Three different photon interactions with tissue occur in the energy range that is relevant to bone densitometry: photoelectric effect, Raleigh scattering and Compton scattering (IAEA 2010).
The photoelectric effect converts the energy of a single photon into that of a single free electron and this type of interaction dominates at low energies. The photon is completely absorbed when it interacts with a bound electron and the electron emerges with kinetic energy equal to the photon energy ℎU minus the electron binding energy. The probability of photoelectric interaction decreases rapidly as the photon energy increases. In general, the mass attenuation coefficient VW for photoelectric absorption varies roughly as 1/(ℎY)Z and with [Z, where [ is the atomic number of the tissue (Hendee et al. 2003).
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Rayleigh scattering occurs when the x-ray photon interacts with the whole atom. Photons are scattered in approximately the same direction as the incident photon with negligible loss of energy. This interaction occurs in media with high atomic number and is important in tissue only for low-energy photons. The importance of Rayleigh scattering is further reduced because little energy is deposited in the attenuating medium.
Compton scattering occurs when an incident x-ray photon ejects an electron from an atom and the photon is scattered with lower energy. The Compton mass attenuation coefficient varies directly with the electron density (electrons per kilogram) of the absorbing medium because Compton interactions occur primarily with loosely bound electrons.
The Compton mass attenuation coefficient is approximately independent of the atomic number of the attenuating medium ([). For this reason, radiographs show very poor contrast when exposed to high-energy photons. When most of the photons in a beam of x-rays interact by Compton scattering, little selective attenuation occurs in materials with different atomic number. The image in a radiograph obtained by exposing a patient to high-energy photons is not the result of differences in atomic number between different regions of the patient, but reflects differences in physical density (kilograms per cubic meter) between the different regions (e.g., bone and soft tissue) (Hendee et al. 2003). The transmission of monoenergetic photons through the body is given by:
\ = \
]K^4 _− ` a
bc(d
ccef
ce)
^
c)g (4.1)
where \ , \] are the incident and transmitted intensities, abc is the mass attenuation coefficient in Rh2 ij) of the absorber material, d
c is the material density, ^c is the material thickness in Rh, N represents an individual tissue and I is the number of different tissues.
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The total attenuation depends on the thickness of the tissue and, in DXA, this will vary as bone mineral and soft tissue thickness varies. As a monoenergetic x-ray beam passes through bone that is surrounded by soft tissue, the transmitted intensity will be given by:
\ = \
]K^4 − ( a
kd
k^
k+ a
ld
l^
l) (4.2)
where m and M represent bone mineral and soft tissue respectively; ^l, ^k are the thickness of bone mineral and soft tissue in the x-ray path and dl, dk are the physical density of bone and soft tissue. The main chemical components of soft tissue are hydrogen, oxygen, nitrogen and carbon and these elements have lower physical density and lower atomic number than calcium and phosphorus that are found in bone mineral. The volume of an attenuating element is given by:
n = A × ^ (4.3)
where A is the projected area perpendicular to the x-ray beam. The physical density is given by:
d = h
n =
h
p × ^ (4.4)
where h is the mass. Combining equations 4.3 and 4.4 gives the area density (M) i.e. mass per unit area as shown in equation 4.5.
q = h
p
= ρ ^ (4.5)
DXA is a two-dimensional projection technique and therefore BMD is an area density measurement not a volumetric density. Substituting for M in equation 4.2 gives:
\ = I
]K^ 4 −(µ
kq
k+ µ
lq
l), (4.6)
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wL x\
*\y = a
kq
k+ a
lq
l(4.7)
The logarithm of the ratio of incident intensity to transmitted intensity ({), can be written as:
J = µ
kq
k+ µ
lq
l(4.8)
In dual energy mode, transmission measurements are made at two different photon energies (low and high) are given by:
Low energy:
{
~= µ
k ~q
k+ µ
l~q
l(4.9)
High energy:{ = a
kq
k+ a
lq
l(4.10)
These simultaneous equations are solved for ql and qk :
q
l=
{
~− Äa
k~a
kÅ Ç {
a
l~− Äa
k~a
kÅ Ç a
l(4.11)
q
k=
xa
l ~a
lÅ y { − {
~xa
l~a
lÅ y a
k− a
k~(4.12)
The ratio of the soft tissue attenuation coefficients at the two energies is defined as Ék and this depends on the soft tissue composition.
É
k=
a
k~51
q
l=
{
~− É
k{
a
l~− É
ka
l(4.14)
In a bone-free region, ql = 0 and so
{
~= µ
k ~q
k(4.15)
{ = a
kq
k(4.16)
É
k=
a
k~a
k=
{
~{ (4.17)
Ék is a measure of the percent fat of the soft tissue (IAEA 2010). It varies with the lean- to fat ratio within soft tissue with an increase in fat reflected by a decrease in Ék. BMD is calculated by assuming that the thickness and composition of the soft tissue in the bone- free region is identical to tissue anterior and posterior to bone and within bone so that they have identical Ékvalues.
The measurement of ql is described in equation 4.14 for a single x-ray path through bone and soft tissue, whereas in practice measurements over a volume of bone are needed. To achieve this, a linear series of measurements across the part of the patient that contains bone must be made. The region of tissue adjacent to bone but within the analysis region is known as the soft tissue baseline. The result is an attenuation profile for each of the x-ray beams. The high-energy absorption profile is multiplied by Ék and then subtracted from the low energy profile to leave bone mineral.
BMD values are presented as a digital image with each pixel corresponding to a measurement point through the patient. In order to cover a larger area of bone, numerous attenuation profiles are acquired along the length of the bone to form an
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image. The Ék value is estimated line-by-line over the image and averaged over soft tissue on either side of bone within the DXA algorithm. The projected area of bone (BA) in Rh2 is the sum of the pixels within the bone edge. Within this area, the BMD of individual pixels is averaged and multiplied by the BA to calculate bone mineral content (BMC) in i.
A lumbar spine DXA image showing analysis regions and an example of BMD, BMC and BA results is shown in Figure 4-1 (IAEA 2010). Lumbar spine BMD measurements are usually made in the antero-posterior (AP) projection although it is also possible to measure BMD in the lateral projection.
Figure 4-1: Lumbar spine DXA image showing analysis regions and an example of BMD, BMC, and BA results