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Unbiased estimation of the liver volume by the Cavalieri principle using magnetic resonance images
European Journal of Radiology 47 (2003) 164 /170 www.elsevier.com/locate/ejrad
Unbiased estimation of the liver volume by the Cavalieri principle using magnetic resonance images Bu¨nyamin Sahin a,*, Mehmet Emirzeoglu a, Ahmet Uzun a, Lu¨tfi Incesu b, Yu¨ksel Bek c, Sait Bilgic a, Su¨leyman Kaplan d a
Department of Anatomy, Medical School, Ondokuz Mayis University, 55139 Samsun, Turkey Department of Radiology, Medical School, Ondokuz Mayis University, 55139 Samsun, Turkey c Department of Biostatistics, Medical School, Ondokuz Mayis University, 55139 Samsun, Turkey d Department of Histology and Embryology, Medical School, Ondokuz Mayis University, 55139 Samsun, Turkey b
Received 19 February 2002; received in revised form 10 May 2002; accepted 13 May 2002
Abstract Objective: It is often useful to know the exact volume of the liver, such as in monitoring the effects of a disease, treatment, dieting regime, training program or surgical application. Some non-invasive methodologies have been previously described which estimate the volume of the liver. However, these preliminary techniques need special software or skilled performers and they are not ideal for daily use in clinical practice. Here, we describe a simple, accurate and practical technique for estimating liver volume without changing the routine magnetic resonance imaging scanning procedure. Materials and methods: In this study, five normal livers, obtained from cadavers, were scanned by 0.5 T MR machine, in horizontal and sagittal planes. The consecutive sections, in 10 mm thickness, were used to estimate the whole volume of the liver by means of the Cavalieri principle. The volume estimations were done by three different performers to evaluate the reproducibility. Results: There are no statistical differences between the performers and real liver volumes (P /0.05). There is also high correlation between the estimates of performers and the real liver volume (r / 0.993). Conclusion: We conclude that the combination of MR imaging with the Cavalieri principle is a non-invasive, direct and unbiased technique that can be safely applied to estimate liver volume with a very moderate workload per individual. # 2002 Elsevier Science Ireland Ltd. All rights reserved. Keywords: Cavalieri principle; Liver volume; MRI; Stereology
1. Introduction Organ volumes can be estimated, without bias, using the Cavalieri principle of volume measurement by means of consecutive serial sections [1,2]. A simple approach to obtain such information was demonstrated over 300 years ago by the Italian mathematician Bonoventura Cavalieri. He pointed out that the volume of any object could be estimated from a set of twodimensional slices through the object, provided that they are parallel, separated by a known distance, and begin randomly within the object, criteria that are met by
* Corresponding author. Tel.: /90-362-457-6000x2262; fax: /90362-457-6041 E-mail address: [email protected] (B. Sahin).
standard magnetic resonance imaging (MRI) techniques [3]. The relevant method has recently been refined and applied in a series of studies involving invasive and noninvasive scanning techniques [1,2,4 /6]. Many recent studies have proven this estimator to be as accurate as digitization-based methods and to correlate closely with displacement volume measurements [7,8]. It is demonstrated that this simple, inexpensive stereological approach is well suited to rapid and an accurate volumetric calculations on the basis of standard MR scans of liver tumors, cirrhosis, hepatitis, hepatomegaly, liver transplantation, hepatectomy and other surgical applications of liver. It is also useful to examine structures that require assessment of changes in volume over time as an indicator of therapeutic effectiveness [9 /14]. The purpose of this study is to describe and adapt the relevant methods of MRI and stereology to estimate
0720-048X/02/$ - see front matter # 2002 Elsevier Science Ireland Ltd. All rights reserved. PII: S 0 7 2 0 - 0 4 8 X ( 0 2 ) 0 0 1 5 2 - 3
B. Sahin et al. / European Journal of Radiology 47 (2003) 164 /170
liver volume and compare the liver volume estimations with the real liver volume measurements. A reliable formula was provided for estimation of the coefficient of error (CE), which allow us to calculate the optimum number of sections required to attain a given precision for a particular scanning direction.
