MODELING OF RESONANCE EFFECTS IN SPINE WITH ADDITIONAL FIXING ELEMENTS

Abstract

Subject of Research. We study the procedure of natural frequencies calculation for a system consisting of a human spine with fixing elements. Resonant effects can occur in the vicinity of such system leading to stability violation or structure destruction. Method. A mathematical model of the conditional vertebral column is proposed, consisting of two anatomically-physiologically isolated columns: the spinal cord (dorsal longitudinal column) and its musculoskeletal “case” (ventral longitudinal column). The model includes complementary boundary condition — an additional fixing element. To solve this problem, the vertebral complex is modeled using a geometric graph. A fourth-order differential operator on the edges of a geometric graph is considered. The graph is a model of a biomechanical system — the spine and metal structure. It is assumed that there are point potentials at the vertices of the graph that model the bond character between the graph elements. A system of differential equations with boundary conditions (conditions for matching solutions on adjacent edges) is solved to find the spectrum of the operator dangerous for the integrity of the mechanical frequency system. Main Results. A technique is proposed for detection of biomechanical system eigenfrequencies that lead to resonant effects. A correct model of a metric graph is created with a fourth-order operator on the edges and the conditions of point interaction at the vertices. Frequency values are obtained for specific values of the system parameters. Practical Relevance. The described method for detection of hazardous frequencies can be used in the treatment of patients with scoliosis to prevent breakage of the installed metal structure and save the patient’s life.

Keywords

• metric graph
• fourth-order operator
• point potential
• operator spectrum
• scoliosis