Simulation of Scoliosis Treatment Using a Brace

Spinal corrective braces (see Fig. 1) are used for treating spine scoliosis in children (pathologic at deformation of chest curvature). The X-ray of the patient from Fig. 1 without and with a brace is shown in Fig. 2. Cheneau-type dynamic corrective braces or according to Cerny’s patent No. 281800CZ (see Fig. 1) are usually used in the Czech Republic. The breast curvature can be classified according to King. A Chenau-type


Introduction
Spinal corrective braces (see Fig. 1) are used for treating spine scoliosis in children (pathologic at deformation of chest curvature).The X-ray of the patient from Fig. 1 without and with a brace is shown in Fig. 2. Cheneau-type dynamic corrective braces or according to Cerny's patent No. 281800CZ (see Fig. 1) are usually used in the Czech Republic.The breast curvature can be classified according to King.A Chenau-type brace of is recommended for spinal curves of type King I, II, and IV types, a Cerny-type brace for spinal curves of King II, III and V types.
The brace constricts the child´s trunk and makes a stress state in the patient's spine.The brace changes the spinal curvature, which means that the pathological spinal form is corrected.After long-term use of the brace, the spinal correction is permanent.The brace is made in the following manner: first, a plaster negative form and then a positive form of the child's trunk are made.According to his experience and the orthopedist's recommendation, the orthopedic assistant deepens the positive form of the plaster's in the place where the brace is to push on the child's trunk.The plastic brace is then made according to this plaster form.
After it has been applicated to child´s trunk, the brace constricts the places where the form has been deepened (the tight shoe principle).
If no computer search is used, the brace force effect is the result of the orthopaedist and his assistant's experience only, and it does not ensure that the form of the designed brace and the manner of treatment are optimal.This paper shows computer algorithms that are able to determine the stress state in vertebra and inter-vertebral discs, and spinal curve changes for specific brace applications.The theoretical conclusions are based on many causes of treatment.The remodelling of a pathological spine curvature depends on the spine stress state, and the time and manner in which the brace is applied.The course of treatment is simulated on the computer.The aim of this study is to determine the ideal brace form and a course of treatment course with the help of computer simulation.The computer program calculates the spine stress state and its curvature changes at each time point.The treatment simulation is now being provided simultaneously with the patient's treatment, and the computer model is being verified.If the computer model and the actual treatment have the same behaviour, then the model can be used for treatment prognosis in orthopaedic practice.Since the course of treatment takes a long time, the simulation model is still being verified so that its prognoses can become as precise as possible.

Spinal curvature
The task is solved using Cartesian coordinates (x -spine axis direction, y, z -frontal and saggital plane).The spinal curvature is stored in the computer as the following three functions ( ) ( ) ( ) where j is the turning based on the x-axis.The spinal curvature can be described if the extreme values of y and/or z are measured from X-ray (the extremes of the white curvature in the X-ray on the left in Fig. 2).The method is also applied for the frontal and sagittal plane.The spinal curvature has 3 extremes of coordinate maxima.The curvature is divided into n sectors between the beginning, the extremes and the end point of the coordinates, respectively (max.n is 4).The ex-Acta Polytechnica Vol.44 No. 2/2004 Fig. 2: Frontal X-ray of the patient from Fig. 1 without and with the corrective brace treme coordinates x i , y i , i = 1, …, n -1 and the coordinate x n of the spine end (spine length where x n = 0) are measured by X-ray (see Fig. 2).The length of segment i is The local coordinate x is considered from the beginning of segment i. Function y is considered as a polynomial.For the first segment (quadratic polynomial function) it is for the inner segments (cubic polynomial function) and for the last segment (quadratic polynomial function) x .

Deformation of the spine
The inertia moment has to be determined for an inter--vertebral disc and a lignum cross-section area (see Fig. 3).The calculation procedure is as follows: the cross-section area is divided into triangles, and one third of triangular areas are concentrated to their side centres.
The spine is treated like a beam in an elastic basis, and the finite element method (deformation variant according to the Lagrange principle) is used for calculating the stress state.It is assumed that the vertebrae have no deformation.The potential energy is calculated for the inter-vertebral disc volume and for the compressed soft tissue region of a child´s trunk.For simplification, the soft tissue is considered constant (rectangular cross-section of the trunk).The displacements and the turning at the vertebral centres are kinematic unknowns: where j x are turnings based on the spine axis.The following algorithm is valid for the frontal and sagittal planes, and planes will not be indicated by the plane index.The stiffness matrix for the part of the spine between the centres of neighbouring vertebra is (torsion and beam influences) The sub-matrices will be determined separately for deformation of the spine and for soft tissue.

Deformation of inter-vertebral discs
The beam and torsion stiffness is where E, I are the modulus of elasticity and the moment of inertia of a cross-section of the inter-vertebral disc and lignums (see Fig. 3), and l is the thickness of the disc.The influence of torsion is: The influence for R-forces (moments) and r-movements (turns) on inter-vertebral disc boundaries are The stiffness matrix K 2 will be searched for inter-vertebral disc and kinematic value at the inter-vertebral borders.Because the vertebra are stiff, they have no potential energy.The inter-vertebral stiffness matrix will be recalculated on matrix K 2 for kinematic unknowns at the vertebra centres.
Acta Polytechnica Vol.44 No. 2/2004 Fig. 3: Inter-vertebral disc and lignums Let the boundary forces Z i , M i , Z i+1 , M i+1 and the kinematic unknowns w i , j i , w i+1 , j i+1 be transformed from vertebra centres to values Z i , M i , Z i+1 , M i+1 , w i , j i , w i+1 , j i+1 at the disc boundary points (see Fig. 4).As there is no deformation between the vertebral centre and the inter-vertebral disc boundary, the central spinal line is straight in this part, the spine movement w has a linear course in the part of length a, and the torsion moment M x and turning j, j x are invariable.
(11) Analogous formulas are valid for the y axis direction.

