Phys. Med. Biol. 44 (1999) 1625-1638. Printed in the UK PH: S0O31-9155(99)01121-5

Semi-automated tabulation of the 3D topology andmorphology of branching networks using CT: application tothe airway tree

V Sauretf, K A Goatmant, J S Flemingt and A G Bailey*t Department of Medical Physics and Bioengineering, Southampton General Hospital,Tremona Rd, Southampton S016 6YD, UK$ Department of Electrical Engineering, University of Southampton, Highfield,Southampton S017 1BJ, UKE-mail: veroQsoton. ac. uk (V Sauret)

Received 20 January 1999, in final form 6 April 1999

Abstract Detailed information on biological branching networks (optical nerves, airways orblood vessels) is often required to improve the analysis of 3D medical imaging data. A semi-automated algorithm has been developed to obtain the full 3D topology and dimensions (directioncosine, length, diameter, branching and gravity angles) of branching networks using their CTimages. It has been tested using CT images of a simple Perspex branching network and applied tothe CT images of a human cast of the airway tree. The morphology and topology of the computerderived network were compared with the manually measured dimensions. Good agreement wasfound. The airways dimensions also compared well with previous values quoted in literature. Thisalgorithm can provide complete data set analysis much more quickly than manual measurements.Its use is limited by the CT resolution which means that very small branches are not visible. Newdata are presented on the branching angles of the airway tree.

1. Introduction

Biological branching networks follow a structure which is usually asymmetrical and similarto that of a tree where one segment (parent) branches out at a point (node) into two or moresegments (daughters) or into none (end segment).

Detailed information on biological branching networks (optical nerves, airways or bloodvessels) is often required to improve the analysis of 3D data from medical imaging in terms ofimage registration. Also morphological models of branching networks are valuable in computermodelling. Since the morphology on which they are based often lacks 3D reality, improvedinformation on the network structure and topology will lead to more accurate modelling.

A well known biological branching network is the lung airway structure. Horsfield andCumming (1967) described an asymmetric branching structure using manual measurementson several lung casts. However, the process was very time-consuming and did not yieldinformation on the topology ofthe airways in 3D space.

This paper describes a semi-automated algorithm for obtaining the full 3D topology anddimensions of a branching network using CT images and its application to the CT images ofa lung airway cast to investigate the bronchial morphometry.

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2. Methods

INPUT : Segmented CT images ofthe branching structure±

Skeletonization ofthe structureIdentification of the branches

Morphology and topology calculation for each branch

OUTPUT : 3D morphology and topology tabulation

Figure 1. Steps of the program for analysing the branching structure.

Figure 2. (a) The thinning operation created spurs, (b) The despuring operation removed most ofthem leaving the segment smooth.

In this paper high-resolution CT images of branching networks of airway casts have beenstudied, but the analysis described here may be applied to different branching structures.Confusion can occur between the words 'segment' and 'branch'. In the context of branchingnetwork we will use 'segment' as the noun and 'branch' as the verb.

The first step in the analysis was to segment out the branching structures. Generally, fora uniform material branching network, small segments had a lower CT value than large onesbecause of the partial volume effect and very small segments, smaller than the size of a voxel,were poorly imaged. Non-linear noise in the high-resolution CT images was removed usinga median filter and an appropriate threshold was chosen to binarize the data. It had, however,the disadvantage of excluding the very small segments.

The subsequent analysis, summarized in figure 1 was implemented using a program writteninC.

2.1. Skeletonization of the structure

To obtain the middle line of each segment, a 3D thinning process (Ma and Sonka 1996) wasused. Erosion of the volume layer by layer is carried out using a connectivity preservingalgorithm until only a 'skeleton' is left. This process must preserve the geometry, i.e. an objectlike 'b' should not be converted in an object like 'o'. This leads to some preserving conditionssuch as end points of segments. It must preserve connectivity as well, i.e. an object like 'o'should not be thinned into an object like 'c'. This is essential to avoid splitting or completelydeleting an object or creating or eliminating a hole in an object.

This thinning process left unwanted spurs (figure 2) and so an algorithm was developedfor spur removal. Considering N(p) the 26-connected neighbourhood of each voxel p, thiscleaning process removed the voxel q 26-connected to p if p had more than two neighboursand if q was not 26-connected to any voxels in N(q) — N(p) — N(N(p) — q). This is efficient

3D topology and morphology of branching networks 1627

to remove spurs located along the segments but does not help in the case of a cluster of spursaround a splitting point. For this latter, a manual correction is needed.