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if the upper right hand corner of the intersection of the cross lines representing them on the test system lies inside the object. After each superimposition, the number of test points hitting the structure of interest on the sections is counted, and the unbiased estimator becomes: est2 V t a=p (P1 P2 Pn ) cm3 ;
(3)
2. Estimation of volume by the Cavalieri principle It is known that the volume of the regular shaped objects (i.e. prism, cube or cylinder) can be estimated by the formula: V t a;
(1)
where (t ) is the height and (a ) is the base area of object. Similar to this principle, using the Cavalieri principle, an unbiased estimate of the volume of an object of arbitrary shape and size may be obtained efficiently and with a known precision [2,5,15,16]. The method requires sectioning the structure with a series of parallel planes. To avoid bias, the first section must be placed at a uniform and random position in a constant interval of length (t ) i.e. to start the scanning always at, for example, 1 cm from the right tip of the object will introduce an unknown amount of bias in general. Moreover, the series of sections must encompass the object entirely. The direction of cutting does not affect the unbiasedness property, but will affect the estimation precision in general [15]. Thus an unbiased estimate of volume can be obtained by multiplying the total area of the section cut surfaces through the structure on all the sections, i.e. est1 V t (a1 a2 an ) cm3 ;
(2)
where (a1/a2/ /an ) denote the section areas in cm2 and (t) is the sectioning interval in cm for the n consecutive sections. Some automatic machines or software can measure the contour of the object to obtain the cut surface area of section. However, several studies have shown that point counting techniques represent a more reliable and efficient approach than planimetric technique for obtaining the required cut surface areas of section [7,17]. The point counting grid, which has some point sets at distinct densities on a transparent sheet, can be used to estimate the cut surface area of the sections [3,15,16,18]. The point counting method consists of overlying each selected section with a regular grid of test points, which is randomly positioned. Test system orientation does not affect unbiasedness, but certain orientation improves the estimation precision. For this reason, the tests system should be superimposed on the section three times and the mean number of points hitting the objects should be used to estimate cut surface area of the section. A test point is a (/) shaped lines and it is said to hit the object
where (P1/P2/ /Pn ) denote the point counts and (a /p ) represents the area associated with each test point, corrected for any change of scale in the images as it is printed on the hardcopy films. The area of each section (ai ) is now estimated by [(a /p ) /Pi ]. The subscript 2 in est2 V indicates that a two-stage process, namely sectioning and point counting, estimates the volume. To avoid the over estimation due to MR scanning, adding a correction limit the bias of overprojection of the liver slices through the scan plan, the Cavalieri principle provides a simple estimator [19]. Thereby, the formula can be changed as follows: est2 V t a=p (P1 P2 Pn Pmax ) cm3 ;
(4)
where (Pmax) is the maximal number of points counted on a single scan plane of the subject. In the Cavalieri principle, the researcher obtains a data called CE to evaluate the reliability of the point density of the grids and sectioning intervals. The CE or relative standard error represent the precision of the volume estimate obtained using the Cavalieri principle. Since consecutive section cut surface areas are not independent quantities, conventional statistical formulae of CE cannot be applied to determine the variance of their sum. Gundersen and Jensen [1] developed a wellknown CE prediction formula for the Cavalieri estimation method. The new formula not only provides the CE but also gives information on the required number of slices and density of the point counting grid. For this purposes, the calculations are being done to estimate the CE and other data by means of the following consecutive formulas. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X ffi pffiffiffi Noise 0:0724 (b= a) n P:
(5)
Noise gives information on the complexity of the examined cut surface area of the specimen. Calculation of the CE using these equations requires knowledge of pffiffiffi the value of a dimensionless shape of coefficient (b= a); which is equivalent to the mean boundary length of the profiles divided by the square root of their mean area, and is a measure of the average shape of the profiles through the structure of interest on the sections [3 / 5,18].
B. Sahin et al. / European Journal of Radiology 47 (2003) 164 /170
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VarSRS
X n a
required to obtain an appropriate variation for section samples. A , B , and C are the total numerical values for the data in the related column of Table 1.
i 1
X X 3 (P2i Noise) 4 Pi Pi1
X
Pi Pi2
=
12:
(6)
This formula can be simplified with the aid of Table 1 as follows: X n VarSRS a i 1
[3 (A Noise) 4 BC]=12:
(7)
Whereas,/VarSRS (ani 1 a) indicates variance of total area in the systematic random sampling (SRS). These data give information on the sufficient section number
(8)
Total Var NoiseVarSRS ; X pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Total Var : CE P P P
(9)
CE is the last calculated value. The generally accepted highest limit of CE is 5% [1,16].
3. Materials and methods Five normal livers were obtained from cadavers stored in the Department of Anatomy. The MR scans were performed by 0.5 T machine (GE Signa, Milwaukee, WI). The scanning was done with the following
Table 1 An example of the estimate of volume and CE of one liver sectioned horizontally and its explanation Section no.
1st Sup
2nd Sup
3rd Sup
1 2 3 4 5 6 7 8 9 10 11 12 13 14
3 21 25 31 32 32 29 31 25 20 17 12 7 2
3 19 26 31 33 33 30 30 27 20 16 13 8 2
2 21 26 31 33 33 30 28 27 21 16 12 7 3
Total
Pi
Pi PI
Pi PI1
Pi Pi2
2.7 20.3 25.7 31.0 32.7 32.7 29.7 29.7 26.3 20.3 16.3 12.3 7.3 2.3
7.1 413.4 658.8 961.0 1067.1 1067.1 880.1 880.1 693.4 413.4 266.8 152.1 53.8 5.4
54.2 521.9 795.7 1012.7 1067.1 969.1 880.1 781.2 535.4 332.1 201.4 90.4 17.1 0.0
68.4 630.3 838.4 1012.7 969.1 969.1 781.2 603.2 430.1 250.8 119.8 28.8 0.0 0.0
289.3
7519.8 A
7258.6 B
6702 C
X pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Noise 0:0724 (b= a) n P 0:0724 5 289:3 14 23:03: VarSRS
X n a [3 (A Noise) 4 BC]=12 [3 (7519:8 23:03) 4 7258:66702]=12 13:16: i 1
Total Var NoiseVarSRS 23:0313:16 36:2: X pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi 36:2 Total Var P 0:021: CE P 289:3 P X Liver volume t a=p [( P) (1=2Pmax )] 1 3:642 (289:3 16) 995:48 cm3 : pffiffiffi pffiffiffi 0.0724/(b= a); the value showing the complexity of the section view. (b= a) is predicted as 5 for livers; Noise, gives information about the n complexity of the examined cut surface of the specimen; VarSRS (ai 1 a); variance of total area in SRS. This data gives information on the sufficient section number to obtain appropriate variation for section samples. A , B , C total numerical values for the data in the related column of the table; CE, the CE is the final data of the estimation. It is generally accepted that the CE of the study must be lower than 5%; a P , total number of the point hitting the sections cut surface area; 1/2Pmax, the half of the maximal number of points counted on a single scan plane of the subject; t , the section thickness. It was 1 cm for this study; a /p , the area of the each point on the point counting grid; Sup, superimposing.
