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Stereotaxic localization of intracranial targets
Inl. J. Radiation Oncology Lliol. Phys.. Vol. 13, pp. 1241-1246 Printed in the U.S.A. All rights reserved.
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0 Technical Innovations and Notes STEREOTAXIC LOCALIZATION OF INTRACRANIAL TARGETS I~OBERT L. SIDDON, PH.D. AND NORMAN H. BARTH, PH.D. Departmentof Radiation Therapy,Joint Centerfor Radiation Therapy,HarvardMedical School, 50 Binney Street,Boston, MA 02 115 We report on a useful clinical method for precisely locating intracranial targets. Utilizing the BRW system, the technique is currently used in stereotaxic irradiation of arteriovenous malformations. An intracranial localizer box, with four radio-opaque markers on each face, surrounds the patient’s head and is attached to the BRW Head Ring. Two localization films are required. One film includes the target and the eight anterior and posterior markers, whereas the other film includes the target and the eight right and left markers. There are no constraints that the films be orthogonal or parallel to the box faces, only that the target and radio-opaque markers appear on the films. In addition, knowledge of the source-image and source-target distances are not required. Analysis of the projected target and radio-opaque markers gives both the target location and magnification. Simulation with the BRW Phantom Base demonstrates that point targets can be located with respect to the BRW system to within 0.3 mm and magnification determined to within 0.5%. Stereotaxic, Intracranial, Radiography.
of Vandermeulen et al.,* does not require projected radio-opaque scales,3 and does not require that the films be parallel to the box faces. 2,4The removal of the parallel film constraint considerably expedites the angiography procedure, particularly when both films are taken simultaneously. The technique, generally applicable for use with any radiographic equipment, may also be used for other intracranial target localization problems, for example, stereotaxic I- 125 brain implants.
INTIRODUmION Previous reports describe the method used at the Joint Center for Radiation Therapy for stereotaxic small-field irradiation of arteriovenous malformations.4,5*6 The irradiation technique uses the Brown-Robert-Wells (BRW) Stereotaxic System* and a modified 6 MV linear accelerator. The linear accelerator is equipped with specially designed circular collimators of diameters from 12.5 mm to 30.0 mm. Target localization is accomplished with a specially designed intracranial localizer box that rigidly attaches to the BRW Head Ring (Fig. 1). The mounting system for the intracranial localizer box is similar to that used for other BRW devices. Four radio-opaque markers, in a rectangular configuration, are fixed to the anterior, posterior, right, and left faces of the localizer box. The coordinates of the sixteen markers are precisely known with respect to .the BRW coordinate system. Two localization films are required for reconstruction of the target position and determination of the target magnification. One filrn includes the target and the eight anterior and posterior markers, whereas the other includes the target and the eight left and right markers. An analysis of the projected target and radio-opaque markers permits reconstruction of the intracranial target position and magnification. In contrast to other approaches to intracranial target localization, the method presented here, similar to that
METHODS
AND
MATERIALS
Target localization As indicated previously, two localization films are required to reconstruct the intracranial target position. One film includes the target and the eight radio-opaque markers on the anterior and posterior faces [Fig. 2(a)], whereas the other includes the target and the eight markers on the right and left faces [Fig. 2(b)]. It is not required that the films and central axes be orthogonal or parallel to the box faces. The intersections of the source-target rays with the box are indicated by points A, P, R, and L (Fig. 1). The intracranial target T is the intersection of the lines connecting points A-P and R-L. These points are found by considering each box face independently. For a particular box face, the rectangular radio-opaque marker array and intersection point Q (Q = A, P, R, or
* Radionics, Inc., Burlington,MA.
Reprintrequeststo: RobertL. Siddon, Ph.D. Acceptedfor publication 27 February1987. 1241
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where Y = (1 -
P1,2)IP1,2.
