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The influence of geometrical factors in 131I-Hippuran renography
This content has been downloaded from IOPscience. Please scroll down to see the full text. 1975 Phys. Med. Biol. 20 67 (http://iopscience.iop.org/0031-9155/20/1/006) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 137.132.123.69 This content was downloaded on 03/10/2015 at 00:59
Please note that terms and conditions apply.
PHYS. MED. BIOL.,
1975, VOL. 20, NO. 1, 67-79. 0 1975
The Influence of Geometrical Factors in 13'I-Hippuran Renography C. C. NIMMOK, B.SC., JOAK M. McALISTER, xsc., F.INST.P., and W . R. CATTELL, M.D., F.R.C.P.,F.R.C.P.E. Radioisotope Department and the Department of h-ephrology, St. Bartholomew's Hospital, London, England.
Receiced 5 March 1974, in Jinal f o r m 20 August 1974 ABSTRACT.Using a particular collimated S a 1 scintillationdetector and a kidney phantom containing 1311, the dependence of the resulting count rates on collimatorkidney geometry has been determined. These results have been used to calculate the geometrical contribution t o the error in the measurement of relative effective renal plasma flow (RFP) by '31I-Hippuran renography. When radiographic and ultrasonic methods of localizing the kidneys are employed, this error has been found to follow a normal distribution with a SD of 2.6% in the case of equally divided function. Combination of this error with that from natural movement and statistical fluctuations, as observed using a dose of 10 yCi 131I-Hippuran, has led to the estimation of a corresponding potential error of 11%. Values of the potentialerror, which is defined as the 9976 probability range, have been calculated covering the range of RFP.
1. Introduction
I n recent years, renography with blood background subtraction has been shown to be a technique capable of providing a measurement of indices of relative kidney function (Britton and Brown 1968). During the uptake phase, before excretion from either kidney has taken place, the ratio of the renogram traces aft'er blood background subtraction is a measure of relative effective which is renalplasma flow. It is thisparameter of relativerenalfunction dealt with in this study and it is subsequently referred t'o as RFP. In order to obtain an accurate measurement of this paramet'er,it is necessary that the blood backgroundsubtractedcurvesshould reflect the radioactive content of the respective kidneys at any given time. However, when using a conventionalsystemwith three collimatedscintillationdetectors, the count rates are dependent on the collimator-kidney geometry. The accuracy of the method is affected by thisdependence on geometry,and by the extent which to this geometry is unknown. It is possible to reduce the geometrical uncertainty by using radiographic and ultrasonic methods of localizing the kidneys. The aim of t'his work has been to estimate theremaining error when such techniques of localization are employed. The dependence of the count rates on relative collimator-kidney geometry has been investigatedexperimentally.Amathematical expression for the parameter error has been formulated in terms of the geometrical errors. Using the experimental results in conjunction with this formula, an expected parameter error distribution is obtained.
C.
68
C.
Nimmon et al.
2. Equipmentandmethods
The detector used in this studywas a 5 x 5 cm NaI(Th) crystal, with aconed surface-to-end-of-collead collimator with a circular aperture.Thecrystal limator distance was 5 cm, the aperture at the end of the collimator 7 cm diameter, and the thickness of the collimation at the crystal was 1.75 cm lead and 0.75 cm steel. The point source response curve obtained with this detector for it 1311 point source in water is shown in fig. 1. Pulse height analysis was used to detect the 0.36 MeV photopeak of l3II.
Fig. 1.
I3lI point
source response of the scintillation detector. Isocount lines are expressed
as a percentage of the count rate obtained at the centre of the collimator face.
The kidney phantom used is shown in fig. 2(a). It is made of a thin-walled plastic material and has the following dimensions : pole-to-pole length 11 cm, midpole thickness 4 cm and lateral width5.5 cm. It was mounted in a perspexwalled water tank and was capable of being accurately moved relative to the face of the collimator, which was positioned against one wall of the tank. The
Fig. 2. (a) The plastic kidney phantom. (b) The mathematically generated phantom and the directions of displacement X , Y and 2 with respect to the centre of the collimator face.
Geometrical Factors in 1311-HippuranRenography
69
phantom was filled with an aqueoussolutioncontaining 10 pCi 1311. The variations in count rates were determined for movement of the phantom along each of the axes X , Y , Z of the right-handed rectangular coordinate system shown in fig. 2(b). Anatomically,displacements inthe directions X , Y , Z correspond to lateral, craniocaudal and posterior-anterior displacements respectively, with respect to the centre of the collimator face. For the study of these variations the kidney pole-to-pole axis was parallel to the Y axis. The effect of rotation of the pole-to-pole axis in the YZ plane was also studied for fixed X, Y,Z displacements. For comparison withthe kidney phantom results, a mathematically simulated kidney model was also used. ,4s shown in fig. 2(b) the model was an ellipsoid with a cut-off section. With the mathematical model the kidney size could be varied and also a volume representing the kidney pelvis could be generated. For the mathematical calculations the model, a t aparticularposition, was split up intosegmental volumes, each correspondingto anelement of the point source response matrix. The required count rate from the whole model was then obtained by summation. The determination of the segmentalvolumes essentially involved solving the mat.hematica1 equations for an ellipse and a circle. Approximate roots were obtained by a grid search technique and more accurate values then obtained using Newton's method of successive approximation.
Experimentalresultsandcomparisonwiththemathematicalmodel.The dependence of count rate on relative collimator-kidney position 3.1. Movement of the kidney centrealongthe Z axis with the pole-to-pole axis
3.
parallel to the Y axis The variation obtained with the phantom is shown in fig. 3, and is in fact similar to that obtained using a point source. Movement of the phantom by 1 cm from t'he position Z = 7 cm alters the count rate by approximately 23%.
I 0
2
4 6 8 1 0 1 2 Distance Z ( c m 1
Fig, 3. Variation of count rate with displacement in the 2 (posterior-anterior) direction.
C . C. Nimmon et al.
