the application of total vertical projections for the unbiased estimation of the length of blood...

Magnetic Resonmce Imaging, Vol. 9, pp. 917-925, 1991 0730-725X/91 $3.00 + .OO Printed in the USA. All rights reserved. Copyright 0 1991 Pergamon Press plc

l Original Contribution

THE APPLICATION OF TOTAL VERTICAL PROJECTIONS FOR THE UNBIASED ESTIMATION OF THE LENGTH OF BLOOD VESSELS AND OTHER

STRUCTURES BY MAGNETIC RESONANCE IMAGING

NEIL ROBERTS,* C. VYVYAN HOWARD,? Lurs M. CRUZ-ORIVE,$

AND RICHARD H.T. EDWARDS*

*Magnetic Resonance Research Centre, University of Liverpool, P.O. Box 147, Liverpool, L69 3BX, UK, tDepartment of Human Anatomy and Cell Biology, University of Liverpool, P.O. Box 147, Liverpool, L69 3BX, UK,

and SStereology Unit, Department of Anatomy, University of Bern, Postfach 139, CH-3000, Bern 9, Switzerland

A new stereological method has recently been developed to estimate the total length of a bounded curve in’3D from a sample of projections about a vertical axis. Unlike other methods based on serial section reconstrucGons, the new method is unbiased (i.e., it has zero systematic error). A basic requirement, not difficult to fulfill in many cases, is that the masking of one structure by another is not appreciable. The application of the new method to real curvilinear structures using a clinical magnetic resonance (MR) imager is illustrated. The first structure measured was a twisted water-filled glass tube of known length. The accuracy of the method was assessed: With six vertical projections, the tube length was measured to within 2% of the true value. The second example was a living bonsai tree, and the third was a clinical application of MR angiography. The possibility of applying the method to other scientific disciplines, for example, the monitoring of plant root growth, is discussed.

Keywords: Angiography; Blood vessels; Curve length; Cycloids; MRI; Plant roots; Stereology; Tree branches; Vertical projections.

INTRODUCTION

A stereological method for estimating feature length without bias from total vertical projections (TVPs) has recently been presented by Cruz-Orive and How- ard’ elaborating on an idea by Gokhale.’ Current length estimation methods based on feature tracking in serial section reconstructions are generally biased, because the linear feature of interest is at best approx- imated by a polygonal curve of variable link segments. The TVPs method, however, is not approximate, but mathematically unbiased provided that each point of the curve can be identified after orthogonal projection on a vertical projection plane.

The unbiasedness property of an estimator depends

exclusively on the sampling regime adopted; it cannot be proved or disproved by looking at sampled data alone. On the other hand, the precision of an estimator is difficult to predict mathematically. Thus, the purpose of this paper is twofold. First, to explore the possibility of using MR imaging to collect the data, and second, to assess estimation precision (which amounts to accuracy if the estimator is unbiased; see, for example, Cochran3). To do this, we used a syn- thetic structure and two real applications. The syn- thetic structure was a twisted 3-mm bore glass tube, of known length, filled with water. The required set of MRTVPs was obtained in two ways; one involved rotation of the object, while in the other method the scanner was preprogrammed to collect MRTVPs with

RECEIVED 1 l/14/90; ACCEPTED 613191. Acknowledgments- We are grateful to Dr. Peter Martin

for assistance in programming the control of the gradients in the clinical imager, Mr. Bert Chappel, the glassblower, and Ms. Barbara Krueger and Mr. David Adkins for assistance in preparing the diagrams. MR angiography data have kindly been supplied by Professor Peter Vock of Bern Uni- versity, Switzerland and Dr. Jim Siebert of Michigan State University, USA.

NR acknowledges support from Shell Research Ltd. and LMC-0 acknowledges support from the Swiss National Sci- ence Foundation grant No. 31-28610.90.

Address correspondence to Dr. Neil Roberts, Magnetic Resonance Research Centre, University of Liverpool, P.O. Box 147, Liverpool, L69 3BX, UK.

917

918 Magnetic Resonance Imaging 0 Volume 9, Number 6, 1991

the required orientations, via appropriate control of its three mutually orthogonal localizing gradients. The correspondence between the length values obtained from the two imaging methods was checked.