B. Sahin et al. / European Journal of Radiology 47 (2003) 164 /170
parameters: spin-echo axial and sagittal repetition time (TR), 500; echo-time (TE), 27; field of view (FOV), 32; number of excitation (NEX), 3; Matrix, 256 /192; and mean scanning time, 9.28 min. Each specimen was scanned in both sagittal and horizontal planes and consecutive sections were taken in 10 mm slice thickness. The scan-plane thickness is constant due to the stepwise movement of the scanner. The images of the liver sections were printed on films in square frames of 8 /7.5 cm side length. The magnification ratios were between 0.24/1 and 0.29/1 cm for each individual livers and section plans of printed films. A square grid test system with d /0.5 cm between test points i.e. a/p /0.25 cm2 representing area per point (Fig. 1) were used to estimate the cut surface area of the slices. The representing area per point in the grid was corrected with the magnification of printed sections and thereby; the real area per point was calculated between 2.97 and 4.16 cm2 for livers in both section planes. The films were placed, in turn, on a light bow and the transparent square grid test system was superimposed uniformly, randomly covering the entire image frame. The superimposing of the test system was repeated three times for each image frame and the points hitting the liver section cut surface area were counted for each section. The mean of the three repeated number of points counted for any section was used to estimate the section cut surface area for sagittal and horizontal scan plans and then the volume of the entire liver was estimated by a modification of the 4th formula as: est2 V t a=p [P1 P2 Pn (1=2Pmax )];
(10)
where (1/2Pmax) is the half of the maximal number of points counted on a single scan plane of the subject.
Fig. 1. A liver slice sectioned in horizontal plan and a point counting grid superimposed on the MR scan.
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The CE for liver volume estimation was determined by means of the 7th /9th formulas. Three performers estimated the liver volumes sectioned in sagittal and horizontal planes using the same sets of printed sections of livers to compare the validity of the estimates according to the performers. Calculation of liver volume, CE of estimates and other related data were simply performed using Microsoft Excel as a spreadsheet. After initial setup and preparation of the formula in a small macro program, the point counts were entered for each scan and user did the calculations automatically. The exact liver volumes were measured using Archimedes’ principle, also known as ‘fluid displacement’ in a measuring cylinder. One-way ANOVA test was performed to compare statistically the repeatability and accuracy of the estimates of three performers and the real measured volume of the livers in both sectioning planes. The associations between the Cavalieri estimates of three performers and real volume of the livers were also analyzed by means of the Pearson correlation coefficient test.
4. Results Liver volumes of five specimens obtained from cadavers were measured by three observers and by means of the water displacement technique. The volumes of five livers estimated by means of Cavalieri principle changed between 857 and 1592 cm3. Cavalieri estimation results of the livers’ volume using sagittal and horizontal plan sections and water displacement technique are summarized in Table 2. The mean of liver volume determined by the Cavalieri estimator and water displacement technique was highly correlated (r / 0.993). The coefficient of correlation (r) for three performers’ liver volume estimates in sagittal direction with the real liver volume was estimated as 0.991, 0.997 and 0.996 for the performers 1, 2 and 3, respectively. The coefficient of correlation of three performers’ liver volume estimates in horizontal direction with the real liver volume was estimated as 0.981, 0.983 and 0.983 for the performers 1, 2 and 3, respectively. The means of three performers’ volume estimates in sagittal and horizontal directions are also correlated well with real liver volume and that values (r) were 0.095 and 0.983 for sagittal and horizontal section planes, respectively (Fig. 2). The Cavalieri estimates of liver volumes for each performer were compared by using one-way ANOVA test. The real liver volumes and the estimations of performers were also compared by using ANOVA technique. The estimates of three performers in sagittal and horizontal section plan and real volume values were not statistically different and significance of the test values (P ) was 0.996 and 1.000 for sagittal and
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Table 2 The volume of the livers (cm3) obtained by three performers on sagittal and horizontal section planes and water displacement technique and the correlation coefficients of real volume between performers’ estimates
Liver 1 Liver 2 Liver 3 Liver 4 Liver 5 r (estxreal)
Performer 1
Performer 2
Performer 3
Sagittal
Horizontal
Sagittal
Horizontal
Sagittal
Horizontal
968 902 1592 952 902 0.991
995 954 1544 950 856 0.981
936 935 1579 950 913 0.997
989 971 1530 901 864 0.983
900 897 1547 911 866 0.996
992 966 1551 923 857 0.983
MTPE
MSSPE
MHSPE
Real Vol
963.3 937.5 1557.0 931.0 876.2
934.5 911.2 1572.5 937.3 893.5
992 963.7 1541.5 924.6 858.9
914 937 1625 904 921
MTPE, mean of three performers’ estimate; MSSPE, mean of sagittal section plane estimates; MHSPE, mean of horizontal section plane estimates; Real Vol, measured volume of the livers by water displacement technique; r(estxreal), coefficient of correlation between three performers’ estimates and real volume of livers.
horizontal section planes, respectively. The estimates of three performers did not showed interobserver differences (P /0.05). The total liver volume estimates obtained by hydrostatic weighing agreed well with the Cavalieri estimate. The mean CE of volume estimates was 1.7 and 2.0% for sagittal and horizontal section planes, respectively. The details of CE values were summarized in Table 3.