70
3.2. Movement of the kidney centre in the Y direction with the pole-to-pole axis
parallel to the Y axis The results are shown in fig. 4 for the phantom centre a t depths of 5 , 7 and 9 cm. For a displacement of the centre by less than 2 cm from t’he collimator axis, the count rate remains greater than 90% of the undisplaced value. This is followed by an almost linear fall-off, dropping to 50% for 2 = 7 cm a t a displacement of 6 cm.
Distance Y ( c m )
Fig. 4. Variation of count rate withdisplacement in the Y (craniocaudal)direction, for the phantom centre at depths 5 , 7 and 9 cm.
3.3. Movement of the kidney centre in the X direction with the pole-to-pole axis
parallel to the Y axis The results are shown in fig. 5 for the phantom centre a t depths of 5 , 6, 7 and 8 cm. At a displacement of X = 14 cm, the count rate drops below 1yo of the undisplaced value. of cross-talk betweenthe two kidney These results are relevantto the amount detectors and are discussed later. 7r
Zb
:S:5
-5
Z(crn)
-6
-7
-a
W
2
g 4
c
23 W &
2 2 &
c 3
G I ‘11
I2
13 14 IS Distance X ( c m )
16
17
Fig. 5 . Variation of count rate with displacement in the X (lateral) direction, for the phantom centre at depths 5 , 6, 7 and 8 cm.
Geometrical Factors in 1311-HippuranRenography
71
3.4. Rotation (e) of the pole-to-pole axis in the YZ plane The results are shown in fig. 6. When the centre of the kidney lies on the collimator axis a rotation of 40" is needed to produce a change in count rateof 20%, and the count rate is increased independently of the direction of rotation.
S U
I -60
, -40
, -20
, 0
, 20
, 40
, 60
Angle 0 (deg 1
Fig. 6. Variation of count rate with rotation6" of the pole-to-poleaxis of the phantom in the Y Z plane. 6 = 0 corresponds to the pole-to-pole axis being parallel to the Y axis. Results are shown for Y = 0 and 5 cm.
When the kidney centre is displaced from the collimator axis in the Y direction the count rate is increased or decreased as the upper pole is rotated in the negative (posteriorly) or positive direction respectively. For Y = 5 cm, a 20" rotation produces a change in count rate of 10%. 3.5. Calculations using the mathematically simulated model The calculated variation in count rate withdisplacement of the kidney in the Y direction was similar to theresults of the phantom measurements. A volume representing the renal pelvis was also generated. With the centre of the model at the position X = 0, Y = 0 , Z = 7 cm, the difference in count rate calculated for a given activity uniformly distributed in thepelvis (volume 38 cm3) and for the same activity in the remainder of the kidney (volume 107 cm3) was less than 2%. Calculations 4.1. Calculation of the error on the function parameter produced by variation in the relative collimator-kidney position Assuming that each kidney is identically placed with respect to its detector and that bothdetectors have equal sensitivity, then the relative function para) meter (i.e. the relative effective renal plasma flow) for the left kidney ( R F P ~ is given by the equation: 4.
1OOC,
(yoof total function) = -
RFP~
C L -tC,
where C, and C, are values of the left and right renogram tracesrespectively a t a time after the injection of labelled Hippuran. RFP is usually calculated by
C. C. Nirnmon et al.
72
sampling the traces over a period of time starting 1 min after injection and continuing up to the timeof the first occurring peak. For the purpose of calculation, consider that the kidneys are positioned with their centres on the respective collimator axis and that the pole-to-pole axes are parallel tothevertebral column.Displacement from this position will modify the count rates CL and c, by factorsf L and f R respectively. RFPL
(observed) =
1 0 0 f LC L R ‘ fL +fR
The percentage error onthe trueRFP value dueto thisdisplacement is given by :
where y = - f. L
fR
Similarly,
If CL = C,, then RFP
(Yo) = 50
and 100(1 - r ) E L (Yo) =
r+l
= -ER
(YO).
From eqn (1) it can be seen that the percentage error E depends both on RFP and on the ratio r , which is determined by the relativedisplacement of the two kidneys. A graph giving the relationship between E L and RFP for values of r = 0.9, 0.82 and 0.67 is shown in fig. 7. These values of r result in an error of 5 ; 10 and 20% respectively for RFP = 50%. Inthe following discussion onerrors,resultshave been calculated for RFP = joyb. The corresponding errors for any value of RFP may be obtained by calculation from eqns (1) and ( 2 ) . 4.2. Combination of errors due to displacement The resultant error on RFP arising from two independent displacements, with r = rl, r2 respectively, can be calculated by substituting r = r 1 r 2 in eqn ( 2 ) .
Geometrical Factors in 1311-Hippuran Renography
RFP
73
(*/e
Fig. 7. Dependence of the percentage error E parameter RFP (yo).
(yo)on the value of the relative function
For small displacements,
To a first approximation,the craniocaudal, depth and rotationdisplacement's will be regarded as independentwhen the craniocaudal displacement is small. 4 . 3 . Estimation of the parameter error distribution using a Monte Carlo method
If the distributions of collimator-kidney position were known, probability density distributions of the error on RFP arising from each displacement could be calculated from the graphs shown in figs 3, 4, 6 and from eqn (2). For small displacements the resultant probability error distributioncould then be obtained using eqn ( 3 ) . Consider the situationwhere the kidneys are localized using a urogram taken with the patient in the renographyposition and where the depthsof the kidney centres below the posterior skin surface are measured using an A-type ultrasonictechnique.Thecraniocaudal collimator-kidney distributioncanbe (SD) dependexpected to follow a normal distribution with a standard deviation ing on the accuracy of transferring the measurements from the radiograph onto the skin of the patient. Similarly, it can be expected that the measured depths will show a normal distribution about the true values with a SD depending on the accuracy of making measurements from a Polaroid film. This latter distribution will give rise to a corresponding error contribution to RFP when the l/fL,l/fR obtainedfrom fig. 3 areapplied. appropriate correctionfactors However, the error will also depend on the actual depths encountered,since the relationship between the depth and thecorrection factor is not linear. The distribution of kidney depth found by Nimmon, McAlister and Cattell (1974) can
