A nonlinear model for predicting radiographic contrast

A nonlinear model for predicting radiographic contrast R. L. Webber, D.D.S., Ph.D.,” Ph.D.,*** Bethesdn, Vd.

If. D. Youmans,

NATIONAL

RESEARCH

INSTITUTE

OF DENT$L

B.A.,**

and R. N. Nagel,

A computerized model has been developed which permits ideal maximum radiographic contrast, associated with clinically meaningful changes in x-ray attenuation for any combination of tissues, to be predicted from the spectral energy of the primary beam and the characteristics of the film. Results suggest that these factors influence availnlde contrast in ways which depend on the specific diagnostic task to be accomplished. Hence, it appears that no single technique is optimal for all tissue configurations.

E

arly work by Jacobson and Stuart-Mackay,l followed by Henrikson’s classic thesis, has shown that x-ray photon energy can be optimized for specific diagnostic tasks. Unfortunately, the approach used by these investigators is not easily extended to the analysis of many potentially interesting radiographic systems because of nonlinearities associated with specific elements. Henrikson* and later Richards and associates” also approached the problem on an experimental basis by actually generating dental radiographs using sources with various spectral distributions. The problem with studies of this type is the large number of variables which must be controlled in any given experiment. Under these conditions, the cascaded effects of individual factors influencing image quality preclude meaningful extrapolation of findings even to similar systems. In order to relate known theoretical factors to a variety of existing and other potentially interesting radiographic systems, a more comprehensive basis for analysis is required. The purpose of this study is to predict the over-all effect of such factors on image contrast for a variety of diagnostic applications of dental interest. RATIONALE

A digital computer permits modeling of a variety of radiographic applications, SO that the effects of nonlinear elements can be taken into account in a comprehensive fashion. *Chief, Clinical Investigations Branch. ““Chief, Biophysic Study Section, Division Health, Food and Drug Administration. ***Senior Staff Fellow.

790

of Biological

Effects,

Bureau

of Radiological

Volume -43 Number 5

Nonlinectr model

predicting

for

radiographic

contmst

799

A)

0

ENERGY

(keV)

ENERGY

(k&d

25

100

BJI, , , j , ( ] , ( , 0 I

40 ENERGY

100 (keV)

q , , , ,, ( ,, , ( 0

100 ENERGY

IkeV)

Fig. 1. X-ray energy spectra evaluatedfor relative effect on radiographiccontrast associatedwith specificdiagnostictasks. If one ignores the effects of scattered radiation (which can only decrease radiographic contrast), it is possible to calculate ideal maximum contrast associated with known changes in radiopacity from knowledge of the following factors : 1. The x-ray energy distribution of photon fiuence in the primary radiation. 2. The bulk density, attenuation characteristics, and thicknesses of the irradiated tissues. 3. The photon energy dependence of the film exposure sensitivity. 4. The H and L) optical density responsecharacteristic of the film/processing combination. The interplay between these various factors can be demonstrated by selecting tissue configurations which correspond to detection tasks of diagnostic interest. MATERIALS

AND

METHODS

A conventional broad-range spectrum and four selected monoenergetic peaks, shown in Fig. 1, were selected as input data for the mathematical model which is described in the Appendix. The continuous distribution was generated by a Gen-

0ral surg. May, 1977

0 Fig.

fluenee.

1A.

90

m

KVP

---------

EXPOSURE

-

PHOTON FLUENCE

---

ENERGY FLUENCE

40

80 80 loo ENERGY (keV1 spectrum expressed in terms of exposure, photon fluence, and energy

era1 Electric 90 II dental unit operated at 90 kVp with an added filtration of 1.5 mm. of aluminum. It was measured with an Ortec liquid-nitrogen-cooled germanium detector and a Hewlett-Packard multichannel analyzer system. The data were stored directly on magnetic tape via the analyzer system which included a tape recorder and a plotter controlled by a programmable calculator. This system was used to develop the basic formulation of the model. The model was later generalized and rewritten in Fortran IV, so that it could be implemented with more speed and flexibility by means of a PDP DECLAB 11/40 computer. The monoenergetic spectra were chosen to span the useful range of diagnostic energies. It was anticipated, from the work of Henrikson2 that these would demonstrate the effects of photon energy on image contrast associated with specific diagnostic tasks. It is important to point out that 90 kVp was selected for comparison with the monoenergetic spectra only because it contains photons in a very broad range of energies. Its inclusion should not be considered a general endorsement of high-kVp radiographic techniques. Since optimal modes of display depend on the diagnostic task to be accomplished, the model was adjusted to predict the contrast associated with lesions of reasonable size in both hard and soft tissues inside the mouth. That is to say that radiographic contrast was computed for the following hypothetical clinical situations wherein the complex of hard and soft tissues of the jaws are considered equivalent to appropriate thicknesses of water, homogenized bone, and hydroxyapatite : 1. A 10 per cent increase in thickness of soft tissue which normally