Compressed Soft Tissue (Elastic Ground)
The compressed soft tissue of a child's trunk round the spine is considered as an elastic ground according to [1] p. 86-113, and the final formulas will be used here.The width of the ground is considered constant.Let us calculate the parameters where E P , h, b are the modulus of elasticity, thickness and width of the compressed soft tissue.The torsion stiffness sub-matrix is: The beam stiffness sub-matrix is:

The first algorithm
The brace constricts the trunk in the place where the plaster positive form of the child´s trunk has been deepened; this means that the trunk surface (soft tissue surface) has non-zero prescribed displacements in these places.Let us assume that the prescribed displacement acts from above for a lying patient, and the z-axis direction is from below.The compression of the soft tissue up the spine is w 0w and below it is w, where w is the spine displacement and w 0 is the prescribed trunk surface displacement.Let matrices K above , K below be calculated according to formulas (12), (13) for the part of the trunk above and below the spine.The potential energy of the soft tissue part is [ ] The term K above r 0 can be calculated, and its negative form can be considered as a load vector (the right side of linear algebraic equations of the finite element method).In this way, the potential energy can be considered in the compressed parts of the soft tissue only; this means that the terms K above (r 0r) and/or K below r are taken into cosideration only if they are positive.An iteration calculation is necessary for correct results; this means that the load vector is calculated for Fig. 4: Spine deformation is linear in the vertebra parts and curvilinear in the inter-vertebral part of the disc the compressed soft tissue part above and/or below the spine according to the results from the last iteration step.
The oblique load will be searched.Let y, z be coordinates of the point of the center where the positive plaster form was deepened, and let D be the depth to which the plaster positive form has been deepened in the perpendicular direction to the child's trunk surface.Now, D is a prescribed trunk surface displacement and y, z are its coordinates (positive displacement is in the direction from the trunk surface to the spine).Let us consider that the transversal cross-section of the trunk has a half elliptic form with radiuses a, b for z > 0 and a, b for z < 0. The following formulas can be written for the ellipse If formula (13) is derived, angle j of the tangent with axis y can be calculated; the negative value of angle j is the angle of the normal with the z axis. .
The prescribed surface displacements v 0 , w 0 in y, z directions are The problem can be solved in the x, y or x, z plane with prescribed displacements v 0 , or w 0 or more correctly as a space problem with a space spine and soft tissue elements.The stiffness matrix for the space spine element can be considered in the same way as formulas ( 4), ( 5) and ( 11), but the matrix K 2 (see ( 11)) also has the elements for the direction of the axis y.As the vertebra have no deformations, the kinematic variables at the vertebra surface can be calculated from the kinematic variables of the centre of gravity of the vertebra (see (3)).The normal and tangential stresses on the boundary between a vertebral and an inter-vertebral disc are then calcu-lated from the resulting joint forces and moments.The normal force of the x-axis load has to be taken into account in the normal stress calculation, too, and the influence of shear and torsion should be taken into account in the tangent stress calculation.
The parameters and calculation algorithm are being verified with values observed in the X-ray of a child with and without a brace, i.e., the calculated function values y, z and y+v, z+w and their extremes are compared with the patient's X-ray.

The second algorithm
The spinal curvature extremes are measured on X-rays taken without and with a brace.The spinal curvature coordinates are determined for the X-rays, and the spine deformation function r 2 is the difference of the two curvature coordinates.The joint force (moment) vector R 2 can be calculated from (10) and beam theory can be used to calculate the spine stress state.If displacement vector w is known, we can calculate from ( 14) vector w 0 -the necessary displacement of the trunk surface and the most suitable place for deepening the brace.

Simulation of treatment
If the brace is removed from the child's trunk after being applied for some time, then the spine does not return to its previous position, but the pathological spine form is partly corrected.An example of the result of scoliosis treatment of a King I type spinal defect is shown in Fig. 5.The maximum of the angles between the spine axis has been measured at the thoracic and lumbar parts of spine.The measured angles were 27 grads and 34 grads before treatment; 2 and 5 grads with the brace; 9 and 12 grads after treatment.Statistical data for various spinal curve types according to King-Moe are given in Table 1.The second column is for the thoracic part of the spine, and the third column is for the lumbar part of the spine.The first set of data is for Ch neau and the second set of data for Cerny brace types.The percentages of angle correction are given first, followed by the number of patients treated, in parenthesses.The prognosis prognosis and speed of the treatment effect are made according to the statistical data and the spine stress state.The condition for succesful treatment is periodical use of the brace in accordance with orthopaedic advice.

Conclusion
Many child patients have been observed within this project, and the dependence between spinal curvature correction, the spine stress state and the time interval of applying the brace have been studied.Theoretical conclusions about spine remodelling have been sought.The computer simulation model and its parameters are being verified to ensure that the behaviour of the model is the same as the child´s course of treatment.Since the treatment takes a long time, the theoretical conclusions can only be determined after a sufficient number of comparison have been made between observed treatment courses and their computer simulations.

Fig. 5 :
Fig. 5: X-rays of a patient with King I type spinal curvature, before treatment, with a brace, and after 2.5 years

Table 1 :
Statistical data for spinal angle correction after scoliosis treatment with braces