(2)
A similar analysis along the perpendicular direction gives point Q relative to the other side of the rectangle, and
/
Fig. 1. The intracranial
localizer box attaches to the BRW Head Ring. Four radio-opaque markers in a rectangular configuration are fixed to each face of the box. Two localizer films are required for target reconstruction. One film includes the target T and the anterior and posterior markers, whereas the second film includes the target T and the right and left markers. The intersections of the source-target rays with the box are indicated by points A, P, R, and L. L) are illustrated in Figure 3. The radio-opaque markers are located at the corners of the rectangle. The diagonals and bisectors are indicated by dot-dash and dash lines, respectively. A directed parametric axis, parallel to one of the sides, which intersects point Q is indicated as the c-w-axis.Point Q is located along this parametric axis by the parameter (Y.The parameter cxis zero on one side of the rectangle and unity on the other. The parameter CY is less than zero or greater than unity for points outside the rectangle. The perspective projection of the rectangle onto an arbitrary plane film is illustrated in Figure 4. In particular, the projection of the a-axis onto the film is denoted as the P-axis. In general, the rectangle projects as a figure with no sides parallel and no sides of equal length. The sides of the projected rectangle define the so called vanishing points of the perspective projection. As in perspective drawing, lines parallel to the sides of the rectangle must intersect one of the vanishing points. Following this parallel-line rule, the projected bisectors (dash lines passing through the projected center of the rectangle) and the directed parametric /?-axis (passing through the projected point Q’), are easily constructed. The parameter /3is zero on one side of the projected rectangle and unity on the other. The parameter /3 is less than zero or greater than unity for points outside the rectangle. The point on the &axis at which /3 = /31,2corresponds to the projection of the point on the a-axis at which CY = $. The quantity & does not in general equal 4. As shown in the Appendix, the parameter LYfor any point Q is given in terms of the parameter p for the corresponding point Q’by the relation o! = rPl[
1 + (7 - 1)Pl,
(1)
Fig. 2. Two films are required for intracranial target reconstruction. One film [Fig. 2(a)] includes the target and the anterior and posterior markers, whereas the other [Fig. 2(b)] includes the target and the right and left markers. The target and markers are indicated by arrows on the two films. Analysis of the projected target and markers gives both the target location and magnification.
Stereotaxiclocalization0 R. L.SIDDONAND N. H. BARTH
1243
Fig. 3. Each face of the intracranial localizer box and corresponding rectangular array of radio-opaque markers is considered independently. The markers are located at the comers of the rectangle. The diagonals and bisectors are indicated by the dot-dash and dash lines, respectively. Point Q denotes the source-target ray intersection with the box face. The a-axis, passing through point Q, is constructed parallel to one of the sides.
thus determines the location of point Q on the box face. For the simple case when the film is parallel to the box face, 81,2 = 1 and Eq. 1 reduces to the expected result CY= ,&. The quantities p an,d p1,2 are obtained directly from the film, for example, /3 is the ratio of the length of the line (O-Q’) to the length of the line (O-l), and similarly for k&,2. Again, it is not necessary that the target appear within the projected rectangle for the above analysis to apply.
For targets outside the projected rectangle, the parameters CY and B take on values less than zero or greater than unity with no loss of generality. Having obtained points A, P, R, and L on the intracranial localizing box, the target point T is the intersection of the lines connecting points A-P and R-L. Because of digitizing errors or difficulty in identifying a unique projected target point on the two films, the lines (A-P) and (R-L) rarely intersect exactly. The target position is therefore defined as the mid-point of the shortest line segment connecting the two lines.’
Target magn$cation
Fig. 4. The rectangular marker array projects onto an arbitrary plane film with no sides parallel, and no sides of equal length. The sides of the rectangle define the vanishing points of the perspective projection. As lines parallel to the sides of the rectangle must intersect one of the vanishing points, the bisectors (dash lines) and the P-axis are easily constructed. The point /3 = @,,2corresponds to the point (Y= 1 (Fig. 3). Projective geometry relates the parameters (Yand B.
In addition to the target location, it is also necessary to determine the target magnification on the two films. The target magnification is required to determine the diameter of the AVM, which in turn determines the collimator diameter necessary for irradiation. The target magnifications, denoted as MT,APand MT,RL , are defined in the usual way as the ratio of the corresponding source-image distance to the source-target distance. However, because of the arbitrary orientation of the two films, the sourceimage and source-target distances can not be conveniently measured directly. Rather, the magnifications are obtained indirectly as follows. For an arbitrary plane film, there exists a line passing through point Q’which is parallel to the box face. From projective geometry,’ it can be shown that such a line through Q’ must be parallel to the line connecting the vanishing points of the projection (Fig. 4). The magnification Mo is obtained by reconstructing, by the method illustrated in the last section, any arbitrary point W’on
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vides a convenient consistency check on the localization set-up. The calculated source and image positions are compared to the actual positions recorded at angiography. For example, an analysis of the AP film must produce an anterior source position and posterior image position as a consistency check. At the source point S, the magnification is infinite. Therefore, from Eqs. 8 and 9, the parameters &, and es are given by Fig. 5. The magnification Mo is obtained by constructing a line passing through point Q’which is parallel to the box face. Such a line can be shown to be parallel to the line connecting the vanishing points of the perspective projection (Fig. 4). The distance from point Q’to an arbitrary point w’ on such a line is denoted as 1’.The distance between the corresponding reconstructed points Q and W is 1. The magnification Mo is simply the ratio l’/l. Also illustrated: the intersections of such a line
passing through the center C’with the projected rectangle must also be equidistant from point C’, that is, point C’bisects the line segment connecting the intersections.
such a line (Fig. 5). As indicated, the distance between points Q’and W’is 1’,and the corresponding distance between reconstructed points Q and W is 1.As the lines (Q’W’) and (Q-W) are parallel in space, the magnification Mo is simply, M, = l’/l.
