Ultrasound in Med. & Biok Vol. 17, No. 3, pp. 291-296, 1991 Printed in the U.S.A. 0301-5629/91 $3.00 + .00 © 1991 Pergamon Press plc OOriginal Contribution VOLUME MEASUREMENT BY ULTRASONIC TRANSVERSE SAGITTAL CROSS-SECTIONAL SCANNING OR O. BASSET, t G . GIMENEZ, f J. L. MESTAS, ¢ D. CATHIGNOL ¢ a n d M . DEVONEC* *Laboratoire Traitement du Signal et Ultrasons, INSA, 69621 Villeurbanne, Cedex; *INSERM, U. 281, Lyon; and *Hrpital de l'Antiquaille, Service d'Urologie, Lyon, France (Received 15 March 1990; in final form 25 September 1990) Abstract--A technique is described that provides an accurate estimation of the volume of an organ from its ultrasonic cross-sectional images. The technique is applied to two types of ultrasonic investigation, one providing transverse and the other sagittal images. The organ outline has to be traced on each scan. The computer first calculates the area and then the volume from the vector areas and the centroids of a series of sections. The technique has been tested with phantoms of various shapes and volumes made with agar gel. These experiments show that the error in the volume estimation is less than 10% and the variability of measurements is less than 2%. Key Words: Acoustics, Ultrasonics, Tomographic scans, Volume measurement, Prostatic ultrasound. INTRODUCTION 1982), but to our knowledge it does not exist on scanners using linear probes. The device and the volume measurement method are described in the first part of this paper. Then, the results obtained from measurements on phantoms are presented. In the last part, the measurement accuracy is studied. In medicine, the volume measurement of an organ or a tumor is of major diagnostic importance, particularly in following the evolution of a disease. Several methods have been developed to measure the volume of an organ noninvasively, estimated from its images. Certain techniques, which are fast but inaccurate, require only one or two sectional images of the organ. They generally assume that the organ is an ellipsoid (Styles et al. 1988; Griffiths et al. 1986). More accurate "planimetric" techniques have also been developed that use serial cross-sectional images (Tamura et al. 1985; Bartsch et al. 1982). In the present study we consider the measurement of prostatic volumes. In this field, ultrasound scanning is the most appropriate imaging technique. Two types of scanning probes are currently used, each "slicing" the organdifferently: sector scanning transducers, providing transverse (radial) sections, and linear array transducers, generating sagittal sections. An accurate volume measurement system has been developed that processes either type of scan using a planimetric approach. Note that such a device can already be found on some scanners using a sector scanning transducer (Bartsch et al. 1982; Hastak et al. DATA ACQUISITION AND PROCESSING Principle and materials The planimetric technique of volume determination requires serial cross-sectional scans. The series of scans are regularly distributed over the investigated object. Figure 1 presents the acquisition processes corresponding to the two types of scanning probes. When using a radial probe, the successive scans are performed by a translation of the transducer, resulting in parallel sections. With sagittal images, the probe is rotated to generate the successive cross-sectional images. Each scan of a series is digitized in 512 X 512 pixels, with 256 grey levels. Then, the operator draws the contours of the object on the resulting image, with a digitizer tablet. This intervention by the operator introduces a subjective element in the results. Then, a computer program calculates the area inside the contour, In practice, this program considers only a limited number of points (20 points), evenly distributed about the contour, so the calculated area is bounded Address correspondence to: O. Basset, INSA, Laboratoire Traitement du Signal et Ultrasons, 20, Avenue Albert Einstein, 69621 Viileurbanne, Cedex, France. 