- Email: [email protected]
A practical method for estimating enclosed volumes using 3D ultrasound
EUROPEAN JOURNAL OF
ELSEVIER
European Journal of Ultrasound 3 (1996) 83-92
Scientific paper
A practical method for estimating enclosed volumes using 3D ultrasound Nils Thune a, Odd Helge Gilja b, Trygve Hausken b, Knut Matre *b aChristian Michelsen Research, N-5036 Fantoft, Bergen, Norway bMedical Department A, University of Bergen, Haukeland Hospital, N-5021 Bergen, Norway Received 13 April 1994; revision received 14 September 1995; accepted 20 September 1995
Abstract
Objective: This paper describes an algorithm which resulted in a practical method for estimating an enclosed volume using three-dimensional ultrasound. It was tested in vitro on thin-walled phantoms. Method." Data was acquired with a standard mechanical sector transducer which was tilted through a given angle (26 °, 51 ° or 88°) by a motorized mechanical holder. A total of 81 frames was captured, stored digitally on a Unix workstation, and scanned and converted into a three-dimensional volumetric data set. Planar contours were drawn manually to indicate an enclosing volume. From the given set of contours, a polyhedron was reconstructed and the volume calculated. Four different principles of manual contour indication; outside, inside, center, and leading edge to leading edge, were investigated. The volume estimation method was tested on a condom filled with water with varying volume. Results: The experiments demonstrated lowest error for the center edge contour indication, mean difference -0.47 ml ± 2.15 ml (mean ± 2 S.D.) in the volume range 1.15-45.90 ml. An inter-observer error of 0.60 ± 5.00 was found using the center edge contour indication. The error for leading edge contour identification was similar, 0.78 ml ± 2.65 ml. Both inside and outside contour identification gave much larger errors. Conclusion: It was concluded that this volume estimation method was accurate, contributed to low inter-observer error, and that the results from the different tracing indications demonstrated the necessity to standardize these procedures for thin-walled organs. Keywords: 3D ultrasound; Volume estimation; Volume reconstruction; Ultrasonography; Computer graphics
1. Introduction Volume estimation o f organs is an i m p o r t a n t tool in diagnostic procedures. Two-dimensional * Corresponding author, Tel.: ÷47 55972986; Fax: +47 55972950.
(2D) ultrasound methods for volume estimation are exposed to errors in geometric shape assumptions, cross-sectional image plane positioning, a n d measurements o f region-of-interest areas and diameters. By using three-dimensional (3D) ultrasound, m o r e accurate volume estimation m e t h o d s can be devised. Papers reporting the ongoing re-
0929-8266/96/$15.00 © 1996 Elsevier Science Ireland Ltd. All rights reserved SSDI 0929-8266(95)00136-F
84
N. Thune et al./ European Journal of Ultrasound 3 (1996) 83-92
search in volume estimation using 3D ultrasound differ in the 3D acquisition technology, the shape reconstruction, and the volume calculation methods used. Cook et al. (1980) presented a volume estimation algorithm based on a complete surface reconstruction (Fuchs et al. 1977) of the object. The volume was calculated by decomposing a polyhedral reconstruction of the object into slices, calculating the volume of each slice, and adding to obtain the total volume. This volume algorithm was used by King et al. (1990, 1991) in their work with a 3D ultrasound scanner system. This system consisted of a real-time conventional 2D scanner with a 3D acoustical spatial locater attached to it, enabling acquisition of arbitrarily oriented image slices. Another approach was taken by Brinkley et al. (1982). They described a method for reconstructing an organ and estimating its volume from arbitrarily oriented ultrasonic scans by summing the areas of parallel cross-sectional interpolated scans according to Simpson's rule. Their system also consisted of a 3D acoustical position locating system attached to a conventional 2D scanner. Watanabe (1982) described a volume calculation technique which did not require the surface of the organ to be reconstructed. The method is also valid for cross-sections which are not parallel and with region-of-interest which are multiplyconnected. This method was used by Basset et al. (1991) to estimate the volume of an organ from its ultrasonic cross-sectional images using either transverse or sagittal scanning. The aims of this study were to develop and validate a new algorithm for volume estimation by means of a phantom and a 3D mechanical ultrasound sector scanner, and to evaluate different tracing principles with respect to the accuracy of volume estimation. The algorithm has been tested earlier for accuracy in volume estimation in thickwalled and homogenous organs like the stomach and the kidney (Gilja et al. 1994, Gilja et al. 1995), but has not been presented in detail. When a thinwalled organ like the gall bladder and the urinary bladder is imaged by ultrasound, the real wall thickness is often less than the spatial resolution of the ultrasound scanner, resulting in only one echo
contour whose thickness highly depends on the spatial resolution and on the gain setting. In a thick-walled organ, echoes from both outer transition and inner transition are usually imaged. To test the influence of different contour tracings, four different principles of manual contour indication; outside, inside, center and leading edge were investigated, all at the same gain level. 2. Materials and methods
The volume estimation method presented is based on acquisition of 3D ultrasound data by tilting a sector transducer, scan conversion of the sector data into a regular volume of voxels, surface reconstruction of the object from planar contours of this regular volume, and volume calculation of the reconstructed polyhedral. 2.1. Data acquisition The equipment used was a commercially available mechanical ultrasound sector scanner, CFM 750 (Vingmed Sound A/S, Horten, Norway), with a mechanical holder attached to it. The holder contained a stepping motor, which was triggered by the frame pulse of the CFM 750 unit, and allowed tilting a standard sector scanner (3.25 MHz) through a given preset angle (26°, 51 ° or 88°). The sector scan plane was changed by the mechanical holder (the changed angle was perpendicular to the scan plane) and was rotated through the preset sector. The captured frames were stored in the local memory on the CFM 750. Depending on the current frame rate - 3 s were needed to capture 81 frames. If the captured frames were visually acceptable, the raw format of the sector data was transferred via a Macintosh IIx (Apple Computer Inc., Cupertino, California, USA) to a Unix workstation (Iris Indigo Elan, Silicon Graphics, Mountain View, California, USA), where all the data were processed using AVS (Application Visualization System from Advanced Visual Systems Inc., Waltham, Massachusetts, USA). Each captured volume contained 81 frames of sector data, where each frame consisted of a number of scan lines as shown in Fig. 1. The sector for-
85
N, Thune et al./ European Journal of Ultrasound 3 (1996) 83-92
Tilting sectorscanner
ReadEchoData ]
~
~ 128 beams
V
ScanConvert [ 81
..........
aw sectordata Visual Slicer ] v DrawContour [
~can conversionprocess)
z
//';x
I
r
ConcatContours ]
3D volumetricdata set
r ResampleContoursJ
+
I II
y
p,
Contoursto SurfaceGeometry ]
Fig. i. Scan conversion of 3D raw sector data into a 3D volumetric data set which 'resides' in 3D discrete voxel space, by intet~lating between the captured ultrasound beams. This voxel space is a 3D integer grid of unit volume cells, or elements called voxels, and is a regular volume reflecting the true sampling resolution of the transducer. Each voxel in the volume was interlmlated from its 8 closest neighbours.