Volume Number

Nonlinear

43 5

model for predicting Xcnl

radiographic

contrast

801

Zcm

-H-l

Collimated X-Rays

Density Difference Which Determines Contrast <

-F’

km 0.3cm

(A)

l-II---

Density Difference Which Detbrmines Contrast

Fig.

Collimated X-Rays i= ?i rJY

<

8. Models

of specific

tissue

configurations

simulated

by computer.

totals 2 cm., such as might be produced by a hyperplastic lesion in the region of the parotid. 2. A similar change in soft-tissue thickness overlying 0.5 cm. of structurally homogeneous bone. 3. A similar change in buccal soft-tissue thickness overlying 0.3 cm. of structurally homogeneous tooth enamel. 4. A 0.02 em. cavity in tooth enamel normally 0.3 cm. thick lying behind buccal soft tissues which total 1.0 cm. in thickness. The method and values used to compute appropriate functional attenuations for the three tissues involved are discussed in the Appendix. Suffice it to say that the attenuation of x-radiation by tissues is essentially independent of the sequence of tissues involved, so that the four tissue configurations listed above can be considered equivalent to models of the types shown in Fig. 2.4 This simplification is permissible, provided that the effects of scatter are ignored and the primary beam is assumed to be nondivergent (parallel). The effects of these assumptions are such that computed contrast will be somewhat greater than that actually produced by real radiographic systems, particularly near edges. The scatter data compiled by Bruce and Johns5 suggest that these changes are relatively insensitive to spectral concentrations in relation to other factors influencing contrast for the tissue thicknesses involved and over the range of energies examined in this investigation. 6 This observation is substantiated by the fact that the film is located in close proximity to the teeth and that scatter from deeper tissues is largely eliminated by the lead backing in the film packet. In any case, it is assumed that contrast computed by the model for different spectra and tissue compositions of dental interest is relatively uninfluenced by scatter. Work is

Oral Burg. May, JOT’;

0.8 s 2 0.7 g 0.6 c u 0.5 s4 0.4 g 0.3 -

::I , , , , , , , , , , 0

20

40

60

ENERGY Fig. bA. Speed expressed as a function (Courtesy of Eastman Kodak Company.)

80

loo

(keV1

of photon energy for Kodak

Ultraspeed

film.

currently underway to further clarify the implications of this assumption. Contrast in this context is defined by the following equation: c = I(12 - 11)1,-‘1

where C is contrast, I, is the intensity of transmitted light through the radiograph in the area of interest, and I, is the intensity of transmitted light through an adjacent region used as a reference. Since contrast defined in this way has been shown to be preceived by the human eye as being relatively constant over a large range of photopic light intensities, 7 it is suggested that this parameter is a reasonable measure of detectability by human interpreters. That is to say that for reasonable brightnesses, and all other factors being equal, the greater the contrast, the more detectable the difference in gray level on the film and hence the more detectable the associated lesion. RESULTSAND DISCUSSION Fig. 3 shows the results of the model in terms of contrast expressed as a function of optical density for the 0.2 cm. change in soft tissue alone and overlying 0.5 cm. of bone and 0.3 cm. of tooth, respectively, when the source radiation has a conventional spectrum. Contrast appears to be approximately proportional to optical density for all three tasks. It is interesting that this result is not an obvious consequence of the relation between contrast and density difference (equation 11A). However, it is consistent with previous finding9 and independent data relating the detectability of lesions within a single tissue to the slope of the associated H and D curve.” The latter conclusion follows from the observation that the slope of the H and D curve for Kodak Ultraspeed film, to a very close approximation, is directly proportional to the optical density. Hence, the predicted detectability of lesions measured in terms of contrast can be unequivo-

Volume Number

Nonlinear

43 5

model for predicting

radiographic

0 B: Same

as “A”

Plus 0.5 cm of bone

A C: Same

as “A”