291 292 Ultrasound in Medicine and Biology Volume 17, Number 3, 1991 TRANSVERSE SCANS /z.x i x' z y' , Y M (x',y') M (x,y,z) , v X M (x',y') M (x,y,z) Y I I Probe rotational axis translation direction Probe , i Fig. 1. Transverse and sagittal acquisition processes. M by a polygon of 20 vertices. The volume between two successive scans is then computed. This process is repeated for each cross-sectional image, after rotation or translation of the transducer probe. The computer used for calculations and control is an IBM PC AT. A MATROX PIP 1024 card is required for image digitizing. It is connected to a highresolution monitor (Mitsubishi H F 1400). The digitizer tablet is a Summagraphic MM 1201. Volume calculation The volume v of an object described by a series of N contours is expressed as (Watanabe 1982): N o = lY~ (~, + s - ~ - 0 ( ~ - ~ H ) / 2 1 (1) i=2 where ~i is the position vector of the centroid of S,., which is the ith contour of the series of flat cross sections, and ~, is the vector area of S~ (i.e., a vector normal to S~, with a magnitude equal to the area of S,-). When the contour of S~ is given as a polygon, its area s and the coordinates of its centroid (w,,, %) in the plane of the considered section are given by eqns: s = IZ j=l (yj+,xj - yjxj+,)'/2l, (2) M vex = Z (Xj+l + xj)(xffj+t -'Xj+IYj)/6 s, j=l (3) M Wy = ~ (Yj+l -k yj)(yjXj÷, -- Yj+lXj)/6 S (4) j=l where (xj, yj) are the coordinates o f t h e j t h vertex of the polygon, which comprises M vertices. As in our application the successive cross-sectional images are taken at a constant longitu~nal or rotational step, eqn ( 1 ) can be simplified. The modifications are presented hereafter for each slicing mode (transverse or sagittal). T r a n s v e r s e c r o s s - s e c t i o n a l s c a n s . With the notation of Fig. 2 the development of eqn ( 1 ) gives: l N v = 5 I Y~ siwicos 0, - siwi_,cos 0i-1 i=2 ql- S i _ l W i C O s Oi -- S i W i _ i C O s 0i+1 [. Volume measurement • O. BASSETet al. 293 In the system of section i - 1, it becomes: ~ti Ie = = si i Fig. 2. Transverse slicing mode. O / 0) sisin a \ sicos a , =/wyic°s . \-wyisin a / As the v a r i e s slice planes are parallel, we get / / // The partial volume Vl contained "between" the two sections i and i - 1 is given by: cos Oi = sin ai, wisin ai - wi_~sin ai_~ = e vi = I -wyi_lsisin a - WyiSi_lsin ~1/2. where e is the distance between two successive slice planes. Finally, the volume of an object represented by radial cross-sectional images is given by: 1 And the entire volume v is N t) = N El)i, i=2 N v =-~ I E ( s i + s i - , ) e l . (5) v = i~2 ~ (-wyi_lS i - wyisi_l)sin a l / 2 . Note that v is independent of the centroid coordinates. Consequently the calculation does not depend on the location of the origin point 0. Sagittal cross-sectional scans. In this case, the origin point 0 is located on the rotational axis of the probe. So a vector ~i is always perpendicular to ~, so that s~ w~ = O. Given an orthogonal coordinate system ( ~ , ~ , ~ ) associated with section i [(~ ~) is the plane of the section], we get ,(s!) Given a similar coordinate system (~_~, y~_~, ~_~) associated with section i - 1, it comes Contrary to the previous case, the volume calculation from serial sagittal scans depends on the centroid coordinates. As the origin 0 is on the rotational axis, the exact location of this axis on the images must be known. In practice, this means that the distance between the axis of rotation and the transmitting surface of the transducer has to be measured. Figure 4 illustrates a simple method for measuring this distance from two ultrasonic scans of a thin wire. The first scan is in a plane perpendicular to the wire. The distance b between the side o f the scan (which corresponds to the transmitting surface of the transducer) and the wire can be measured on the image. Rotating the probe by a known angle 3' gives the second scan. Then, the distance between the side of the scan and the wire is c. So the distance a is given by: a = (c cos 3' - b ) / ( 1 + cos 3"). By rotating the/-referenced coordinate system by an angle - a (Fig. 3) about the probe rotational axis (which supports x), the vectors st and wi can be expressed in the system associated with section i - 1. The rotational matrix is: R= 1 0 0 cos-a 0 sin-a (6) i=2 0 ) -sin-a . cos-a / O y Fig. 3. Sagittal slicing mode. 294 Ultrasound in Medicineand Biology Wire Probe rotational axis a+b b cos 7 a+c a= c . cos ~f - b 1 - cos 7 Volume17, Number 3, 1991 15 scans were made every 10 degrees. In both cases, the areas are calculated from 20 points per contour. For image construction, the B-scan imagingsystem uses a mean velocity of ultrasound propagation in tissues: 1540 ms -~. Since our experiments on phantoms were made in water at 20°C, sound velocity is theoretically 1486 ms-I; therefore, this difference leads to a geometric distortion of the images. Because of lower sound velocity in water at 20°C, the objects appear larger on the images. The velocities ratio a equals the sizes ratio oL = l/tissuJVwaterat 20oc = Fig. 4. Method measuring the distance between the rotational