The first real structure analysed was a leafless bonsai tree; the problem was to estimate its total branch length. The tree was first imaged to produce a set of MRTVPs. A second set of TVPs of the bonsai tree was obtained by photographing the tree from a prescribed range of viewing angles. Finally, the total branch length was estimated by a direct, nonstereolog- ical method using cotton strands. The correspondence between the three length estimates was checked.

In the second real application, the clinical useful- ness of the MRTVP technique for determining feature length was illustrated using MR angiography data. MR angiography has the potential to not only reveal the blood vessels but to provide information on the rate and direction of blood flow. Quantitative information on the relative length of vessel containing blood flowing at various rates could be of value in better understanding diseases of the cardiovascular system.

STEREOLOGICAL DETERMINATION OF FEATURE LENGTH

Gokhale* has described a method for determining feature length that uses vertical slices4 of known thickness obtained from the solid containing the feature of interest. The method does not yield the total length of a bounded curve, but the mean length per unit reference volume. Two additional drawbacks are (a) having to slice the reference space and (b) having to know the slice thickness precisely. Elaborating on Gokhale’s idea, however, Cruz-Orive and Howard’ have recently developed a stereological method for determining total feature length which does not suffer from the preceding shortcomings and is easier to im- plement. The more recent method requires images to be obtained in the form of TVPs. First, a convenient vertical axis is fixed with respect to the object under study. Then, with a random starting position and con- tinuing at uniform intervals, the whole object is pro- jected onto a plane in a systematic set of directions between 0” and 180”, all perpendicular to the vertical axis. Each image is thus a sum of the information on the object in a particular direction; the requirement re- garding slice thickness being solely that for each orientation it should be so large as to contain the whole of the object. On each TVP, a cycloid test system4 (Fig. 1) is manually superimposed with uniform random position, and with the shorter principal axes of the cycloids always perpendicular to the fixed vertical direction, as in Gokhale.2 Intersections of the cycloid

>

( >

( > / L

>

( >

: > (

Fig. 1. Cycloid test systems for estimating the length of a curve in 3D from TVPs using Eq. (1). On each curve projection, the arrow labeled vertical should be aligned with the direction of the rotation axis. The only quantity pertaining to each test system which enters in Eq. (1) is the ratio of test area to test line length. The length of an isolated cycloid arc from the lower test system is four times its shorter (horizontal) axis, or 4/a times its longer (vertical) axis. For this test system a// equals ?r times the length of a short cycloid axis, or equivalently, ~14 times the horizontal distance between consecutive arcs. In the upper test system the test length density is exactly twice of that for the lower test system, so that a/l is half of that for the lower test system.

grid with the feature of interest in the TVP are counted. The length of the curve is then estimated ac- cording to the formula

est L = 2(~/l)M-~n- CZi (1)

where estL is the unbiased estimator of length (in cm), a/l is the known ratio (in cm) of test area to test line length for the test system used, M is the linear magnification of the vertical projections, n the number of vertical projections analysed and CZi the total number of intersections counted.

Estimation of blood vessel length 0 N. ROBERTS ET AL. 919

The mathematical theory justifying the unbiasedness of the preceding estimator is given in Cruz-Orive and Howard.’ In the same paper, the method is applied to measure a twisted steel wire of known length from TVPs recorded with an ordinary camera. With six systematic TVPs (taken every 30”) the length estimate obtained was within 3% of the true value.

The above procedure can be used on a set of TVPs produced not only by photography, but by a number of imaging techniques. In particular, MR imagers can provide a set of MRTVPs.

PROJECTION MR IMAGING

There are three ways in which a set of MRTVPs can be produced for various orientations of an object imaged on the General Electric SIGNA 1.5 Tesla whole-body clinical imaging system in use at the Uni- versity of Liverpool.