74
C. C . Nimmon et al.
beused. In the absence of available data on the distribution of post'erioranterior kidney rotation, a hypot'hetical distribution must be used. A possible choice would be a normal distribution with mean 9 = 0". Assuming normaldistributions for the craniocaudaldisplacements, depth SDS of 1.0 cm, 0.25 cm and 10" respectively, and errors a'nd kidney rotation with using the depth distribution reported by Nimmon et al. (1974), a parameter error distribution was calculated by a Monte Carlo method (James 1968). The results are shown in fig. 8, together withthe individual craniocaudal, depth and rotat'ion contributions. f (€1
Rotation Craniocaudal Depth -Total " "
E 010 1 Fig. 8. Calculated paramet'er error distribution, together with the individual craniocaudal, depth and rotation contributions.
The resultant parametererror distribution is closely normal with SD = 2.60,/,. The individual contributions have a SD of 1.5, 1.9 and 0.2% respectively. 4.4. Further possible sources of error on RFP
4.4.1. Cross-talk. It is possible for each detector to record an amount of pickup (cross-talk) from the activity in the contralateral kidney. Consider the crosstalk to be x% of the count rate recorded by the contralateral detector. The error in RFP due to this effect FL will be :
For
Geometrical Factors in 1311-Hippuran Renography
75
Over the possible range of RFP the maximum error incurred will be f x. The error due to cross-talkis most significant in the measurement of a low RFP. For example, for RFP = loyo,if the cross-talk is x%, the measured RFP will be RFP +x. For the cases where the kidney centres lie in t'he same lateral plane, i.e.have the same Y coordinate, the extent of the cross-talkexpectedfor different lateral separations is given in fig. 5 , and is less than 2.5% for separations greater than 12 cm. From the distribution of lateral position reported by Nimmon et al. (1974), it may be concluded that this magnitude of cross-talk will be realized in only a few per cent of cases. 4.4.2. Rotation in the X Y plane. It is known that the kidney pole-to-pole axes do not ingeneral lie parallel to the vertebral column, but that they are rotated in the XY plane (Moell 1963). However, the point source response curves of detectors withcircular collimatorsare radially symmetrical about the collimator axes. Therefore, rotation in t h e X Yplane should not' produce a contributionto the error on RFP, provided that the collimator axes are aligned with the kidney centres and that' cross-t'alk is not significant'ly increased. 4.4.3. Natural movement and statistical Jluctuations.
Errors can be expected to arise from the natural movement of the kidneys during respiration over the period of measurement. An estimate of t'his error combined with the error due to statistical fluctuation is given by the SD on the mean of the measurement's taken over the sampling period. Results from 30 renograms, in which a dose of 10 pCi 1311-Hippuranwas used and measurementswere made a t 10 S intervals over the sampling period, show a mean error of & 1.7 for RFP values in the range 10-90yo. This error is particularly significant in t'he measurement of a low value of RFP. 4.4.4.Systematic errors in the background subtraction ratios. One of the assumptions in the technique of blood background subtraction using 1311-HSAis that t'he ratioof intravascular to extravascular fluid volume in the regions viewed by all three detectors is the same. There is evidence from studies on nephrectomized patient's (Farmelant, Sachs and Burrows 1970, Nagnusson 1972) t'hat this assumption is not correct and that theabove ratio is smaller for the kidney region than for the subclavicular region. Consequently, in using the HSA subtraction technique, the Hippuran blood background is slightly underestimated (Brit'ton and Brown1971), thus giving rise to aresidualbackground. It is realized that this residual background will not follow the shape of the curve obtained from the subclaviculardetector.Xevertheless,forsimplicityin estimating the order of magnitude of the error in RFP one can consider the residualbackgroundasarising from asystematicerror in the background subtraction ratios. It has been suggested (Britton and Brown 1971) that the effective error in the subtraction ratios may be 10%. The resultant contribution ( E L )to the error on RFP may be calculated as follows: C, = KL -X, H C, = KX-SRH
C. C. Nimmon et al.
76
where K, and KR are the values of the left and right renogram traces before blood background subtraction. S , and S , are the HSA subtraction factors for the left and right kidney respectively. H is the count rate observed by the subclavicular detector.
-
”
where AS, and AS, are the errors on the fact’ors S , and S , respectively. Using the approximation S , = S, = 1, and assuming the error in the subtraction factors tobe loyo,the values of the factor W expected over the range of RFP are shown in fig. 9. The graph shows the rapid increase of this factor as 200 r
l
20
40
60
80
100
RFP~(%)
Fig. 9. Variation of the parameter W
(so) with RFP (76).
RFP decreases below 20%. The error on RFP due t o the possible systematic error in the subtraction ratios depends also on the ratioH/(CL+ CR). The mean value of this ratio over the period of relative function measurement depends on total renal function. It has been shown (Britton and Brown 1971) that this ratio is such that the resultant contribution to the error on RFP is only likely to become significant for total creatinine clearances less than 30 mlmin-l. In this range the error becomes particularly significant for values of RFP less than 20%.
Geometrical Factors in 1311-Hippuran Renography
77
Discussion It has been found that using the described technique of kidney localization, the two main sources of error in the measurement of the relative function parameterare the residual uncertaintyin collimator-kidney geometry and the statistical error. When RFP = 50%, the resultant potential error from these two sources is 11%, calculated as 2.58 x the resultant SD (i.e. the expected 99% range for a normal distribution). The potential error varies with RFP and some values are given in table 1. The geometrical contribution at anyvalue of 5.