axis of the probe and its transmitting surface. A wire is used in this experiment because it presents a small section that can be located precisely on the scans. RESULTS The polygon vertices, used for the area calculation, are given on the digitized images from their coordinates, expressed in pixels. So the calculated volumes are first expressed in voxels (elementary volumes). These can be translated into conventional units (cubic centimeter) using the horizontal and vertical graduated axis (in centimeters) available on the scans. Note that the pixel density is specific to each scanner and even to each magnification of each scanner. In addition, in some cases the densities may be different in the horizontal and vertical directions. The volume calculation method has been tested from transverse and sagittal scans of phantoms of different shapes and volumes. These phantoms are solid objects made with a gelatin solution (Agar-Agar) and having an acoustical impedance approximately equal to that of water. Scanned in water, the phantom sections appear plainly in the ultrasonic images. It is then easy to draw their contours (Fig. 5). Many measurements were made with different objects (cylinder, sphere, bar), in order to assess the accuracy and the variability of the volume measurements. Here, variability means the differences between repeated volume measurements on the same object by the same observer. It is given by the standard deviation of the measurements series divided by the mean volume. Table l presents some repeated volume calculations for a cylinder. With the radial probe, 11 scans were performed every 5 mm. With the sagittal probe, DimagelDactual = 1540/1486 = 1.036. Consequently, the volume measurements must be corrected. When volumes are measured from transverse scans, this error occurs on two linear dimensions (in the slice plane). The third dimension is independent of the image reconstruction (the probe translation step). That means that the correction of the volume value consists in dividing the results by a 2. When sagittal scans are used, the error occurs on the three linear dimensions, and volumes must be divided by a 3. As shown by the measurements reported in Table 1, the volume can be estimated with an error of less than 7% and a variability less than 1%. A lot of other trials were performed to assess the m a x i m u m error (10%) on the volume estimation and the maxim u m variability (2%). Remark that in this case the accuracy is slightly better for the transverse slicing mode. This is justified by the following. Another volume measurement experiment was Fig. 5. Example of a phantom scan with the corresponding contour drawn by the operator. 295 Volume measurement • O. BASSETet aL Table 1. Results of volume calculations for a cylinder for transverse and sagittal slicing modes. Corrected volumes are reported. Shape : Cylinder ; actual Volume= 26.5 cm3 Slicing : T r a n s v e r s e images Sagittal estimated volume (in cm3) corrected volume V/co 2 % difference estimated volume (in cm3) corrected volume V/a 3 % difference 27.04 27.07 27.23 26.83 27.2 25.19 25.22 25.37 24.99 25.34 - 4.9% - 4.8% - 4.2% - 5.6% - 4.4% 27.96 27.39 28.19 27.9 27.49 25.14 24.63 25.35 25.09 24.72 - 5.1% - 7% - 4.3% - 5.3% - 6.7% olume 25.22 - 4,8 % mean 27.78 volume 24.98 - 5.7% mean 27.07 variability 0.55 % variability 1% conducted on water-filled balloons in order to assess the correlation of m e a s u r e m e n t s between the two slicing modes. The balloons were filled with a k n o w n volume of water varying from 5 to 70 c m 3. Figure 6 c o m p a r e s the " s e c t o r i a l " v o l u m e s (transverse images) with the "linear" volumes (sagittal images). Table 2 reports prostatic v o l u m e m e a s u r e m e n t s on two patients for each slicing mode. The first case, n a m e d prostate 1, corresponds to a particularly small prostate with low contrast images. Despite these unfavourable conditions, the two slicing modes give close values (difference: 8.2%). In the second case, pros- y = 0,10074 + 1,0266x R ^ 2 = 0,999 80 60 40 20 0 0 images 20 40 60 Sagittal v o l u m e (cm3) Fig. 6. Transverse volumes measurements versus sagittal volumes. 