2.2. Segment identificationThe constraints to the identification were the following:

• each segment must be assigned a unique identification number (ID) from which thesegment can be located on the network;

• the identity of the parent must be contained in each segment ID;• the voxels of the same segment must be assigned the segment ID;• each node is either bi- or trifurcation;• each segment ID refers also to the full morphological and topological parameters of the

segment;• since the list of segments is not known in advance, the set seen so far must be sorted at all

times, by placing each segment into its proper position in the order it arrives;• each segment terminates when no more points are found (terminal segment) or when a

node is found.A binary tree data structure to track down the tree each segment right and left of the parentseemed natural to use. However, an adaptation had to be made to allow trifurcation in thedata structure (right, left or middle daughters). Each segment was then described by the 3Dcoordinates of its middle and end points and by three pointers to the child nodes. No segmentmay have only one child; it may have zero or two, or even three.

A tree-structured algorithm followed the lines of the skeleton from the trunk of the treeby determining the 26-connectivity of each point moving down the tree. The fact that thethinning process does not create discontinuities in the skeleton is essential for the tree-structuredalgorithm to be able to track down all the segments of the tree. A node was created each time abifurcation was found (connectivity ofthe current segment voxel ̂ 2) starting from the 'root'node (the top of the tree). To prevent back-tracking, the node was marked but the points inthe path to it were deleted. The nodes were maintained so that at any node, the left and rightsubtrees contained respectively the first and the second child segment found. If trifurcating,the third child segment was defined as the middle subtree. This search process was inherentlyrecursive to methodically descend all the segments of the branching network and form thetree structure of nodes. Accordingly a recursive routine for printing was used, the directionleft being defined as 1, right as 2 and middle as 3. The output segment names followed aconvention close to the Lovelace convention (Phalen et al 1978) but slightly different becauseof the introduction of the middle direction.

2.3. Morphology and topology calculation for each segment

The complete 3D morphology of the branching network was given by the position andidentification of each segment with the following algorithm. The information about eachsegment was treated as a unit instead of as separate entities. Therefore the algorithm waswritten using an array of segment structures; each segment was described with the followingcomponents:

• record number of the segment;• name;• generation number, defined as the number of digits in the name;• record of the parent segment (set to —1 for the seed point, top of the tree);

1628 VSauretetal

Branch.. JI: lengthB

Parent (Erection

I I pointsR

Figure 3. Morphometric information obtained for each segment: (left) length AB, branchingangle /J, gravity angle y and (right) diameter.

• number of daughters;• 3D coordinates of the first, middle and last points A, M and B ofthe segment;• 3D cosine direction of the segment;• morphometric information (figure 3):

lengthdiameterbranching anglegravity angle;

• an error flag, set in case of a cluster around the splitting point. Manual correction of thenode neighbourhood was needed to correct that.

Also, in the view of the application to the airway tree the lobe to which the segment belonged(right upper, right middle, right lower, left upper or left lower) was added to the segmentstructure.

The length of a segment starting at a point A and ending at a point B, was defined as theCartesian distance AB between the point A and the point B:

AB = [(x« - xA)2 + (yB - yA)2 + (zB - ZA)2]l/2. (1)Its gravity angle (90° for a horizontal segment) was then defined as the inverse cosine of thescalar product between the gravity direction g and the direction AB:

A B c o s y = A B > g . ( 2 )The branching angle (angle formed between the parent direction and the studied segmentdirection) was derived from the scalar product between the parent direction D and AB:

A B c o s B = D - A B . ( 3 )To calculate the diameter, first the object points that were 6-connected to a background pointin the CT images were detected as edge points. For each segment AB and its middle point M,the points />,- were the edge points belonging to the plane IT centred in M and perpendiculartoAB:

Pi eTl: axpf + bypi + czpi = dwherea = xB —xA, b = yB—yA*c = Zb—za and d = axM+byM+czM- The mean diameterd was calculated as:

2 £ , M P,d =EtP i (4)

3D topology and morphology of branching networks 1629

and as long as the number of points Pj (±10%) was greater than the number of points thata circle of diameter d would have (ltd), d was recalculated with only the points />, such asMPi < d.

