seg}}$ is the length of the lines, and $n$ is the total number of the lines.

Fig. 12

Approximation of a curve by using polygonal chains, (a) segmentation on the $z$-axis and (b) parameterization by equal-length line segments

Rectification gives the length ($L$) of a curve by adding up the length of the line segments [63]. For example, for the parameterized curve in Fig. 12b, the length ($L$) of the curve is $n \cdot L_{\mathrm{seg}}$. Indeed, the rectification gives a good approximation of the curve length if $L_{\mathrm{seg}}$ is sufficiently small. Thus, the parameterization of two curves of equal length (i.e. $L_{1} =L_{2}$) can result in the same number of the equal-length line segments (i.e. $n_{1} =n_{2} \rightarrow L_{1} =n_{1} \cdot L_{\mathrm{seg}} = L_{2} =n_{2} \cdot L_{\mathrm{seg}}$) if $L_{\mathrm{seg}}$ is sufficiently small.