Table 1. Calculated values of the potential error Potential error
Measured value O f RFP
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95
(yo)
(yoof the measured value of
RFP)
88.6 45.4 31.3 24.3 20.1 17.3 15.2 13.5 12.2 11.0 10.0 9.1 8.2 7.5 6.8 6.1 5.5 5.1 4.7
99Yo probability range of the true value of RFP (",) 0.6- 9.4 5'5-14'5 10'3-19'7 15.1-24.9 20'0-30.0 24.8-35.2 29'7-40'3 34.6-45'4 39'5-50.5 44.5-55.5 49'5-60'5 54.6-65'4 59'7-70'3 64'8-75.2 69'9-80.1 75.1-84'9 80'3-89.7 85.5-94.5 90'6-99.4
can be calculated using eqns (1) and (2), together with the SD obtained above for RFP = 50%. The corresponding potential error canthen be calculated as above. For example, if the measured value of RFP is loyo,the error will be 45.4% and the true RFP should lie between 5.5 and 14.5%. The corresponding value of the relative function parameter for the contralateral kidney will lie between 85.5 and 94.5%. Two cases of exception, when errors larger than those due to thegeometrical and statistical errors maybe expected, bothoccur in the measurement of a low RFP. Firstly, in a few cases, the contribution of cross-talk may give rise to an additionalcomparable error. This will occur for a lateral separation of the kidney centres of less than 12 cm and when the centres lie on the same lateral plane, If judged to be significant from the localizing radiograph, the error may be est'imated from the graph shown in fig. 5 and from eqn (4). Secondly, at low values of total renal function and for a valueof RFP less than 20% the dominant error may arise from the possible systematic error in the determination of the RFP
78
C. C. Nimmon et al.
subtraction ratios using the HSA technique, An estimat’ion of the possible error, assuming a 10% error in the ratios can be made from the graph shown in fig. 9 and from eqn ( 6 ) . Given these observations regarding the potential error in the calculation of relativefunction by renography,whatistheir significance withrespect to clinical practice T Most often, the clinical question posed is whether t’o conserve or remove a seriously diseased kidney. I n t h e example quoted above, if there is only 10% function being contributed by one kidney, it is of little importance in making a clinical decision whether this is actually 5 or 15%. I n this context, the effect of the additional error arising from the systematic error in the subtractionratiosininstances oflow tot’alkidneyfunction will be of little significance. Thus, the potential error inherent in the measurement of relative kidney function is acceptable when there is a large asymmetry in kidney function, but seriouslylimits the value of renographyindemonst’ratingsmall differences between kidneys with nearly symmetrical function. We gratefully acknowledge the technical assistance ofMiss A. Merry and Miss G. Tout, in carrying out the experimentalwork with the kidney phantom. We also acknowledge the computing facilit’ies provided by the University of London Computer Centre. The kidney phantom used in this study was supplied by Merck, Sharp and Dohme International. The work was supported by the Board of Governors of St. Bartholomew‘s Hospital.
RESUME L‘effet des facteurs geometriques en radiographie renale B l’hippuran 1311 En employant en detecteur special collimate a scintillations (NaI) et un fant6me renal contenant 1311, on a determine la relationentre le taux de comptage resultant et la geometric collimateur-rein. On a employe ces resultats pour la calculation de la contribution de la geomktrie a l’erreur dans la mesure du courant relatif effectif du plasma renal (RFP) aumoyen de la radiographie renale a l’hippuran 1311. Lorsqu’on emploie les methodes radiographiques et ult,rasonores de localisation des reins, on trouve, que cette erreur obeit la distribution normale avec 1’Qcartnormal de ?,S?; pour une function de division &gale. La combinaison de cette erreur avec celle provenant du mouvement natural et des fluctuations statistiques, observee pour une dose de 10 yCi de 1311hippuran, a conduit & l’estimation de l’erreur potentielle correspondante de 11yo. On a calculb pour toute la gamme de R F P les valeurs de l’erreur potentielle, que l’on definit comme la gamme de probabilitb de 99%.
ZUSAMMENFASSUNG Auswirkung geometrischer Faktoren in der Renographie mittels 1311-Hippuran Indem man einen besonders ausgeblendeten SaI-Szintillations-Detektor und ein 1311-enthaltendesXierenphantom anwandte, ist die dbhangigkeitder resultierenden ZBhlraten \-on der Sieren-Kollimator-Geometrie bestimmt worden. hIan benutzte diese Ergebnisse, um den geometrischen Eintrag zum Fehler in der Xessung des relativen effektiven Nierenplasma-Strom (RFP) mitHilfe der Renographie mit 1311-Hippuranzu errechnen. Im Falle, wenn man die radiographischen und Ultraschallverfahren zur Lokalisierung der Sieren anwandte, wurde es gefunden, dass dieser Fehler eine normale Verteilung befolgt, mit einer Standardabweichung von 2,696, falls die Funktion in gleicher Weise geteilt ist. Die Kombinierung dieses Fehlers mit dem von der natiirlichen Bewegung und den statistischen Fluktuationen hervorgerufenen-wie mit einer Dosis van 10 yCi ’311-Hippuran beobachtet-fiihrte zu einer Einschatzung eines entsprechenden potentiellen Fehlers von 11%. Es wurden errechnet-fur den Bereich van RFP-die Werte des potentiellen Fehlers, welcher als das 99~o-Wahrscheinlichkeitsgebiet definiert wird.
Qeometrical Factors in 1311-Hipp~ran Renography
79
REFERENCES BRITTON,K. E., and BROWN,N. J. G., 1968, &i"dical Monograph No. 1 (Edinburgh: Nuclear Enterprises Ltd). BRITTON, K. E., and BROWN, N. J. G., 1971, Clinical Renography (London: Lloyd-Luke). FARMELANT, M. H., SACHS,C. E., and BURROWS, B.A., 1970, J . Nucl. Med., 11, 112. JAMES, F., 1968, Cern Report CERN 68-15. MAGNUSSON,G., 1972, Radionuclides in Nephrology (New York: Grune and Stratton). MOELL,H., 1963, Acta Radiol., 1, 22. NINMON,C. C., MCALISTER, JOAN M., and CATTELL, W . R., 1974, Br. J . Radiol. (to be published).