Plus 0.3 cm of tooth

003

0

0 A: 0.2 cm thick soft tissue lesion overlying 2.0 cm of soft tissue 0.6

contrast

0 A o

/

o

90 KVP Radiation 1 ‘h mm Al. Filtration Kodak Ultra Speed Fil

0

1

2 OPTICAL

Fig. 3. Radiographic contrast expressed tissue configurations using broad-spectrum

4

3 DENSlTY

as a function of optical primary radiation.

density

for

three

different

tally related to actual detectability of induced lesions in related studies which indicate a positive correlation with slope.” Since optical density may be considered the output and radiation exposure the input of the image-forming system, the appropriate relationship for predicting the effect of exposure on contrast is given by the H and D curve modified so that exposure is no longer scaled logarithmically. The resultant curve for Kodak Ultraspeed film is shown in Fig. 4, in which the slope is relatively constant throughout the range plotted. At first glance, the apparent linearity of this function seems to contradict observations based on previous experiments which suggested that diagnostic performance was related to the slope of the unmodified H and D curve.g This apparent impasse is also evident from superficial interpretation of results obtained from the model. All other factors remaining constant, the proportionality between density and exposure shown in Fig. 4 would seem to dictate that a given contrast based on a specified difference in optical density should be independent rather than proportional to exposure as observed in the data from the model displayed in Fig. 5. The fallacy of this argument becomes apparent when one takes into account the fact that the changes in exposure produced by the associated lesions do not remain constant with increases in exposure. On the contrary, these changes are necessarily proportional to exposure since the fractional attention of the tissues determines the proportion of the primary radiation reaching the film (see Appendix) . The fact that all the curves shown in Fig. 5 are linear and different only

Oral May,

Surg. 1977

Kodak Ultra Speed Film 80 KVP With Kodak Processing Chemistry

I

I

I

I

0

0.01

0.02

0.03

Fig. 4. Optical density expressed broad-spectrum primary radiation,

0.4

I I I 0.04 0.05 0.06 EXPOSURE (RI

as a function

of exposure

I 0.07 for

I 0.08 Kodak

I 0.09

I 0.10

Ultraspeed

film

using

0 A: 0.2 cm thick soft tissue lesion overlying 2.0 cm of soft tissue 0 B: Same as “A” Plus 0.5 cm of bone D C: Same as “A” Plus 0.8 cm of tooth

OA

! 90 KVP Radiation 1% mm Al. Filtration

4II 6 0.2 d lz 8 0.1

t 0 EXPOSURE Pig. 5. Radiographic contrast configurations using broad-spectrum

expressed primary

(RI

as a function radiation.

of exposure

for

three

different

tissue

Volume 43 Number 5

No&near

model for predicting

radiographic contrast

805

Table I. Task : Detect a 2 mm. change in soft-tissue thickness contrast

Technique (90 kVp I .5 mm. aluminum)

A

Three conventional packets

1

B

0.32 0.15 0.26 One 3-speed packet 0.32 0.26 A = 0.2 cm. change in 2 em. of soft tissue. B = A superimposed over 0.5 cm. of bone. C = A superimposed over 0.3 em. of tooth.

I

C

0.07 0.12 0.20 0.20

Relative (atJIm

exposure plane)

1X ::: 3x

in slope, irrespective of the task, suggests that detectability of all the lesions studied are predictably influenced by the exposure reaching the plane of the film. An interesting characteristic of the particular film used in this study” is that the relationship between contrast and exposure is linearly increasing for all readily visualized densities (Fig. 5). As a result, one would expect lesion detectability to be always improved by increasing the exposure over a relatively large range, irrespective of the output density. This prediction is consistent with data obtained in an earlier experiment wherein the radiographic detectability of induced carious lesions in interproximal enamel was found to be improved 29 per cent by increasing the optical density of the enamel image from 0.6 to 2.5 by means of increased exposure.7oHowever, interpretation of such dark radiographs required the use of illumination intensities significantly above those obtainable from conventional light boxes. Granted that interpretation is limited to density extremes amenable to visual interpretation with conventional light sources, a reasonable upper limit for optical density can be defined. Under normal conditions, visually detectable changes in contrast are limited to densities less than 2.’ In terms of Fig. 3, this means that the curves may be considered meaningless for densities greater than 2 unless special viewing conditions are employed. To the extent that this is true, it would be advantageous to consider using a new multiple film packet which would contain individual films having a variety of photographic speeds.For example, if such a packet contained three different films, each having the same shape of H and D curve but differing in speed, it would be possible to increase contrast simultaneously for all three tasks without exceeding the density range suitable for interpretation by conventional methods (Table I). An additional bonus would be the 50 per cent reduction in exposure required at the film plane by comparison with that resulting from the three separate exposures which would be required to obtain the same information in a conventional manner using only high-speed film. Such an approach also could reap additional benefit from the higher resolution obtainable from slower films. The effect of changing the primary x-ray spectrum from a conventional distribution to that of selected monoenergetic sources is shown in Tables II to V for selected tissue configurations. Table II shows the optical densities and corresponding exposures required to produce equal contrast of 0.35 for a 0.2 cm. change in soft tissue normally 2 cm. thick. This corresponds to the model shown *Kodak Ultraspeed