mat consisted of lines (usually 128), where the number of lines depended on the current sector angle of the transducer. Each line consisted of 512 samples and each sample had an intensity value in the range 0-255. The 81 captured frames of sector data were scanned and converted into a 3D volumetric data set (Kaufman 1991), which 'resides' in 3D discrete voxel space, by interpolating between the captured ultrasound lines. This space is a 3D integer grid of unit volume cells, or elements called voxels, and is a regular volume reflecting the sampling resolution of the transducer (Fig. 1). Each voxel in the volume was interpolated from its 8 closest neighbours using tri-linear interpolation. 2.2. Manual contour indication
The object was reconstructed based on a set of planar contours which were manually drawn in the volumetric data set. To aid in reconstructing the
GeometryViewer
] ComputePolyhedralVolume ]
I
Fi& 2. The pipefine with the different modules which were used for manual contour indication, polyhedron reconstruction, volume calculation and visualization of reconstructed objects.
correct object, the volume data were visualized, allowing user interaction during the process. A pipeline of the different modules used for manual contour indication, polyhedron reconstruction, volume calculation, and visualization of the reconstructed object is shown in Fig. 2. The following description of this pipeline refers to this figure. First, the volume of sector data from the CFM 750 system was fetched from a file and scan convetted into a regular volume (Fig. 1) by the Read Echo Data and Scan Convert modules. The Visual Slicer module (Fig. 3) displayed all slices along either one of the three major axis of the regular volume according to user choice. The extent of the object in the 3D data set could now easily be identiffed and the user decided which contours were necessary to draw in order to reconstruct the object properly. The following step was to draw contours manually in the image slices containing the object. This was done by first selecting one of the
86
N. Thune et al./ European Journal of Ultrasound 3 (1996) 83-92
Fig. 3. The Visual Slicer module showing all the slices of a condom along the z-axis. The user chose a major axis and each slice along this axis in the regular volume was then shown in this window. The user could select any of these slices for further viewing and contour drawing in the Draw Contour module. Any contours the user had drawn was superimposed on the respective slices in the Visual Slicer window for visual feedback.
slices in the Visual Slicer window and then displaying this slice with higher resolution in the Draw Contour window. In this window, the user outlined the contour on top of the chosen image using a mouse. If the drawn contour was acceptable to the user, the contour advanced to the Concat Contours module which concatenated the contours into a linked list, and next advanced to the Resample Contours module. Finally, a surface was reconstructed in the Contours to Surface Geometry module and the volume was computed in the Compute Polyhedral Volume module. This was done interactively each time the
user drew a new contour or edited an existing one. The number of tracings for a specific volume was 10-12. This procedure took approximately 2 rain for one volume estimation and - 5 rain including data transfer. In the Geometry Viewer module (Fig. 4), the reconstructed object was visualized concurrently, and allowed the user to rotate, translate or scale the object in real time. This visual feedback permitted the user to comprehend the geometrical structures of the object more easily. This was important for being certain that the volume was closed at its extremities and to identify tracing errors.
N. Thane et al./ European Journal of Ultrasound 3 (1996) 83-92
87
Fig. 4. The GeometryViewermoduledisplayinga reconstructedmodelof a water-filledcondomwith the calculatedvolumeindicated.
2.3. Polyhedron contours
reconstruction
from
planar
We were given a set of planar contours of an object and our task was to create a polyhedron; a set of polygons (usually triangles) which described the object. In general, there was (and is) no way to join contours together correctly since all the connectivity information is limited to the contour plane and there exists no information about the relationship between adjacent contours. Several authors have addressed this problem of reconstructing a 3D solid from serial cross sections. Fuchs et al. (1977) reconstructed the surface between contiguous serial contours by minimizing the surface area. Such an optimal surface cannot be found using only local decision making and therefore efficient methods of global graph sear-
ching were needed. The method described by Fuchs et al. required computational steps in the order of 2 mn[log2m], (m < n) (where m and n were the number of distinct contour points on two neighbouring contours). However, if geometrical constraints were known about the object to be modelled, then heuristic methods of triangulation could be used to reduce the number of steps to
(re+n). We based our work on the heuristic method described by Ganapathy and Dennehy (1982), in which they applied inter-contour coherence. The reconstruction of the polyhedron from the set of contour data was accomplished in three separate steps: (1) Resampling of the contours. (2) Choosing the starting point for each contour.
88
N. Thune et al./ European Journal of Ultrasound 3 (1996) 83-92
(3) Triangulation of contour pairs. Step 1: The first step was to resample each contour so that it consisted of vertices which were evenly spaced (the vertex spacing was set by the user and could be as small as the sampling resolution of the transducer). This resarapling step was necessary because the number of vertices and their individual spacing along the contour depended on the speed by which the user moved the drawing device (in our case a mouse). The resampling was done continuously and hence the user verified that the chosen vertex spacing was adequate for representing the drawn contour. Step 2: The second step was to find a starting point on each contour from where to begin the process of triangulation. If neighbouring contours were known to have similar shape, a simple solution was to select that point on each contour which had the minimum x value. Step 3: The third step was the triangulation of contour pairs. The heuristic rule in the method presented by Ganapathy and Dennehy was based on neighbouring pairs of contours having similar size and shape. The triangulation process is described in detail in the Appendix.
2.4. Experimental setup A phantom of known volume was constructed using a condom. The condom was filled with water ranging in volume from 1.15 ml to 45.90 ml and was placed in a water-filled tank padded with foam rubber. The water was heated to a temperature of - 38°C to provide a medium to which the ultrasound system was calibrated (1560 m/s in tissue at 37°C). The condom was initially filled with 45.90 ml water. After imaging the condom, water was extracted, based on a pre-determined scheme, using a syringe and a catheter. This procedure was repeated 16 times giving 17 samples of each series. The gain of the scanner was kept low and constant to avoid saturation at the echo interfaces. Eighty-one frames were captured through an angle of 51 ° and scan converted into a regular volume of voxels. Then, contours were drawn manually using a principle of either outside, inside,
Tram
/
/
! Outside
/C
Inside
Center
J Leading edge
Tracing pnnciple i
Surface wall of condom
Fig. 5. The four principles of contour indication which were investigated after data aequsition with 3D ultrasound.
center or leading edge contour indication (see Fig. 5) and the volume computed as described earlier. The thickness of the wall in a condom is very small, and data acquisition using ultrasound introduced an artifact of thin condom walls being depicted as thick, even at low gain. To sidestep this problem, the leading edge relative to the direction of sound transmission, as illustrated in Fig. 5, is normally used. All the volume measurements were performed by one person (NT), except for the central contour indication principle where inter-observer error was measured between two observers (NT and OHG).