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My IOPscience
The influence of geometrical factors in 131I-Hippuran renography
This content has been downloaded from IOPscience. Please scroll down to see the full text. 1975 Phys. Med. Biol. 20 67 (http://iopscience.iop.org/0031-9155/20/1/006) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 137.132.123.69 This content was downloaded on 03/10/2015 at 00:59
Please note that terms and conditions apply.
PHYS. MED. BIOL.,
1975, VOL. 20, NO. 1, 67-79. 0 1975
The Influence of Geometrical Factors in 13'I-Hippuran Renography C. C. NIMMOK, B.SC., JOAK M. McALISTER, xsc., F.INST.P., and W . R. CATTELL, M.D., F.R.C.P.,F.R.C.P.E. Radioisotope Department and the Department of h-ephrology, St. Bartholomew's Hospital, London, England.
Receiced 5 March 1974, in Jinal f o r m 20 August 1974 ABSTRACT.Using a particular collimated S a 1 scintillationdetector and a kidney phantom containing 1311, the dependence of the resulting count rates on collimatorkidney geometry has been determined. These results have been used to calculate the geometrical contribution t o the error in the measurement of relative effective renal plasma flow (RFP) by '31I-Hippuran renography. When radiographic and ultrasonic methods of localizing the kidneys are employed, this error has been found to follow a normal distribution with a SD of 2.6% in the case of equally divided function. Combination of this error with that from natural movement and statistical fluctuations, as observed using a dose of 10 yCi 131I-Hippuran, has led to the estimation of a corresponding potential error of 11%. Values of the potentialerror, which is defined as the 9976 probability range, have been calculated covering the range of RFP.
1. Introduction
I n recent years, renography with blood background subtraction has been shown to be a technique capable of providing a measurement of indices of relative kidney function (Britton and Brown 1968). During the uptake phase, before excretion from either kidney has taken place, the ratio of the renogram traces aft'er blood background subtraction is a measure of relative effective which is renalplasma flow. It is thisparameter of relativerenalfunction dealt with in this study and it is subsequently referred t'o as RFP. In order to obtain an accurate measurement of this paramet'er,it is necessary that the blood backgroundsubtractedcurvesshould reflect the radioactive content of the respective kidneys at any given time. However, when using a conventionalsystemwith three collimatedscintillationdetectors, the count rates are dependent on the collimator-kidney geometry. The accuracy of the method is affected by thisdependence on geometry,and by the extent which to this geometry is unknown. It is possible to reduce the geometrical uncertainty by using radiographic and ultrasonic methods of localizing the kidneys. The aim of t'his work has been to estimate theremaining error when such techniques of localization are employed. The dependence of the count rates on relative collimator-kidney geometry has been investigatedexperimentally.Amathematical expression for the parameter error has been formulated in terms of the geometrical errors. Using the experimental results in conjunction with this formula, an expected parameter error distribution is obtained.
C.
68
C.
Nimmon et al.
2. Equipmentandmethods
The detector used in this studywas a 5 x 5 cm NaI(Th) crystal, with aconed surface-to-end-of-collead collimator with a circular aperture.Thecrystal limator distance was 5 cm, the aperture at the end of the collimator 7 cm diameter, and the thickness of the collimation at the crystal was 1.75 cm lead and 0.75 cm steel. The point source response curve obtained with this detector for it 1311 point source in water is shown in fig. 1. Pulse height analysis was used to detect the 0.36 MeV photopeak of l3II.
Fig. 1.
I3lI point
source response of the scintillation detector. Isocount lines are expressed
as a percentage of the count rate obtained at the centre of the collimator face.
The kidney phantom used is shown in fig. 2(a). It is made of a thin-walled plastic material and has the following dimensions : pole-to-pole length 11 cm, midpole thickness 4 cm and lateral width5.5 cm. It was mounted in a perspexwalled water tank and was capable of being accurately moved relative to the face of the collimator, which was positioned against one wall of the tank. The
Fig. 2. (a) The plastic kidney phantom. (b) The mathematically generated phantom and the directions of displacement X , Y and 2 with respect to the centre of the collimator face.
Geometrical Factors in 1311-HippuranRenography
69
phantom was filled with an aqueoussolutioncontaining 10 pCi 1311. The variations in count rates were determined for movement of the phantom along each of the axes X , Y , Z of the right-handed rectangular coordinate system shown in fig. 2(b). Anatomically,displacements inthe directions X , Y , Z correspond to lateral, craniocaudal and posterior-anterior displacements respectively, with respect to the centre of the collimator face. For the study of these variations the kidney pole-to-pole axis was parallel to the Y axis. The effect of rotation of the pole-to-pole axis in the YZ plane was also studied for fixed X, Y,Z displacements. For comparison withthe kidney phantom results, a mathematically simulated kidney model was also used. ,4s shown in fig. 2(b) the model was an ellipsoid with a cut-off section. With the mathematical model the kidney size could be varied and also a volume representing the kidney pelvis could be generated. For the mathematical calculations the model, a t aparticularposition, was split up intosegmental volumes, each correspondingto anelement of the point source response matrix. The required count rate from the whole model was then obtained by summation. The determination of the segmentalvolumes essentially involved solving the mat.hematica1 equations for an ellipse and a circle. Approximate roots were obtained by a grid search technique and more accurate values then obtained using Newton's method of successive approximation.
Experimentalresultsandcomparisonwiththemathematicalmodel.The dependence of count rate on relative collimator-kidney position 3.1. Movement of the kidney centrealongthe Z axis with the pole-to-pole axis
3.
parallel to the Y axis The variation obtained with the phantom is shown in fig. 3, and is in fact similar to that obtained using a point source. Movement of the phantom by 1 cm from t'he position Z = 7 cm alters the count rate by approximately 23%.
I 0
2
4 6 8 1 0 1 2 Distance Z ( c m 1
Fig, 3. Variation of count rate with displacement in the 2 (posterior-anterior) direction.
C . C. Nimmon et al.
