Quantitation of articular surface topography and cartilage thickness in knee joints using stereophotogrammetry

.I. Biomechantcs Vol. 24. No. 8, pp. 761-776,

0021-92YO/91 S3.00+.00 Pergamon Press plc

1991.

Printed in Great Britain

QUANTITATION OF ARTICULAR SURFACE TOPOGRAPHY AND CARTILAGE THICKNESS IN KNEE JOINTS USING STEREOPHOTOGRAMMETRY G. A. ATESHIAN,* L. J. SOSLOWSKY* and V. C. Mow?

*Orthopaedic Research Laboratory, Departments of Mechanical Engineering and Orthopaedic Surgery, Columbia University, New York, NY 10032, U.S.A. and tDirector, Orthopaedic Research Laboratory, Departments of Mechanical Engineering and Orthopaedic Surgery, Columbia University, New York, NY 10032, U.S.A. Abstract-An analytical stereophotogrammetry (SPG) technique has been developed based upon some of the pioneering work of Selvik [Ph.D. thesis, University of Lund, Sweden (1974)] and Huiska and coworkers [J. Biomechanics 18, 559-570 (1985)], and represents a fundamental step in the construction of biomechanical models of diarthrodial joints. Using this technique, the precise three-dimensional topography of the cartilage surfaces of various diarthrodial joints has been obtained. The system presented in this paper delivers an accuracy of 90 pm in the least favorable conditions with 95% coverage using the same calibration method as Huiskes et al. (1985). In addition, a method has been developed, using SPG, to quantitatively map the cartilage thickness over the entire articular surface of a joint with a precision of 134 pm (95% coverage). In the present study, our SPG system has been used to quantify the topography, including surface area, of the articular surfaces of the patella, distal femur, tibia1 plateau, and menisci of the human knee. Furthermore, examples of cartilage thickness maps and corresponding thickness data including coefficient of variation, minimum, maximum, and mean cartilage thickness are also provided for the cartilage surfaces of the knee. These maps illustrate significant variations over the joint surfaces which are important in the determination of the stresses and strains within the cartilage during diarthrodial joint function. In addition, these cartilage surface topographies and thickness data are essential for the development of anatomically accurate finite element models of diarthrodial joints. The geometric data obtained in this study may also be useful for other bioengineering applications such as the determination of joint contact areas [Ateshian et al., ASME Biomech. Symp. 98, 105-108 (1989); Soslowsky et al., ASME Adu. Bioengng 15, 129-130 (1989); Soslowsky et al., Riomechanics ofDiarthrodia/ Joints, pp. 243-268. Springer, NY (1990)] and the development of anatomically accurate artificial joints.

INTRODUCTION

Significant advances have recently been made toward the development of analytical and computer models of diarthrodial joints. These advances include the development of: (1) the infinitesimal and finite deformation biphasic theories for articular cartilage (Mow et al., 1980; Kwan et al., 1990) and associated finite element formulations (Spilker et ai., 1987, 1988; Spilker and Maxian, 1990, Suh et al., 1991); (2) the boundary conditions at the cartilage-synovial fluid interface (Hou et al., 1989a,b); (3) the determination of the material coefficients and constitutive equations for synovial fluid (Lai et al., 1978; Schurz and Ribitsch, 1987); and (4) analytical stereophotogrammetry (SPG) methods for the determination of joint surface topographies (Ghosh, 1983; Huiskes et al., 1985; Ateshian et al., 1988). The development of a technique for obtaining accurate and quantitative descriptions of the geometry of the articular cartilage surfaces is essential to our development of accurate computer models for stress--strain analyses of diarthrodial joints. Such a complete model of diarthrodial joints must include information not only on the topography of the

Receioed

in final form 26 February 1991.

