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A computer program to study image formation of small blood vessels
COMPUTER PROGRAMS IN BIOMEDICINE 2 (1972) 118-122. NORTH-HOLLAND PUBLISHING COMPANY
A COMPUTER
PROGRAM
TO STUDY IMAGE FORMATION
OF SMALL BLOOD VESSELS * T. SANDOR and D.F. ADAMS Harvard Medical School and Peter Bent Brigham Hospital, Boston, Massachusetts, USA
A computer program is described which facilitates the rapid assessment of a number of physical parameters affecting the image formation of small blood vessels. These parameters are: the dimensions of the focus, the intensity distribution of the emitted photons over the focus area, the energy spectrum of the emitted photons, the attenuation by the contrast agent used, and the geometry of the exposure. Independently measured or calculated values of the above parameters are used in the program. The output of these values is a map of the intensity distribution of X rays in the image plane during a simulated exposure of a small blood vessel. Blood vessels
1. Introduction The recent increased interest in and knowledge of the microvasculature in disease of various human organ systems has stimulated great rapid developments in the manufacture of X-ray tubes with small focal spots. The radiologist who wants to obtain in vivo information about these vessels is faced with questions such as: What is the minimum iodine concentration necessary within a blood vessel of given diameters to provide a perceptable image? What is the smallest blood vessel size that can be perceived? How does the non-uniform emission of X-ray photons from a focal spot affect the image of a blood vessel? To answer questions like those above the radiologist must know the combined effect of a number o f physical parameters on the blood vessel image. The most important ones are: 1. the dimensions of the focal spot, * This work was supported in part by grants HE11668, HE05832, GM01910, 5-S01-FR 05489-08, and 5-S01-FR 05489-06 from the National Institutes of Health, U.S. Public Healtl/Service.
Image simulation
2. the emission pattern of photons from the focus, 3. the filter employed, 4. the energy spectrum of photons, 5. the applied kilovoltage, 6. the contrast agent used, more specifically its linear attenuation coefficient, ~(E) as a function o f energy, 7. the geometry of the exposure, 8. the effect of photon scattering and 9. patient motion. The experimental assessment of the role of these parameters in the image formation of small blood vessels is complex and time consuming. A rapid analysis can be obtained, however, by using a computer model recently developed [1,2]. The model determines the relative intensity distribution o f X-ray photons which results from a simulated exposure to X rays of a small blood vessel. It employs measured data for these parameters; it can handle all the parameters listed above with the exception of photon scattering and patient motion.
2. The computer model The basic concepts of the model can readily be understood by reference to fig. 1.
IMAGE FORMATIONOF SMALL BLOOD VESSELS FOCAL
i i
/
.
SPOT
,,"
+
A X u
~,~)
~~kM
To determine attenuation by the contrast material, we assumed that the contrast agent filled up the blood vessel completely. In the case of Renografin 76 for instance, the blood vessel contains 37% iodine by weight and the rest has the same attenuation as water. The values of the attenuation coefficients for both water and iodine were obtained from the paper by White Grodstein [4]. The path length of the photon within the blood vessel can be written as L = [(x2-xl )2 + (y2-Yl)2 + (z2-zl)211/2 •
Y
Pt.ANE
119
(1)
The values o f x 1 ,Yl ,Zl andx2 ,Y2,Z2 (for notation see fig. 1) can be computed from the geometry of the system, from the position and diameter of the blood vessel, and from the geometrical data of the emitting subdivision of the focus and of the receiving subdivision on the film. Thus the attenuation of a photon of energy E will be proportional to exp [-~(E)L], where ;k(E) is the linear attenuation coefficient.
i ,,"
¢
,
x ~ xo
X
Fig. 1. Diagrammatic presentation of the model. R is the radius o f the blood vessel;xl, Yt, z 1, and x 2, Ys, z ~ represent the coordinates of the intersection of an X-ray photon emitted from the focus with coordinates x", y", A and arriving at the film at coordinates x', y', and 0.
The upper and lower planes in fig. 1 represent the focus and the film plane, respectively, and the cylinder approximates the blood vessel. The aim of the procedure is to find the distribution of photon intensity on the film plane. This is achieved in the following way: Both the focal plane and the film plane are divided into subdivisions. The X-ray photons emitted by the individual subdivisions of the focus proceed toward the film and either miss the blood vessel or pass through it. In this model, attenuation of the X rays occurs in the blood vessel only. To determine it, the following quantities have to be known: (a) the energy spectrum of the X-ray photons, (b) the attenuation coefficient of the contrast agent filling the blood vessel, and (c) the path length covered by the photon within the blood vessel. The energy spectrum of the photons has been derived from an experimental study by Arnold [3].