80 tate2, corresponding to normal m e a s u r e m e n t conditions, the difference between the two values is very low (2.4%). ACCURACY Let us recall that the stated accuracy o f this syst e m has been evaluated on the basis of n u m e r o u s experiments. Measured volumes are always lower than actual volumes with less than 10% of error. This implies that each linear dimension is measured to an accuracy of less than 3%. So the technique achieves a fair precision. In fact, the accuracy depends on a n u m ber o f independent factors, with each contributing to the total error. These contributions are often hardly quantifiable, such as the operator's skill in outlining the object of the scans, but we m a y assume that they c o m p e n s a t e themselves w h e n m e a s u r e m e n t s are made. The volume estimated also varies with the way the sections are chosen and particularly the first and last sections. The best result is obtained when these two sections are tangential to the object. I f they are not, the volume is underestimated. This is the case when the investigated object is a cylinder. With transverse images, perpendicular to the cylinder axis, the extreme slices m a y be tangential to the object and then the complete volume is considered. Slicing a cylinder with sagittal scans is m o r e difficult. It remains usually the extremities which are not taken into account. 296 Ultrasound in Medicine and Biology Volume 17, Number 3, 1991 Table 2. Results of prostatic v o l u m e measurements for transverse a n d sagittal slicing modes. Prostatel Prostate2 Sagittal volume (cm3) 10.747 27.706 The number of sections used in the calculation and the number of vertices around the polygonalized contours also influence the accuracy. The polygonalized contours always lead to an underestimated volume. The more sections there are and the more points on each contour, the lower the error. The accuracy of this volume measurement method can also be improved by modeling the object with cubic polynomial functions (spline). However, it would be difficult to justify the use of a more accurate but therefore more complex calculation method because the measurement accuracy would still depend primarily upon the precision of the operator's hand-drawn contours with respect to the actual contours. The number of points used to define the contours and the number of sections in the object have to be chosen in such a way as to avoid an "undersampling" of the object. If the object exhibits intricate features, a high number of sections and contour points have to be used. The operator's experience acquired during many measurements will probably improve the overall accuracy. SUMMARY AND CONCLUSION A volume measurement device has been described that is based on cross-sectional (transverse or sagittal) scans of an object. Many trials performed with phantoms have shown that the measurements have good accuracy. In vivo measurements on the prostate confirm the good correlation between the two approaches to volume measurements. A volume can be estimated with an error of less than 10%. This accuracy depends mainly on the skill of the operator in outlining an object on the scans. Transverse volume (cm3) 11.65 27.044 % Difference 8.2 % 2.4 % However, when an actual organ volume is being investigated, the relevant information is often the time-variation of the volume. This can be known very precisely if the successive volume measurements are performed by the same operator, and under the same conditions. It may be advantageous to use this device to record the time-evolution of tumors, for example, to help determine the action of a medicine or a treatment. REFERENCES Bartsch, G.; Egender, G.; Hubscher, H.; Rohr, H. Sonometrics of the prostate. J. Urol. 127:1119-1121; 1982. Grifliths, C. J.; Murray, A.; Ramsden, P. D. Accuracy and repeatability of bladder volume measurement using ultrasound imaging. J. Urol. 138:808-812; 1986. Hastak, S. M.; Gammelgaard, J.; Holm, H. H. Transrectal ultrasonic volume determination of the prostate. A pre-operative and post-operative study. J. Urol. 127:1115-1118; 1982. Styles, R. A.; Neal, D. E.; Powell, P. H. Reproducibility ofmeasuremerit of prostatic volume by ultrasound. Eur. Urol. 14:266269; 1988. Tamura, S.; Nakamo, S.; Matsumoto, M.; Shimazu, T.; Fujiwara, M.; Matsuyama, T.; Hanrath, P. Three-dimensional reconstruction ofechocardiograms based on orthogonal sections. Pattern Recognition 18:115-124; 1985. Watanabe, Y. A method for volume estimation by using vector areas and centroids of serial cross-sections. IEEE Transaction on Biomedical Engineering BME-29:202-205; 1982. APPENDIX Note: The software for volume calculation is fitted to the following materials: • computer: PC AT, • digitizing card: MATROX PIP 1024, • scanner: Hitachi equipped with a bi-plane probe EUP-U322. People working with equivalent materials and interested in the software program can contact the authors.