5. Discussion and conclusion Precise information for liver volume is often required for the assessment of changes in its volume over time as an indicator of therapeutic effectiveness, evaluation of the liver in cirrhosis, before and after liver transplantation and other surgical applications of it [9 /14]. Any accurate information about the actual volume of organ cannot be obtained by routine physical examination of the liver. Its major role in clinical diagnosis remains just to define the characteristics of the lower edge of the organ. Liver volume has been shown to be useful in prognosis of patients with cirrhosis, but its measurement needs to be quantitative and reproducible which can be obtained only by imaging techniques [20]. Several studies have been conducted to estimate liver volume by means of imaging techniques [21 /25]. A brief literature review for both computed tomography (CT) and MRI measurements revealed that most of the studies in which liver volume was measured relied on
either observer or computer delineation of a region of interest and subsequent relevant pixel count, often on a limited number of slices and expressed as a ratio measurement. These methods require the original imaging data set and experienced performers in manipulating it. Such requirements, on a routine basis, are costly in both time and financial resources [3]. The studies comparing the real volume with ultrasound and CT scans have failed to show the superiority of imaging modalities in estimating liver volume. Moreover, in CT volumetric analytic techniques, actual liver volume tends to be under or over estimated [12]. For these reasons, all such methods are biased. We have as far as
Fig. 2. A graphic showing the relation between the real liver volume values (Real Vol) obtained by the water displacement technique and the mean of the Cavalieri principle estimates of three performers (MTP) and standard deviations (SD).
Table 3 The mean CE of volume estimation of three performers on sagittal and horizontal section planes for each liver Liver 1
Section number Mean CE
Liver 2
Liver 3
Liver 4
Liver 5
Sagittal
Horizontal
Sagittal
Horizontal
Sagittal
Horizontal
Sagittal
Horizontal
Sagittal
Horizontal
19 0.016
14 0.021
20 0.016
14 0.017
21 0.015
17 0.016
20 0.021
14 0.022
16 0.019
15 0.024
MTP, mean of three performers’ estimates.
B. Sahin et al. / European Journal of Radiology 47 (2003) 164 /170
seen only one study on fetuses in vivo using the unbiased estimation technique for predicting the liver volume [26]. The volumes of five livers estimated by means of Cavalieri principle changed between 857 and 1592 cm3. The estimates of three performers also correlated well with each other. The total liver volume estimates obtained by hydrostatic weighing agreed well with the Cavalieri estimate. In the light of our findings, we can say that the Cavalieri estimator of volume can be applied within a few minutes to standard MR images to acquire liver volume with little interobserver variability at no additional cost. The mean CE for the liver estimates was 1.9%. The range of CE values changed from 1.5 to 2.4%, which are accepted as reasonable values for the volume estimates [1 /3,16]. Stereological methods provide some data to the researcher making appropriate changes on the sampling or estimating procedures. Therefore, the method presented here supplies a CE of measurements with each volume, giving as a percentage of the potential variability in any given volume measurement. When the CE of these measurements is large, it can generate obvious problems in accuracy and hence interpretation. These problems may arise if too few slices or too few points are taken into account. The observer is eligible to change the spacing of points in the grid or the number of slices available in any MRI study to obtain a reasonable CE value. It is important to note that the appropriate grid size and number of slices required for an object under study is determined at the beginning, there is no need to calculate the CE value for repeated sessions. This reduces the calculations by using just the third equation for point counting. In this case, the volume of the liver can be estimated simply using the 10th formula. In other words, we have to choose a point density in the grid so that 200 /300 points were counted for each studied MR section series to yield coefficients of error at 5% or less [3] as was seen in the present study. The given values of the liver volume are unbiased since the first slice hits the liver randomly followed by systematic sections with a known, fixed interval. The point counting is unbiased, since the set of systematic points is placed randomly on the MR images. However, there are some bias sources for the estimation of liver volume using MRI technique. Systematic sampling methods with non-random start for the sectioning induce a bias of remainder term, well-known in the context of numerical integration, which may decrease but never disappears as the number of sections is increased. It is essential to start from a random point and scan the whole liver. Failure to ensure a random start for the whole series of scan sections will cause bias, which cannot be corrected, no matter how precisely the section areas are measured [6]. Another factor that must be taken into consideration is liver motion due to
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respiration during scanning and partial voluming artifacts. To manage this problem, each cross sectional MR scanning should be taken during the inspiration phase, and the accuracy of the presented method depends on the same inspiratory effort in order to relocate the organ exactly as in the previous scanning [21]. Our purpose was to develop an easy way to evaluate the liver volume on ordinary MR scans without having to change the routine procedure for making such scans in every radiological center. It is not necessary to further standardize the MRI in order to determine the liver volume. The present evaluation of liver volume can be done on any complete set of MR images, where planescan distance and magnification factor is known, which already takes place onto MR images. The Cavalieri estimator of volume is notably independent of both shapes and orientation of the object. The method is inexpensive and fast, since point counting is carried out within 5/10 min per subject.