2.1. The energy spectrum of X rays Of the various spectra published by Arnold [3], the one which represents a filtration of 491.5 mg/cm 2 AI was chosen because this is the closest to the X-ray tubes of practical interest to radiologists (2 mm added filtration). The energy spectrum was sampled at ten values: 25, 30, 33, 40, 50, 60, 70, 80, 90, and 100 keV. It was assumed that each subdivision of the focus emitted partial X-ray intensities with the energy values mentioned above.
2.2. Non-uniform emission of photons In fig. 2 the pinhole camera picture of a 1-mm focal spot is shown. This picture indicates a fairly symmetrical but non-uniform emission pattern of the X rays from the focus. The optical density distribution of the picture was obtained by using a scanner. These measured data are assumed to be proportional to the photon intensity emitted by the focus, and they are incorporated into the model in the form of weights
Fig. 2. Magnified pinhole camera picture of a 1-mm focal spot.
120
T. SANDOR and D.F. ADAMS
assigned to the individual subdivisions on the focus. The model in this present form assumed symmetry perpendicular to an imaginary axis running parallel to the high intensity bands of the focal spot and bisecting it. 2.3. The cosine effect In the computation the cosine effect due to the variation of the angle between the X ray and the recording surface and the variation of solid angle extended by the recording surface has been neglected. 2.4. The procedure of computation The computer selects the first subdivision on the film and begins collecting partial X-ray intensities with energy values mentioned in section 2.1 from the first subdivision on the focus. Each partial-intensity is multiplied before summation (a) by the appropriate amplitude of the energy spectrum, (b) by the appropriate attenuation, exp [-k(E)L], and (c) by the appropriate weight. The procedure is repeated for the second, third, and subsequent subdivisions on the focus, and all these partial intensities are accumulated on the first subdivision on the film. When all subdivisions from the focus are accounted for * the computer steps to the second subdivision on thefilm and the procedure described above is carried out again. This way a map of the distribution of X-ray intensity can be obtained.
4. Sample run
4.1. Sample data listing (1) Distance between focus and film, A. Unit: millimeter × 100. (2) Radius of blood vessel, R. Unit: millimeter X 100. (3) Distance of the symmetry axis of the blood vessel from the film plane, z 0. Unit: millimeter × 100. (4) Distance of the symmetry axis of the blood vessel from the origin of the coordinate system, x 0. Unit: subdivision number. (5) Weights assigned to subdivision on focus, W(/). (6) Amplitudes for energy spectrum. (7) Absorption coefficients for selected energy values. (8) Number of sampling rows (alongy axis), MA, on focus. (9) Number of samples within a row on focus, K. (1 O) Number of samples within a row, I, on film. (11) Number of sampling rows, J, on film. 4.2. Sample output The intensity values computed for subdivisions on the film are provided in four columns in sequence: the data for the first, second, third and fourth subdivisions are printed in the first line, the fifth to eighth data on the second line, and so on.
5. Hardware and software specifications 3. Flow chart
The main steps of the program are shown in the flow chart. The "constants of geometry" are specified in section 4.1 under (1), (2), (3), and (4). ARGU is the discriminant of the quadratic equation, the solution of which indicates whether the X ray hit the blood vessel (ARGU > 0) or missed it (ARGU < 0).
The program is written in double-precision FORTRAN IV on the IBM 370/155 computer at the Harvard Computing Center facilities, Cambridge, Massachusetts. CPU (Central Processing Unit) time for a situation containing 251 subdivisions on the film and 31 subdivisions on the focus is about 14 seconds. Core requirement is 100 k.
6. Mode of availability of the program
* In the case of axial symmetry, computation for one row of subdivisions across the symmetry axis is sufficient,
A copy of the program listing can be obtained from the authors.