(3)
The corresponding magnifications on the box faces at points A, P, R, and L are denoted as MA, Mp, MR , and ML, respectively. _ The po_sition of points A, P, R, and L a;e denoted as RA, Rp, RR, RL, respectively. All points R on the lines (A-P) and (R-L) are given parametrically by the expressions R=R,+6&-I?,)
respectively. At the target point RT, the parameters 6T and tT are therefore
ET =
(10)
es = -M$/(M;’
(11)
- MR’).
From Eqs. 4 and 5, the corresponding for the two films are given by ii, = ii,
.
t
+
source positions
&(I& - Ii,) _
(12)
.
Rs = RR + ts(RL - RR).
(13)
At the image point I, the magnification is unity. Therefore, from Eqs. 8 and 9, the parameters 6’and E’are given by 6’= (1 - MA’)/(Mp’ - M,‘)
(14)
t’ = (1 - MR’)/(M;’ - MR’).
(15)
From Eqs. 4 and 5, the corresponding for the two films are given by R’ = I?, + s&l - R,) _ + _ R’ = RR + t,(Rr - RR).
image positions
(16) (17)
The magnitude of the vectors (Rs - RT) and (Rs - R’) correspond to the source-target and source-image distances, respectively.
(4)
(5)
dT =
& = -MA’/(Mp’ - MA’)
(ii, - ii,)& - ii,)&. - Ei-Al’ (6)
As the quantity M-’ is linear with the distance from the source, the magnifications MT,Ar and MT,RLof the target on the A-P and R-L films are given by MT,ip = MA’+ &(M,’ - MA’)
(8)
M+,k’_= M,’ + ET(ML’- MR’),
(9)
where the parameters 6T and tT are given by Eqs. 6 and 7, respectively. Calculation of the source position S and image position I with respect to the BRW coordinate system pro-
DISCUSSION The current design of the intracranial localizer box is a modification of an original design of Lutz et al4 The only modification was removal of the ‘out-riggers’, which were originally used to ensure that the films were oriented parallel to the box faces. The radio-opaque marker array is a square with 6.0 cm sides. The distance between parallel sides of the box is 29.0 cm. The angiography focal-film distance is approximately 100 cm. With the box approximately centered between the source and film, the object magnification ranges from 1.55 on the film side of the box to 2.82 on the source side. The current angiography protocol requires that the appropriate eight radio-opaque markers must appear on each localizer film. Approximate alignment of the central axis with the centers of the 6.0 cm squares allows this requirement to be easily met. However, it is possible to reconstruct the target point from films with as few as 6 visible markers on each film (either 4 markers from one film and 2 from the other, or 3 markers from each). Re-
Stereotaxiclocalization
0 R.L.SIDDON AND N.H.BARTH
construction from the minimum number of 6 markers, which follows from the .fundamental theorem of projective geometry,’ can be shown as follows. Briefly, as the sides of the A and P rectangles are parallel to each other (likewise for the R and L rectangles), the corresponding projected rectangles must share the same vanishing points. Having constructed the vanishing points of the projection from the six visible points, the two missing points are easily found from inspection. We have implemented the two-film reconstruction technique for intracranial targets in FORTRAN-77 on a VAX 1 l/785.? However, the advantage of such an extensive computer system is one of convenience, not necessity. We have found the technique as described straightforward to implement in computer code, but rather difficult to construct and verify ‘by hand’. The reason for the difficulty is that the vanishing points (Fig. 4), which are used in the reconstruction method, are usually located far from the prqiected rectangle, thus making the ‘by-hand’ construction Iofsuch vanishing points difficult and error prone. An alternative, equivalent approach not relying on the vanishing points, which lends itself to a convenient ‘by hand’ reconstruction, but which is somewhat more difficult to implement in computer code is given below. The projected space is divided into four quadrants, defined by the diagonals of the projected rectangle (Fig. 6). Lines are drawn from point Q’to the two ‘opposite’ radio-opaque markers. The intersection with one of the projected diagonals, indicated by the P-axis, is denoted as point K’. Eqs. 1 and 2 give point K on the corresponding diagonal on the box face. A similar analysis gives point K on the other diagonal and therefore permits reconstruction of point Q. The magnification is obtained by constructing a line passing through point C’whose intersections with the projected rectangle are equidistant from point C’, that is, point C’ bisects the line segment connecting the intersection points (Fig. 5). From projective geometry,’ it can be shown that such a line is also parallel to the box face. As shown in the last section, reconstruction of a line through point Q’parallel to the line passing through point C’yields the magnification Mo. In addition to developing a reconstruction algorithm, it is also necessary to ensure, to the extent possible, that erroneous data input is detected by such an algorithm. Such erroneous data input can result from inverting films before digitization, incorrectly identifying AP-PA or RL-LR films, digitizing the radio-opaque markers in reverse or incorrect order, inconsistent target identifica-
1245
Fig. 6. An alternative method for target reconstruction utilizes the diagonals of the rectangle rather than the vanishing points. A line from point Q’to an opposite marker intersects the diagonal @at point K’. Reconstruction of point K, and a similar analysis of the other diagonal permits reconstruction of point Q.
