Polynomial dual energy inverse functions for bone Calcium/Phosphorus ratio determination and experimental evaluation

An X-ray dual energy (XRDE) method was examined, using polynomial nonlinear approximation of inverse functions for the determination of the bone Calcium-to-Phosphorus (Ca/P) mass ratio. Inverse fitting functions with the least-squares estimation were used, to determine calcium and phosphate thicknesses. The method was verified by measuring test bone phantoms with a dedicated dual energy system and compared with previously published dual energy data. The accuracy in the determination of the calcium and phosphate thicknesses improved with the polynomial nonlinear inverse function method, introduced in this work, (ranged from 1.4% to 6.2%), compared to the corresponding linear inverse function method (ranged from 1.4% to 19.5%). - Highlights: • Determination of bone Ca/P mass ratio using linear and nonlinear XRDE methods. • The method was verified by measuring six test bone phantoms. • Nonlinear quadratic function modeled calcium and phosphate thicknesses. • Improved accuracy in the determination of the Ca/P mass ratio with nonlinear method.