Turning the Object To encode spatial information from a 3D object

with MRI, three mutually orthogonal gradients5*6 are available. The direction Oz can be adopted as the vertical axis. Then, to obtain a 2D total vertical projection of a 3D object in the direction Ox, it suffices to remove the x gradient from the imaging protocol. If the object being imaged is rotated about the vertical axis, Oz, sequentially through a prescribed series of angles, then by imaging at each orientation, the required set of MRTVPs are obtained.

Manipulating the Gradients As an alternative to the above, by appropriate con-

trol of the scanner’s three spatial localizing gradients, it is possible to obtain the required set of MRTVPs without having to rotate the object. The first image is acquired in the same manner as in the preceding para- graph, but for the next image, rather than moving the object, the strengths of the scanner’s three gradients are combined to cancel the gradient in a desired direction (e.g., 10” from the Ox direction). Next the gradients are combined to cancel the gradient at 20” from the Ox direction, and so on until a series of MRTVP images is obtained at the required intervals around the Oz direction. This method has the advantage that sample positioning errors do not arise, and holds the potential for data acquisition to be automated. A de- scription of gradient coils and their control is to be found in Chen and Hoult.7

Projections of a 30 Data Set A third alternative is to obtain MRTVPs by post-

acquisition processing of volume image data. A volume scan is prescribed, producing a 3D data set which

contains localized information about the object in all three directions, Ox, O_Y, and Oz. The MRTVPs are then obtained by subsequently projecting this 3D data set in any required orientation. Ideally, the volume data will have the same spatial resolution in each orthogonal direction. This commonly may not be the case and the data may need to be interpolated within the projection algorithm.

As well as enabling the production of projection images, a volume data set also affords the potential for retrospective reformatting of oblique planes. In view of the convenience this offers for diagnostic ra- diology, assisted by developments in fast scanning techniques (see review by Wehrli8), modern clinical MR systems are being designed to routinely acquire volume as well as multislice data. Also, MR angiography investigations generally produce 3D phase contrast maps.9 The angiography data are particularly interesting in view of the inherent linearity of the features involved; the display, but not the quantitation, of which has been addressed by Cline et al. lo

APPLICATION AND TESTING OF THE METHODS

Water-filled Glass Tube The first MR images to be used in assessing the ap-

plication of the method of Cruz-Orive and Howard’ to MRTVPs, were of a water-filled 3-mm bore glass tube that the glassblower had twisted into the shape in Fig. 2. From three independent measurements of the length of a strand of wire threaded through the test object, the true length along the bore of the structure was estimated as 81.0 cm, (SEM = 0.3 cm). Similarly, the separation between the end points of the test object was measured three times with a ruler and determined to be 15.0 cm, (SEM = 0.3 cm). Next, two series of MRTVPs were obtained of the object. As in Cruz-Orive and Howard, ’ each series contained 18 images acquired at 10” intervals over a range of 180”. The first series was obtained by rotating the object and the second by controlling the field gradients. Fig- ure 1 contains examples of MRTVPs from the first series.

In order to determine the tube length stereologi- tally, the MRTVPs were placed on a light box and a cycloid grid (similar to that shown in Fig. 2, bottom) constrained only so that the short axes of the cycloids should be perpendicular to the vertical direction, was superimposed at random on the images. For each MRTVP, the number of intersections of the cycloids with the feature was counted. To take account of the finite width of the bore of the tube, intersections were counted with both boundaries of each tube projection, and the total count was therefore divided by two be-


Fig. 2. A set of six MRTVPs obtained by rotating the water- filled tube systematically every 30” about a fixed vertical axis. The field of view (FOV) of the MRTVPs is 32 cm. The fixed vertical axis is parallel to the edges of the page. To estimate the true length of the tube in 3D with Eq. (I), the cycloid test system (cf. Fig. 1) is overlaid on each MRTVP with uniform random position, but always with the shorter principal axes of the cycloids perpendicular to the fixed vertical axis (see Table 1).

fore being used in Eq. (1) above. On the other hand, to account for the physical thickness of the cycloid test curves, intersections were counted only with the left hand side boundary line of each curve. Listed in Table 1 are the results from the stereological analysis of the two sets of MRTVPs obtained for the twisted water-filled glass tube. The accuracy of the stereological length estimation is consistent whether the MRTVPs were obtained by rotating the specimen or through control of the magnetic field gradients. The pooled stereological length measurement agrees to within 2% of the physical measurement.