2.5. Statistical methods The correlation coefficient determined by linear regression was used as a measure of association between the true volume (TV) and estimated volume (EV), while limits of agreement were determined as suggested by Bland and Altman (1986). Accuracy of the method was defined by how closely EV agreed with TV and was expressed as the mean of volume differences (d). Mean difference is a measure of bias. Assuming normal distribution of d, 95% of the differences were within the limits given by (d ± 2S.D.), defining the limits of agreement. The correlation coefficient between volume differ-
N. Thune et al. / European Journal of Ultrasound 3 (1996) 83-92
ences and volume means was given as: r2 = Corr ( E V - TV, (EV + TV)/2), and was calculated to test for dependence. Inter-observer variation was described by the differences between the two observers, against the corresponding mean for each volume estimate (Brennan and Silman 1992). The inter-observer error was reported as the overall mean percentage of the inter-observer variation ± the standard deviation of the interobserver variation. 3. Results
The experimental results are summarized in Table 1. For each of the four tracing principles, the first part of the table reports the linear regression coefficients between EV and TV while the second part describes the limits of agreement according to Bland and Altman (1986). The result of tracing the outside surface of the condom gave good correlation between EV and TV (rl = 0.998). However, the EV were consistently over-estimated. This is also reflected by a high mean difference (d = 7.14) and wide limits of agreement (2S.D. = 8.30). This over-estimation was expected since the true surface is not followed by this outside surface tracing technique. Furthermore, the over-estimation seemed to increase as the true volume increased. This was verified by a
89
significant correlation (r 2 = 0.971) between volume differences (EV - TV) and mean volumes ((EV + TV)/2). EV and TV show good correlation (rl = 0.996) also when tracing the inside surface of the condom. However, this tracing principle resulted in increasing under-estimation with increasing volumes. The negative mean difference (d =-5.59) together with high negative correlation (r 2 = -0.971) between volume differences and mean volumes confirmed this under-estimation. This could be expected since the condom wall was depicted much to thick due to artifacts during the signal acquisition. Accordingly, the tracing did not follow the true inside of the condom wall. Centering the tracing between the inner and outer wall contour demonstrated high correlation (rl = 0.998) and displayed EV in close agreement with TV. This was confirmed by low mean difference (d =-0.47). No over- or under-estimation was apparent which also was reflected by the low correlation (rz = 0.493) between volume differences ( E V - TV) and mean volumes ((EV + TV)/2). The final tracing principle, leading edge,
50
4O
Table 1 Results of in vitro validation of 3D ultrasound in volume estimation of phantoms
k5 3O
Tracing principle (n = 17)
E
Linear Agreement regression (rl) d (ml) ± 2 S.D. r 2
(ml) Outside Inside Leading edge Center (NT) Center (OHG)
0.998 0.996 0.996 0.998 0.998
7.14 -5.59 0.78 -0.47 0.58
8.30 5.85 2.65 2.15 1.94
./
./
E~ C
20 ~D
0.971 -0.971 0.334 0.493 0.307
~96J l
0 Tracing principle details, see text; NT and OHG, independent observers; rl, linear correlation coefficient; d, the mean difference between measured and true volumes, ± 2S.D. are the limits of agreement according to Bland and Airman; r 2, the linear regression coefficient between d and the average of the measured and true volumes.
10
l
20
l
30
J__
40
50
True volume (ml) Fig. 6. Scatter plot illustrating the correlation between estimated and true volumes when using the leading edge contour tracing principle. The line of identity is shown.
90
N. Thune et al./ European Journal of Ultrasound 3 (1996) 83-92 4 mean+2SD
3 2 1 I
0
)•
-1
s
mean
-2
2SD
-3 4
I ...... ~
0
10
h
20
30
±
40
J
50
60
(Estimated volume + True volume)/2 (ml}
Fig. 7. Mean difference and limits of agreement when using the leading edge contour tracing principle.
was also included• This method resulted in a high correlation (rl = 0.996) between EV and TV, as seen in Fig. 6. Low mean difference (d = 0.78) and low limits of agreement (2S.D. = 2.65) confirm this method of tracing as being satisfactory regarding accuracy (Fig. 7). There was no apparent over- or under-estimation, and a low correlation (r 2 -- 0.344) between volume differences (EV - TV) and mean volumes ((EV + TV)/2) was found. Inter-observer error between observers NT and O H G was calculated to be 0.6% 4- 5.0% when using the center edge contour tracing principle. Both tracings had low mean difference (d = -0.47) and (d = 0.58), and limits of agreement (2S.D. = 2.15) and (2S.D. = 1.94), respectively. 4. Discussion The presented volume estimation method based on 3D-ultrasound appeared to be accurate and practical. The best result were obtained with the leading edge and center edge contour tracing principles. When the water volumes inside the condom diminished, the condom was deformed and 'wrinkled'• In these cases, the surface was difficult to trace which contributed to the estimation errors. The small volumes are, however, included in
the calculations as we followed the pre-determined setup for water extraction from the condom. In our condom experiments, high image contrast was displayed, which is often not the case in low contrast clinical imaging of focal lesions. The thickness of the imaged condom surface was due to the spatial resolution characteristics of the transducer. In clinical imaging, this kind of error is normally a smaller problem but can be present when imaging thin-walled organs, like the gall bladder. Compared to Cook et al. (1980), who reported phantom studies with an accuracy in volume estimation within 3%, our method competes well• This is also true for other comparable methods that have been reported, such as described by Brinkley et al. (1982). They utilized a system with an accuracy in volume determination of balloons to within 1.8%, of kidneys to within 5.1%, and of left ventricular moulds to within to 5.9% of the true volume. King et al. (1990) reported in their work a 3D ultrasound system with accuracy in volume estimation of 1.6% (mean error). Another comparable method, described by Basset et al. (1991), tested a technique on phantoms of various shapes and volumes made of agar gel or water-filled balloons and reported a maximum volume estimation error less than 10% and a maximum variability less than 2%. Again our method competes well. High accuracy and precision in volume estimation with the ability to detect small changes in volumes, makes 3 D ultrasound promising as a clinical diagnostic tool. For the management of cancer, it will be possible to evaluate the growth or regression of the tumor in different organs. Also, regarding the fol!0w-up of benign structures, as tumors and cysts In the liver, kidneys, and pancreas, 3D ultrasound~may be of clinical value. By vizualization in 3 dimensions, it is possible to evaluate the exact position Of a focal lesion, and its demarcation from the surrounding tissue, and in particular its relation to blood vessels. This is of major significance in surgical decision-making and pre-operatwe planning. Accurate locahsatlon of gallstones by means of a 3D ultrasound system may improve therapies like lithotripsy. Furthermore, exact volume estimation of organs before •
.
•
.
\
\
•
.