Acknowledgements We would like to thank to Dr Mark Lauer, Cleveland Clinical Foundation, Lerner Research Foundation, Department of Biomedical Engineering, Cleveland, USA for language revising of the manuscript.
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Unbiased estimation of the liver volume by the Cavalieri principle using magnetic resonance images Bu¨nyamin Sahin a,*, Mehmet Emirzeoglu a, Ahmet Uzun a, Lu¨tfi Incesu b, Yu¨ksel Bek c, Sait Bilgic a, Su¨leyman Kaplan d a
Department of Anatomy, Medical School, Ondokuz Mayis University, 55139 Samsun, Turkey Department of Radiology, Medical School, Ondokuz Mayis University, 55139 Samsun, Turkey c Department of Biostatistics, Medical School, Ondokuz Mayis University, 55139 Samsun, Turkey d Department of Histology and Embryology, Medical School, Ondokuz Mayis University, 55139 Samsun, Turkey b
Received 19 February 2002; received in revised form 10 May 2002; accepted 13 May 2002
Abstract Objective: It is often useful to know the exact volume of the liver, such as in monitoring the effects of a disease, treatment, dieting regime, training program or surgical application. Some non-invasive methodologies have been previously described which estimate the volume of the liver. However, these preliminary techniques need special software or skilled performers and they are not ideal for daily use in clinical practice. Here, we describe a simple, accurate and practical technique for estimating liver volume without changing the routine magnetic resonance imaging scanning procedure. Materials and methods: In this study, five normal livers, obtained from cadavers, were scanned by 0.5 T MR machine, in horizontal and sagittal planes. The consecutive sections, in 10 mm thickness, were used to estimate the whole volume of the liver by means of the Cavalieri principle. The volume estimations were done by three different performers to evaluate the reproducibility. Results: There are no statistical differences between the performers and real liver volumes (P /0.05). There is also high correlation between the estimates of performers and the real liver volume (r / 0.993). Conclusion: We conclude that the combination of MR imaging with the Cavalieri principle is a non-invasive, direct and unbiased technique that can be safely applied to estimate liver volume with a very moderate workload per individual. # 2002 Elsevier Science Ireland Ltd. All rights reserved. Keywords: Cavalieri principle; Liver volume; MRI; Stereology
1. Introduction Organ volumes can be estimated, without bias, using the Cavalieri principle of volume measurement by means of consecutive serial sections [1,2]. A simple approach to obtain such information was demonstrated over 300 years ago by the Italian mathematician Bonoventura Cavalieri. He pointed out that the volume of any object could be estimated from a set of twodimensional slices through the object, provided that they are parallel, separated by a known distance, and begin randomly within the object, criteria that are met by
* Corresponding author. Tel.: /90-362-457-6000x2262; fax: /90362-457-6041 E-mail address: [email protected] (B. Sahin).
standard magnetic resonance imaging (MRI) techniques [3]. The relevant method has recently been refined and applied in a series of studies involving invasive and noninvasive scanning techniques [1,2,4 /6]. Many recent studies have proven this estimator to be as accurate as digitization-based methods and to correlate closely with displacement volume measurements [7,8]. It is demonstrated that this simple, inexpensive stereological approach is well suited to rapid and an accurate volumetric calculations on the basis of standard MR scans of liver tumors, cirrhosis, hepatitis, hepatomegaly, liver transplantation, hepatectomy and other surgical applications of liver. It is also useful to examine structures that require assessment of changes in volume over time as an indicator of therapeutic effectiveness [9 /14]. The purpose of this study is to describe and adapt the relevant methods of MRI and stereology to estimate
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liver volume and compare the liver volume estimations with the real liver volume measurements. A reliable formula was provided for estimation of the coefficient of error (CE), which allow us to calculate the optimum number of sections required to attain a given precision for a particular scanning direction.