The Branch Length of a Bonsai Tree After having tested the stereological method on a

manufactured test object of known dimensions and water content, an attempt was made to determine the total branch length of a bonsai tree from MRTVPs. Every branch of the small olive tree contained a large number of leaves and this was useful in indicating that water was being actively transported through them. However, preliminary imaging showed that the leaves produced a significant MR signal and this would inter- fere with the recognition of the branches in the

Table 1. Estimating the length of a water-filled glass tube using MR

Rotating Object Rotating Field

Sample Angle I(t) Z(e) I(l) I(e)

1 0 26 2 26 2 30 26 3 28 3 60 25 3 20 3 90 26 3 24 4

120 26 3 22 3 150 21 2 27 1

total/2 = 75 16 73.5 16 e.st(L) = 80.3 17.1 78.7 17.1

(2.4%) (10.6%) (1.8%) (6.8%)

2 10 25 1 29 2 40 30 2 27 3 70 22 2 23 3

100 26 3 24 2 130 26 3 25 3 160 26 3 23 2

total/2 = 77.5 14 75.5 15 est(L) = 83.0 15.0 80.9 16.1

(2.4%) (10.6%) (1.8%) (6.8)

3 20 21 1 22 3 50 28 3 28 3 80 28 1 24 2

110 23 3 21 2 140 27 2 24 2 170 21 3 27 2

total/2 = 74 13 73 14 est(L) = 79.3 13.9 78.2 15.0

(2.4%) (10.6%) (1.8%) (6.8%)

Pool total/2 = 226.5 43 222 45 e&(L) = 80.9 15.4 79.3 16.1

Eighteen MR projections were obtained over a range of 180”; the interval between each projection being 10”. The true length of the tube was L = 81 .O cm, whereas the true distance between the tube end points was 15.0 cm. By taking every third MRTVP, the images were divided into three groups of six. The ratio of area per test line length for the cycloid test system used was a/l = 1.175 cm. The field of view (FOV) was 32 cm, corresponding to a distance on the image of 11.7 cm, so that the magnification M = 0.3656. These values were substituted in Eq. (1) (with n = 6) to give est L = 1.071*1(t) cm, where Z(t) = half the total number of intersections between both linear boundaries of the tube projections and the cycloid test system, for the determination of total feature length, and est L = 1.071*Z(e) cm, where Z(e) = the total number of intersections between a line drawn between the two end points of the feature, and the cycloid test system, for the determination of the distance between the two end points. The coefficient of variation among the three estimates is given within parentheses. See Fig. 2.

MRTVP image. Consequently, the tree was stripped of its leaves before study. A selection of the MRTVP images obtained for the bonsai tree, by control of the scanner gradients about a fixed vertical axis, are shown in Fig. 3. The stereological estimates of the to-


Fig. 3. (A) Photographs of the bonsai tree before and after removal of its leaves. (B) A set of six MRTVPs from which a length estimate was obtained (see Table 2). The FOV of the MRTVPs was 32 cm.

tal branch length of the tree from the MRTVPs are presented in Table 2.

In order to test the accuracy of the estimate from the MRTVPs, two other methods were used to determine the branch length of the bonsai tree. First, a series of 18 photographs were obtained for the bonsai tree at 10” intervals about a fixed axis perpendicular to the focal axis of the camera. An estimate of the branch length was determined from these photographs using the same stereological procedure as was used for the MRTVPs. The scale of the photographs was determined from a ruler placed parallel to the rotation axis and at the same distance from the camera lens. The results of the application of the stereological method of length determination to these photographs are also shown in Table 2.