N. Thaneet al./ European Journal of Ultrasound3 (1996) 83-92
and after physiological or medical intervention facilitates dynamic studies like gastric and gall bladder emptying. From these in vitro experiments, it appeared that the tracing principle was of utmost importance, implying that standardization of the method of tracing for volume comparison is necessary. Furthermore, if possible, the gain setting should be kept low to reduce artifacts at tissue transitions. Even though the presented method demonstrated high accuracy in volume estimation, improvements are possible. Contours were represented with a constant distance between vertices. To improve the shape accuracy, the spacing should be dependent upon the curvature of the contour. Also, methods for automatic edge detection should be investigated to aid the user in improving the consistency of contour indication and in saving time doing the manual outlining. In addition to the tracing error, other sources of error inelude mechanical inaccuracy in transducer movement and resampling inaccuracy. We conelude that this 3D ultrasound system yields high accuracy in volume estimation of phantoms and contributes to low inter-observer error.
91
Starting points
t
i.. [ . . . f ' Y
lyp/C,~I
/+1
vc,.,
Fig. 8. A step in the triangulation of a contour pair C i and Ci+ t. Tiles were added to the polyhedron by proceeding along contour C i and Ci+ I in such a manner that the absolute difference in travelled distance along each contour was minimized at all times.
lute difference between the normalized travelled distance along each contour was minimized at all times. The heuristic rule for adding the next triangle (tile) was as follows:
Begin Acknowledgements eo= This work was supported by the Norwegian Research Council for Science and Technology (NTNF). The authors would like to thank the Institute of Biomedical Engineering, Faculty of Medicine, University of Trondheim, who have designed and constructed the 3D transducer equipment we used. We also thank Vingmed Sound A/S, Horten, Norway, for supporting this project.
PI = V f i+l
Appendix The third step in the reconstruction of a polyhedron was the triangulation of contour pairs. The heuristic rule in the method presented by Ganapathy and Dennehy (1980) was based on neighbouring pairs of contours having similar size and shape appearance. A step in the triangulation process is shown in Fig. 8 and illustrates how tiles were added to the polyhedron by proceeding along contour Ci and Ci+l in such a way that the abso-
Polyhedron
Planar surface s with surface area As,
Fig. 9. This diagram shows how the volume of a polyhedron is calculated by a summation over all surface tiles according to Goldman 1991.
92
N.
Thune et
al./ European Journal of Ultrasound 3 (1996) 83-92
i f I @kCi "t" ~Ok,k+ Ci I __ ¢I~/Ci+lI < I*Ci+l..I.,,Ci+l Wl, l+l -- ¢~Ci k I
Y = lie [ [~jPo, j • S j ) l S j * [~Pk, j x Pk+l,j] [11
then
where (.) is the scalar product and (x) is the cross product. It should be noted that the equation for the surface area (2) and polyhedron volume (4) are also valid for non-convex shapes.
P 2 - - VCkk~l
(4)
else Refereaces O_ _-- l/Ci+ 1 ~2 • /+1
End where P0, PI, P2 (see Fig. 9) were vertices on the new triangle to be added; Vci, V~i+1 were the current visited vertices along contour Ci and Ci+l; o c g o ci+l were the normalized travelled distances so far along contour Ci and Ci+l; and ek.k+PC"¢Ci+ll.l+lwere the normalized lengths of the next contour segments along contour C i and Ci+l. Volume of a Polyhedron The volume v of a polyhedron (Goldman 1991) could now be expressed as
(1) there S0,...Sn were planar polygonal faces on the polyhedron, Qj was any point on Sj; Nj was a outward pointing unit vector normal to Sj; and Asj, was the surface area of Sj (see Fig. 9). If P0,i,...Pmj were vertices of Sj oriented counter clockwise with respect to the outward pointing normal of Sj, then the surface area Asj of Sj could be expressed as Asj = 1/2 I Nj " I F
Pkj X Pk+l.j]I
(2)
where
the first and last vertex coincides, P0d. The point Qj could be chosen to be P0j and the normal vector could be computed as Pm+lj =
I(e~,j - e04) x (e2,j - e0,/)l Nj = I(Pi,J Po,j) x (P2,j - P0,j)l
(3)
Care had to be taken to ensure that the points chosen from the surface Sj to compute the normal vector did indeed give the outward pointing normal of Sj. The following equation then represents the volume of a polyhedron
Basset O, Gimenez G, Mestas JL, Cathignol D, Devonec M. Volume measurement by ultrasonic transverse or sagittal cross-sectioual scanning. Ultrasound Med Biol 1991; 17: 291-296. Bland JM, Altman DG. Statistical methods for assessing agreement between two methods of clinical measurements. Lancet 1986; i: 307-310. Brennan P, Siiman A. Statistical methods for assessing observer variability in clinical measures. Br Meal J 1992; 304: 1491-1494. Brinkley JF, Muramatsu SK, McCallum WD, Popp RL. In vitro evaluation of an ultrasonic three-dimensional imaging and volume system. Ultrason Imaging 1982; 4: 126-139. Cook LT, Cook PN, Lee KR, Batnitzky S, Wong BYS, Fritz SL, Ophir J, Dwyer SJ, Bigongiari LR, Templeton AW. An algorithm for volume estimation based on polyhedral approximation. IEEE Trans Biomed Eng 1980; 27: 493-500. Fuchs H, Kedem ZM, Uselton SP. Optimal surface reconstruction from planar contours. Commun ACM 1977; 20: 693-702. Ganapathy S, Dennehy TG. A new general triangulation method for planar contours. Comput Graph 1982; 16: 69-75. Giija OH, Thune N, Matre K, Hausken T, ~iegaard S, Berstad A. In vitro evaluation of three-dimensional ultrasonography in volume estimation of abdominal organs. Ultrasound Med Biol 1994; 20: 157-165. Gilja OH, Smievoll AI, Thune N, Matre K, Hausken T, ~legaard S, Berstad A. In vivo comparison of 3D ultrasonography and magnetic resonance imaging in volume estimation of human kidneys. Ultrasound Med Biol 1995; 21: 25-32. Goldman RN. Area of planar polygons and volume of polyhedra. In: Arvo J, ed. Graphics Gems II. New York: Academic Press Inc., 1991; 170-171. Kaufman A. Introduction to volume visualization, ln: Kaufman A, ed. Volume Visualization. IEEE Computer Society Press Tutorial, 1991; 1-18. King DL, King DL Jr, Shao MYC. Three-dimensional spatial registration and interactive display of position and orientation of real-time ultrasound images. J Ultrasound Med 1990; 9: 525-532. King DL, King DL Jr, Shao MYC. Evaluation of in vitro measurement accuracy of a three-dimensional ultrasound scanner. J Ultrasound Med 1991; 10: 77-82. Watanabe Y. A method for volume estimation by using vector areas and centroids of serial cross sections. IEEE Trans Biomed Eng 1982; 29: 202-205.