thereof. As mentioned in the previous section, the target position is at the midpoint of the shortest line segment connecting the lines A-P and R-L. We have found that the length of this shortest segment is a very sensitive indicator of possible erroneous data input. For correct and consistent data entry, the length of the shortest segment is typically less than 1.O mm. For the various erroneous cases mentioned above, the shortest segment assumes values typically much larger than this. Any input data for which the length of the shortest segment is larger than 1.0 mm is therefore assumed suspect and is reexamined. The BRW Phantom Base offers a convenient system to determine the accuracy of the digitization procedure and intracranial localizer box construction. The intracranial localizer box attaches to the BRW Phantom Base with the same mounting system as the BRW Head Ring. The BRW Phantom Base has a radio-opaque pointer that can be positioned to within 0.1 mm inside the localizer box. The end of the pointer projects sharply onto film, thus permitting unique target identification on the two films. Repeated trials with the BRW Phantom Base over a large range of source-film distances and extreme film orientations demonstrate that the target (pointer) can be reconstructed to within 0.3 mm and its magnification obtained to within 0.5%. These residual small errors are mainly caused by digitization error in locating exact centers of the radio-opaque markers. We have described a general reconstruction technique for intracranial target location and magnification, which when used in conjunction with the localizer box and BRW System offers an excellent solution to stereotaxic localization.
tion on the two films, or any combination
REFERENCES 1. Ayres, F.: Theory and Problems of Projective Geometry. New York, McGraw Hill. 1967, pp. 155-197. 2. Bergstrom, M., Greilz, T., Ribbe, T.: A method of stereotaxic localization adopted for conventional and digital radiography. Neuroradiology 28: 1OO- 104, 1986. t Digital Equipment
Corp., Maynard, MA.
3. Bergstrom, M., Greitz, T., Steiner, L.: An approach to stereotaxic radiography. Acta Neurochir. 54: 157-165, 1980. 4. Lutz, W., Winston, K.R., Maleki, N.: A system for stereotactic radiosurgery with a linear accelerator. Med. Phys. 13: 611, 1986.
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5. Rice, R.K., Hansen, J.L., Svensson, G.K., Siddon, R.L.:
Dose distributions for small 13: 585, 1986. 6. Saunders, W.M., Winston, R.L., Svensson, G.K.: Small teriovenous malformations.
field irradiations.
Med. Phys.
K.R., Kijewski, P., Siddon, field radiation therapy for ar-
Radiology 161P: 247, 1986.
August 1987, Volume 13, Number 8
I. Siddon, R.L., Chin, L.M.: Two-film brachytherapy reconstruction algorithm. Med. Phys. 12: 77-83, 1985. 8. Vandermeulen, D., Suetens, P., Gybels, J., Oosterlinck, A.: A new software package for the microcomputer based BRW stereotactic system: integrated stereoscopic views of CT data and angiograms. SPZE 593: 103-l 14,1985.
APPENDIX The relation between LYand /3 (Eq. 1-2) is derived by the “invariance of cross-ratio” method, described as follows. The plane containing a-axis and its projection, denoted as the P-axis, is shown in Figure 7. From the “law
of sines”, the following relations are obtained by inspection: cyZ,/sin 19= Xi /sin 6 [( 1 - a)Z,]/sin 4 = Y, /sin 6 PZ,/sin 0 = (Xi + X&in [( 1 - @Z&sin f$ = (Y, + Y#.in
t E,
where Z, and Z, are the lengths of the line (O-l) along the a-axis and p-axis, respectively. These four expressions are combined to yield: ]a(1 -
PWCP(1- 41 = 7,
(A.11
where Y = [X,(Y, + y2ww,w,
+ X2)1.
64.2)
For a given orientation of the source, box face (a-axis), and film (P-axis), the values of Xi, X2, Y ,, and Y2 are constant. From Eq. A.2, the quantity y is also constant. Therefore, from Eq. A. 1, the ratio [LY( 1 - @)I/[@(l-a)] is likewise constant (invariant), and may be evaluated at any (Yand 0. The convenient choice LY= 1 and /3 = Pllz yields the simple result: Fig. 7. The relationship between LYand /3 is derived by the “invariance of cross-ratio” method. The projection of the a-axis onto an arbitrary plane film is denoted as the p-axis. The plane containing the a-axis and the p-axis is illustrated in this figure. The ratio y = [a( 1 - @)I/[@(1 - a)] is constant for all a! and ,6.
a = rPl]l
+(Y - 1)Pl
(A.3)
where Y = (1 - P1,2M31,2.