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if the upper right hand corner of the intersection of the cross lines representing them on the test system lies inside the object. After each superimposition, the number of test points hitting the structure of interest on the sections is counted, and the unbiased estimator becomes: est2 V t a=p (P1 P2 Pn ) cm3 ;
(3)
2. Estimation of volume by the Cavalieri principle It is known that the volume of the regular shaped objects (i.e. prism, cube or cylinder) can be estimated by the formula: V t a;
(1)
where (t ) is the height and (a ) is the base area of object. Similar to this principle, using the Cavalieri principle, an unbiased estimate of the volume of an object of arbitrary shape and size may be obtained efficiently and with a known precision [2,5,15,16]. The method requires sectioning the structure with a series of parallel planes. To avoid bias, the first section must be placed at a uniform and random position in a constant interval of length (t ) i.e. to start the scanning always at, for example, 1 cm from the right tip of the object will introduce an unknown amount of bias in general. Moreover, the series of sections must encompass the object entirely. The direction of cutting does not affect the unbiasedness property, but will affect the estimation precision in general [15]. Thus an unbiased estimate of volume can be obtained by multiplying the total area of the section cut surfaces through the structure on all the sections, i.e. est1 V t (a1 a2 an ) cm3 ;
(2)
where (a1/a2/ /an ) denote the section areas in cm2 and (t) is the sectioning interval in cm for the n consecutive sections. Some automatic machines or software can measure the contour of the object to obtain the cut surface area of section. However, several studies have shown that point counting techniques represent a more reliable and efficient approach than planimetric technique for obtaining the required cut surface areas of section [7,17]. The point counting grid, which has some point sets at distinct densities on a transparent sheet, can be used to estimate the cut surface area of the sections [3,15,16,18]. The point counting method consists of overlying each selected section with a regular grid of test points, which is randomly positioned. Test system orientation does not affect unbiasedness, but certain orientation improves the estimation precision. For this reason, the tests system should be superimposed on the section three times and the mean number of points hitting the objects should be used to estimate cut surface area of the section. A test point is a (/) shaped lines and it is said to hit the object
where (P1/P2/ /Pn ) denote the point counts and (a /p ) represents the area associated with each test point, corrected for any change of scale in the images as it is printed on the hardcopy films. The area of each section (ai ) is now estimated by [(a /p ) /Pi ]. The subscript 2 in est2 V indicates that a two-stage process, namely sectioning and point counting, estimates the volume. To avoid the over estimation due to MR scanning, adding a correction limit the bias of overprojection of the liver slices through the scan plan, the Cavalieri principle provides a simple estimator [19]. Thereby, the formula can be changed as follows: est2 V t a=p (P1 P2 Pn Pmax ) cm3 ;
(4)
where (Pmax) is the maximal number of points counted on a single scan plane of the subject. In the Cavalieri principle, the researcher obtains a data called CE to evaluate the reliability of the point density of the grids and sectioning intervals. The CE or relative standard error represent the precision of the volume estimate obtained using the Cavalieri principle. Since consecutive section cut surface areas are not independent quantities, conventional statistical formulae of CE cannot be applied to determine the variance of their sum. Gundersen and Jensen [1] developed a wellknown CE prediction formula for the Cavalieri estimation method. The new formula not only provides the CE but also gives information on the required number of slices and density of the point counting grid. For this purposes, the calculations are being done to estimate the CE and other data by means of the following consecutive formulas. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X ffi pffiffiffi Noise 0:0724 (b= a) n P:
(5)
Noise gives information on the complexity of the examined cut surface area of the specimen. Calculation of the CE using these equations requires knowledge of pffiffiffi the value of a dimensionless shape of coefficient (b= a); which is equivalent to the mean boundary length of the profiles divided by the square root of their mean area, and is a measure of the average shape of the profiles through the structure of interest on the sections [3 / 5,18].
B. Sahin et al. / European Journal of Radiology 47 (2003) 164 /170
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VarSRS
X n a
required to obtain an appropriate variation for section samples. A , B , and C are the total numerical values for the data in the related column of Table 1.
i 1
X X 3 (P2i Noise) 4 Pi Pi1
X
Pi Pi2
=
12:
(6)
This formula can be simplified with the aid of Table 1 as follows: X n VarSRS a i 1
[3 (A Noise) 4 BC]=12:
(7)
Whereas,/VarSRS (ani 1 a) indicates variance of total area in the systematic random sampling (SRS). These data give information on the sufficient section number
(8)
Total Var NoiseVarSRS ; X pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Total Var : CE P P P
(9)
CE is the last calculated value. The generally accepted highest limit of CE is 5% [1,16].
3. Materials and methods Five normal livers were obtained from cadavers stored in the Department of Anatomy. The MR scans were performed by 0.5 T machine (GE Signa, Milwaukee, WI). The scanning was done with the following
Table 1 An example of the estimate of volume and CE of one liver sectioned horizontally and its explanation Section no.
1st Sup
2nd Sup
3rd Sup
1 2 3 4 5 6 7 8 9 10 11 12 13 14
3 21 25 31 32 32 29 31 25 20 17 12 7 2
3 19 26 31 33 33 30 30 27 20 16 13 8 2
2 21 26 31 33 33 30 28 27 21 16 12 7 3
Total
Pi
Pi PI
Pi PI1
Pi Pi2
2.7 20.3 25.7 31.0 32.7 32.7 29.7 29.7 26.3 20.3 16.3 12.3 7.3 2.3
7.1 413.4 658.8 961.0 1067.1 1067.1 880.1 880.1 693.4 413.4 266.8 152.1 53.8 5.4
54.2 521.9 795.7 1012.7 1067.1 969.1 880.1 781.2 535.4 332.1 201.4 90.4 17.1 0.0
68.4 630.3 838.4 1012.7 969.1 969.1 781.2 603.2 430.1 250.8 119.8 28.8 0.0 0.0
289.3
7519.8 A
7258.6 B
6702 C
X pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Noise 0:0724 (b= a) n P 0:0724 5 289:3 14 23:03: VarSRS
X n a [3 (A Noise) 4 BC]=12 [3 (7519:8 23:03) 4 7258:66702]=12 13:16: i 1
Total Var NoiseVarSRS 23:0313:16 36:2: X pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi 36:2 Total Var P 0:021: CE P 289:3 P X Liver volume t a=p [( P) (1=2Pmax )] 1 3:642 (289:3 16) 995:48 cm3 : pffiffiffi pffiffiffi 0.0724/(b= a); the value showing the complexity of the section view. (b= a) is predicted as 5 for livers; Noise, gives information about the n complexity of the examined cut surface of the specimen; VarSRS (ai 1 a); variance of total area in SRS. This data gives information on the sufficient section number to obtain appropriate variation for section samples. A , B , C total numerical values for the data in the related column of the table; CE, the CE is the final data of the estimation. It is generally accepted that the CE of the study must be lower than 5%; a P , total number of the point hitting the sections cut surface area; 1/2Pmax, the half of the maximal number of points counted on a single scan plane of the subject; t , the section thickness. It was 1 cm for this study; a /p , the area of the each point on the point counting grid; Sup, superimposing.