ELSEVIER
European Journal of Ultrasound 3 (1996) 83-92
Scientific paper
A practical method for estimating enclosed volumes using 3D ultrasound Nils Thune a, Odd Helge Gilja b, Trygve Hausken b, Knut Matre *b aChristian Michelsen Research, N-5036 Fantoft, Bergen, Norway bMedical Department A, University of Bergen, Haukeland Hospital, N-5021 Bergen, Norway Received 13 April 1994; revision received 14 September 1995; accepted 20 September 1995
Abstract
Objective: This paper describes an algorithm which resulted in a practical method for estimating an enclosed volume using three-dimensional ultrasound. It was tested in vitro on thin-walled phantoms. Method." Data was acquired with a standard mechanical sector transducer which was tilted through a given angle (26 °, 51 ° or 88°) by a motorized mechanical holder. A total of 81 frames was captured, stored digitally on a Unix workstation, and scanned and converted into a three-dimensional volumetric data set. Planar contours were drawn manually to indicate an enclosing volume. From the given set of contours, a polyhedron was reconstructed and the volume calculated. Four different principles of manual contour indication; outside, inside, center, and leading edge to leading edge, were investigated. The volume estimation method was tested on a condom filled with water with varying volume. Results: The experiments demonstrated lowest error for the center edge contour indication, mean difference -0.47 ml ± 2.15 ml (mean ± 2 S.D.) in the volume range 1.15-45.90 ml. An inter-observer error of 0.60 ± 5.00 was found using the center edge contour indication. The error for leading edge contour identification was similar, 0.78 ml ± 2.65 ml. Both inside and outside contour identification gave much larger errors. Conclusion: It was concluded that this volume estimation method was accurate, contributed to low inter-observer error, and that the results from the different tracing indications demonstrated the necessity to standardize these procedures for thin-walled organs. Keywords: 3D ultrasound; Volume estimation; Volume reconstruction; Ultrasonography; Computer graphics
1. Introduction Volume estimation o f organs is an i m p o r t a n t tool in diagnostic procedures. Two-dimensional * Corresponding author, Tel.: ÷47 55972986; Fax: +47 55972950.
(2D) ultrasound methods for volume estimation are exposed to errors in geometric shape assumptions, cross-sectional image plane positioning, a n d measurements o f region-of-interest areas and diameters. By using three-dimensional (3D) ultrasound, m o r e accurate volume estimation m e t h o d s can be devised. Papers reporting the ongoing re-
0929-8266/96/$15.00 © 1996 Elsevier Science Ireland Ltd. All rights reserved SSDI 0929-8266(95)00136-F
84
N. Thune et al./ European Journal of Ultrasound 3 (1996) 83-92
search in volume estimation using 3D ultrasound differ in the 3D acquisition technology, the shape reconstruction, and the volume calculation methods used. Cook et al. (1980) presented a volume estimation algorithm based on a complete surface reconstruction (Fuchs et al. 1977) of the object. The volume was calculated by decomposing a polyhedral reconstruction of the object into slices, calculating the volume of each slice, and adding to obtain the total volume. This volume algorithm was used by King et al. (1990, 1991) in their work with a 3D ultrasound scanner system. This system consisted of a real-time conventional 2D scanner with a 3D acoustical spatial locater attached to it, enabling acquisition of arbitrarily oriented image slices. Another approach was taken by Brinkley et al. (1982). They described a method for reconstructing an organ and estimating its volume from arbitrarily oriented ultrasonic scans by summing the areas of parallel cross-sectional interpolated scans according to Simpson's rule. Their system also consisted of a 3D acoustical position locating system attached to a conventional 2D scanner. Watanabe (1982) described a volume calculation technique which did not require the surface of the organ to be reconstructed. The method is also valid for cross-sections which are not parallel and with region-of-interest which are multiplyconnected. This method was used by Basset et al. (1991) to estimate the volume of an organ from its ultrasonic cross-sectional images using either transverse or sagittal scanning. The aims of this study were to develop and validate a new algorithm for volume estimation by means of a phantom and a 3D mechanical ultrasound sector scanner, and to evaluate different tracing principles with respect to the accuracy of volume estimation. The algorithm has been tested earlier for accuracy in volume estimation in thickwalled and homogenous organs like the stomach and the kidney (Gilja et al. 1994, Gilja et al. 1995), but has not been presented in detail. When a thinwalled organ like the gall bladder and the urinary bladder is imaged by ultrasound, the real wall thickness is often less than the spatial resolution of the ultrasound scanner, resulting in only one echo
contour whose thickness highly depends on the spatial resolution and on the gain setting. In a thick-walled organ, echoes from both outer transition and inner transition are usually imaged. To test the influence of different contour tracings, four different principles of manual contour indication; outside, inside, center and leading edge were investigated, all at the same gain level. 2. Materials and methods
The volume estimation method presented is based on acquisition of 3D ultrasound data by tilting a sector transducer, scan conversion of the sector data into a regular volume of voxels, surface reconstruction of the object from planar contours of this regular volume, and volume calculation of the reconstructed polyhedral. 2.1. Data acquisition The equipment used was a commercially available mechanical ultrasound sector scanner, CFM 750 (Vingmed Sound A/S, Horten, Norway), with a mechanical holder attached to it. The holder contained a stepping motor, which was triggered by the frame pulse of the CFM 750 unit, and allowed tilting a standard sector scanner (3.25 MHz) through a given preset angle (26°, 51 ° or 88°). The sector scan plane was changed by the mechanical holder (the changed angle was perpendicular to the scan plane) and was rotated through the preset sector. The captured frames were stored in the local memory on the CFM 750. Depending on the current frame rate - 3 s were needed to capture 81 frames. If the captured frames were visually acceptable, the raw format of the sector data was transferred via a Macintosh IIx (Apple Computer Inc., Cupertino, California, USA) to a Unix workstation (Iris Indigo Elan, Silicon Graphics, Mountain View, California, USA), where all the data were processed using AVS (Application Visualization System from Advanced Visual Systems Inc., Waltham, Massachusetts, USA). Each captured volume contained 81 frames of sector data, where each frame consisted of a number of scan lines as shown in Fig. 1. The sector for-
85
N, Thune et al./ European Journal of Ultrasound 3 (1996) 83-92
Tilting sectorscanner
ReadEchoData ]
~
~ 128 beams
V
ScanConvert [ 81
..........
aw sectordata Visual Slicer ] v DrawContour [
~can conversionprocess)
z
//';x
I
r
ConcatContours ]
3D volumetricdata set
r ResampleContoursJ
+
I II
y
p,
Contoursto SurfaceGeometry ]
Fig. i. Scan conversion of 3D raw sector data into a 3D volumetric data set which 'resides' in 3D discrete voxel space, by intet~lating between the captured ultrasound beams. This voxel space is a 3D integer grid of unit volume cells, or elements called voxels, and is a regular volume reflecting the true sampling resolution of the transducer. Each voxel in the volume was interlmlated from its 8 closest neighbours.