(4.4)
Copyri&t
0360-3016/87 0 1987 Pergamon
$3.00 + 00 Journals Ltd.
0 Technical Innovations and Notes STEREOTAXIC LOCALIZATION OF INTRACRANIAL TARGETS I~OBERT L. SIDDON, PH.D. AND NORMAN H. BARTH, PH.D. Departmentof Radiation Therapy,Joint Centerfor Radiation Therapy,HarvardMedical School, 50 Binney Street,Boston, MA 02 115 We report on a useful clinical method for precisely locating intracranial targets. Utilizing the BRW system, the technique is currently used in stereotaxic irradiation of arteriovenous malformations. An intracranial localizer box, with four radio-opaque markers on each face, surrounds the patient’s head and is attached to the BRW Head Ring. Two localization films are required. One film includes the target and the eight anterior and posterior markers, whereas the other film includes the target and the eight right and left markers. There are no constraints that the films be orthogonal or parallel to the box faces, only that the target and radio-opaque markers appear on the films. In addition, knowledge of the source-image and source-target distances are not required. Analysis of the projected target and radio-opaque markers gives both the target location and magnification. Simulation with the BRW Phantom Base demonstrates that point targets can be located with respect to the BRW system to within 0.3 mm and magnification determined to within 0.5%. Stereotaxic, Intracranial, Radiography.
of Vandermeulen et al.,* does not require projected radio-opaque scales,3 and does not require that the films be parallel to the box faces. 2,4The removal of the parallel film constraint considerably expedites the angiography procedure, particularly when both films are taken simultaneously. The technique, generally applicable for use with any radiographic equipment, may also be used for other intracranial target localization problems, for example, stereotaxic I- 125 brain implants.
INTIRODUmION Previous reports describe the method used at the Joint Center for Radiation Therapy for stereotaxic small-field irradiation of arteriovenous malformations.4,5*6 The irradiation technique uses the Brown-Robert-Wells (BRW) Stereotaxic System* and a modified 6 MV linear accelerator. The linear accelerator is equipped with specially designed circular collimators of diameters from 12.5 mm to 30.0 mm. Target localization is accomplished with a specially designed intracranial localizer box that rigidly attaches to the BRW Head Ring (Fig. 1). The mounting system for the intracranial localizer box is similar to that used for other BRW devices. Four radio-opaque markers, in a rectangular configuration, are fixed to the anterior, posterior, right, and left faces of the localizer box. The coordinates of the sixteen markers are precisely known with respect to .the BRW coordinate system. Two localization films are required for reconstruction of the target position and determination of the target magnification. One filrn includes the target and the eight anterior and posterior markers, whereas the other includes the target and the eight left and right markers. An analysis of the projected target and radio-opaque markers permits reconstruction of the intracranial target position and magnification. In contrast to other approaches to intracranial target localization, the method presented here, similar to that
METHODS
AND
MATERIALS
Target localization As indicated previously, two localization films are required to reconstruct the intracranial target position. One film includes the target and the eight radio-opaque markers on the anterior and posterior faces [Fig. 2(a)], whereas the other includes the target and the eight markers on the right and left faces [Fig. 2(b)]. It is not required that the films and central axes be orthogonal or parallel to the box faces. The intersections of the source-target rays with the box are indicated by points A, P, R, and L (Fig. 1). The intracranial target T is the intersection of the lines connecting points A-P and R-L. These points are found by considering each box face independently. For a particular box face, the rectangular radio-opaque marker array and intersection point Q (Q = A, P, R, or
* Radionics, Inc., Burlington,MA.
Reprintrequeststo: RobertL. Siddon, Ph.D. Acceptedfor publication 27 February1987. 1241
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August 1987. Volume 13, Number 8
where Y = (1 -
P1,2)IP1,2.