cartilage surface, but also on that of the subchondral bone, thus creating the solid geometric model required for finite element analysis. This information will aid in: (1) the calculation of stresses and strains in the cartilage due to joint contact; (2) the determination of modes of lubrication during joint function; and (3) the assessment of biomechanical effects of subtle changes in joint surface congruence on biomechanical function. Many methods have been used to quantify diarthrodial joint surfaces. Mechanical techniques include the production of plastic moldings (S&horn et al., 1972), the production of a silicone rubber mold used to make a plaster casting (Scherrer, 1977; Scherrer and Hillberry, 1979), and the use of a mechanical measuring pin attached to a dial gauge (Wismans et al., 1980). Other methods such as slicing (Shiba et al., 1988) and ultrasound have also been described (Rushfeldt et al., 1981). More recently, optical techniques have been used such as close-range photogrammetry (Ghosh, 1983) and analytical stereophotogrammetry (Huiskes et al., 1985; Ateshian et al., 1988; Soslowsky et al., 1989a). In addition, CT and MRI technologies have also been used to measure anatomical structures of diarthrodial joints (Moon er al., 1983; Belsole et al., 1988; Feldkamp et al., 1989; Garg and Walker, 1990). Similarly, for the determination of articular cartilage 141

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G. A. ATESHIANet al.

several methods have been used including measurements obtained from radiographs (Armstrong and Gardner, 1977; Hall and Wyshak, 1980), slicing specimens fixed in formalin (Meachim et al., 1977), successively photographing specimen slices (McLeod et al., 1977), and ultrasound (Rushfeldt et al., 1981; Modest et al., 1989). In this paper, we will describe in detail our SPG system and mathematical method required for the determination of cartilage thickness maps. Analytical stereophotogrammetry is a technique used to obtain precise three-dimensional measurements of an object through the process of recording and measuring two-dimensional photographic images, combined with appropriate mathematical computations using both perspective and projective geometry. From the SPG technique, a three-dimensional mathematical model is constructed based on information obtained from digitized points on two twodimensional photographs of the object surface (Selvik, 1974; Ghosh, 1979). Our SPG system is specifically designed for diarthrodial joint studies and is capable of measuring cartilage and underlying bone surface topography, surface areas, and cartilage thickness. This paper presents a description of our SPG system, its calibrated precision and accuracy, as well as its use to determine these anatomic quantities for the human distal femur, retropatellar surface, menisci, and tibia1 plateau. thickness,

MATERIALSAND EXPERIMENTALMETHODS

The SPG apparatus consists of several components necessary for data acquisition (Fig. 1). These include a calibration frame [Figs 2(a) and (b)], a specimen mounting system, a high intensity optical spotlight, two precision large format cameras, a film processor, an X-Y coordinate digitizer with video camera and monitor, and supporting computer facilities, all of which are described in detail below. Specimens

Twelve normal patellae, twelve tibia1 plateaux with menisci, and three distal femora were harvested from freshly frozen human cadaver specimens. The patellae (four paired, four single) were obtained from four females, two males (average 34 yr old), and two specimens of undetermined age and sex; the tibia1 plateaux and menisci (six pairs) came from three female and three male specimens (average 34 yr old); the femoral surfaces (one paired, one single) were obtained from one female and one male specimen (average 44 yr old). Specimens were stored at - 25 “C and thawed prior to dissection. During dissection and preparation, all specimens were kept moist using a physiologic saline enzyme-inhibitor solution to retard dehydration and specimen degradation. Only specimens appearing normal upon visual inspection were included in the study.

Precision Calibration Frame

Laboratory Reference

Film Plane

High Intensity Optical Spotlight Fig. 1. The stereophotogrammetry apparatus consists of several components including the precision calibration frame, two cameras, and a high intensity optical spotlight (computer, digitizer, and film processor not shown). The precision calibration frame defines a laboratory reference frame, the collimation angle is 4, and the distance between the lens plane and film plane isf:

Quantitation of articular surface topography Calibration frames

Two precision calibration frames of different dimensions are used for a wide range ofjoint sizes. The larger frame has outer dimensions of 24.4 x 20.3 x 5.1 cm with a working space of 12.7 x 12.7 x 5.1 cm, large enough to accommodate any diarthrodial joint surface in the human body. The smaller frame has outer dimensions of 12.2 x 10.2x2.5 cm, and a working space of 6.4 x 6.4 x 2.5 cm, which is used for smaller joints such as those of the hand. The frames are made from a reaction-bonded silicon nitride ceramic material (S&N,) which has a significantly lower coefficient of thermal expansion than steel or aluminum. Sixteen custom-made optical (calibration) targets are affixed to each frame and serve as control points for the stereophotogrammetric procedure [Figs 2(a) and (b)]. These targets are made of a white opaque film, and consist of a black crossline surrounded by concentric circles used for optical alignment (line width= 100 pm on targets used for the large frame, 50 pm for the small frame). The three-dimensional coordinates of the targets affixed to the calibration frame have been obtained to an accuracy of + 6 pm using a non-contacting optical coordinate measuring machine for location of two dimensions and a contacting height gauge for the measurement of the third dimension (GageLine Technology Inc., Rochester, NY). Specimen mounting system