mat consisted of lines (usually 128), where the number of lines depended on the current sector angle of the transducer. Each line consisted of 512 samples and each sample had an intensity value in the range 0-255. The 81 captured frames of sector data were scanned and converted into a 3D volumetric data set (Kaufman 1991), which 'resides' in 3D discrete voxel space, by interpolating between the captured ultrasound lines. This space is a 3D integer grid of unit volume cells, or elements called voxels, and is a regular volume reflecting the sampling resolution of the transducer (Fig. 1). Each voxel in the volume was interpolated from its 8 closest neighbours using tri-linear interpolation. 2.2. Manual contour indication
The object was reconstructed based on a set of planar contours which were manually drawn in the volumetric data set. To aid in reconstructing the
GeometryViewer
] ComputePolyhedralVolume ]
I
Fi& 2. The pipefine with the different modules which were used for manual contour indication, polyhedron reconstruction, volume calculation and visualization of reconstructed objects.
correct object, the volume data were visualized, allowing user interaction during the process. A pipeline of the different modules used for manual contour indication, polyhedron reconstruction, volume calculation, and visualization of the reconstructed object is shown in Fig. 2. The following description of this pipeline refers to this figure. First, the volume of sector data from the CFM 750 system was fetched from a file and scan convetted into a regular volume (Fig. 1) by the Read Echo Data and Scan Convert modules. The Visual Slicer module (Fig. 3) displayed all slices along either one of the three major axis of the regular volume according to user choice. The extent of the object in the 3D data set could now easily be identiffed and the user decided which contours were necessary to draw in order to reconstruct the object properly. The following step was to draw contours manually in the image slices containing the object. This was done by first selecting one of the
86
N. Thune et al./ European Journal of Ultrasound 3 (1996) 83-92
Fig. 3. The Visual Slicer module showing all the slices of a condom along the z-axis. The user chose a major axis and each slice along this axis in the regular volume was then shown in this window. The user could select any of these slices for further viewing and contour drawing in the Draw Contour module. Any contours the user had drawn was superimposed on the respective slices in the Visual Slicer window for visual feedback.
slices in the Visual Slicer window and then displaying this slice with higher resolution in the Draw Contour window. In this window, the user outlined the contour on top of the chosen image using a mouse. If the drawn contour was acceptable to the user, the contour advanced to the Concat Contours module which concatenated the contours into a linked list, and next advanced to the Resample Contours module. Finally, a surface was reconstructed in the Contours to Surface Geometry module and the volume was computed in the Compute Polyhedral Volume module. This was done interactively each time the
user drew a new contour or edited an existing one. The number of tracings for a specific volume was 10-12. This procedure took approximately 2 rain for one volume estimation and - 5 rain including data transfer. In the Geometry Viewer module (Fig. 4), the reconstructed object was visualized concurrently, and allowed the user to rotate, translate or scale the object in real time. This visual feedback permitted the user to comprehend the geometrical structures of the object more easily. This was important for being certain that the volume was closed at its extremities and to identify tracing errors.
N. Thane et al./ European Journal of Ultrasound 3 (1996) 83-92
87
Fig. 4. The GeometryViewermoduledisplayinga reconstructedmodelof a water-filledcondomwith the calculatedvolumeindicated.
2.3. Polyhedron contours
reconstruction
from
planar
We were given a set of planar contours of an object and our task was to create a polyhedron; a set of polygons (usually triangles) which described the object. In general, there was (and is) no way to join contours together correctly since all the connectivity information is limited to the contour plane and there exists no information about the relationship between adjacent contours. Several authors have addressed this problem of reconstructing a 3D solid from serial cross sections. Fuchs et al. (1977) reconstructed the surface between contiguous serial contours by minimizing the surface area. Such an optimal surface cannot be found using only local decision making and therefore efficient methods of global graph sear-
ching were needed. The method described by Fuchs et al. required computational steps in the order of 2 mn[log2m], (m < n) (where m and n were the number of distinct contour points on two neighbouring contours). However, if geometrical constraints were known about the object to be modelled, then heuristic methods of triangulation could be used to reduce the number of steps to
(re+n). We based our work on the heuristic method described by Ganapathy and Dennehy (1982), in which they applied inter-contour coherence. The reconstruction of the polyhedron from the set of contour data was accomplished in three separate steps: (1) Resampling of the contours. (2) Choosing the starting point for each contour.
88
N. Thune et al./ European Journal of Ultrasound 3 (1996) 83-92
(3) Triangulation of contour pairs. Step 1: The first step was to resample each contour so that it consisted of vertices which were evenly spaced (the vertex spacing was set by the user and could be as small as the sampling resolution of the transducer). This resarapling step was necessary because the number of vertices and their individual spacing along the contour depended on the speed by which the user moved the drawing device (in our case a mouse). The resampling was done continuously and hence the user verified that the chosen vertex spacing was adequate for representing the drawn contour. Step 2: The second step was to find a starting point on each contour from where to begin the process of triangulation. If neighbouring contours were known to have similar shape, a simple solution was to select that point on each contour which had the minimum x value. Step 3: The third step was the triangulation of contour pairs. The heuristic rule in the method presented by Ganapathy and Dennehy was based on neighbouring pairs of contours having similar size and shape. The triangulation process is described in detail in the Appendix.
2.4. Experimental setup A phantom of known volume was constructed using a condom. The condom was filled with water ranging in volume from 1.15 ml to 45.90 ml and was placed in a water-filled tank padded with foam rubber. The water was heated to a temperature of - 38°C to provide a medium to which the ultrasound system was calibrated (1560 m/s in tissue at 37°C). The condom was initially filled with 45.90 ml water. After imaging the condom, water was extracted, based on a pre-determined scheme, using a syringe and a catheter. This procedure was repeated 16 times giving 17 samples of each series. The gain of the scanner was kept low and constant to avoid saturation at the echo interfaces. Eighty-one frames were captured through an angle of 51 ° and scan converted into a regular volume of voxels. Then, contours were drawn manually using a principle of either outside, inside,
Tram
/
/
! Outside
/C
Inside
Center
J Leading edge
Tracing pnnciple i
Surface wall of condom
Fig. 5. The four principles of contour indication which were investigated after data aequsition with 3D ultrasound.
center or leading edge contour indication (see Fig. 5) and the volume computed as described earlier. The thickness of the wall in a condom is very small, and data acquisition using ultrasound introduced an artifact of thin condom walls being depicted as thick, even at low gain. To sidestep this problem, the leading edge relative to the direction of sound transmission, as illustrated in Fig. 5, is normally used. All the volume measurements were performed by one person (NT), except for the central contour indication principle where inter-observer error was measured between two observers (NT and OHG).