70
3.2. Movement of the kidney centre in the Y direction with the pole-to-pole axis
parallel to the Y axis The results are shown in fig. 4 for the phantom centre a t depths of 5 , 7 and 9 cm. For a displacement of the centre by less than 2 cm from t’he collimator axis, the count rate remains greater than 90% of the undisplaced value. This is followed by an almost linear fall-off, dropping to 50% for 2 = 7 cm a t a displacement of 6 cm.
Distance Y ( c m )
Fig. 4. Variation of count rate withdisplacement in the Y (craniocaudal)direction, for the phantom centre at depths 5 , 7 and 9 cm.
3.3. Movement of the kidney centre in the X direction with the pole-to-pole axis
parallel to the Y axis The results are shown in fig. 5 for the phantom centre a t depths of 5 , 6, 7 and 8 cm. At a displacement of X = 14 cm, the count rate drops below 1yo of the undisplaced value. of cross-talk betweenthe two kidney These results are relevantto the amount detectors and are discussed later. 7r
Zb
:S:5
-5
Z(crn)
-6
-7
-a
W
2
g 4
c
23 W &
2 2 &
c 3
G I ‘11
I2
13 14 IS Distance X ( c m )
16
17
Fig. 5 . Variation of count rate with displacement in the X (lateral) direction, for the phantom centre at depths 5 , 6, 7 and 8 cm.
Geometrical Factors in 1311-HippuranRenography
71
3.4. Rotation (e) of the pole-to-pole axis in the YZ plane The results are shown in fig. 6. When the centre of the kidney lies on the collimator axis a rotation of 40" is needed to produce a change in count rateof 20%, and the count rate is increased independently of the direction of rotation.
S U
I -60
, -40
, -20
, 0
, 20
, 40
, 60
Angle 0 (deg 1
Fig. 6. Variation of count rate with rotation6" of the pole-to-poleaxis of the phantom in the Y Z plane. 6 = 0 corresponds to the pole-to-pole axis being parallel to the Y axis. Results are shown for Y = 0 and 5 cm.
When the kidney centre is displaced from the collimator axis in the Y direction the count rate is increased or decreased as the upper pole is rotated in the negative (posteriorly) or positive direction respectively. For Y = 5 cm, a 20" rotation produces a change in count rate of 10%. 3.5. Calculations using the mathematically simulated model The calculated variation in count rate withdisplacement of the kidney in the Y direction was similar to theresults of the phantom measurements. A volume representing the renal pelvis was also generated. With the centre of the model at the position X = 0, Y = 0 , Z = 7 cm, the difference in count rate calculated for a given activity uniformly distributed in thepelvis (volume 38 cm3) and for the same activity in the remainder of the kidney (volume 107 cm3) was less than 2%. Calculations 4.1. Calculation of the error on the function parameter produced by variation in the relative collimator-kidney position Assuming that each kidney is identically placed with respect to its detector and that bothdetectors have equal sensitivity, then the relative function para) meter (i.e. the relative effective renal plasma flow) for the left kidney ( R F P ~ is given by the equation: 4.
1OOC,
(yoof total function) = -
RFP~
C L -tC,
where C, and C, are values of the left and right renogram tracesrespectively a t a time after the injection of labelled Hippuran. RFP is usually calculated by
C. C. Nirnmon et al.
72
sampling the traces over a period of time starting 1 min after injection and continuing up to the timeof the first occurring peak. For the purpose of calculation, consider that the kidneys are positioned with their centres on the respective collimator axis and that the pole-to-pole axes are parallel tothevertebral column.Displacement from this position will modify the count rates CL and c, by factorsf L and f R respectively. RFPL
(observed) =
1 0 0 f LC L R ‘ fL +fR
The percentage error onthe trueRFP value dueto thisdisplacement is given by :
where y = - f. L
fR
Similarly,
If CL = C,, then RFP
(Yo) = 50
and 100(1 - r ) E L (Yo) =
r+l
= -ER
(YO).
From eqn (1) it can be seen that the percentage error E depends both on RFP and on the ratio r , which is determined by the relativedisplacement of the two kidneys. A graph giving the relationship between E L and RFP for values of r = 0.9, 0.82 and 0.67 is shown in fig. 7. These values of r result in an error of 5 ; 10 and 20% respectively for RFP = 50%. Inthe following discussion onerrors,resultshave been calculated for RFP = joyb. The corresponding errors for any value of RFP may be obtained by calculation from eqns (1) and ( 2 ) . 4.2. Combination of errors due to displacement The resultant error on RFP arising from two independent displacements, with r = rl, r2 respectively, can be calculated by substituting r = r 1 r 2 in eqn ( 2 ) .
Geometrical Factors in 1311-Hippuran Renography
RFP
73
(*/e
Fig. 7. Dependence of the percentage error E parameter RFP (yo).
(yo)on the value of the relative function
For small displacements,
To a first approximation,the craniocaudal, depth and rotationdisplacement's will be regarded as independentwhen the craniocaudal displacement is small. 4 . 3 . Estimation of the parameter error distribution using a Monte Carlo method
If the distributions of collimator-kidney position were known, probability density distributions of the error on RFP arising from each displacement could be calculated from the graphs shown in figs 3, 4, 6 and from eqn (2). For small displacements the resultant probability error distributioncould then be obtained using eqn ( 3 ) . Consider the situationwhere the kidneys are localized using a urogram taken with the patient in the renographyposition and where the depthsof the kidney centres below the posterior skin surface are measured using an A-type ultrasonictechnique.Thecraniocaudal collimator-kidney distributioncanbe (SD) dependexpected to follow a normal distribution with a standard deviation ing on the accuracy of transferring the measurements from the radiograph onto the skin of the patient. Similarly, it can be expected that the measured depths will show a normal distribution about the true values with a SD depending on the accuracy of making measurements from a Polaroid film. This latter distribution will give rise to a corresponding error contribution to RFP when the l/fL,l/fR obtainedfrom fig. 3 areapplied. appropriate correctionfactors However, the error will also depend on the actual depths encountered,since the relationship between the depth and thecorrection factor is not linear. The distribution of kidney depth found by Nimmon, McAlister and Cattell (1974) can
