Part 1 The linear attenuation equation (Eqn. 1) describes the attenuation of X-rays, of a given energy (E), after passing through a homogenous material and is related to the linear attenuation coefficient, μ(E), and the thickness of the material that the X-rays transverse, t,whereby the intensity after the beam passes through a thickness t is given as I(E; t) and the initial intensity of the X-ray beam is given as I0(E) (Eqn. 1). I(E; t) = I0(E) exp(-μ(E) t) (1)
DEXA assumes an n = 2 model for the attenuation of an X-ray beam, of energy E, where it is approximated that the soft tissue is considered one layer (with thickness ts and linear attenuation coefficient μs(E)) and the bone a second layer (with thickness tb and linear attenuation coefficient μb(E)), which is an assumption from the reality of three layers. The total thickness of the measurement site, that the X-rays need to transverse prior to interacting with the film or detector, is thus defined as x = ts + tb.
Derive an expression for I(E; x), the intensity of the X-ray beam with an initial intensity I0 passing through a material which follows this n = 2 model, considering an arbitrary monochromatic X-ray beam of energy E. Hint: The final solution to this problem has been provided to you in the DEXA lecture for your reference. Be sure to show all steps.
Part 2 Define bone mineral density (BMD) in relation to the density of bone and the apparent thickness of bone (tb) as determined by DEXA and discuss the unit. DEXA requires that two photons of different energy, one higher than the other, be used in order to compute the BMD from the fundamental DEXA equation. Derive, showing all work, the fundamental DEXA expression for the computation of BMD, based on your solution from Part 1. You will need to use two equations, one for high energy, the other for low energy and you will need to initially solve for the thickness of bone tb before you can solve for BMD. Hint: There is no need to define R, the tissue correction factor. Please ensure that you show all of your work.