After disarticulation and removal of the surrounding soft tissue, the joint surfaces are positioned within the calibration frame using a specially designed specimen mounting system. This system consists of a disposable Plexiglas specimen plate to which the specimen is rigidly fixed using a cancellous screw and cyanoacrylate cement [Figs 2(a) and (b)]. This specimen plate is rigidly fixed to the inner alignment frame [Figs 2(a) and (b)]. Optical alignment targets, of the type described above, are affixed to 12 Acetal round rods of various lengths (l-5 cm) which are rigidly mounted on the alignment frame. These alignment targets and rods are distributed in a non-uniform fashion around the periphery of the specimen on the alignment frame. During photography, the specimen, fixed to the alignment frame, is inserted into the calibration frame. The entire assembly is supported on an aluminum base resting on vibration pads. Optical spotlight and grid

A high intensity optical spotlight, equipped with an F2.8 7&125 mm zoom lens and a flash tube rated at 1200 W. s, projects a fine grid pattern on the joint surface. These grid intersections serve as recognizable target points (nodes) on the joint surface which will be digitized from photographs [Fig. 2(b)]. The grid pattern, produced with line width of 12.5 pm and line spacing of 72.5 pm, is projected from a 35 mm glass slide. When projected on the joint surface, the line width varies from 5&150pm with line spacing of

763

approximately 0.7-2.3 mm depending on the size and curvature of the joint surface under consideration. Photography and digitizing

The frames and specimen are photographed from two camera positions separated by a collimation angle 4%400, using two 20 x 25 cm (8 x 10 in) format cameras equipped with high quality F9 240 mm lenses set at F90 to provide the necessary depth of field (Fig. 1). Black and white 20 x 25 cm (8 x 10 in) Polaroid print film is used in the cameras and is processed in the laboratory. Calibration targets, alignment targets, and grid intersections appearing in each stereogram (the pair of photographs) are digitized using the Aristomat 102D X-Y digitizer which has a resolution of 5 pm and a rated accuracy of 20 ,um. The digitizer consists of a 86 x 86 cm working table, an expansion module with a serial port for on-line connection to a host computer, and a closed-circuit television system. An interactive graphics program displays the digitized points during data acquisition and saves their coordinates on a computer file. Digitizing time for an experienced operator varies from 2 to 4 h for an average size joint surface. Bone surface

For quantifying the underlying bone surface of the joint and measuring the cartilage thickness, the specimen and alignment frame are removed from the calibration frame after being photographed, and are carefully submerged in a solution of 5.25% sodium hypochlorite (Boyde and Jones, 1983; Bullough and Jagannath, 1983). This solution gently dissolves the cartilage layer over a period of 3-6 h down to the level of the tidemark. This surface is then carefully sprayed with a thin layer of white paint (measured to be < 15 pm) to improve grid contrast. The specimen and alignment frame are subsequently repositioned inside the calibration frame and the bone surface is photographed and digitized using the same procedure as for the cartilage surface. The alignment targets, which do not move relative to the bone substrate of the specimen, are used to realign the bone and cartilage surfaces into a common frame of reference [Figs 2(a) and (b)]. Accuracy and precision experiments

Accuracy and precision measurements were performed on a stainless steel cylinder (radius = 50 mm, height = 120 mm). Thirty marks, engraved on the cylinder surface and calibrated with a high-accuracy mechanical digitizer, provided check points for these measurements. In addition, one of the patellae (both cartilage and bone surfaces) and its associated set of alignment targets was also used for precision measurements. The three experiments performed to assess the accuracy and precision of our SPG system are outlined below. Experiment 1. The calibration cylinder was photographed once with its long axis horizontal and once

764

G. A. ATESHIANet al.

with its long axis vertical. The coordinates of the check points were digitized six times from the same stereogram for each position. Experiment 2. Each of the cartilage and bone surfaces were photographed six times. The surface and projected grid were repositioned each time, and thus the same surface was described by a different set of nodal grid points on each of its six stereograms. All visible nodes and alignment targets were digitized from these stereograms. Experiment 3. One stereogram for each surface was selected from Experiment 2, from which a subset of 30 nodes were digitized six times each.

