PROjECTION RADIOGRAPHY
6.2.3. Effects of projection geometry 1. Superposition
as noted in section 6.2.2, radiographs are a 2-D representation of a 3-D object. this superposition leads to a significant loss of image contrast, which provided one of the prime motivations for the development of ct scanners.
superposition also leads to loss of all depth information, and ambiguity in the relative sizes of objects at different depths. furthermore, it directly overlays objects in such a way that it can become difficult or impossible to distinguish one from the other, or even to identify some of the objects.
(a) (b)
FIG. 6.2. (a) Effect of depth of objects on their projected size; (b) effect of angulation on the projected length of an angled object.
6.2.3.2. Geometrical distortion
geometrical distortion can be considerable and confusing in projection radiographs. the first effect is that all objects are magnified in the image. the further from the image receptor the object is placed, the greater the oiD and the greater the magnification. the image size of objects, therefore, depends on their actual size and on the oiD and projection direction, leading to ambiguity. this effect is illustrated in fig. 6.2(a). the three spheres a, b and c are the same size, but are projected at different sizes owing to their oiDs. furthermore, projection leads to shape distortion. in fig. 6.2(b), a tilted object is shown projected at a range of angles, illustrating the increasing degree of foreshortening as the angle increases.
6.2.3.3. Inverse square law
for an isotropic point source, the X ray beam intensity is inversely proportional to the square of the distance from the source. an X ray tube with its attached collimator is a good approximation to a point source for distances greater than about 50 cm from the focal spot, and obeys the inverse square law (isl) almost exactly at distances greater than this. only at low kV settings, such as those typical of mammography, does air attenuation affect the inverse square relationship. this is illustrated in fig. 6.3, where the air kerma per unit mas is shown over the fiD range of 50–250 cm. figure 6.3 also presents the calculated curve assuming the isl.
FIG. 6.3. Deviation from the ISL due to air attenuation for a tungsten target X ray beam with 0.5 mm Al added filtration at a voltage setting of 30 kV and no compression paddle.
the isl results in the need for an increase in the mas as the fiD is increased in order to maintain the same air kerma at the image plane. the increase required is given by:
2 1
2 2 FID
1 FID
mAs mAs
d d
= (6.6)
furthermore, the air kerma at the patient entrance surface is greater than that at the image receptor (neglecting attenuation), by the ratio:
2 FID FSD
d d
(6.7)
in these expressions, dfiD is the fiD and dfsD is the focus to skin distance (fsD). it is easy to show that as the fiD is increased, the incident air kerma (Ki) may be decreased, keeping the same kerma at the image plane; the formula for this is:
2 1
2 1
1 2
2
FID FSD
i i
FID FSD
d d
K K
d d
= (6.8)
this relationship can be used to prevent excessive skin doses; generally, an fiD of 100 cm or greater is sufficient. it does not, however, result in a similar reduction in the overall dose to the patient because an increase in the entrance surface X ray beam size is required as the fiD is increased in order to prevent cut-off of the region of clinical interest. the effective dose is approximately proportional to the dose–area product. the dose reduces at longer fiD according to eq. (6.8), but the area increases. therefore, there is little or no change in effective dose [6.1].
6.2.3.4. Geometrical unsharpness
ideal image sharpness would be produced by a point source, the spatial resolution in such a case being limited by the image receptor factors such as phosphor layer thickness, lateral spread of light in scintillators, and the image matrix. however, owing to the restriction on the permissible temperature of the focal spot and the focal track of the anode, typical focal spot sizes of 0.6–2.0 mm
images of small body parts; typically, the fine focal spots are 0.3–1.0 mm, but must be operated at lower mas to protect the X ray tube from heating effects.
the spatial resolution depends on the focal spot size and the image receptor, and both need to be considered. for the demagnified image, the width of the penumbra, or more correctly the edge gradient, caused by a focal spot of size Xf, is given by the geometric unsharpness (Ug) divided by the magnification, where Ug is given by:
g F OID FID
U X d
= d (6.9)
where doiD is the oiD. since the magnification, m, of the object at the image receptor is given by:
FID
FID OID
m d
d d
= − (6.10)
then eq. (6.9) is equivalent to:
Ug = Xf (m – 1)/m (6.11)
if the fiD were to be changed, then to maintain the same focal spot resolution, the new focal spot size may be determined using eq. (6.9) for the old and new cases and equating. this gives:
new
new old
old
FID
F F
FID
X X d
= d (6.12)
however, the change in fiD will change the magnification, which will affect the overall image sharpness because of the effect of the image receptor blur. the overall unsharpness is given by Dance et al. [6.2] as:
1/2
2 2 2
2 F
2 2 r 2
r
1 1 FS
1 m 1 1 1 X
U F U U
m U
m F m
−
= + = + −
(6.13)
in this expression, Ur is the intrinsic image receptor unsharpness (that for m = 1) and it is assumed that the geometric and receptor unsharpness can be added in quadrature. the overall unsharpness U is scaled to a magnification of 1.
optimization of projection radiographs involves choosing an appropriate focal spot size. this requires a compromise between the exposure time and the resolution. for example, a very small focal spot will provide good spatial resolution, but only permit a low tube current, therefore requiring a long exposure time, leading to increased risk of motion blur. While it may be considered that quantum noise limits the detectability of fine details, there is some evidence that smaller focal spots than are currently employed may lead to improved spatial resolution. this is because the system detective quantum efficiency (see chapter 4) is affected by the focal spot modulation transfer function (Mtf).
the focal spot Mtf may be measured using a pinhole to determine the point spread function, or a slit to determine the line spread function, and calculating the normalized modulus of the fourier transform of the spread function. figure 6.4 shows a pinhole image of a focal spot and a 2-D representation of the Mtf.
FIG. 6.4. Typical distribution of the X ray intensity of a 2.0 mm focal spot (left) and the corresponding 2-D MTF (right).
note that the Mtf of a focal spot is given by convention for a magnification factor of 2.0. to correct the Mtf for the true magnification, the frequency axis must be scaled as follows (where the symbols have the obvious meanings):
old new
new old
old new
1
1
m m
f f
m m
−
= × − (6.14)