The lobe to which each segment belongs was identified by looking at the relative valuesof its direction vector coordinates. The right and left side of the picture and its descendingaxis were requested from the operator and the lobe name was stored in the segment structure.

2.4. VerificationA simple branching network phantom made of Perspex was CT scanned. It consisted of amain vertical 20 mm diameter cylinder from which 11 smaller cylinders branched out at anglesranging from 10 to 130°. Their diameters ranged from 2 to 10 mm and their lengths from 1 to4 cm. The branching angles were the same as the gravity angles. 3D visualization of all theimages was done on a Sun station using the software Analyze version 7. The contrast betweenthe background voxels and the object voxels was good, therefore the phantom segments weresegmented out by thresholding.

To assess the correctness of the thinning process it was applied to the phantom CT images.The calculation of the morphological parameters was checked by comparing them with themanual measurements of the phantom segments.

2.5. Practical investigation

High-resolution CT scanning was performed on a human lung airway cast made of plasticmaterial. Each scanned slice was a coronal view of the lungs. The parameters used were thefollowing: 273 slices, 1 mm slice thickness, 0.781 mm x 0.781 mm pixel size, axial acquisition.These data were reformatted into 1 mm cubic voxels with an 8-bit grey scale. Since the contrastin the cast images between the background and the airways was high, the segmentation ofthenetwork was achieved by thresholding. With an absolute grey-level threshold of 60 there were3079 airways identified by the tree-structured algorithm in the skeleton formed by the thinningprocess. A higher threshold would remove some of the smallest ones identified. A lowerthreshold would not allow them to be differentiated from the noise. Figures and tables shownbelow are related to the CT images threshold at 60.

To assess if all the airways were included in the tree formation process, the number of segments per generation was computed. The voxels defined as object points by thinning but not bythe tree formation process were also saved in a file. If a voxel was missed by the tree formationalgorithm, manual examination of the bifurcation neighbourhood was used for correction.

To evaluate the accuracy and precision of the computed morphological dimensions, thediameters of the few easily accessible airways were manually measured and compared withthe computed ones. Also the mean values ofthe lengths, branching angles, gravity angles anddiameters of same generation airways were compared with those found in the literature.

3. Results

3.1. Verification

Following thresholding the thinning algorithm detected all the segments of the phantom(figure 4). It did not break the continuity of the network skeleton. The behaviour of thethinning algorithm depended on the ratio of parent-to-daughter diameter. The skeleton of adaughter segment which was small compared with the parent deviated from the path of a largecylinder less than a larger one (figure 5). Therefore an alternative algorithm was used to revise

V Sauret et al

Figure 4. CT scan ofthe Perspcx phantom and its skeleton (black lines overlaid). The perspectiveeffect explains why some segments seem inside the main vertical cylinder. They are in fact in anoblique plane to the paper.

theoreticalthinning algorithm

Figure 5. Behaviour of the thinning algorithm when the ratio parent-to-daughter diameter is (a)about 1 and (/;) much greater than 1. In the case (/•). the position of A should be changed to A' inthe angle calculation equations.

the location of the starting point A of the daughter. The length and branching angle of thedaughter segment were evaluated as described above using this corrected starting point.

To compare the dimensions calculated from the images ofthe phantom segments with theirmanual measurements, Bland-Altman graphs (difference between the two methods against theaverage) were plotted and significance tests were undergone (Bland 1996).

For the length and angle calculations the mean of the difference between the computedand manual measurements ofthe phantom segments was found to be equal to 1.1 mm and 2.96°respectively, the standard error of the mean to 1.2 mm and 4.46° respectively. On the basisof the P value (12 degrees of freedom, T = 0.92 and T = 0.66 respectively), we concludedthat no significant difference between the computed and the manual methods was shown in thelength and angle calculations.

The Bland-Altman graph for the diameter calculations (figure 6) shows an over-estimationof the small diameters. This is certainly due to the partial volume effect and to the fact thatthe number of points taken into account in the diameter calculation (4) is very small (oneor two points). The T value calculated lay outside the 95% limits of agreement when the2 mm diameter cylinders were included in the statistics (T = 2.4, 15 degrees of freedom) butinside when they were excluded {T = 1.3, 12 degrees of freedom) showing then no significantdifference between the computed and the manual dimensions.