MATHEMATICAL PROCEDURES

forms of the nonlinear equations resulting from the collinearity condition. Some of the more sophisticated models include terms which account for lens distortion effects. For the present application, a modification of the methods presented by Faig and Moniwa (1973), and Ghosh (1979) is used. A least-squares analysis is performed on the governing equations since the number of control points (calibration targets on the ceramic calibration frame) is redundant for the determination of the camera interior and exterior orientation parameters. After the two camera positions are obtained, the three-dimensional coordinates of object points (targets or nodes) are reconstructed using space intersection (Fig. 3). This method has been described in detail by other authors (e.g. Huiskes et al., 1985).

Three-dimensional reconstruction

Surface realignment

The purpose of this technique is to create threedimensional geometric models of diarthrodial joint cartilage and bone surfaces and to use these two geometric surfaces to calculate cartilage thickness. More specifically, the spatial coordinates of the projected grid intersection points on the joint surface, or nodes, are to be reconstructed from their respective photographic coordinates (i.e. from the two-dimensional photographic coordinates of each node digitized from the stereogram). The first step in the reconstruction procedure consists of determining the two camera positions accurately (camera calibration and space resection) using the collinearity condition. Numerous investigators have described the various

The reconstructed cartilage surface of a joint and its corresponding underlying bone surface have two sets of nodes, each belonging to a different reference frame. In order to obtain a common reference frame, a rigid body rotation and translation are performed on the bone nodes, which transform the bone surface reference frame into the cartilage surface reference frame. The rotation matrix and translation vector used in this transformation are evaluated from the alignment targets which form a common set to both reference frames. Since there are more than three alignment targets, either a least-squares or an eigenvalueeigenvector analysis is used (Spoor and Veldpaus, 1980). precision calibration frame

alignment frame

specimen

plate

n

alignment targets calibration targets

Fig. 2(a). A human patella is rigidly fixed to a disposable specimen plate which is rigidly fixed to an alignment frame with alignment targets and placed inside a precision calibration frame with calibration targets.

Fig. 2(b). A grid is projected on the cartilage surface defining a large number of recognizable nodes (e.g. 336 on the surface shown).

765

P

(a)

‘l’hicknrtee

(b) Fig. 6. (a, b).

766

Thicknero 7.00

6.00 5.00 4.00 3.00 2.00 1.00 0.00

M

mm

Fig. 6. (c, d).

167

(e)

'l'hicknesa 4.00 3.20 2.40 1.60 0.m 0.00

Fig. 6. (e, f). Fig. 6. Cartilage thickness is displayed as grey shades superposed on the topographical map of the paired bone surfaces of typical (a) left and (b) right patellae; (c) left and (d) right tibia1 plateaux; and (e) left and (f) rigbt distal femora demonstrating the variation of thickness over the various surfaces (M=medial, A = anterior, P = proximal).

768

169

Quantitation of articular surface topography

photograph 2

photograph 1

Fig. 3. Space intersection is used to determine the three-dimensional coordinates of object points pL(targets or nodes). In theory, pk is located at the intersection of the two lines originating from the perspective centers of the camera Oi in the two different locations and orientations, and passing through the corresponding photographic coordinates of the node (xi. y&f, andf, are the focal lengths of the two cameras. In practice, the two lines do not intersect in space due to digitizing inaccuracies and unaccounted lens distortion. The intersection point is approximated as the midpoint of the shortest segment (residual parallax)joining the two lines.