74
C. C . Nimmon et al.
beused. In the absence of available data on the distribution of post'erioranterior kidney rotation, a hypot'hetical distribution must be used. A possible choice would be a normal distribution with mean 9 = 0". Assuming normaldistributions for the craniocaudaldisplacements, depth SDS of 1.0 cm, 0.25 cm and 10" respectively, and errors a'nd kidney rotation with using the depth distribution reported by Nimmon et al. (1974), a parameter error distribution was calculated by a Monte Carlo method (James 1968). The results are shown in fig. 8, together withthe individual craniocaudal, depth and rotat'ion contributions. f (€1
Rotation Craniocaudal Depth -Total " "
E 010 1 Fig. 8. Calculated paramet'er error distribution, together with the individual craniocaudal, depth and rotation contributions.
The resultant parametererror distribution is closely normal with SD = 2.60,/,. The individual contributions have a SD of 1.5, 1.9 and 0.2% respectively. 4.4. Further possible sources of error on RFP
4.4.1. Cross-talk. It is possible for each detector to record an amount of pickup (cross-talk) from the activity in the contralateral kidney. Consider the crosstalk to be x% of the count rate recorded by the contralateral detector. The error in RFP due to this effect FL will be :
For
Geometrical Factors in 1311-Hippuran Renography
75
Over the possible range of RFP the maximum error incurred will be f x. The error due to cross-talkis most significant in the measurement of a low RFP. For example, for RFP = loyo,if the cross-talk is x%, the measured RFP will be RFP +x. For the cases where the kidney centres lie in t'he same lateral plane, i.e.have the same Y coordinate, the extent of the cross-talkexpectedfor different lateral separations is given in fig. 5 , and is less than 2.5% for separations greater than 12 cm. From the distribution of lateral position reported by Nimmon et al. (1974), it may be concluded that this magnitude of cross-talk will be realized in only a few per cent of cases. 4.4.2. Rotation in the X Y plane. It is known that the kidney pole-to-pole axes do not ingeneral lie parallel to the vertebral column, but that they are rotated in the XY plane (Moell 1963). However, the point source response curves of detectors withcircular collimatorsare radially symmetrical about the collimator axes. Therefore, rotation in t h e X Yplane should not' produce a contributionto the error on RFP, provided that the collimator axes are aligned with the kidney centres and that' cross-t'alk is not significant'ly increased. 4.4.3. Natural movement and statistical Jluctuations.
Errors can be expected to arise from the natural movement of the kidneys during respiration over the period of measurement. An estimate of t'his error combined with the error due to statistical fluctuation is given by the SD on the mean of the measurement's taken over the sampling period. Results from 30 renograms, in which a dose of 10 pCi 1311-Hippuranwas used and measurementswere made a t 10 S intervals over the sampling period, show a mean error of & 1.7 for RFP values in the range 10-90yo. This error is particularly significant in t'he measurement of a low value of RFP. 4.4.4.Systematic errors in the background subtraction ratios. One of the assumptions in the technique of blood background subtraction using 1311-HSAis that t'he ratioof intravascular to extravascular fluid volume in the regions viewed by all three detectors is the same. There is evidence from studies on nephrectomized patient's (Farmelant, Sachs and Burrows 1970, Nagnusson 1972) t'hat this assumption is not correct and that theabove ratio is smaller for the kidney region than for the subclavicular region. Consequently, in using the HSA subtraction technique, the Hippuran blood background is slightly underestimated (Brit'ton and Brown1971), thus giving rise to aresidualbackground. It is realized that this residual background will not follow the shape of the curve obtained from the subclaviculardetector.Xevertheless,forsimplicityin estimating the order of magnitude of the error in RFP one can consider the residualbackgroundasarising from asystematicerror in the background subtraction ratios. It has been suggested (Britton and Brown 1971) that the effective error in the subtraction ratios may be 10%. The resultant contribution ( E L )to the error on RFP may be calculated as follows: C, = KL -X, H C, = KX-SRH
C. C. Nimmon et al.
76
where K, and KR are the values of the left and right renogram traces before blood background subtraction. S , and S , are the HSA subtraction factors for the left and right kidney respectively. H is the count rate observed by the subclavicular detector.
-
”
where AS, and AS, are the errors on the fact’ors S , and S , respectively. Using the approximation S , = S, = 1, and assuming the error in the subtraction factors tobe loyo,the values of the factor W expected over the range of RFP are shown in fig. 9. The graph shows the rapid increase of this factor as 200 r
l
20
40
60
80
100
RFP~(%)
Fig. 9. Variation of the parameter W
(so) with RFP (76).
RFP decreases below 20%. The error on RFP due t o the possible systematic error in the subtraction ratios depends also on the ratioH/(CL+ CR). The mean value of this ratio over the period of relative function measurement depends on total renal function. It has been shown (Britton and Brown 1971) that this ratio is such that the resultant contribution to the error on RFP is only likely to become significant for total creatinine clearances less than 30 mlmin-l. In this range the error becomes particularly significant for values of RFP less than 20%.
Geometrical Factors in 1311-Hippuran Renography
77
Discussion It has been found that using the described technique of kidney localization, the two main sources of error in the measurement of the relative function parameterare the residual uncertaintyin collimator-kidney geometry and the statistical error. When RFP = 50%, the resultant potential error from these two sources is 11%, calculated as 2.58 x the resultant SD (i.e. the expected 99% range for a normal distribution). The potential error varies with RFP and some values are given in table 1. The geometrical contribution at anyvalue of 5.