2.5. Statistical methods The correlation coefficient determined by linear regression was used as a measure of association between the true volume (TV) and estimated volume (EV), while limits of agreement were determined as suggested by Bland and Altman (1986). Accuracy of the method was defined by how closely EV agreed with TV and was expressed as the mean of volume differences (d). Mean difference is a measure of bias. Assuming normal distribution of d, 95% of the differences were within the limits given by (d ± 2S.D.), defining the limits of agreement. The correlation coefficient between volume differ-
N. Thune et al. / European Journal of Ultrasound 3 (1996) 83-92
ences and volume means was given as: r2 = Corr ( E V - TV, (EV + TV)/2), and was calculated to test for dependence. Inter-observer variation was described by the differences between the two observers, against the corresponding mean for each volume estimate (Brennan and Silman 1992). The inter-observer error was reported as the overall mean percentage of the inter-observer variation ± the standard deviation of the interobserver variation. 3. Results
The experimental results are summarized in Table 1. For each of the four tracing principles, the first part of the table reports the linear regression coefficients between EV and TV while the second part describes the limits of agreement according to Bland and Altman (1986). The result of tracing the outside surface of the condom gave good correlation between EV and TV (rl = 0.998). However, the EV were consistently over-estimated. This is also reflected by a high mean difference (d = 7.14) and wide limits of agreement (2S.D. = 8.30). This over-estimation was expected since the true surface is not followed by this outside surface tracing technique. Furthermore, the over-estimation seemed to increase as the true volume increased. This was verified by a
89
significant correlation (r 2 = 0.971) between volume differences (EV - TV) and mean volumes ((EV + TV)/2). EV and TV show good correlation (rl = 0.996) also when tracing the inside surface of the condom. However, this tracing principle resulted in increasing under-estimation with increasing volumes. The negative mean difference (d =-5.59) together with high negative correlation (r 2 = -0.971) between volume differences and mean volumes confirmed this under-estimation. This could be expected since the condom wall was depicted much to thick due to artifacts during the signal acquisition. Accordingly, the tracing did not follow the true inside of the condom wall. Centering the tracing between the inner and outer wall contour demonstrated high correlation (rl = 0.998) and displayed EV in close agreement with TV. This was confirmed by low mean difference (d =-0.47). No over- or under-estimation was apparent which also was reflected by the low correlation (rz = 0.493) between volume differences ( E V - TV) and mean volumes ((EV + TV)/2). The final tracing principle, leading edge,
50
4O
Table 1 Results of in vitro validation of 3D ultrasound in volume estimation of phantoms
k5 3O
Tracing principle (n = 17)
E
Linear Agreement regression (rl) d (ml) ± 2 S.D. r 2
(ml) Outside Inside Leading edge Center (NT) Center (OHG)
0.998 0.996 0.996 0.998 0.998
7.14 -5.59 0.78 -0.47 0.58
8.30 5.85 2.65 2.15 1.94
./
./
E~ C
20 ~D
0.971 -0.971 0.334 0.493 0.307
~96J l
0 Tracing principle details, see text; NT and OHG, independent observers; rl, linear correlation coefficient; d, the mean difference between measured and true volumes, ± 2S.D. are the limits of agreement according to Bland and Airman; r 2, the linear regression coefficient between d and the average of the measured and true volumes.
10
l
20
l
30
J__
40
50
True volume (ml) Fig. 6. Scatter plot illustrating the correlation between estimated and true volumes when using the leading edge contour tracing principle. The line of identity is shown.
90
N. Thune et al./ European Journal of Ultrasound 3 (1996) 83-92 4 mean+2SD
3 2 1 I
0
)•
-1
s
mean
-2
2SD
-3 4
I ...... ~
0
10
h
20
30
±
40
J
50
60
(Estimated volume + True volume)/2 (ml}
Fig. 7. Mean difference and limits of agreement when using the leading edge contour tracing principle.
was also included• This method resulted in a high correlation (rl = 0.996) between EV and TV, as seen in Fig. 6. Low mean difference (d = 0.78) and low limits of agreement (2S.D. = 2.65) confirm this method of tracing as being satisfactory regarding accuracy (Fig. 7). There was no apparent over- or under-estimation, and a low correlation (r 2 -- 0.344) between volume differences (EV - TV) and mean volumes ((EV + TV)/2) was found. Inter-observer error between observers NT and O H G was calculated to be 0.6% 4- 5.0% when using the center edge contour tracing principle. Both tracings had low mean difference (d = -0.47) and (d = 0.58), and limits of agreement (2S.D. = 2.15) and (2S.D. = 1.94), respectively. 4. Discussion The presented volume estimation method based on 3D-ultrasound appeared to be accurate and practical. The best result were obtained with the leading edge and center edge contour tracing principles. When the water volumes inside the condom diminished, the condom was deformed and 'wrinkled'• In these cases, the surface was difficult to trace which contributed to the estimation errors. The small volumes are, however, included in
the calculations as we followed the pre-determined setup for water extraction from the condom. In our condom experiments, high image contrast was displayed, which is often not the case in low contrast clinical imaging of focal lesions. The thickness of the imaged condom surface was due to the spatial resolution characteristics of the transducer. In clinical imaging, this kind of error is normally a smaller problem but can be present when imaging thin-walled organs, like the gall bladder. Compared to Cook et al. (1980), who reported phantom studies with an accuracy in volume estimation within 3%, our method competes well• This is also true for other comparable methods that have been reported, such as described by Brinkley et al. (1982). They utilized a system with an accuracy in volume determination of balloons to within 1.8%, of kidneys to within 5.1%, and of left ventricular moulds to within to 5.9% of the true volume. King et al. (1990) reported in their work a 3D ultrasound system with accuracy in volume estimation of 1.6% (mean error). Another comparable method, described by Basset et al. (1991), tested a technique on phantoms of various shapes and volumes made of agar gel or water-filled balloons and reported a maximum volume estimation error less than 10% and a maximum variability less than 2%. Again our method competes well. High accuracy and precision in volume estimation with the ability to detect small changes in volumes, makes 3 D ultrasound promising as a clinical diagnostic tool. For the management of cancer, it will be possible to evaluate the growth or regression of the tumor in different organs. Also, regarding the fol!0w-up of benign structures, as tumors and cysts In the liver, kidneys, and pancreas, 3D ultrasound~may be of clinical value. By vizualization in 3 dimensions, it is possible to evaluate the exact position Of a focal lesion, and its demarcation from the surrounding tissue, and in particular its relation to blood vessels. This is of major significance in surgical decision-making and pre-operatwe planning. Accurate locahsatlon of gallstones by means of a 3D ultrasound system may improve therapies like lithotripsy. Furthermore, exact volume estimation of organs before •
.
•
.
\
\
•
.