Surface patches To obtain a geometric model of a joint, it is necessary to provide a mathematical description of its entire articular surface. The three-dimensional coordinates of the nodes of a surface (either bone or cartilage), expressed in the fixed laboratory reference frame, may be curve-fitted using a set of piecewisecontinuous parametric bicubic surface patches (Coons, 1967; Scherrer and Hillberry, 1979). In the method outlined in this paper, the corner points of the surface patches correspond to the projected nodes on the joint surface. The twist vectors at each node are set to zero producing Ferguson- or F-patches. Hence, the patches have first order (C’) continuity across their boundary curves and second order (C’) continuity at the corner nodes. The spline curves, fitted along the two parametric directions of the surface, are natural splines, i.e. the curvature at the end points is set to zero. Piecewise Cl-continuous curve-fits pass through all the data points, thus ensuring that complex geometric shapes are faithfully reproduced. Cartilage

thickness

Cartilage thickness over a joint surface can be obtained from the quantitative knowledge of the cartilage surface (S,), as well as the underlying bone surface (S,). Thickness at a point on the bone surface can be defined as the length of the normal vector originating at that point and ending at the cartilage surface (Fig. 4).* Given the equations of the bicubic patches constituting the two surfaces in a common reference frame, this ‘thickness’ vector is evaluated as outlined below.

lAlternatively, thickness can be defined as the length of the normal vector originating at a point on the cartilage surface and ending at the bone surface.

Fig. 4. Cartilage thickness is defined as the length of the vector originating on and perpendicular to the bone surface, and ending on the cartilage surface.

Cartilage thickness is a continuous function defined over the entire bone surface. However, since it is not possible to obtain a closed-form expression for this function, the value of the function is determined at a discrete set of points (e.g. thickness is evaluated at each node of the bone surface). Each individual bicubic patch is defined by parametric coordinates (u, w) varying from zero to one while adjoining patches form a network defining the anatomy of the joint surface mathematically. Each patch on S, may be defined in its algebraic form (Mortenson, 1985) by: xk =

aiQu 3-iw3-j

i, j, k=U, 1, 2, 3

(1)

where x0 = 1 and xl, x2, xj, are the spatial coordinates of a point on the bicubic patch at the parametric coordinates (u, w) (repeating indices indicate summations). Similarly, the equations of the patches on S, have the form: x,=bij,u3-‘w3-j

i, j, k=O, 1,2, 3

(2)

110

G.

A. ATESHIAN

where the coefficients nijt and b, are obtained from curve-fits of the anatomical data. The tangents along the u- and w-directions at a point on S, are given by the directional derivatives: x;=(3-i)bjj,ur-iW3-j

i=O, 1,2; j, k=O, 1, 2, 3

x;=(3-j)bijkU3-iW2-j

j=o, 1,2; i, k=O, 1,2, 3. (3)

The outward-pointing normal to the surface nk at the point (u, w) can be evaluated from the normalized cross-product of these two tangents. The equation of a line L, originating at a node pk on Si,, and normal to the surface may be expressed as: x,=p,+tn,

(n,=O)

k=O, 1,2, 3

(4)

where t is a parametric coordinate which represents the physical distance from plr to some point xlr on L,. The mathematical analysis for cartilage thickness measurements reduces to finding the intersection of the line L, with one of the parametric patches of the cartilage surface, and then calculating the length of the segment bounded by the two surfaces (Fig. 4). Ray tracing algorithms which are common to solid modeling analyses in computer graphics perform just that kind of calculation and are easily adapted to biomechanical applications (Scherrer and Hillberry, 1979). One possible method (Kajiya, 1982) consists of using an alternate form for the equation of L,, which can be obtained from the intersection of the two planes passing through pk and normal to x; and x;, respectively: [;x,=O

k=O, 1,2, 3; do= 1,2

et al.

which can be solved simultaneously to yield initial values of u and w. In order to increase the chances of finding a good initial guess, the range of acceptable values of (u, w) is extended to C-E, 1 + E], where E is a suitably chosen constant usually less than 0.5. Cartilage thickness map

The result of the previous set of calculations is the value of the thickness of cartilage at each node of the discrete set of nodal points of Sb. These thickness values are displayed on the bone surface in the form of a contour map. The map displays different grey levels corresponding to ranges of thickness values much like a geographic map indicates elevation from sea level using color contours. Joint surface area

The surface area of a patch is given by the double integral (Mortenson, 1985): 1

1

ss0

0

A=

f(u, w)dudw

(9)

where f(u, w) is the norm of the cross-product of x; and xr. The double integral in equation (9) is evaluated using Gauss quadrature. The total joint surface area is obtained by summing the area over all the patches of the surface.