IMAGE FORMATION OF SMALL BLOOD VESSELS
READ AND PRINT | WEIGHTS, AM- | PLITUDES OF | ENERGY SPEC- I TRY, X's
J
1 CnNSTANTS OF GEOMETRY
CALCULATE COORDINATES OF INTERSECTION
1
I
ON FILN
CALCULATE PATH LENGTtl
J
1
1
DO LOOP (1) TO J STEP TO NEXT SUBDIVISION ON | FILM ALONG I
I
X-AXIS
| J
l
$
DO LOOP (J) TO | STEP TO NEXT | ROW OF SUBDI-| VISIONS ON | FILq ALONG | Y-AXI~ I
I i STEP TO NEXT ROW OF SUBDIVISIONS ON FOCUS ALONG Y-AXIS I
i DO LOOP (K) TO STEP TO NEXT SUBDIVISION ON FOCUS ALONG X-AXIS
I
|
|
I
CALCULATE I UNATTENUATED INTENSITY
i
I
S~t ATTENUATED AND UNATTENUATED INTENSITIES
CALCULATE ATTENUATED INTENSITY
O Q :a Q PRINT INTENSITIES
Fig. 3. Flow chart.
121
122
T. SANDOR and D.F. ADAMS image formation of blood vessels of subfocal dimensions, presented at the 19th Annual Meeting of University Radiologists, Durham, N.C., May 1971 (Abstract: Invest. Radiol. 6 (1971) 359). [2] T. Sandor and D.F. Adams, A computer model to study the effect of the X-ray energy spectrum on the image formation of blood vessels of subfocal dimensions, presented at the 57th Annual Meeting of the Radiological Society of North America, Chicago, November, 1971. [3] B.A. Arnold, Experimental study of the photon energy spectrum of primary diagnostic X rays using a lithium drifted detector, (Thesis), Yale University (1969). [4] G. White Grodstein, NBS Circular 583 (1957).
Acknowledgements It is a pleasure to acknowledge the assistance of Mr. Glen D. Wilson, Information Design, Inc., for his work on developing the program. We are indebted to Dr. Bjarngard for valuable discussions.
References [ 1 ] T. Sandor and D.F. Adams, A computer model to study
ERRATUM E. Marubini and L.F. Resele, Computer program for fitting the logistic and the Gompertz function to growth data, Computer Programs in Biomedicine 2 (1972) 16-23. Eq. (1.3) should read:
Yu = P ÷ c e x p ( - ea - btu). Eq. (2.7) should read: Zu = In [ l n ( y u - - ~ )
]
=a-btu.
A COMPUTER
PROGRAM
TO STUDY IMAGE FORMATION
OF SMALL BLOOD VESSELS * T. SANDOR and D.F. ADAMS Harvard Medical School and Peter Bent Brigham Hospital, Boston, Massachusetts, USA
A computer program is described which facilitates the rapid assessment of a number of physical parameters affecting the image formation of small blood vessels. These parameters are: the dimensions of the focus, the intensity distribution of the emitted photons over the focus area, the energy spectrum of the emitted photons, the attenuation by the contrast agent used, and the geometry of the exposure. Independently measured or calculated values of the above parameters are used in the program. The output of these values is a map of the intensity distribution of X rays in the image plane during a simulated exposure of a small blood vessel. Blood vessels
1. Introduction The recent increased interest in and knowledge of the microvasculature in disease of various human organ systems has stimulated great rapid developments in the manufacture of X-ray tubes with small focal spots. The radiologist who wants to obtain in vivo information about these vessels is faced with questions such as: What is the minimum iodine concentration necessary within a blood vessel of given diameters to provide a perceptable image? What is the smallest blood vessel size that can be perceived? How does the non-uniform emission of X-ray photons from a focal spot affect the image of a blood vessel? To answer questions like those above the radiologist must know the combined effect of a number o f physical parameters on the blood vessel image. The most important ones are: 1. the dimensions of the focal spot, * This work was supported in part by grants HE11668, HE05832, GM01910, 5-S01-FR 05489-08, and 5-S01-FR 05489-06 from the National Institutes of Health, U.S. Public Healtl/Service.
Image simulation
2. the emission pattern of photons from the focus, 3. the filter employed, 4. the energy spectrum of photons, 5. the applied kilovoltage, 6. the contrast agent used, more specifically its linear attenuation coefficient, ~(E) as a function o f energy, 7. the geometry of the exposure, 8. the effect of photon scattering and 9. patient motion. The experimental assessment of the role of these parameters in the image formation of small blood vessels is complex and time consuming. A rapid analysis can be obtained, however, by using a computer model recently developed [1,2]. The model determines the relative intensity distribution o f X-ray photons which results from a simulated exposure to X rays of a small blood vessel. It employs measured data for these parameters; it can handle all the parameters listed above with the exception of photon scattering and patient motion.