dental x-ray film.

Oral May,

Table II. Exposures of soft tissue

Technique

cstimatctl

for cyual detcctabilit~-

Optical density

90 kVp mm. aluminum (conventional) 2.25 10 keV (monoenergetic) 0.36 40 keV (monoenergetic) 3.00 100keV (monoenergetic) 4.50 “Weighted for equal filmblackening potential.

Contrasr 0.35 0.35 0.35 0.35

of an outgrowth Maximum exposure* alfilm plane (mR) 100 10.4 140 243

Surg. 7977

of 0.2 cm.

En trance exposure* (mRJ 238 2:; 349

Estimated exposures and contrast associated with an outgrowth of 0.2 cm. in 2 cm. of soft tissue when the density of the tissue is held constant Table

III.

Maximum Technique

Optical density

90 kVp mm. aluminum (conventional) 10 keV (monoenergetic) 25 keV (monoenergetic) 40 keV (monoenergetic) IO0 keV (monoenergetic) *Weighted for equal film-blackening

Contras1

1.0 I .o 1.0

0.14 2.69 0.92

t ::

0.07

0.10

exposure* atfilm plane (mR) 37.3 37.3 37.3 37.3 37.3

Entrance exposure* (mRI 87 ,toO t:

potential.

in Fig. 2, where the thickness of hard tissue (X) may he zero. In this case, exposure is defined as roentgens required to produce the observed film density on Kodak Ultraspeed film using Kodak processing chemistry under standardized conditions. Notice that for equal contrast the estimated skin exposure is highest at the low photon energy when the monoenergetic spectra arc compared. On the other hand, the estimated maximum exposure at the film plane, and beyond, is least from 10 keV radiation. Clearly, the optimum in terms of integral dose should reflect exposures measured both .ways. Another interesting observation is that for the contrast selected, all but the 10 keV spectrum require that the optical density of the soft-tissue image be greater than 2 which is the limit the eye can appreciate when conventional view boxes are used. Table III shows the effect on contrast when all exposures are adjusted to produce the same optical density and the results for a 25 keV monoenergetic source are included to provide more insight regarding the trade-off between exposure and contrast for monoenergetic sources. Since the optical density is uniquely determined by the maximum exposure at the film plane and the thickness of the cheek is assumedto be uniform (2 cm.) over the entire film, exposure in this location is necessarily the same for all sources (see Appendix). One should notice, however, the tremendous differences in entrance exposure and the large range of contrasts produced under these conditions. Of particular interest are the data associated with the 25 keV monoenergetic source. Such a beam increases contrast 500 per cent over that produced with the conventional 90 kVp, 1.5 mm. aluminum spectrum while increasing skin exposure only 26 per cent. Less exposure would be required at slightly higher energies which exceed the k absorption edge for silver halide film (about 25.5 keV) .

Nonlinear

model

for

predicting

radiographic

contrast

807

IV. Exposure estimated for equal contrast associated with a 0.2 mm. cavity in a tooth 3.0 mm. thick through 1 cm. of cheek Table

Technique

90 kVp mm. aluminum (conventional) IO keV (monoenergetic)

Optical density

Contrast

Maximum exposure* at film plane (mR)

0.60 0.02

0.07 0.07

80 101’

40 0.60 100keV keV (monoenergetic) (monoenergetic) 3.43 *Weighted for equal film-blackening potential.