B. Sahin et al. / European Journal of Radiology 47 (2003) 164 /170
parameters: spin-echo axial and sagittal repetition time (TR), 500; echo-time (TE), 27; field of view (FOV), 32; number of excitation (NEX), 3; Matrix, 256 /192; and mean scanning time, 9.28 min. Each specimen was scanned in both sagittal and horizontal planes and consecutive sections were taken in 10 mm slice thickness. The scan-plane thickness is constant due to the stepwise movement of the scanner. The images of the liver sections were printed on films in square frames of 8 /7.5 cm side length. The magnification ratios were between 0.24/1 and 0.29/1 cm for each individual livers and section plans of printed films. A square grid test system with d /0.5 cm between test points i.e. a/p /0.25 cm2 representing area per point (Fig. 1) were used to estimate the cut surface area of the slices. The representing area per point in the grid was corrected with the magnification of printed sections and thereby; the real area per point was calculated between 2.97 and 4.16 cm2 for livers in both section planes. The films were placed, in turn, on a light bow and the transparent square grid test system was superimposed uniformly, randomly covering the entire image frame. The superimposing of the test system was repeated three times for each image frame and the points hitting the liver section cut surface area were counted for each section. The mean of the three repeated number of points counted for any section was used to estimate the section cut surface area for sagittal and horizontal scan plans and then the volume of the entire liver was estimated by a modification of the 4th formula as: est2 V t a=p [P1 P2 Pn (1=2Pmax )];
(10)
where (1/2Pmax) is the half of the maximal number of points counted on a single scan plane of the subject.
Fig. 1. A liver slice sectioned in horizontal plan and a point counting grid superimposed on the MR scan.
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The CE for liver volume estimation was determined by means of the 7th /9th formulas. Three performers estimated the liver volumes sectioned in sagittal and horizontal planes using the same sets of printed sections of livers to compare the validity of the estimates according to the performers. Calculation of liver volume, CE of estimates and other related data were simply performed using Microsoft Excel as a spreadsheet. After initial setup and preparation of the formula in a small macro program, the point counts were entered for each scan and user did the calculations automatically. The exact liver volumes were measured using Archimedes’ principle, also known as ‘fluid displacement’ in a measuring cylinder. One-way ANOVA test was performed to compare statistically the repeatability and accuracy of the estimates of three performers and the real measured volume of the livers in both sectioning planes. The associations between the Cavalieri estimates of three performers and real volume of the livers were also analyzed by means of the Pearson correlation coefficient test.
4. Results Liver volumes of five specimens obtained from cadavers were measured by three observers and by means of the water displacement technique. The volumes of five livers estimated by means of Cavalieri principle changed between 857 and 1592 cm3. Cavalieri estimation results of the livers’ volume using sagittal and horizontal plan sections and water displacement technique are summarized in Table 2. The mean of liver volume determined by the Cavalieri estimator and water displacement technique was highly correlated (r / 0.993). The coefficient of correlation (r) for three performers’ liver volume estimates in sagittal direction with the real liver volume was estimated as 0.991, 0.997 and 0.996 for the performers 1, 2 and 3, respectively. The coefficient of correlation of three performers’ liver volume estimates in horizontal direction with the real liver volume was estimated as 0.981, 0.983 and 0.983 for the performers 1, 2 and 3, respectively. The means of three performers’ volume estimates in sagittal and horizontal directions are also correlated well with real liver volume and that values (r) were 0.095 and 0.983 for sagittal and horizontal section planes, respectively (Fig. 2). The Cavalieri estimates of liver volumes for each performer were compared by using one-way ANOVA test. The real liver volumes and the estimations of performers were also compared by using ANOVA technique. The estimates of three performers in sagittal and horizontal section plan and real volume values were not statistically different and significance of the test values (P ) was 0.996 and 1.000 for sagittal and
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Table 2 The volume of the livers (cm3) obtained by three performers on sagittal and horizontal section planes and water displacement technique and the correlation coefficients of real volume between performers’ estimates
Liver 1 Liver 2 Liver 3 Liver 4 Liver 5 r (estxreal)
Performer 1
Performer 2
Performer 3
Sagittal
Horizontal
Sagittal
Horizontal
Sagittal
Horizontal
968 902 1592 952 902 0.991
995 954 1544 950 856 0.981
936 935 1579 950 913 0.997
989 971 1530 901 864 0.983
900 897 1547 911 866 0.996
992 966 1551 923 857 0.983
MTPE
MSSPE
MHSPE
Real Vol
963.3 937.5 1557.0 931.0 876.2
934.5 911.2 1572.5 937.3 893.5
992 963.7 1541.5 924.6 858.9
914 937 1625 904 921
MTPE, mean of three performers’ estimate; MSSPE, mean of sagittal section plane estimates; MHSPE, mean of horizontal section plane estimates; Real Vol, measured volume of the livers by water displacement technique; r(estxreal), coefficient of correlation between three performers’ estimates and real volume of livers.
horizontal section planes, respectively. The estimates of three performers did not showed interobserver differences (P /0.05). The total liver volume estimates obtained by hydrostatic weighing agreed well with the Cavalieri estimate. The mean CE of volume estimates was 1.7 and 2.0% for sagittal and horizontal section planes, respectively. The details of CE values were summarized in Table 3.