CALIBRATION

(5) Calibration cylinder

where 1: =p;;

1; = -1;m

and 1;=pr;

If=-lfpk

k=l,2,3.

(6)

To find the intersection of L, with S, , equation (5) is substituted into equation (1): I;eijku3-W-j=o

a=l,2.

(7)

Equation (7) represents two curves which lie in S, and intersect at the tip qk of the thickness vector. By substituting c;= l;aijk in equation (7), the parametric coordinates (u, w) of qk can be solved from the following set of nonlinear equations: cr,u3-iw3-j,0 1, c?,u3-iwJ-j=0 1,

Huiskes and co-workers (1985) have tested the accuracy of their SPG system on a set of check points engraved on a stainless steel cylinder which was photographed in two positions (with its axis in both a vertical and a horizontal position). The spatial COordinates of the check points were previously determined with a high-accuracy (2 pm), three-dimensional mechanical digitizer. In our current study, the same calibration cylinder was used in conjunction with the same calibration procedure to determine the overall accuracy of our SPG system. Additional measurements were also performed to assess the precision of the system. Accuracy of measurement

i, j=O, 1, 2,3.

(8)

From the solution of this system of equations, it is possible to evaluate qk which is then substituted in equation (4), with xk =qkr to solve for t (Fig. 4). An initial guess for the solution of equation (8) is first evaluated by fitting S, with parametric triangular plane patches in place of the bicubic patches. The problem of solving for the initial guess is conceptually similar to the steps described above. However, the resulting equations now constitute a linear system

The measured coordinates of a check point on the cylinder are given by pi, where i = 1,2,3 indicates the coordinate direction and j = l-n, refers to the jth point in a set of size n,. Similarly, the ‘true’ or calibrated coordinates of these points are given by Pi. The measurement error at each point is then obtained from: c{=pi-Pi

i=

l-3;

j=

l-n,.

The average and sample standard deviation of the error in each direction are denoted by Ei and ei

771

Quantitation of articular surface topography

Then, the sample variance for each set is given by:

respectively, where: q= f I=1

Ej;

& ,t

CT{= [

c

I

1

11’2.

(&i-&i)2

(10)

The accuracy of the system is defined as the uncertainty in measurements for a 95% coverage (Abernethy et al., 1985; Huiskes et al., 1985). Thus, along each coordinate direction, the uncertainty Ui is obtained from:

(11) where the two-tailed Student’s c-value is evaluated at a level of significance of 0.05, with v = n, - 1 degrees of freedom. Precision of measurements

Two types of repeated measurements have been performed in this study, both of which require the same mathematical analysis for evaluating their related precision indices. Type I. This type of precision measurement applies to points digitized N times from the same stereogram, i.e. Experiments 1 and 3. The precision index resulting from these measurements is an indication of how precisely a node or target can be digitized. Type II. In this type of measurement, the same object is measured N times, but in a different position each time. This case applies to the cartilage and bone surfaces of Experiment 2 (N = 6). The alignment targets are used for converting all coordinates to a common reference frame. However, even after realignment, the surface nodal points from each set are not related in any specific manner. To compare the surfaces, it is first necessary to curve-fit them with bicubic splines. Then ‘thickness’ is measured for each pair of surfaces to compare how closely these surfaces match. In an ideal situation, this ‘thickness’ would be zero everywhere, indicating that the surfaces are identical and thus were reconstructed with infinite precision each time. Type II measurements indicate the combined precision of digitizing, curve-fitting, and realigning. In both cases, the same principle applies for measuring the precision index (Jaech, 1985). There are n points on the object, each of which is measured N times. It is assumed that all coordinates have been transformed to a common reference frame, where necessary. Let pi” be defined as the ith coordinate (i= 1,2,3) of the jth object point (j= l-n), where k indicates the kth set of measurements (k= 1-N). The following N(N - 1)/2 sets of differences are calculated:t

and the average variance over all sets is:

s;=

N(N-1)

’

The precision index, along the ith coordinate direction, is then given by si, and its associated number of degrees of freedom v =(n- 1) (N - 1). To retain a consistent notation, precision indices resulting from Types I and II measurements are denoted by s: and sf’, respectively. Precision indices relating to the vertical cylinder, horizontal cylinder, bone surface nodes, and cartilage surface nodes are similarly denoted by s,, , s,,,, sbr,and s,,, respectively. Thus, s:, denotes the digitizing precision of a bone node measured N times from the same stereogram, while sf, is the digitizing, curvefitting, and realigning precision of bone surface points measured from multiple stereograms. Precision of cartilage thickness measurements