N. Thaneet al./ European Journal of Ultrasound3 (1996) 83-92
and after physiological or medical intervention facilitates dynamic studies like gastric and gall bladder emptying. From these in vitro experiments, it appeared that the tracing principle was of utmost importance, implying that standardization of the method of tracing for volume comparison is necessary. Furthermore, if possible, the gain setting should be kept low to reduce artifacts at tissue transitions. Even though the presented method demonstrated high accuracy in volume estimation, improvements are possible. Contours were represented with a constant distance between vertices. To improve the shape accuracy, the spacing should be dependent upon the curvature of the contour. Also, methods for automatic edge detection should be investigated to aid the user in improving the consistency of contour indication and in saving time doing the manual outlining. In addition to the tracing error, other sources of error inelude mechanical inaccuracy in transducer movement and resampling inaccuracy. We conelude that this 3D ultrasound system yields high accuracy in volume estimation of phantoms and contributes to low inter-observer error.
91
Starting points
t
i.. [ . . . f ' Y
lyp/C,~I
/+1
vc,.,
Fig. 8. A step in the triangulation of a contour pair C i and Ci+ t. Tiles were added to the polyhedron by proceeding along contour C i and Ci+ I in such a manner that the absolute difference in travelled distance along each contour was minimized at all times.
lute difference between the normalized travelled distance along each contour was minimized at all times. The heuristic rule for adding the next triangle (tile) was as follows:
Begin Acknowledgements eo= This work was supported by the Norwegian Research Council for Science and Technology (NTNF). The authors would like to thank the Institute of Biomedical Engineering, Faculty of Medicine, University of Trondheim, who have designed and constructed the 3D transducer equipment we used. We also thank Vingmed Sound A/S, Horten, Norway, for supporting this project.
PI = V f i+l
Appendix The third step in the reconstruction of a polyhedron was the triangulation of contour pairs. The heuristic rule in the method presented by Ganapathy and Dennehy (1980) was based on neighbouring pairs of contours having similar size and shape appearance. A step in the triangulation process is shown in Fig. 8 and illustrates how tiles were added to the polyhedron by proceeding along contour Ci and Ci+l in such a way that the abso-
Polyhedron
Planar surface s with surface area As,
Fig. 9. This diagram shows how the volume of a polyhedron is calculated by a summation over all surface tiles according to Goldman 1991.
92
N.
Thune et
al./ European Journal of Ultrasound 3 (1996) 83-92
i f I @kCi "t" ~Ok,k+ Ci I __ ¢I~/Ci+lI < I*Ci+l..I.,,Ci+l Wl, l+l -- ¢~Ci k I
Y = lie [ [~jPo, j • S j ) l S j * [~Pk, j x Pk+l,j] [11
then
where (.) is the scalar product and (x) is the cross product. It should be noted that the equation for the surface area (2) and polyhedron volume (4) are also valid for non-convex shapes.
P 2 - - VCkk~l
(4)
else Refereaces O_ _-- l/Ci+ 1 ~2 • /+1
End where P0, PI, P2 (see Fig. 9) were vertices on the new triangle to be added; Vci, V~i+1 were the current visited vertices along contour Ci and Ci+l; o c g o ci+l were the normalized travelled distances so far along contour Ci and Ci+l; and ek.k+PC"¢Ci+ll.l+lwere the normalized lengths of the next contour segments along contour C i and Ci+l. Volume of a Polyhedron The volume v of a polyhedron (Goldman 1991) could now be expressed as
(1) there S0,...Sn were planar polygonal faces on the polyhedron, Qj was any point on Sj; Nj was a outward pointing unit vector normal to Sj; and Asj, was the surface area of Sj (see Fig. 9). If P0,i,...Pmj were vertices of Sj oriented counter clockwise with respect to the outward pointing normal of Sj, then the surface area Asj of Sj could be expressed as Asj = 1/2 I Nj " I F
Pkj X Pk+l.j]I
(2)
where
the first and last vertex coincides, P0d. The point Qj could be chosen to be P0j and the normal vector could be computed as Pm+lj =
I(e~,j - e04) x (e2,j - e0,/)l Nj = I(Pi,J Po,j) x (P2,j - P0,j)l
(3)
Care had to be taken to ensure that the points chosen from the surface Sj to compute the normal vector did indeed give the outward pointing normal of Sj. The following equation then represents the volume of a polyhedron
Basset O, Gimenez G, Mestas JL, Cathignol D, Devonec M. Volume measurement by ultrasonic transverse or sagittal cross-sectioual scanning. Ultrasound Med Biol 1991; 17: 291-296. Bland JM, Altman DG. Statistical methods for assessing agreement between two methods of clinical measurements. Lancet 1986; i: 307-310. Brennan P, Siiman A. Statistical methods for assessing observer variability in clinical measures. Br Meal J 1992; 304: 1491-1494. Brinkley JF, Muramatsu SK, McCallum WD, Popp RL. In vitro evaluation of an ultrasonic three-dimensional imaging and volume system. Ultrason Imaging 1982; 4: 126-139. Cook LT, Cook PN, Lee KR, Batnitzky S, Wong BYS, Fritz SL, Ophir J, Dwyer SJ, Bigongiari LR, Templeton AW. An algorithm for volume estimation based on polyhedral approximation. IEEE Trans Biomed Eng 1980; 27: 493-500. Fuchs H, Kedem ZM, Uselton SP. Optimal surface reconstruction from planar contours. Commun ACM 1977; 20: 693-702. Ganapathy S, Dennehy TG. A new general triangulation method for planar contours. Comput Graph 1982; 16: 69-75. Giija OH, Thune N, Matre K, Hausken T, ~iegaard S, Berstad A. In vitro evaluation of three-dimensional ultrasonography in volume estimation of abdominal organs. Ultrasound Med Biol 1994; 20: 157-165. Gilja OH, Smievoll AI, Thune N, Matre K, Hausken T, ~legaard S, Berstad A. In vivo comparison of 3D ultrasonography and magnetic resonance imaging in volume estimation of human kidneys. Ultrasound Med Biol 1995; 21: 25-32. Goldman RN. Area of planar polygons and volume of polyhedra. In: Arvo J, ed. Graphics Gems II. New York: Academic Press Inc., 1991; 170-171. Kaufman A. Introduction to volume visualization, ln: Kaufman A, ed. Volume Visualization. IEEE Computer Society Press Tutorial, 1991; 1-18. King DL, King DL Jr, Shao MYC. Three-dimensional spatial registration and interactive display of position and orientation of real-time ultrasound images. J Ultrasound Med 1990; 9: 525-532. King DL, King DL Jr, Shao MYC. Evaluation of in vitro measurement accuracy of a three-dimensional ultrasound scanner. J Ultrasound Med 1991; 10: 77-82. Watanabe Y. A method for volume estimation by using vector areas and centroids of serial cross sections. IEEE Trans Biomed Eng 1982; 29: 202-205.