Table 1. Calculated values of the potential error Potential error
Measured value O f RFP
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95
(yo)
(yoof the measured value of
RFP)
88.6 45.4 31.3 24.3 20.1 17.3 15.2 13.5 12.2 11.0 10.0 9.1 8.2 7.5 6.8 6.1 5.5 5.1 4.7
99Yo probability range of the true value of RFP (",) 0.6- 9.4 5'5-14'5 10'3-19'7 15.1-24.9 20'0-30.0 24.8-35.2 29'7-40'3 34.6-45'4 39'5-50.5 44.5-55.5 49'5-60'5 54.6-65'4 59'7-70'3 64'8-75.2 69'9-80.1 75.1-84'9 80'3-89.7 85.5-94.5 90'6-99.4
can be calculated using eqns (1) and (2), together with the SD obtained above for RFP = 50%. The corresponding potential error canthen be calculated as above. For example, if the measured value of RFP is loyo,the error will be 45.4% and the true RFP should lie between 5.5 and 14.5%. The corresponding value of the relative function parameter for the contralateral kidney will lie between 85.5 and 94.5%. Two cases of exception, when errors larger than those due to thegeometrical and statistical errors maybe expected, bothoccur in the measurement of a low RFP. Firstly, in a few cases, the contribution of cross-talk may give rise to an additionalcomparable error. This will occur for a lateral separation of the kidney centres of less than 12 cm and when the centres lie on the same lateral plane, If judged to be significant from the localizing radiograph, the error may be est'imated from the graph shown in fig. 5 and from eqn (4). Secondly, at low values of total renal function and for a valueof RFP less than 20% the dominant error may arise from the possible systematic error in the determination of the RFP
78
C. C. Nimmon et al.
subtraction ratios using the HSA technique, An estimat’ion of the possible error, assuming a 10% error in the ratios can be made from the graph shown in fig. 9 and from eqn ( 6 ) . Given these observations regarding the potential error in the calculation of relativefunction by renography,whatistheir significance withrespect to clinical practice T Most often, the clinical question posed is whether t’o conserve or remove a seriously diseased kidney. I n t h e example quoted above, if there is only 10% function being contributed by one kidney, it is of little importance in making a clinical decision whether this is actually 5 or 15%. I n this context, the effect of the additional error arising from the systematic error in the subtractionratiosininstances oflow tot’alkidneyfunction will be of little significance. Thus, the potential error inherent in the measurement of relative kidney function is acceptable when there is a large asymmetry in kidney function, but seriouslylimits the value of renographyindemonst’ratingsmall differences between kidneys with nearly symmetrical function. We gratefully acknowledge the technical assistance ofMiss A. Merry and Miss G. Tout, in carrying out the experimentalwork with the kidney phantom. We also acknowledge the computing facilit’ies provided by the University of London Computer Centre. The kidney phantom used in this study was supplied by Merck, Sharp and Dohme International. The work was supported by the Board of Governors of St. Bartholomew‘s Hospital.
RESUME L‘effet des facteurs geometriques en radiographie renale B l’hippuran 1311 En employant en detecteur special collimate a scintillations (NaI) et un fant6me renal contenant 1311, on a determine la relationentre le taux de comptage resultant et la geometric collimateur-rein. On a employe ces resultats pour la calculation de la contribution de la geomktrie a l’erreur dans la mesure du courant relatif effectif du plasma renal (RFP) aumoyen de la radiographie renale a l’hippuran 1311. Lorsqu’on emploie les methodes radiographiques et ult,rasonores de localisation des reins, on trouve, que cette erreur obeit la distribution normale avec 1’Qcartnormal de ?,S?; pour une function de division &gale. La combinaison de cette erreur avec celle provenant du mouvement natural et des fluctuations statistiques, observee pour une dose de 10 yCi de 1311hippuran, a conduit & l’estimation de l’erreur potentielle correspondante de 11yo. On a calculb pour toute la gamme de R F P les valeurs de l’erreur potentielle, que l’on definit comme la gamme de probabilitb de 99%.
ZUSAMMENFASSUNG Auswirkung geometrischer Faktoren in der Renographie mittels 1311-Hippuran Indem man einen besonders ausgeblendeten SaI-Szintillations-Detektor und ein 1311-enthaltendesXierenphantom anwandte, ist die dbhangigkeitder resultierenden ZBhlraten \-on der Sieren-Kollimator-Geometrie bestimmt worden. hIan benutzte diese Ergebnisse, um den geometrischen Eintrag zum Fehler in der Xessung des relativen effektiven Nierenplasma-Strom (RFP) mitHilfe der Renographie mit 1311-Hippuranzu errechnen. Im Falle, wenn man die radiographischen und Ultraschallverfahren zur Lokalisierung der Sieren anwandte, wurde es gefunden, dass dieser Fehler eine normale Verteilung befolgt, mit einer Standardabweichung von 2,696, falls die Funktion in gleicher Weise geteilt ist. Die Kombinierung dieses Fehlers mit dem von der natiirlichen Bewegung und den statistischen Fluktuationen hervorgerufenen-wie mit einer Dosis van 10 yCi ’311-Hippuran beobachtet-fiihrte zu einer Einschatzung eines entsprechenden potentiellen Fehlers von 11%. Es wurden errechnet-fur den Bereich van RFP-die Werte des potentiellen Fehlers, welcher als das 99~o-Wahrscheinlichkeitsgebiet definiert wird.
Qeometrical Factors in 1311-Hipp~ran Renography
79
REFERENCES BRITTON,K. E., and BROWN,N. J. G., 1968, &i"dical Monograph No. 1 (Edinburgh: Nuclear Enterprises Ltd). BRITTON, K. E., and BROWN, N. J. G., 1971, Clinical Renography (London: Lloyd-Luke). FARMELANT, M. H., SACHS,C. E., and BURROWS, B.A., 1970, J . Nucl. Med., 11, 112. JAMES, F., 1968, Cern Report CERN 68-15. MAGNUSSON,G., 1972, Radionuclides in Nephrology (New York: Grune and Stratton). MOELL,H., 1963, Acta Radiol., 1, 22. NINMON,C. C., MCALISTER, JOAN M., and CATTELL, W . R., 1974, Br. J . Radiol. (to be published).