3D topology and morphology of branching networks

mean(phys. meas., computed)

diameter manually measured( m m )

mean--0.3 mm SD=0.5 rrmT--2.4mean=0.2mm SD=0.56mm T=1.3 (2mm diam excluded)

Figure 6. (a) The computed diameter of each segment ofthe phantom was plotted versus its manualmeasurement. (/?) Bland-Altman graph for the phantom segment diameters.

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Figure 7. Superimposed I mm thick CT images of the airway cast and its skeleton (black linesoverlaid). The perspective effect explains why some segment middle lines seem to cross otherairways.

3.2. Practical investigation

The skeleton picture ofthe airway cast is shown in figure 7. Good agreement with the originalCT image is apparent. However, clusters appeared at bifurcation sites as shown by the errorflags. The identification process could not then distinguish several daughter segments andmissed them in the tree description (figure 8). In the first nine generations 150 segments wererecorded with the error flag set. Figure 9 shows the new residual data file after their manualcorrection.

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3D topology and morphology of branching networks 1633Table 1. Number of airways per generation number.

Generation Computed number Number of segmentsnumber of airways (always bi-furcating tree)12

12

12

3 4 44 8 85 16 166 33t 327 66 648 128* 1289 249§ 25610 345 51211 430 102412 420 204813 337 409614 266 819215 216 1638416 159 3276817 119 6553618 105 13107219 77 26214420 and more 133 IO6

t One trifurcation.X Two airways terminate at generation 7.§ One trifurcation, four airways terminate at generation 8.

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mean (manual mess., computed meas.) (mm)

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Figure 10. Bland-Altman graph of 13 airway diameters of the cast.

The statistics on the diameter of the few first accessible airways of the cast (13 airways,mean = 0.8 mm, SE = 0.4 mm) did not show a significant difference between manual andcomputed measurements (95% confidence interval = [—0.07 mm, 1.67 mm]).

Figure 10 shows the Bland-Altman graph ofthe diameters. The arrow points at the biggestdifference between manual and computed diameter dimensions. It is relative to the first leftmain bronchus that is very curved unlike all the other airways of the cast. This difference isexplained by the interpolation of AB as the normal vector to the plane n (figure 3).

The computed diameters were averaged for each generation n and plotted against thegeneration number from generation 1 to 9 (figure 11). Table 2 (first column) summarizesthe averaged relationship between daughter diameters and lengths and daughter and parentdiameters, and branching angle values for all the airways up to generation 9.

1 6 3 4 V S a u r e t e t a l

Table 2. Airway diameters and lengths ratios, and averaged branching angle.

Computed ratios up to generation 9 Published ratiosL e n g t h / d i a m e t e r t 3 . 4 9 2 . 8 - 3 . 2 5 [ |Smaller to larger conjugate daughters diameters* 0.81; range: [0.38,1.00] 0.86; range: [0.5,1]1R a t i o o f c o n j u g a t e d a u g h t e r s l e n g t h s * 0 . 5 9 0 . 6 2 5Ratio of daughter diameter to parent diameter§ 0.75 0.79^; range: [0.74,0.82]+A v e r a g e d b r a n c h i n g a n g l e 4 2 . 5 ° r a n g e : [ 3 0 ° - 4 3 ° ] +

t For 1 mm < diameters < 4.6 mm.* Up to generation 7.§ For diameters > 1 mm.|| Weibel (1963) for 1 mm < diameters < 4.6 mm.«| Weibel (1963) averaged up to generation 5-7.+ Horsfield and Cumming (1967).

Mean gravity and branching angles per generation up to generation 9 are given in table 3for the right and left upper lobes. It was found that generally minor daughters branch out at agreater angle from the parent than the major ones.

