Theoretical analysis of error propagation in triple-energy absorptiometry: Application to measurement of lead in bone in vivo

We propose a three component tissue decomposition for quantifying lead in bone from a mixture of bone and muscle in viva using a triple-energy absorptiometric method. The theoretical optimization of this method, by relating signal uncertainty to radiation dose, requires an expression of the signal variance. The error propagation was therefore theoretically modeled for a counting detector, assuming noise dominance by quantum statistics and neglecting covariance between energy levels. A final expression for the lead signal variance at each energy level was obtained via a Jacobian matrix. The Jacobian was maximized by choosing the first energy as low as permissible by dose constraints below the lead K edge. A second optimum was achieved when the upper energy was just above and the middle energy was just below the lead K edge. While the signal-to-noise ratio (SNR) had similar behavior to that of the Jacobian as a function of middle and upper energies, the SNR was almost constant as a function of lower energy in the 40-60 keV range. Hence, dose could be reduced without SNR loss. A simulated clinical measurement on an adult tibia using a 50 mCi ¹⁵⁵Eu source and a 10 min acquisition time resulted in a standard deviation of 4 μg Pb/g bone mass. This approach can be applied to other systems containing three components, provided there is a K edge within the counting energy range.