X-ray attenuation at multiple energy levels As an X-ray beam passes through a homogeneous material, the intensity at depth t within the material can be described with the
Beer-Lambert law: I(t) = I_0 e^{-\mu t}, where I_0 is the initial beam intensity and \mu is the
linear attenuation coefficient. For DXA, the above equation is often expressed in terms of the
mass attenuation coefficient (the attenuation coefficient divided by the density of the material \rho): I(t) = I_0 e^{-\mu t} = I_0 e^{-(\frac{\mu}{\rho}) t \rho} = I_0 e^{-(\frac{\mu}{\rho}) \sigma}, where (\mu / \rho) is the mass attenuation coefficient, and \sigma is the
area density. For a non-homogeneous material with n separate materials, the absorption relationship is given by: I = I_0 e^{-\sum^{n}_{i=1}(\frac{\mu}{\rho})_i \sigma_i}, where (\mu / \rho)_i and \sigma_i are the mass attenuation coefficient and area density of material i, respectively. Soft tissue and bone have different attenuation coefficients for X-rays. A single X-ray beam passing through the body is attenuated by both soft tissue and bone, and it is not possible to determine from a single beam how much attenuation is attributable to the bone. However, attenuation coefficients vary with the energy of the X-rays, {{NumBlk|:|I^{H} = I^{H}_0 e^{-((\frac{\mu}{\rho})^{H}_{b}\sigma_b + (\frac{\mu}{\rho})^{H}_{s}\sigma_s)},|}} {{NumBlk|:|I^{L} = I^{L}_0 e^{-((\frac{\mu}{\rho})^{L}_{b}\sigma_b + (\frac{\mu}{\rho})^{L}_{s}\sigma_s)}|}} where the superscripts H and L refer to the high and low energy beams, respectively, and the subscripts b and s refer to bone and soft tissue, respectively.
Direct density measurement The area density of bone can be obtained by first re-arranging the two equations for I^{L} and I^{H}, defined in and , in terms of \sigma_s, setting them equal to each other, and re-arranging the resulting equation for \sigma_b: {{NumBlk|:|R =\frac{(\frac{\mu}{\rho})^{L}_{1}f_1 + (\frac{\mu}{\rho})^{L}_{2}f_2}{(\frac{\mu}{\rho})^{H}_{1}f_1 + (\frac{\mu}{\rho})^{H}_{2}f_2}.|}} In the special case where only material 1 is present in the sample (i.e. f_1=1, f_2=0), then R is just the ratio of the mass attenuation coefficients at those energy levels R =\frac{(\frac{\mu}{\rho})^{L}_{1}}{(\frac{\mu}{\rho})^{H}_{1}}=R_1, which is a constant value R_1 that can be calculated using known values of the mass attenuation coefficients of material 1 at the two energy levels. Likewise, the constant value R_2 can be calculated as the ratio of mass attenuation coefficients of material 2 at both energy levels, which can be seen by substituting f_1=0, f_2=1 into . In
body composition measurements, pixels in regions of the DXA image with no bone are considered to be a mixture of fat and lean soft tissue compartments. At higher energy levels (e.g. 70 KeV) the mass attenuation coefficients of both fat and lean soft tissue compartments are similar. Making the approximation that {{NumBlk|:|(\frac{\mu}{\rho})^{H}_{1} \approx (\frac{\mu}{\rho})^{H}_{2} = (\frac{\mu}{\rho})^{H}|}} and substituting into gives R \approx\frac{(\frac{\mu}{\rho})^{L}_{1}f_1 + (\frac{\mu}{\rho})^{L}_{2}f_2}{(\frac{\mu}{\rho})^{H}(f_1 + f_2)} = \frac{(\frac{\mu}{\rho})^{L}_{1}f_1 + (\frac{\mu}{\rho})^{L}_{2}f_2}{(\frac{\mu}{\rho})^{H}} = R_1 f_1 + R_2 f_2, since by definition f_1 + f_2 = 1. Substituting f_2 = 1 - f_1 into the above and re-arranging gives f_1 = \frac{R-R_2}{R_1-R_2} and f_2 = 1 - f_1 = \frac{R_1-R}{R_1-R_2}. Thus, when the assumption in holds the relative fractions of the two materials can be estimated using the measured value of R and known values of R_1 and R_2.
Scanner characteristics One type of DXA scanner uses a
cerium filter with a
tube voltage of 80
kV, resulting in effective photon energies of about 40 and 70
keV. Another type of DXA scanner uses a
samarium filter with a tube voltage of 100 kV, which produces effective energies of 47 and 80 keV.
Limitations compared to 3D techniques Because DXA calculates BMD using area, it is not an accurate measurement of true bone mineral density, which is
mass divided by a
volume. To distinguish DXA BMD from
volumetric bone-mineral density, researchers sometimes refer to DXA BMD as an areal bone mineral density (aBMD). The confounding effect of differences in bone size is due to the missing depth value in the calculation of bone mineral density. A consequence of dividing by area instead of volume is that BMD is overestimated in taller subjects, or those with larger bones, and underestiamted in shorter subjects, or those with smaller bones. Three-dimensional imaging modalities such as
quantitative computed tomography (QCT), which measure volumetric bone mineral density (vBMD), have been shown to correlate more strongly with fracture risk than aBMD; however, both measures exhibit significant associations with fracture risk and are capable of discriminating between individuals with and without fractures. Compared with QCT, DXA scans are both cheaper and impart a lower dose of ionizing radiation to the patient. To correct for the lack of depth information when caclulating aBMD by dividing the bone mineral content by area, methods to correct for this by calculating a bone mineral apparent density (BMAD) score have been proposed. These attempt to account for bone size using geometric assumptions to account for various bone sizes, for example by dividing by Area^{3/2} for the spinal vertebrae, and Area^{2} for the femur, femoral neck and forearm. ==Bone density measurement==