A further estimate of the branch length was found by a physical method. Strands of cotton were laid

Table 2. Estimating the branch length of a bonsai tree using MR (Fig. 3) and photography

Sample

1

2

3

Pool

MR Photography Angle I(tr) I(tr)

0 25 2.5 30 27 19 60 23 25 90 19 32

120 25 22 150 27 28

total/2 = 73.0 75.5 est(L) = 78.2 73.9

(5.1%) (6.7%)

10 28 26 40 17 22 70 23 20

100 25 22 130 22 26 160 30 22

total/2 = 72.5 69.0 est(L) = 77.6 67.5

(5.1%) (6.7%)

20 25 27 50 28 26 80 16 22

110 15 29 140 25 26 170 24 28

total/2 = 66.5 79 es?(t) = 71.2 77.3

(5.1%) (6.7%)

total/2 = 212.0 223.5 est(L) = 75.7 72.9

The vertical projection samples were analogous to those obtained for the water-filled glass tube (Table 1 and Fig. 2). The cycloid test system used to estimate total branch length by formula (1) was also the same. The final magnification was A4 = 0.3656 for the MR images and M = 0.4000 for the photographic images. The total branch length, estimated directly by a very time consuming method involv- ing cutting cotton strands, was 77.2 cm (SEM = 1.1 cm).

along, and cut to, the length of each branch of the tree in turn. From three such trials a mean value of 77.2 cm (SEM = 1.1 cm) was obtained for the total branch length.

The two stereological estimates of branch length agreed to within 2% of their mean, and their mean was within 4% of the value obtained by physical measurements.

The above investigation has shown the potential for obtaining reliable values for the length of a living structure from stereological methods applied to MRTVPs. The in-situ determination of root length is of course likely to be a more interesting application of


the MR technique than the determination of branch length. However, there are a number of extra consid- erations when investigating a botanical specimen in a growing medium other than air. Next, we report a first attempt to determine the length of two structures seen in MRTVPs obtained in the course of clinical investigation.

Length of the Blood Vessels in Clinical MR Angiography Data

A set of MRTVPs of abdomen and one of brain from adult males, were obtained by taking appropriate projections of 3D phase contrast angiograms. A selection of the abdominal MRTVP angiograms is shown in Fig. 4. The splenic vein is a vessel that can be clearly recognized in the images,” and the results of the application of the above stereological method to determine the length of this vessel, between its junc- tion with the superior mesenteric vein and the splenic hilum, are shown in Table 3. The length estimate of 15.2 cm agrees well with values found in anatomical textbooks. With six projection images, the coefficient of error of the length estimate (namely CEM = SEM/ mean) was 1.5%. With all 18 projections taken to- gether, the CEM cannot be predicted as (1.5%)/a = 0.9% because the three sets of six projections are ev- idently not independent. However, since systematic sampling is more efficient than independent random sampling, the value of 0.9% is likely to constitute an upper bound for the CEM of the pooled estimate of length.

Table 3. Estimating the length of splenic vein from MR abdominal ansGograms (Fig. 4)

Sample Angle JSV)

0 4 6 30 8 6 60 8 8 90 6 5

120 4 5 150 4 4

(mean total)/2 = 17.7 est(L) = 15.0 (1.5%)

10 10 8 40 8 9 70 10 6

100 4 6 130 2 3 160 4 2 (mean total)/2 = 18.2

est(L) = 15.4 (1.5%)

3 20 6 8 50 9 10 80 7 7

110 6 8 140 2 2 170 4 4

(mean total)/2 = 18.2 est(L) = 15.4 (1.5%)

Pool (mean total)/2 = 54 est(L) = 15.2

6 6

10

6 6 4

8 8

10

5 2 4

MRTVP samples were obtained as in the preceding two examples, and analyzed with the same cycloid test system. The final magnification was M = 0.4625. For each projection, three independent superpositions of the cycloid test system were made, to obtain a rea- sonable number of intersection counts.

Fig. 4. Set of MRTVPs obtained by projecting 3D phase contrast MR angiography data of the abdomen at 30” intervals about a fixed vertical axis. The FOV of the images is 48 cm. Length estimates of the splenic vein (identified with the initials SV) are displayed in Table 3.

923

Fig. 5. Set of MRTVPs obtained by projecting 3D phase contrast MR angiography data, of a 5-cm thick slice through the head of an adult male, at 180”/31 intervals about a fixed vertical axis. The FOV of the images is 20 cm. Length estimates of the Circle of Willis (CW) and of the vessels are displayed in Table 4.