4. Discussion

The thinning process provides a skeleton of the original CT images that represents well themain structure of the branching network (figures 4 and 7). Other image processing on airwaytrees had some segment middle lines laying outside their actual segment (Wood et al 1995).This justifies the choice of this thinning process. It is quite good at finding a one voxelwide skeleton line for simple branching networks (figure 4). It does not create discontinuoussegments but can generate clusters at bifurcation for complex branching networks, which thenrequire manual correction (figures 8 and 9). It is unlikely that a fully automatic algorithmcapable of dealing with the large variety of anatomy found in the airway tree is feasible. Acombination of resolution and noise also limits the number of segments detected by the thinningalgorithm. The choice of segmentation method, which is generally application dependent, isalso important. The images used in this study were CT images of solid plastic cast andphantom. Due to the high contrast between the background and the branching network, astraightforward thresholding method proved to be adequate. However, the contrast in imagesof solid plastic objects is higher and the noise lower than they would be in clinical practice.In the case of in vivo images, thresholding techniques are very limited to segment out objectsand more complex automatic segmentation methods should be applied. Initial results using anautomatic region-based algorithm (Burt et al 1981) were encouraging. Better quality imagescould be obtained by imaging for longer and storing the data in a larger matrix. However, verydetailed CT images will also occupy huge memory space. Although this is possible with castmeasurements, if this were to be extended to in vivo human measurements the subject wouldreceive a higher x-ray dose than for a lower-resolution scan.

The tree reconstruction algorithm developed here identifies each segment of a branchingnetwork from its skeleton and gives the full information on its location in the 3D space.Extensive work on simulated images showed that it is not limited by the degree of curvatureof the segments except for the instance where the segment becomes a closed loop.

The morphology computation algorithm gives statistically satisfactory results (figures 6and 10) for the phantom and for the airway cast when compared with manual measurementsof the segments. It has been shown in the simulated images that the calculation method is notapplicable in the case of a significantly curved segment as the length is underestimated and

3 D t o p o l o g y a n d m o r p h o l o g y o f b r a n c h i n g n e t w o r k s 1 6 3 5

Tfcble 3. Mean gravity angles y and mean branching angles fi per generation for the right upperlobe, right middle lobe, right lower lobe, left upper lobe and left lower lobe.

Generation K(°) fi(°) Number of airways

Right upper lobe 1 62 t I t2 53 37 13 99 48 14 108 49 25 104 35 46 105 37 87 93 35 168 96 40 329 94 47 64

Right middle lobe 1 62 t I t2 53 37 13 36 17 14 71 40 15 78 23 26 90 47 57 93 43 108 84 47 169 80 47 33

Right lower lobe 1 62 t I t2 53 37 13 35 17 14 43 48 15 45 72 16 41 46 27 48 49 48 60 38 89 71 36 16

Left upper lobe 1 62 t I t2 62 42 13 HI 54 14 103 34 25 HI 37 46 113 34 87 112 36 168 104 45 329 101 48 55

Left lower lobe 1 62 t I t2 62 42 13 57 17 14 74 44 25 74 45 46 77 37 87 79 39 168 78 43 329 83 48 65

t Trachea.

the direction of the normal vector to the plane in which the diameter is calculated is wrong.An alternative algorithm could be used to measure length and diameter in this instance by alsosaving the number and the coordinates of all the voxels belonging to the skeleton of the curvedsegment when reconstructing the tree structure. The coordinates of only the first, middle and

1 6 3 6 V S a u r e t e t a l

end voxels of the skeleton of straight segments need to be saved to compute their morphology.Resolution influences the accuracy ofthe computation ofthe morphometry.

Applied to the lung, the advantage of this method is clear when compared with the waythe human lung morphometry has been found in the literature: it was based on the dimensionsof a few human lung casts measured manually; an extremely time-consuming process. Thecurvature limitation ofthe dimensions computing algorithm did not affect our measurements onthe airways since they are not significantly curved, as shown in the airways skeleton (figure 7).The long first left main bronchus after the trachea is an exception and the small error in itsdiameter calculation was noted (figure 10).

The published lung morphometries were based on the following casts: Weibel (1963) usedresin and silicon rubber casts, Horsfield and Cumming (1967) used the cast of a 25-year-oldmale, the Lovelace group (Phalen et al 1978) used the silicon rubber casts of two males, 50and 60 years old. Menache and Graham (1987) related results from work done by Mortensenand co-workers in 1983 on 11 casts of children and young adults aged 0 to 21 years. The datawere always gathered by airway generation: the trachea was called generation 0 by Weibeland Horsfield, the first branching point being then the carina, but was called generation 1 in theLovelace data and Mortensen's. Different systems of identification of the airways have alsobeen used. We used the Lovelace identification scheme for its adaptability to the tree-structuredalgorithm.

The number of airways present in the casts used in previous work was estimated to bebetween approximately 30 000 and 60 000 if they were all bifurcating (Menache and Graham1997) but the number of single airways actually measured varied from one author to an other.The number of airways found by our computed method was limited by the resolution of theCT scanners (1 mm) and the casting procedure. It becomes very apparent from generation 10and onwards (table 1). Therefore we did not process any calculations using data from thosegenerations.