(2)
A similar analysis along the perpendicular direction gives point Q relative to the other side of the rectangle, and
/
Fig. 1. The intracranial
localizer box attaches to the BRW Head Ring. Four radio-opaque markers in a rectangular configuration are fixed to each face of the box. Two localizer films are required for target reconstruction. One film includes the target T and the anterior and posterior markers, whereas the second film includes the target T and the right and left markers. The intersections of the source-target rays with the box are indicated by points A, P, R, and L. L) are illustrated in Figure 3. The radio-opaque markers are located at the corners of the rectangle. The diagonals and bisectors are indicated by dot-dash and dash lines, respectively. A directed parametric axis, parallel to one of the sides, which intersects point Q is indicated as the c-w-axis.Point Q is located along this parametric axis by the parameter (Y.The parameter cxis zero on one side of the rectangle and unity on the other. The parameter CY is less than zero or greater than unity for points outside the rectangle. The perspective projection of the rectangle onto an arbitrary plane film is illustrated in Figure 4. In particular, the projection of the a-axis onto the film is denoted as the P-axis. In general, the rectangle projects as a figure with no sides parallel and no sides of equal length. The sides of the projected rectangle define the so called vanishing points of the perspective projection. As in perspective drawing, lines parallel to the sides of the rectangle must intersect one of the vanishing points. Following this parallel-line rule, the projected bisectors (dash lines passing through the projected center of the rectangle) and the directed parametric /?-axis (passing through the projected point Q’), are easily constructed. The parameter /3is zero on one side of the projected rectangle and unity on the other. The parameter /3 is less than zero or greater than unity for points outside the rectangle. The point on the &axis at which /3 = /31,2corresponds to the projection of the point on the a-axis at which CY = $. The quantity & does not in general equal 4. As shown in the Appendix, the parameter LYfor any point Q is given in terms of the parameter p for the corresponding point Q’by the relation o! = rPl[
1 + (7 - 1)Pl,
(1)
Fig. 2. Two films are required for intracranial target reconstruction. One film [Fig. 2(a)] includes the target and the anterior and posterior markers, whereas the other [Fig. 2(b)] includes the target and the right and left markers. The target and markers are indicated by arrows on the two films. Analysis of the projected target and markers gives both the target location and magnification.
Stereotaxiclocalization0 R. L.SIDDONAND N. H. BARTH
1243
Fig. 3. Each face of the intracranial localizer box and corresponding rectangular array of radio-opaque markers is considered independently. The markers are located at the comers of the rectangle. The diagonals and bisectors are indicated by the dot-dash and dash lines, respectively. Point Q denotes the source-target ray intersection with the box face. The a-axis, passing through point Q, is constructed parallel to one of the sides.
thus determines the location of point Q on the box face. For the simple case when the film is parallel to the box face, 81,2 = 1 and Eq. 1 reduces to the expected result CY= ,&. The quantities p an,d p1,2 are obtained directly from the film, for example, /3 is the ratio of the length of the line (O-Q’) to the length of the line (O-l), and similarly for k&,2. Again, it is not necessary that the target appear within the projected rectangle for the above analysis to apply.
For targets outside the projected rectangle, the parameters CY and B take on values less than zero or greater than unity with no loss of generality. Having obtained points A, P, R, and L on the intracranial localizing box, the target point T is the intersection of the lines connecting points A-P and R-L. Because of digitizing errors or difficulty in identifying a unique projected target point on the two films, the lines (A-P) and (R-L) rarely intersect exactly. The target position is therefore defined as the mid-point of the shortest line segment connecting the two lines.’
Target magn$cation
Fig. 4. The rectangular marker array projects onto an arbitrary plane film with no sides parallel, and no sides of equal length. The sides of the rectangle define the vanishing points of the perspective projection. As lines parallel to the sides of the rectangle must intersect one of the vanishing points, the bisectors (dash lines) and the P-axis are easily constructed. The point /3 = @,,2corresponds to the point (Y= 1 (Fig. 3). Projective geometry relates the parameters (Yand B.
In addition to the target location, it is also necessary to determine the target magnification on the two films. The target magnification is required to determine the diameter of the AVM, which in turn determines the collimator diameter necessary for irradiation. The target magnifications, denoted as MT,APand MT,RL , are defined in the usual way as the ratio of the corresponding source-image distance to the source-target distance. However, because of the arbitrary orientation of the two films, the sourceimage and source-target distances can not be conveniently measured directly. Rather, the magnifications are obtained indirectly as follows. For an arbitrary plane film, there exists a line passing through point Q’which is parallel to the box face. From projective geometry,’ it can be shown that such a line through Q’ must be parallel to the line connecting the vanishing points of the projection (Fig. 4). The magnification Mo is obtained by reconstructing, by the method illustrated in the last section, any arbitrary point W’on
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vides a convenient consistency check on the localization set-up. The calculated source and image positions are compared to the actual positions recorded at angiography. For example, an analysis of the AP film must produce an anterior source position and posterior image position as a consistency check. At the source point S, the magnification is infinite. Therefore, from Eqs. 8 and 9, the parameters &, and es are given by Fig. 5. The magnification Mo is obtained by constructing a line passing through point Q’which is parallel to the box face. Such a line can be shown to be parallel to the line connecting the vanishing points of the perspective projection (Fig. 4). The distance from point Q’to an arbitrary point w’ on such a line is denoted as 1’.The distance between the corresponding reconstructed points Q and W is 1. The magnification Mo is simply the ratio l’/l. Also illustrated: the intersections of such a line
passing through the center C’with the projected rectangle must also be equidistant from point C’, that is, point C’bisects the line segment connecting the intersections.
such a line (Fig. 5). As indicated, the distance between points Q’and W’is 1’,and the corresponding distance between reconstructed points Q and W is 1.As the lines (Q’W’) and (Q-W) are parallel in space, the magnification Mo is simply, M, = l’/l.
