4.2.1 3D Geometric Model Reconstruction of the Spine

The first modeling step is to acquire the geometry of the anatomy to be included in the model, which is generally based on medical imaging techniques (i.e., coronal and sagittal radiographs), ^{19 ,​ 20} computed tomography (CT) scan, ^{21 ,​ 22} and magnetic resonance imaging (MRI). ²³ A CT-scan is a 3D imaging method with high accuracy of bony structure, but it induces a high radiation dose for the patient. Methods based on MRI do not have the problem of high radiation dose, but are more for soft tissues than for bony structures; they are more expensive and not appropriate for patients with implants of ferromagnetic materials. The modeling technique presented in this chapter used patient-specific 3D data of the spine and pelvis from biplanar radiographs acquired for routine clinical assessment of spinal pathologies.

3D spine geometry is built using coronal and lateral plain radiographs and 3D multiview reconstruction techniques. ¹⁹ On the two digital radiographs acquired with patients wearing a calibration object, key anatomical landmarks on the spine and pelvis (e.g., pedicles, vertebral endplate middle and corner points, transverse and spinous process extremities, femoral head centers, and iliac crests) are identified and their 3D coordinates are computed using an optimization algorithm. ¹⁹ The reconstruction process was completed by registering detailed vertebral models using a free-form deformation technique. ¹⁹ Average reconstruction accuracies for pedicles and vertebral bodies were 1.6 mm (SD 1.1 mm) and 1.2 mm (SD 0.8 mm), respectively. ²⁴ Reconstruction variations for a given patient are 0.8° or less (Cobb angles), 5.3° or less (sagittal curves), and 4°–8° (vertebral axial rotation), which are within the error levels reported for equivalent 2D measurements used by clinicians. ^{24 ,​ 25}

4.2.2 Multibody Modeling of the Spine

MBMs have been developed to investigate the biomechanics of the spine. In our long-standing experience, MBMs were developed to particularly assess biomechanical indices, such as spinal balance, deformity corrections, and forces at the bone–implant interface. The MBM of the spine and pelvis was built using their reconstructed geometries. Vertebrae from T1 through L5 and the pelvis were modeled as rigid parts, and intervertebral tissues and connections were modeled using multiple flexible elements with appropriate mechanical properties. For each functional spinal unit (FSU), six cable-like elements, two 6D general springs, and one primary general spring were defined to connect a pair of vertebrae (Fig. 4‑1). The cable-like elements represent the anterior longitudinal ligament, posterior longitudinal ligament, ligamentum flavum, intertransverse ligament, and the combined effect of the interspinous ligament (ISL) and the supraspinous ligament. The biomechanical behavior of the facet joints is more complex compared to the other intervertebral ligaments ²⁶ ; they were represented by two 6D springs. The primary general spring represents the intervertebral disk also incorporating the combined effect of all elements not explicitly modeled (e.g., the rib cage and surrounding muscles and their interconnections).

Stiffness of model component was defined in four complementary processes. In the first method, the stiffness of cable-like elements was defined based on experiments on cadaveric specimens ^{27 ,​ 28} ; the stiffness matrices of the three general springs were defined such that the load-displacement simulations reproduced the reported load displacements. ^{29 ,​ 30} In the second approach, the stiffness of the modeling elements was modified to adjust their percent contribution to the overall stiffness of the FSU using data from biomechanical tests. ^{31 ,​ 32 ,​ 33 ,​ 34} In the third method, a weighting factor was applied to the stiffness matrix of the primary general spring to account for the contribution of the rib cage to the overall stiffness of the thoracic spine (i.e., 40%, 35%, and 31%, respectively, in flexion/extension, lateral bending, and axial rotation). ³⁵ Finally, all model element stiffness could be further adjusted such that side bending (or traction) simulations reproduced the Cobb angles measured on the patient’s side bending (or traction or supine) radiographs using an optimization technique reported in Refs. 17 and 36. Osteotomy procedures were modeled by removing the modeling elements involved in the procedure.

4.2.3 Multibody Modeling of Spinal Instrumentation and Postoperative Physiological Loads

The intraoperative surgical position of the patient was modeled by applying boundary conditions such that the pelvis had a fixed position and T1 was constrained on a caudocranially oriented line and free to rotate in all directions. The rods were modeled as flexible beams whose definitions were based on their geometry and material properties. Hooks and fixed-angle screws (monoaxial screws) were modeled as single-component rigid bodies; uniaxial and multiaxial (polyaxial) screws were modeled as two-component rigid bodies whose components were connected through hinge joint and ball-socket joint, respectively. Implant models were positioned and aligned with the anchoring vertebrae according to the respective surgical techniques. ³⁷ The implant–vertebra connection was modeled as a nonlinear general spring (represented with a flexible connector available on the computer-aided engineering platform Adams/View, Version MD Adams 2010). These springs used parametric force-displacement and moment-angle curves to relate the implant-vertebra relative displacement to the implant-vertebra load. Mechanical properties acquired from experimental tests on instrumented cadaveric vertebrae were used to define these parametric curves. ^{17 ,​ 38} The commonly used deformity correction maneuvers were modeled, which included rod reduction, rod derotation, vertebral derotation, compression/distraction, and set-screw tightening. This was done by applying appropriate forces, moments, and kinematic constraints on and between the instrumentation constructs. ³⁸

Change from an intraoperative position to a postoperative upright position was modeled by applying new boundary conditions for an upright position. The T1 plumb line and pelvic tilt (PT) were estimated using linear regression equations reported in Ref. 39 (ΔPT = –0.185ΔLL–7.299 and ΔSVA = –1.52ΔLL–11.45, where ΔPT is PT change [°],ΔLL is L1-S1 lumbar lordosis change [°], and ΔSVA is sagittal balance change [mm]). The T1 plumb line and the pelvis were repositioned using the estimations and the pelvis was fixed. A downward force was applied to each vertebra whose magnitude was determined based on the weight of the specific body section at each vertebral level, as reported in Pearsall’s anthropometric model. ⁴⁰ The application point was positioned anteriorly with respect to the vertebral center of mass as reported by Kiefer et al. ⁴¹ Between each pair of adjacent vertebrae, follower loads were applied between the transverse and spinous processes, respectively, and their magnitudes were determined such that gravity forces were balanced at the estimated T1 plumb line position.

Functional flexion/extension, side bending, and transverse plane rotation were simulated with forward/backward, side translation of T1 plumb line, and T1 rotation in the transverse plane; additional external force and/or moment were simulated by applying force and/or moment on T1. Validation works of the modeling techniques have been done by simulating surgical spinal instrumentations and comparing the simulation results with the actual surgery results, with the former being ±5° to the latter in terms of Cobb angles in the coronal and sagittal planes. ¹⁷