5. Discussion and conclusion Precise information for liver volume is often required for the assessment of changes in its volume over time as an indicator of therapeutic effectiveness, evaluation of the liver in cirrhosis, before and after liver transplantation and other surgical applications of it [9 /14]. Any accurate information about the actual volume of organ cannot be obtained by routine physical examination of the liver. Its major role in clinical diagnosis remains just to define the characteristics of the lower edge of the organ. Liver volume has been shown to be useful in prognosis of patients with cirrhosis, but its measurement needs to be quantitative and reproducible which can be obtained only by imaging techniques [20]. Several studies have been conducted to estimate liver volume by means of imaging techniques [21 /25]. A brief literature review for both computed tomography (CT) and MRI measurements revealed that most of the studies in which liver volume was measured relied on
either observer or computer delineation of a region of interest and subsequent relevant pixel count, often on a limited number of slices and expressed as a ratio measurement. These methods require the original imaging data set and experienced performers in manipulating it. Such requirements, on a routine basis, are costly in both time and financial resources [3]. The studies comparing the real volume with ultrasound and CT scans have failed to show the superiority of imaging modalities in estimating liver volume. Moreover, in CT volumetric analytic techniques, actual liver volume tends to be under or over estimated [12]. For these reasons, all such methods are biased. We have as far as
Fig. 2. A graphic showing the relation between the real liver volume values (Real Vol) obtained by the water displacement technique and the mean of the Cavalieri principle estimates of three performers (MTP) and standard deviations (SD).
Table 3 The mean CE of volume estimation of three performers on sagittal and horizontal section planes for each liver Liver 1
Section number Mean CE
Liver 2
Liver 3
Liver 4
Liver 5
Sagittal
Horizontal
Sagittal
Horizontal
Sagittal
Horizontal
Sagittal
Horizontal
Sagittal
Horizontal
19 0.016
14 0.021
20 0.016
14 0.017
21 0.015
17 0.016
20 0.021
14 0.022
16 0.019
15 0.024
MTP, mean of three performers’ estimates.
B. Sahin et al. / European Journal of Radiology 47 (2003) 164 /170
seen only one study on fetuses in vivo using the unbiased estimation technique for predicting the liver volume [26]. The volumes of five livers estimated by means of Cavalieri principle changed between 857 and 1592 cm3. The estimates of three performers also correlated well with each other. The total liver volume estimates obtained by hydrostatic weighing agreed well with the Cavalieri estimate. In the light of our findings, we can say that the Cavalieri estimator of volume can be applied within a few minutes to standard MR images to acquire liver volume with little interobserver variability at no additional cost. The mean CE for the liver estimates was 1.9%. The range of CE values changed from 1.5 to 2.4%, which are accepted as reasonable values for the volume estimates [1 /3,16]. Stereological methods provide some data to the researcher making appropriate changes on the sampling or estimating procedures. Therefore, the method presented here supplies a CE of measurements with each volume, giving as a percentage of the potential variability in any given volume measurement. When the CE of these measurements is large, it can generate obvious problems in accuracy and hence interpretation. These problems may arise if too few slices or too few points are taken into account. The observer is eligible to change the spacing of points in the grid or the number of slices available in any MRI study to obtain a reasonable CE value. It is important to note that the appropriate grid size and number of slices required for an object under study is determined at the beginning, there is no need to calculate the CE value for repeated sessions. This reduces the calculations by using just the third equation for point counting. In this case, the volume of the liver can be estimated simply using the 10th formula. In other words, we have to choose a point density in the grid so that 200 /300 points were counted for each studied MR section series to yield coefficients of error at 5% or less [3] as was seen in the present study. The given values of the liver volume are unbiased since the first slice hits the liver randomly followed by systematic sections with a known, fixed interval. The point counting is unbiased, since the set of systematic points is placed randomly on the MR images. However, there are some bias sources for the estimation of liver volume using MRI technique. Systematic sampling methods with non-random start for the sectioning induce a bias of remainder term, well-known in the context of numerical integration, which may decrease but never disappears as the number of sections is increased. It is essential to start from a random point and scan the whole liver. Failure to ensure a random start for the whole series of scan sections will cause bias, which cannot be corrected, no matter how precisely the section areas are measured [6]. Another factor that must be taken into consideration is liver motion due to
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respiration during scanning and partial voluming artifacts. To manage this problem, each cross sectional MR scanning should be taken during the inspiration phase, and the accuracy of the presented method depends on the same inspiratory effort in order to relocate the organ exactly as in the previous scanning [21]. Our purpose was to develop an easy way to evaluate the liver volume on ordinary MR scans without having to change the routine procedure for making such scans in every radiological center. It is not necessary to further standardize the MRI in order to determine the liver volume. The present evaluation of liver volume can be done on any complete set of MR images, where planescan distance and magnification factor is known, which already takes place onto MR images. The Cavalieri estimator of volume is notably independent of both shapes and orientation of the object. The method is inexpensive and fast, since point counting is carried out within 5/10 min per subject.
Acknowledgements We would like to thank to Dr Mark Lauer, Cleveland Clinical Foundation, Lerner Research Foundation, Department of Biomedical Engineering, Cleveland, USA for language revising of the manuscript.
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