2. The computer model The basic concepts of the model can readily be understood by reference to fig. 1.
IMAGE FORMATIONOF SMALL BLOOD VESSELS FOCAL
i i
/
.
SPOT
,,"
+
A X u
~,~)
~~kM
To determine attenuation by the contrast material, we assumed that the contrast agent filled up the blood vessel completely. In the case of Renografin 76 for instance, the blood vessel contains 37% iodine by weight and the rest has the same attenuation as water. The values of the attenuation coefficients for both water and iodine were obtained from the paper by White Grodstein [4]. The path length of the photon within the blood vessel can be written as L = [(x2-xl )2 + (y2-Yl)2 + (z2-zl)211/2 •
Y
Pt.ANE
119
(1)
The values o f x 1 ,Yl ,Zl andx2 ,Y2,Z2 (for notation see fig. 1) can be computed from the geometry of the system, from the position and diameter of the blood vessel, and from the geometrical data of the emitting subdivision of the focus and of the receiving subdivision on the film. Thus the attenuation of a photon of energy E will be proportional to exp [-~(E)L], where ;k(E) is the linear attenuation coefficient.
i ,,"
¢
,
x ~ xo
X
Fig. 1. Diagrammatic presentation of the model. R is the radius o f the blood vessel;xl, Yt, z 1, and x 2, Ys, z ~ represent the coordinates of the intersection of an X-ray photon emitted from the focus with coordinates x", y", A and arriving at the film at coordinates x', y', and 0.
The upper and lower planes in fig. 1 represent the focus and the film plane, respectively, and the cylinder approximates the blood vessel. The aim of the procedure is to find the distribution of photon intensity on the film plane. This is achieved in the following way: Both the focal plane and the film plane are divided into subdivisions. The X-ray photons emitted by the individual subdivisions of the focus proceed toward the film and either miss the blood vessel or pass through it. In this model, attenuation of the X rays occurs in the blood vessel only. To determine it, the following quantities have to be known: (a) the energy spectrum of the X-ray photons, (b) the attenuation coefficient of the contrast agent filling the blood vessel, and (c) the path length covered by the photon within the blood vessel. The energy spectrum of the photons has been derived from an experimental study by Arnold [3].
2.1. The energy spectrum of X rays Of the various spectra published by Arnold [3], the one which represents a filtration of 491.5 mg/cm 2 AI was chosen because this is the closest to the X-ray tubes of practical interest to radiologists (2 mm added filtration). The energy spectrum was sampled at ten values: 25, 30, 33, 40, 50, 60, 70, 80, 90, and 100 keV. It was assumed that each subdivision of the focus emitted partial X-ray intensities with the energy values mentioned above.
2.2. Non-uniform emission of photons In fig. 2 the pinhole camera picture of a 1-mm focal spot is shown. This picture indicates a fairly symmetrical but non-uniform emission pattern of the X rays from the focus. The optical density distribution of the picture was obtained by using a scanner. These measured data are assumed to be proportional to the photon intensity emitted by the focus, and they are incorporated into the model in the form of weights
Fig. 2. Magnified pinhole camera picture of a 1-mm focal spot.
120
T. SANDOR and D.F. ADAMS
assigned to the individual subdivisions on the focus. The model in this present form assumed symmetry perpendicular to an imaginary axis running parallel to the high intensity bands of the focal spot and bisecting it. 2.3. The cosine effect In the computation the cosine effect due to the variation of the angle between the X ray and the recording surface and the variation of solid angle extended by the recording surface has been neglected. 2.4. The procedure of computation The computer selects the first subdivision on the film and begins collecting partial X-ray intensities with energy values mentioned in section 2.1 from the first subdivision on the focus. Each partial-intensity is multiplied before summation (a) by the appropriate amplitude of the energy spectrum, (b) by the appropriate attenuation, exp [-k(E)L], and (c) by the appropriate weight. The procedure is repeated for the second, third, and subsequent subdivisions on the focus, and all these partial intensities are accumulated on the first subdivision on the film. When all subdivisions from the focus are accounted for * the computer steps to the second subdivision on thefilm and the procedure described above is carried out again. This way a map of the distribution of X-ray intensity can be obtained.