(3)
The corresponding magnifications on the box faces at points A, P, R, and L are denoted as MA, Mp, MR , and ML, respectively. _ The po_sition of points A, P, R, and L a;e denoted as RA, Rp, RR, RL, respectively. All points R on the lines (A-P) and (R-L) are given parametrically by the expressions R=R,+6&-I?,)
respectively. At the target point RT, the parameters 6T and tT are therefore
ET =
(10)
es = -M$/(M;’
(11)
- MR’).
From Eqs. 4 and 5, the corresponding for the two films are given by ii, = ii,
.
t
+
source positions
&(I& - Ii,) _
(12)
.
Rs = RR + ts(RL - RR).
(13)
At the image point I, the magnification is unity. Therefore, from Eqs. 8 and 9, the parameters 6’and E’are given by 6’= (1 - MA’)/(Mp’ - M,‘)
(14)
t’ = (1 - MR’)/(M;’ - MR’).
(15)
From Eqs. 4 and 5, the corresponding for the two films are given by R’ = I?, + s&l - R,) _ + _ R’ = RR + t,(Rr - RR).
image positions
(16) (17)
The magnitude of the vectors (Rs - RT) and (Rs - R’) correspond to the source-target and source-image distances, respectively.
(4)
(5)
dT =
& = -MA’/(Mp’ - MA’)
(ii, - ii,)& - ii,)&. - Ei-Al’ (6)
As the quantity M-’ is linear with the distance from the source, the magnifications MT,Ar and MT,RLof the target on the A-P and R-L films are given by MT,ip = MA’+ &(M,’ - MA’)
(8)
M+,k’_= M,’ + ET(ML’- MR’),
(9)
where the parameters 6T and tT are given by Eqs. 6 and 7, respectively. Calculation of the source position S and image position I with respect to the BRW coordinate system pro-
DISCUSSION The current design of the intracranial localizer box is a modification of an original design of Lutz et al4 The only modification was removal of the ‘out-riggers’, which were originally used to ensure that the films were oriented parallel to the box faces. The radio-opaque marker array is a square with 6.0 cm sides. The distance between parallel sides of the box is 29.0 cm. The angiography focal-film distance is approximately 100 cm. With the box approximately centered between the source and film, the object magnification ranges from 1.55 on the film side of the box to 2.82 on the source side. The current angiography protocol requires that the appropriate eight radio-opaque markers must appear on each localizer film. Approximate alignment of the central axis with the centers of the 6.0 cm squares allows this requirement to be easily met. However, it is possible to reconstruct the target point from films with as few as 6 visible markers on each film (either 4 markers from one film and 2 from the other, or 3 markers from each). Re-
Stereotaxiclocalization
0 R.L.SIDDON AND N.H.BARTH
construction from the minimum number of 6 markers, which follows from the .fundamental theorem of projective geometry,’ can be shown as follows. Briefly, as the sides of the A and P rectangles are parallel to each other (likewise for the R and L rectangles), the corresponding projected rectangles must share the same vanishing points. Having constructed the vanishing points of the projection from the six visible points, the two missing points are easily found from inspection. We have implemented the two-film reconstruction technique for intracranial targets in FORTRAN-77 on a VAX 1 l/785.? However, the advantage of such an extensive computer system is one of convenience, not necessity. We have found the technique as described straightforward to implement in computer code, but rather difficult to construct and verify ‘by hand’. The reason for the difficulty is that the vanishing points (Fig. 4), which are used in the reconstruction method, are usually located far from the prqiected rectangle, thus making the ‘by-hand’ construction Iofsuch vanishing points difficult and error prone. An alternative, equivalent approach not relying on the vanishing points, which lends itself to a convenient ‘by hand’ reconstruction, but which is somewhat more difficult to implement in computer code is given below. The projected space is divided into four quadrants, defined by the diagonals of the projected rectangle (Fig. 6). Lines are drawn from point Q’to the two ‘opposite’ radio-opaque markers. The intersection with one of the projected diagonals, indicated by the P-axis, is denoted as point K’. Eqs. 1 and 2 give point K on the corresponding diagonal on the box face. A similar analysis gives point K on the other diagonal and therefore permits reconstruction of point Q. The magnification is obtained by constructing a line passing through point C’whose intersections with the projected rectangle are equidistant from point C’, that is, point C’ bisects the line segment connecting the intersection points (Fig. 5). From projective geometry,’ it can be shown that such a line is also parallel to the box face. As shown in the last section, reconstruction of a line through point Q’parallel to the line passing through point C’yields the magnification Mo. In addition to developing a reconstruction algorithm, it is also necessary to ensure, to the extent possible, that erroneous data input is detected by such an algorithm. Such erroneous data input can result from inverting films before digitization, incorrectly identifying AP-PA or RL-LR films, digitizing the radio-opaque markers in reverse or incorrect order, inconsistent target identifica-
1245
Fig. 6. An alternative method for target reconstruction utilizes the diagonals of the rectangle rather than the vanishing points. A line from point Q’to an opposite marker intersects the diagonal @at point K’. Reconstruction of point K, and a similar analysis of the other diagonal permits reconstruction of point Q.