A selection of the MR angiograms obtained for the brain are shown in Fig. 5. The results of the stereological determination of the length of the Circle of Willis from 31 regularly spaced MRTVPs covering a range of 185.8” are presented in Table 4. The starting projection angle was not uniform random between 0” and 180”/31 = 5.8”. However, because the latter range is relatively small, the resulting bias is likely to be negligible for all practical purposes. By taking every fourth, the 31 MRTVPs were divided into three groups of eight and one of seven. The pooled estimate of 6.8 cm obtained from these images is again in good agree- ment with values to be found in anatomical textbooks, and is likely to represent an accurate in-vivo determi-

nation of the length of the Circle of Willis (see Ta- ble 5). Table 5 also displays estimates of the total length of blood vessel contained within the MR-imaged volume.

DISCUSSION

The new method described here to estimate the length of a curvilinear feature in 3D enjoys the advan- tages of being noninvasive and unbaised with a mini- mum of requirements, namely a fiied curve shape and not-too-high length density, so that overlapping ef- fects are negligible after projection. No external knowledge is needed other than the direction of the vertical axis about which projections are obtained, the


Table 4. Estimating the length of the Circle of Willis (CW in Fig. 5) and of brain blood vessels from

MR brain angiograms

Sample Angle I (CW) I (VU

1 ox 4 4 6 75 4x 8 6 4 107 8X 2 8 6 109

12x 8 10 4 140 16x 4 8 7 127 20x 8 4 8 126 24x 4 6 4 112 28x 2 2 4 80

(mean total)/2 = 21.8 438.0 est(L) = 6.9 (4.4%) 180.0 (O.SCro)

lx 4 4 4 81 5X 4 4 4 113 9x 8 6 6 120

13x 8 7 10 143 17x 4 6 6 135 21x 4 4 6 113 25x 4 2 4 110 29x 4 4 6 70

(mean total)/2 = 20.5 442.5 e&(L) = 6.5 (4.4%) 181.9 (O.SCro)

2x 2 2 6 93 6x 8 6 4 120

10x 8 8 6 116 14x 6 6 8 134 18x 6 6 8 128 22x 4 4 4 115 26x 4 4 4 98 30x 4 4 4 a7

(mean total)/2 = 21 .O 445.5 est(L) = 6.7 (4.4%) 183.1 (0.8%)

3x 4 6 3 100 7x 4 8 6 107

11x 4 5 10 115 15x 8 8 8 124 19x 4 6 6 116 23x 4 4 8 123 27x 4 4 4 96

(mean total)/2 = 19.7 390.5 esf(L) = 7.2 (4.4%) 183.4 (0.8%)

Pool (mean total)/2 = 85.7 1771.9 e&(L) = 6.8 182.1

Thirty-one MRTVPs were obtained at x = 180”/31 = 5.8” intervals over the range (O”, 185.8”), around a transverse axis of the patient’s head. The data were divided into three samples of eight and one of seven MRTVPs as shown. The cycloid test system used to analyse each MRTVP was the same as in the preceding examples. The final magnification was A4 = 0.9250 for the CW and A4 = 0.7150 for the blood vessel analysis. The bias inherent in sampling over the range (0”, 185.8”) instead of (0”, 180”) is likely to be negligible.

curve length per unit area for the cycloid test system used, and the final magnification. Existing alternative methods based on serial section information, except for the vertical slices method of Gokhale,2 are biased to a variable, unknown extent.

Total vertical projections of a 3D curve can be obtained in a variety of ways, perhaps the most obvious one being the use of ordinary photography (Cruz- Orive and Howard,’ Fig. 2). Digital imaging and computer aided design may also be convenient ways of obtaining and analyzing vertical projections of a 3D curve, for example, a 3D network, a chromosome model, a macromolecule model, and so forth. For the automatic analysis of digitized projections the use of cycloid test curves may not be optimal. Here it may be preferable to use instead a test array of parallel lines making a sine-weighted angle with the vertical direction. The details of this, and of a third method called the weighted rose of directions method are given in Cruz-Orive and Howard. ’

In biology and biomedicine, a powerful way to obtain vertical projections of a curvilinear feature is by MRI, as demonstrated in the foregoing sections. Our last two examples pertain to clinical angiography, but the methodology is likely to be applicable to features other than blood vessels, such as plant roots, provided that the latter are not too congested.