The precision indices for surface points (i.e. not necessarily nodes) are denoted by SF,for the bone, and sii for the cartilage (i= 1, 2, 3). Cartilage thickness is calculated by simply subtracting the coordinates of a cartilage surface point from the coordinates of a bone surface node (or, more generally from a bone surface point) as in equation (4). Using elementary uncertainty analysis, it can be shown that the precision index of a difference is equal to the root-sum-square of the indices of the minuend and the subtrahend, respectively. Thus, the precision index for cartilage thickness measurements along each coordinate direction sri is given by: s,,=[s~i2+s~~*]1/2

i=l,2,3.

(14)

To obtain a 95% coverage, s,, is multiplied by the Student’s t-value evaluated at vlt degrees of freedom, where v,~ is obtained from the Welch-Satterthwaite formula (Abernethy et al., 1985): (15)

where vf and v! are the number of degrees of freedom associated with $ and SE, respectively.

(12) SYSTEM

t For Type II measurements, pik refers to the jth node of

the kth surface, and pi’ is the tip of the thickness vector, as defined in equation (4), when thickness is measured from surface k to surface 1. In effect, ep’ represents the ith component of the thickness vector.

Calibration results

Six sets of measurements were performed on the check points of the calibration cylinder (Experiment 1,

G. A. ATESHIANet 01.

112

N = 6). The standard deviations criof the error for each set were evaluated from equation (10) for both the vertical and horizontal positions, and are listed in Tables 1 and 2, respectively. The calibration results improved when the axis of the cylinder was rotated from a vertical to a horizontal position as has been demonstrated previously by Huiskes and co-workers (1985). These authors have attributed the change in error to surface curvature effects in the plane defined by the two camera optical axes. The average errors for the check points si were zero in all cases. This result was expected since the least-squares procedure, used to align the coordinate systems of the calibrated and measured coordinates of the check points, sought to minimize that average error. Tables 1 and 2 also display a quantity so, called the residual parallax (Faig and Moniwa, 1973) or the quality of measurement (Huiskes et af., 1985). This quantity, which measures the lack of intersection of rays in the three-dimensional reconstructions of nodal coordinates, has been used by these and other authors as an indication of system accuracy (Fig. 3). The bottom row of each table contains the root-meansquare values (RMS) of the measurements in the corresponding column, which are taken as the global value for the N = 6 sets of measurements. The measurement uncertainties are evaluated by substituting the RMS values into equation (ll), with v=(6-1) (28-1)=135 and t=2, and are listed in Table 3. Tables 4 and 5 summarize the measurement precision analyses performed on the cylinder (Experiment 1) and a typical human retropatellar surface for which

Table 1. Calibration measurements for vertical cylinder (pm) n,=28,4*40 Set No.

Cl

02

03

%

1 2 3 4 5 6

33 29 36 33 34 33

34 33 36 :; 34

48 48 42 48 41 45

50 52 47 51 44 51

RMS

33

34

45

50

Table 2. Calibration measurements for horizontal cylin-

der (pm) n,=28,4-40” Set No.

=1

02

03

SO

32 30 26 31 24 28

29

: 3 4 5 6

:: 24 24 28

31 48 36 39 31 39

42 49 31 52 69 34

RMS

29

26

38

47

Table 3. Measurement uncertainties (pm)

Vertical cylinder Horizontal cylinder

u1

u*

u,

66 58

68 52

90 76

the cartilage and bone surface topography, as well as the cartilage thickness, were quantified (Experiments 2 and 3). Since large samples of data points were used (Table 5), all precision indices can be multiplied by t=2 to obtain their corresponding 95% coverage. Discussion