thereof. As mentioned in the previous section, the target position is at the midpoint of the shortest line segment connecting the lines A-P and R-L. We have found that the length of this shortest segment is a very sensitive indicator of possible erroneous data input. For correct and consistent data entry, the length of the shortest segment is typically less than 1.O mm. For the various erroneous cases mentioned above, the shortest segment assumes values typically much larger than this. Any input data for which the length of the shortest segment is larger than 1.0 mm is therefore assumed suspect and is reexamined. The BRW Phantom Base offers a convenient system to determine the accuracy of the digitization procedure and intracranial localizer box construction. The intracranial localizer box attaches to the BRW Phantom Base with the same mounting system as the BRW Head Ring. The BRW Phantom Base has a radio-opaque pointer that can be positioned to within 0.1 mm inside the localizer box. The end of the pointer projects sharply onto film, thus permitting unique target identification on the two films. Repeated trials with the BRW Phantom Base over a large range of source-film distances and extreme film orientations demonstrate that the target (pointer) can be reconstructed to within 0.3 mm and its magnification obtained to within 0.5%. These residual small errors are mainly caused by digitization error in locating exact centers of the radio-opaque markers. We have described a general reconstruction technique for intracranial target location and magnification, which when used in conjunction with the localizer box and BRW System offers an excellent solution to stereotaxic localization.
tion on the two films, or any combination
REFERENCES 1. Ayres, F.: Theory and Problems of Projective Geometry. New York, McGraw Hill. 1967, pp. 155-197. 2. Bergstrom, M., Greilz, T., Ribbe, T.: A method of stereotaxic localization adopted for conventional and digital radiography. Neuroradiology 28: 1OO- 104, 1986. t Digital Equipment
Corp., Maynard, MA.
3. Bergstrom, M., Greitz, T., Steiner, L.: An approach to stereotaxic radiography. Acta Neurochir. 54: 157-165, 1980. 4. Lutz, W., Winston, K.R., Maleki, N.: A system for stereotactic radiosurgery with a linear accelerator. Med. Phys. 13: 611, 1986.
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I. J. Radiation Oncology 0 Biology 0 Physics
5. Rice, R.K., Hansen, J.L., Svensson, G.K., Siddon, R.L.:
Dose distributions for small 13: 585, 1986. 6. Saunders, W.M., Winston, R.L., Svensson, G.K.: Small teriovenous malformations.
field irradiations.
Med. Phys.
K.R., Kijewski, P., Siddon, field radiation therapy for ar-
Radiology 161P: 247, 1986.
August 1987, Volume 13, Number 8
I. Siddon, R.L., Chin, L.M.: Two-film brachytherapy reconstruction algorithm. Med. Phys. 12: 77-83, 1985. 8. Vandermeulen, D., Suetens, P., Gybels, J., Oosterlinck, A.: A new software package for the microcomputer based BRW stereotactic system: integrated stereoscopic views of CT data and angiograms. SPZE 593: 103-l 14,1985.
APPENDIX The relation between LYand /3 (Eq. 1-2) is derived by the “invariance of cross-ratio” method, described as follows. The plane containing a-axis and its projection, denoted as the P-axis, is shown in Figure 7. From the “law
of sines”, the following relations are obtained by inspection: cyZ,/sin 19= Xi /sin 6 [( 1 - a)Z,]/sin 4 = Y, /sin 6 PZ,/sin 0 = (Xi + X&in [( 1 - @Z&sin f$ = (Y, + Y#.in
t E,
where Z, and Z, are the lengths of the line (O-l) along the a-axis and p-axis, respectively. These four expressions are combined to yield: ]a(1 -
PWCP(1- 41 = 7,
(A.11
where Y = [X,(Y, + y2ww,w,
+ X2)1.
64.2)
For a given orientation of the source, box face (a-axis), and film (P-axis), the values of Xi, X2, Y ,, and Y2 are constant. From Eq. A.2, the quantity y is also constant. Therefore, from Eq. A. 1, the ratio [LY( 1 - @)I/[@(l-a)] is likewise constant (invariant), and may be evaluated at any (Yand 0. The convenient choice LY= 1 and /3 = Pllz yields the simple result: Fig. 7. The relationship between LYand /3 is derived by the “invariance of cross-ratio” method. The projection of the a-axis onto an arbitrary plane film is denoted as the p-axis. The plane containing the a-axis and the p-axis is illustrated in this figure. The ratio y = [a( 1 - @)I/[@(1 - a)] is constant for all a! and ,6.
a = rPl]l
+(Y - 1)Pl
(A.3)
where Y = (1 - P1,2M31,2.
(4.4)