4. Sample run
4.1. Sample data listing (1) Distance between focus and film, A. Unit: millimeter × 100. (2) Radius of blood vessel, R. Unit: millimeter X 100. (3) Distance of the symmetry axis of the blood vessel from the film plane, z 0. Unit: millimeter × 100. (4) Distance of the symmetry axis of the blood vessel from the origin of the coordinate system, x 0. Unit: subdivision number. (5) Weights assigned to subdivision on focus, W(/). (6) Amplitudes for energy spectrum. (7) Absorption coefficients for selected energy values. (8) Number of sampling rows (alongy axis), MA, on focus. (9) Number of samples within a row on focus, K. (1 O) Number of samples within a row, I, on film. (11) Number of sampling rows, J, on film. 4.2. Sample output The intensity values computed for subdivisions on the film are provided in four columns in sequence: the data for the first, second, third and fourth subdivisions are printed in the first line, the fifth to eighth data on the second line, and so on.
5. Hardware and software specifications 3. Flow chart
The main steps of the program are shown in the flow chart. The "constants of geometry" are specified in section 4.1 under (1), (2), (3), and (4). ARGU is the discriminant of the quadratic equation, the solution of which indicates whether the X ray hit the blood vessel (ARGU > 0) or missed it (ARGU < 0).
The program is written in double-precision FORTRAN IV on the IBM 370/155 computer at the Harvard Computing Center facilities, Cambridge, Massachusetts. CPU (Central Processing Unit) time for a situation containing 251 subdivisions on the film and 31 subdivisions on the focus is about 14 seconds. Core requirement is 100 k.
6. Mode of availability of the program
* In the case of axial symmetry, computation for one row of subdivisions across the symmetry axis is sufficient,
A copy of the program listing can be obtained from the authors.
IMAGE FORMATION OF SMALL BLOOD VESSELS
READ AND PRINT | WEIGHTS, AM- | PLITUDES OF | ENERGY SPEC- I TRY, X's
J
1 CnNSTANTS OF GEOMETRY
CALCULATE COORDINATES OF INTERSECTION
1
I
ON FILN
CALCULATE PATH LENGTtl
J
1
1
DO LOOP (1) TO J STEP TO NEXT SUBDIVISION ON | FILM ALONG I
I
X-AXIS
| J
l
$
DO LOOP (J) TO | STEP TO NEXT | ROW OF SUBDI-| VISIONS ON | FILq ALONG | Y-AXI~ I
I i STEP TO NEXT ROW OF SUBDIVISIONS ON FOCUS ALONG Y-AXIS I
i DO LOOP (K) TO STEP TO NEXT SUBDIVISION ON FOCUS ALONG X-AXIS
I
|
|
I
CALCULATE I UNATTENUATED INTENSITY
i
I
S~t ATTENUATED AND UNATTENUATED INTENSITIES
CALCULATE ATTENUATED INTENSITY
O Q :a Q PRINT INTENSITIES
Fig. 3. Flow chart.
121
122
T. SANDOR and D.F. ADAMS image formation of blood vessels of subfocal dimensions, presented at the 19th Annual Meeting of University Radiologists, Durham, N.C., May 1971 (Abstract: Invest. Radiol. 6 (1971) 359). [2] T. Sandor and D.F. Adams, A computer model to study the effect of the X-ray energy spectrum on the image formation of blood vessels of subfocal dimensions, presented at the 57th Annual Meeting of the Radiological Society of North America, Chicago, November, 1971. [3] B.A. Arnold, Experimental study of the photon energy spectrum of primary diagnostic X rays using a lithium drifted detector, (Thesis), Yale University (1969). [4] G. White Grodstein, NBS Circular 583 (1957).
Acknowledgements It is a pleasure to acknowledge the assistance of Mr. Glen D. Wilson, Information Design, Inc., for his work on developing the program. We are indebted to Dr. Bjarngard for valuable discussions.
References [ 1 ] T. Sandor and D.F. Adams, A computer model to study
ERRATUM E. Marubini and L.F. Resele, Computer program for fitting the logistic and the Gompertz function to growth data, Computer Programs in Biomedicine 2 (1972) 16-23. Eq. (1.3) should read:
Yu = P ÷ c e x p ( - ea - btu). Eq. (2.7) should read: Zu = In [ l n ( y u - - ~ )
]
=a-btu.