0.07

2::

Entrance exposure* ImR)

120 > IO’03 2::

Table V. Estimated exposures and contrasts associated with a 0.2 mm. cavity in a tooth 3.00 mm. thick through 1 cm. of cheek when the optical density of the tooth is held constant

Technique

Optical density

90 kVp mm. aluminum (conventional) 0.60 0.60 10 keV (monoenergetic) 40 keV (monoenergetic) 0.60 0.60 100 keV (monoenergetic) *Weighted for equal film-blackening potential.

Contrast

0.07 IO” 0.07 0.01

Maxim urn exposure* atfilm plane (mR)

Entrance exposure* (ml?)

>;&3

120 > 10’03

:i

;;t

Tables IV and V relate to a more familiar task, that is, detection of an incipient lesion in a tooth as simulated by the model shown in Fig. 2. Again, when contrast is held constant, Table IV showsinteresting variations in exposure both at the film plane and at the skin surface. In this casecontrast and optical density were made comparable to the respective values associated with a typical, conventional bitewing technique using 90 kVp radiation. At high mean energies, densities can range well beyond the usual 0 to 2 values when the model is constrained in this way. When the density is held constant at the same level for all techniques and the contrast is allowed to vary, Table V likewise shows large differences in exposure measured both at the skin surface and at the plane of the film. In this case, the maximum exposure at the film plane is not determined by the hard tissues per se, since someof the radiation through the cheek reaches the film without passing through the teeth. Hence, the maximum exposure at the film plane varies with the technique when the density of the particular tissue configuration (tooth @US cheek) is held constant. Again, very large differences in contrast and exposure are apparent. In the case of 10 keV monoenergetic radiation, exposure values exceeded the capacity of the computer. Likewise, the incredibly large value of the resulting contrast is not meaningful except to suggest that 10 keV radiation is grossly unsuited for this type of diagnostic task. The huge exponential exposures associatedwith 10 keV monoenergetic spectra in Tables IV and V are the result of arbitrarily constraining contrast and density to the values shown. This requires extrapolation of the fitted curves to unrealistic extremes. Such values should be interpreted only as indicating the physical impossibility of attaining the clamped

808

ll’ehbcr,

I’ollm(ols,

tr,rd

Oral May,

*V,,p?l

Hurg. 1977

values with realistic exposures. Conversely, 40 key radiation appears to be clearly superior to a conventional 90 kVp beam, since the monoenergctic x-rays provide the same contrast with considerably less exposure measured either at the film plane or at the skin surface. This finding confirms earlier wnrk 1,~ IIcnrikson* and later. by Richards and associates’
AND

CONCLUSIONS

d computerized model has been developed which permits ideal maximum radiographic contrast, associated with clinically meaningful changes in x-ray attenuation for any combination of tissues, to be predicted from the spectral energy of the primary beam and the characteristics of the film. Results suggest that these factors influence available contrast in ways which depend on the specific diagnostic task to be accomplished. Hence, it appears that no single technique is opt,imal for all tissue configurations. Although contrast comnutetl in this way docx not take into account other factors of diagnostic interest, such as image resolution and context-dependent cues to lesion identity, it is an independent factor which can profoundly influence the detectability of specific changes in tissue attenuation and, as such, it may be considered a reasonable basis for predicting diagnostic performance.

APPENDIX A radiography

model

The radiation from a diagnosticx-ray tube comprises photons of many energies, and a plot of the measured number of photons at each energy is commonly called a “spectrum.” The maximum energy that a photon may have is determined by the voltage applied to t,he x-ray tube. The shape of the spectral distribution is primarily determined by the strongly nonlinear attenuation of the beam by the xvindows of the tube and its housing and hy filters that may have been added. When an x-ray beam passes through a heterogeneous set of tissues, it is attenuated primarily by the amount and chemical composition of each tissue in the set. This produces an x-ray image which is varied in both quantity and spectral quality. If this spatial pattern of x-ray spectra is partially ahsorbed by a silver bromide film, a latent image mill be produced which can be made visihle by chemical processing. The semitransparent radiographic image consists of a variety of gray levels, and it is the contrast of the differences in gray levels which makes possible diagnostic interpretation of the structure of the set of tissues. Clearly, the physics of radiography is dominated by strongIy nonlinear functions of photon energy, and it is this aspect to which the model presented here speaks. Spectral energy fluencr (q) is the first moment of the photon fluence distribution (G) : * = E,+l 1 = 1, 2, . . ., (IA) where a repeated subscript in a term indicates summation. The exposure, X, is proportional to the sum of the product of the energy absorption coefficient of air, pa,, and the energy fluence distribution, x= (2A) w.i*k, where CY is a constant which accounts for geometry and the conversion of units. Fig. 1A demon straten the different spectral shapes of these quantities. The j-th. member of a set of tissues may have the k-th. density (P,~), l-th. thickness (tj,), mass thickness (rjLl), and an attenuation coefficient (r,,). In terms of these, the fracctional attenuation, f ,,,,, by the m-th. suhset of tissues is