Our computed data on airways agree with previous published work summarized in table 2(Weibel 1963, Horsfield and Cumming 1967).

The mean ratio length/diameter in this study was calculated for airways with a diameterbetween 1 mm and 4.6 mm to be directly comparable with Weibel's average. Reasonableagreement is obtained; however, our calculation was limited to the first nine generations asexplained above. It is considered that an average including higher-generation data would givea value within Weibel's range.

The computed ratio of the daughter diameter and parent diameter, calculated for theconductive airways, averages at 0.75 which shows that the correction in the angle calculationmade for the phantom segments was not necessary in the airway structure. This value concurswith Weibel's relationship between diameter and generation number (n) in his theory ofdichotomy (Weibel 1963) (5) and is also within the range that Horsfield and Cumming (1967)found (table 2). Koblinger and Hofmann (1985) plotted this same relationship rearranged (6)versus the generation number

d { r i ) = 2 -n ' 3d (0 ) & < / ( "V ) = 2 "1 '3 = 0 .79 (5 )din)< & d ( n ) = 2 ~ l / 3 d ( n - 1 ) . ( 6 )

Figure 11 shows good agreement between theory (6) and the results of the algorithm. Itdemonstrates that at each bifurcation the airway diameter decreases, which agrees with Weibeland Horsfield's general observations. This also generally holds for the airway length.

The computed ratios of the diameters and of the lengths of conjugate daughter segmentsare respectively 0.81 and 0.59 for larger airways (table 2). These agree with Weibel's average

3D topology and morphology of branching networks 1637

EE

13i -IOESvcCOoS

4 5 6 7 8 9generation n

Figure 11. Mean diameters d(n) in mm computed from the CT images of the cast for eachgeneration n are plotted against the generation number (triangles). The equation d{n) =2",/3<i(« - 1) versus the generation number is also shown (curve).

measurements of the airways up to generation 5-7 and are also confirmed by Phalen et al(1978).

In the literature, branching angles and angles of inclination to gravity have not beenmeasured as systematically as the diameter and length values, which were measured for moreor less each airway of the lung by several authors. The most complete set of angular dataappears to be from Yeh and Schum (1980). However, surprisingly it did not show any segmentgoing upwards in any lung lobe (gravity angle >90°), whereas in our cast the mean gravityangle per generation and per lobe, calculated up to generation 9, was greater than 90° in 7generations in the right and left upper lobes (table 3).

Our results also show the branching angle proportional to the diameter of the daughter inthe sense that the minor daughter branches at a greater angle than conjugate major one. Thisagrees with previous observations (Phalen etal 1978). The theory for dichotomy forecasts anideal branching angle figure of 37°28\ Horsfield and Cumming (1967) found that the meanbranching angles varied between 30° and 43°. We found it averaging at 42.5°.

This algorithm contributes to our knowledge of lung morphology by providing a largeset of data, especially on branching angle values and 3D topology, on which there is littleinformation in the literature. Analysis of CT images of more casts using this algorithm couldquickly provide intersubject data sets as well. Generally, detailed branching networks willalso be useful for 3D viewing and teaching purposes.

Future use of the morphological data obtained with this algorithm includes thedevelopment of a computed 3D morphological airway tree model. This will be used for betteranalysis of radionuclide images of aerosol deposition in terms of deposition in the differentgenerations or segments ofthe airway structure. It will also be valuable in improving computermodelling of deposition.

5. Conclusion

This algorithm calculates semi-automatically both the location in 3D space and themorphometry of branching networks from their CT images. The method has been successfullyapplied to casts ofthe human airway tree where it has been shown to be a great improvement onthe very time-consuming manual technique used previously. Information on airway branching

1 6 3 8 V S a u r e t e t a l

angles and 3D topology are especially valuable since the literature has little information onthem. This algorithm may also be used to provide intersubject data sets. The algorithm ismainly limited by the CT resolution. In application to in vivo data, this would be related to thex-ray dose delivered to the subject.

Acknowledgments

The authors wish to thank Dr Milan Sonka for providing the thinning algorithm code and theAnatomy Department at Guy's Hospital, London for the loan of the cast. The project wasfinanced by a University of Southampton, Faculty of Engineering studentship.

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