PEACEM * Field Theory for Soft Tissue and Application to The Intervertebral Disc Soft tissue is characterized as a poroelastic material containing a solid matrix frame, fluid (water), mobile and immobile ions, electric potential variations and an imposed magnetic field. The theoretical equations are formulated in terms of field variables amenable to a finite element computational model. 2D and 3D Finite element models of the intervertebral disc have been developed to study the response of the disc due to external mechanical loading and imposed electric and magnetic fields. Jeffrey P. Laible Department of Civil and Environmental Engineering University of Vermont * PEACEM = Poroelastic and Chemical, Electric, Magnetic

Disc Vertebral Body Neck vertebrae Thoratic vertebrae Lumber vertebrae

Disc Above VERTEBRAE Below Not Shown

Nucleus Pulposus Annulus

In mixture theory, the disc is considered to have four phases, 1) the solid matrix phase 2) the fluid (water) phase 3) the negative ion phase 4) the positive ion phase. The momentum equations include the forces acting on each phase from the other phases in terms of friction and body forces due to 1) inertia (zero for slow motion) 2) external electric field 3) external magnetic field.

Potentials, s=solid, w=fluid (water), p,n= + and - ions f = friction coefficients, = absolute velocity, c = concentration n = porosity, F=Faraday Constant, E=Electrical Field Strength E = - ,  = Electric Potential B = Magnetic flux density vector 1) Solid, 2)Fluid, 3) Positive Ions, 4) Negative Ions

Body force =

In terms of the relative velocities, the electric field vector and the magnetic field matrix the equations are:

Flux in terms of Potentials and Magnetic Field Magnetic Field. Using the current equation and the expressions for the electric and chemical potential, the fluxes in terms of potentials and magnetic body forces are obtained from the fluid and chemical momentum equations (equations 2-4). Mobility Matrix K

Inverse Permeability

Fluid Pressure Osmotic Pressure Reference Potential Change in Water Content Chemical Repulsion Mechanical Fluid Stress Chemical Stress

Summing the four Momentum Equations Yields Solid Potential Mechanical Fluid Potential

Solid Stress Fluid Stress Total Fluid Stress

Solid Material Stiffness Matrix Solid Stress

Relative Fluid Displacement Absolute Solid Displacement

U=solid displacement W=relative fluid displacement Cn=negative (CL) ion concentration =electric potential The Poroelastic Equations, The Conservation of Mass of the Ions The Mobility Equations and The Kinematic Equations are Manipulated to Produce: In terms of :

Time derivatives of the Field variables Field variables Mass Matrix Stiffness Matrix Load Terms

The value of a field variable at any point inside the element at natural coordinates r,s,t is given by an interpolation of the values at the nodes r s t C(r,s,t) = N 1 C 1 + N 2 C 2 + N 3 C 3 +N 4 C 4 Nodal Values Basis Functions = f(r,s,t)

Differential Equation: Finite Element Approximation Galerkin Weighted Residual Statement Weight Functions = Basis Functions Basis functions Nodal Values

Substituting the finite element approximations and integrating the second order terms by parts yields: Matrix Equation