Volume Number

Table

Nodimar

4’: 5

IA. Data used to calculate

Tissue

model

for predicti?lg radiographic contrast

fractional

attenuation,

Bulk density gm./cm.”

Cheek (soft tissue) Bone

1.00 1.65

Dental enamel (hydroxy apatite)

3.00

809

fill, Per cent atomic constituents (by weight) H 11.1 0 88.9 H 6.4 C 21.8 N0 4::: Mg 0.2 P 7.0 s 0.2 Ca 14.7 H 0.2 0 41.4 P 18.5 Ca 39.9

ft, = exP(-bwjrtji~jrlm)

= eXP(-/hj~jtl~jklm)~ (3A) j, k, 1, m = 1, 2, 3, . . . , where (Us,,,,) is a nonphysical indicator function whose elements have the values 0 or 1. In this investigation j ranged from 1 to 3 corresponding to soft tissue, bone, and dental enamel, respectively. The bulk density and relative atomic composition of these tissues, used to calculate their fractional attenuations, are listed in Table 1.4. The respective attenuation coefficients for these constituents were obtained from National Bureau of Sandards Circular 583, X-ray Attenmtion Coefficients From 10 keV to 100 VeV. Values were calculated for each tissue at seven different energies ranging from 15 to 80 kev. These values were then used to compute a beet-fit curve for an empirically matched function having the general form: In p = A + B(ln E) + C(ln E)a, (46) where p is the attenuation coefficient at energy E, and A, B, and C are empirically determined constants. The exposure (roentgen) of an attenuated x-ray beam can he expressed as the product of Equations ZA and 3A, (5A) Xl” = %*,f,“. The speed, S, reciprocal of the exposure required to produce an optical densit,y of 1, of a film depends upon photon energy as shown in Fig. 2A. However, in this figure n, represents film speed normalized to the maximum value in the range and, as such, t] may be considered a measure of the relative radiographic effectiveness of different photon energies. This function may be calculated with fair accuracy with knowledge of the physicochemical properties of film, lmt here the following function was fitted to data for Kodak Ultraspeed dental x-ray film supplied by Eastman Kodak Company: p = 0, 1, 2, . . . ? S = exp[Er(lnE)r], Cf.=) where r0 = -1.235 and c, = 1.269 for 16 < E 5 24, F~ = -17.16, Ed = 10.69, and Ed = -1.385 for 24< E 2 35, to = -15.66 cl = 10.86, and E? = -1.550 for 35< E 5 100. The dependence of film density, S, on exposure, X, i.e. the H and D curve for Ultraspeed dental film, was also supplied by Eastman Kodak Company. The following function was fitted: 6 = antilog [en(log,,X)‘t], q = 0, 1, 2, . . . , (7.4) where F” = 0.07945 and Ed = 0.2945 for -2.82
6 = log,,(b/I).

(8A)

i:: defined in tams of light transmitted (I) through two (liffcwnt 2 is cwnsitlcred thtb twt area and 1 is the rrfcrc~nw area : (’ = (T, - I,,/], In the special case for t ransmittwl light corrcspond~ng to gray li>vcJla 1 antI can lw expressed rcspectiwly :

and

Contrast 2 whfm~

By optical

combining density:

Equation

9h

with

6, = log,,, t iii/l,) 6, = log,, (To/r:) Equation lOA, contrast

gray

lcvrls

I

( 9.1 1 ‘I, lklu:ltiorl

X.1

(lO;\j

can lw now

expressed

in terms

of

C = antilog,, (6, - 6,) -1 (11X) The equations developed to this point contain all of the information necessary for R geonw try- and time-independent radiography model. The principal difficulties to be overcome are that 7 and 6 arc not independent and the (Y may be either unknown or of doubtful accuracy. These constraints are removed as follows: The dependence of radiographic effectiveness of 9 on energy may be accounted for by multiplying the terms of the energy fluence by those of 1). An accommodation of the H and D transfer function is more subtle because it requires the imposition of a parametric constraint, 6,(X,). This is not entirely unfortunate, since it forces conscious attention to the task-dependent nature of the analysis. The accommodation takes the form of a factor of ~l~*~,p....,: (12A) Pr...~ = ff~r.,,/L, where the subscripts, m and n, signify different indicator functions. Subscript r indicates a particular value of optical density, and this is chosen first to fix both the task-dependent reference gray level and its functionally concomitant exposure. Should one elect to have m = 0, the null indicator function would be implied and there would be no tissue attenuation. This definition of /3 also removes (Y from further concern. To summarize, we may write the following expression for optical density which includes the basic list of radiographic variables: Thus, any combination of tissues other than that chosen to correspond to 6, will result in a 6 M-T, and image contrast can then be determined by the use of Equation 11A. If desired, the radiographic task can be specified in terms of a contra$ value for a given pair of tissue combinations. However, one optical density-tissue set must he identified as the reference condition. Because of this, the model predicts contrast to be dependent on the optical density of the reference image. Clearly, the model can be used inversely to determine the most desirable transducer characteristics for any specification of the remaining variables. Finally, it should be noted that skin and film exposures, independent of geometry, are readily obtainable from the model, and that, since the film/air exposure ratio analogue of the therapeutic tumor/air ratio is accessible, suitable depth dose data would make possible the estimation of actual skin and integral doses. The authors gratefully acknowledge Dr. Richard P. Chiacchierini, Bureau of Radiological Health, Food and Drug Administration, for his most valuable consultation on the mathematics and for fitting appropriate functions to the tabular data base used in the model. Thanks also are due Drs. Margarete Ehrlich, Center for Radiation Research, National Bureau of Standards, and Mario Nylen, Laboratory of Biological Structure, National Institute of Dental Research, for providing insight and data resource information. The helpful suggestions by Mr. Wah Lrc of the Division of Biological Effects, Bureau of Radiological Health, Food and Drug Administration, and the programming efforts of Mr. Ed Mangiaratti of the Diagnostic Methodology Section, National Institute of Dental Research, National Institutes of Health, are also appreciated. REFERENCES

1. Jacobson, B., and Stuart-Mackay, Biol. Med. Phys. 6: 201-261, 1958.

R. S.:

Radiological

Contrast

Enhancing

Methods,

Adv.

Volume Number

43 5

Noalizear

model for predic&lg

radiographic

contrast

811

2. Henrikson, C. 0.: Iodine 125 as a Radiation Source for Odontological Roentgenology, Acta Radiol. Supp. 269, 1967. A. G., Barber, G. L., Bader, J. D., and Hale, J. D.: Samarium Filters for Dental 3. Richards, Radiography, ORAL SURG. 29: 704-715,197O. 4. Hiibner, TV.: ifrber den Einfluss von Fremdstoffen in Filtersubstanzen bei Schwachungsund Halbwerte schichtmessunaen. Fortschr. Geb. Roentgen&r. Nuklearmed. 89: 629, 1958. Scattered in Low Atomic Number 5. Bruce,, IV. R., and Johns, H. E.: ‘The Spectra of X-ray; Materials, Br. J. Radio]., Rupp. 9: 1, 1960. M. M.: The Physical Aspects of Diagnostic Radiology, New York, 1967, 6. Ter-Pogossian, Harper & Row, pp. 217, 252-263. C. H. (editor) : Vision and Visual Perception, New York, 1965, John Wiley L 7. Graham, Sons, Inc., pp. 68-75. 8. Ritehey, B., Feldman, A., and Greer, W.: Roentgenography of Enamel; Apatite as a Phantom Material; Contrast as a Function of Exposure Factors, ORAL SURG. 13: 188-193, 1960. 9. Webber, R. L., and Ryge, G.: The Significance of Exposure Parameters in Dental Radiog raphy, ORAL SZ~G. 27: 740-753, 1969. Significance of Intra10. Webher, R,. L., Benton, P. A., Cvar, J. F., and Ryge, G.: Diagnostic tissue Contrast in Bitewing Radiographs, ORAL SURG. 28: 352-358, 1969. Beprint requests to: Dr. R. L. Webber Chief, Clinical Investigations Bldg. No. 10 Rm. ZB09 National Institutes of Health Bethesda, Md. 20014

Branch,

NIDR