For the first two objects analyzed, namely the twisted glass tube and the bonsai tree, the corresponding true lengths could be measured directly, whereby it was possible to assess the accuracy of our method. Six vertical projections 30” apart yielded estimates within 4% of the true length. Accuracy may be worse for curves exhibiting strong planar anisotropy, but nevertheless three to six systematic vertical projections covering the range (0’) 180’) are likely to yield fairly accurate length estimates in many applications.

Counting intersections between the TVPs of a linear feature and a system of test lines (e.g., cycloids, or sine-weighted straight lines) may be achieved for each projection by analyzing consecutive thin slices which scan the linear feature exhaustively in a direction perpendicular to the projection plane. The test system is kept independently in focus on the observation plane, and intersections are counted between the test lines and the feature projection as soon as the latter appears in focus. A similar technique (called the optical disec- tar) is currently applied to count particles directly in 3D by optical transmission, or by confocal scanning microscopy (Gundersen et al.,12 Cruz-Orive and Wei- be113). Improved technology is likely to bring the preceding procedure accessible to MRI in the next few years.


1.

2.

3.

4.

5.

6.

7.

8.

REFERENCES

Cruz-Orive, L.M.; Howard, C.V. Estimating the length of a bounded curve in three dimensions using total vertical projections. .I. Microsc. 163:101-113; 1991. Gokhale, A.M. Unbiased estimation of curve length in 3D using vertical slices. .I Microsc. 159:133-141; 1990. Cochran, W.G. Sampling Techniques, 3rd Edition. New

York: John Wiley; 1977. Baddeley, A.J.; Gundersen, H.J.G.; Cruz-Orive, L.M. Estimation of surface area from vertical sections. J. Microsc. 142:259-276; 1986. Lauterbur, P.C. Image formation by induced local in- teractions: examples employing NMR. Nature 242: 190- 191; 1973. Edelstein, W.A.; Hutchison, J.M.S.; Johnson, G.; Red- path, T.W. Spin warp NMR imaging. Phys. Med. Biol. 25:756-759; 1980. Chen, C-N.; Hoult, D.I. Biomedical Magnetic Reso- nance Technology. New York: Adam Hilger; 1989:~. 340. Wehrli, F.W. Fast-scan Magnetic Resonance: Principles and Applications. Bristol and Philadelphia: Raven Press; 1991:~. 164.

9.

10.

11.

12.

13.

Dumoulin, C.L.; Souza, S.P.; Walker, M.F.; Wagle, W. Three-dimensional phase contrast angiography. Mag. Res. Med. 9:139-149; 1989. Cline, H.E.; Lorensen, W.E.; Herfkens, R. J.; Johnson, G.A.; and Glover, G.H. Vascular morphology by three- dimensional magnetic resonance imaging. Mag. Reson. Imaging 7:45-54; 1989. Vock, P.; Terrier, F.; Wegmuller, H.; Mahler, F.; Gertsch, Ph.; Souza, S.P.; Dumoulin, C.L. Magnetic resonance angiography of abdominal vessels: Early ex- perience using three-dimensional phase-contrast technique. Br. .Z. Radio/. 64:10-16; 1991. Gundersen, H.J.G.; Bagger, P.; Bendtsen, T.F.; Evans, S.M.; Korbo, L.; Marcussen, N.; Moller, A.; Nielsen, K.; Nyengaard, J.R.; Pakkenberg, B.; Sorensen, F.B.; Vesterby, A.; West, M.J. The new stereological tools: Disector, fractionator, nucleator and point sampled in- tercepts and their use in pathological research and diag- nosis. APMZS 96:857-881; 1988. Cruz-Orive, L.M.; Weibel, E.R. Recent stereological methods for cell biology: a brief survey. Am. J. Phys- iol. 258:L148-L156: 1990.

the application of total vertical projections for the unbiased estimation of the length of blood...

Documents