The accuracy of the SPG system when measuring the calibration cylinder check points is given by oi in Tables 1 and 2, or Ui in Table 3 (95% coverage) and the measurement precision indices sf for these same quantities are given in Table 4. The differences between cri and si are due to bias or systematic errors in the system. Possible sources of these systematic errors are the distortions in the photographic paper, the grain resolution of the photographic paper, and the distortion of the camera lens (Ghosh, 1979). These error sources come in various forms, some of which, such as linear distortions, were compensated for in the governing equations. The systematic errors which were not or could not be modeled, were assumed to have been minimized, based on careful choices made in the experimental apparatuses. For example, temperature effects were kept to a minimum by operating the system in a stable environment and using a calibration frame with a very low coefficient of thermal expansion. The use of an ‘instant’ photographic processor allowed development to be done quickly in our laboratory. Using large format photography permitted digitizing of the original, rather than requiring an enlargement from negatives which would produce additional unaccounted (i.e. nonlinear) emulsion and lens distortion. In fact, when using the large calibration frame, the ratio of the image size on the photographs to the original object size was very close to one, thereby introducing minimal lens distortion effects (camera lenses used were manufactured for optimal reproduction at 1: 1 magnification). The accuracy of the cartilage and bone nodal coordinates cannot be measured directly, but may be inferred by comparing their respective si values to those of the cylinder listed in Table 4. Thus, it appears that the bone nodes are obtained to a better accuracy than the check points of either the vertical or the horizontal cylinder. The cartilage nodes, on the other hand, are slightly less accurate than the check points of the vertical cylinder when all three coordinate directions are considered together. Superior results for bone over cartilage are obtained since the projected grid appears with a better contrast on the stereogram of the bone surface.

773

Quantitation of articular surface topography Table 4. Measurement precision indices (pm) Surface Vertical cylinder Horizontal cylinder Patella cartilage Patella bone

50 41 50 29

11 12 17 13

11 14 20 11

37 31 35 20

21 11

NA* NA 23 6

58 34

* NA: not applicable. Table 5. Precision of thickness measurements (sti in pm) V;

1930

II

VC

1910

VII

2528

“12

2169

VI3

3081

St1

St2

St3

29

24

67

The second type of precision measurement, given by sf’ in Table 4, is the most informative since it incorporates all the measurement, data processing, curve-fitting, and propagated errors of the system. Thus, for the bone, the precision of the coordinate values of any point on the surface is comparable to that of the check points of the cylinder. This result indicates that bone surfaces can be measured to within a precision of 68 pm in the least favorable coordinate direction (95% coverage). For the cartilage surface, precision in the three-direction is calculated to be 116 pm (95% coverage) based on the results displayed in Table 4. Thus, the precision in cartilage thickness measurements is 134 pm (95% coverage) in the least favorable direction as obtained from equation (14) and listed in Table 5. In addition, loss of precision for points within a patch, compared with the precision at the corner nodes, supports the observations of Scherrer and Hillberry (1979) and illustrates the inherent error introduced by any curve-fitting technique. For individual nodes specifically, the measurement precision is 40 pm for the bone surface, and 70 pm for the cartilage surface, with a 95% coverage. These results are an improvement over those obtained with other optical techniques used for anatomical measurements (Ghosh, 1983; Huiskes et al., 1985). Furthermore, while all of the above results refer to the large calibration frame, preliminary results suggest that precision measurements improve by a factor of two with the small calibration frame. This improvement may be attributed to the 1: 2 magnification between the small calibration frame and its stereogram.

JOINT ANATOMY

Results and discussion

Accurate three-dimensional geometric models of the cartilage surface of the distal femora, retropatellar surfaces, and tibial plateaux, as well as the menisci have been generated (Fig. 5). Topographical maps of the patella distinctly show anatomic features such as the mid-sagittal and transverse ridges, and the odd

facet. The tibia1 plateau is represented by two separate networks of patches which constitute the lateral and medial compartments. The concavity of the medial plateau and convexity of the lateral plateau are faithfully reproduced by these models. The superior surfaces of the menisci were obtained from stereograms of the tibia1 plateau prior to the removal of the menisci by careful dissection. These meniscal surfaces were realigned with the cartilage surface of the tibial plateau using the alignment target data in the same way that the bone surfaces were realigned with the cartilage surfaces using alignment targets. Because of the narrow and highly curved outline of the menisci, it is generally necessary to use a finer grid projection in order to catch the features of these surfaces. The models of the distal femoral surfaces show the patellar groove and anterior aspect of the condyles. Due to the high curvature of this surface, it is only possible to capture the entire femoral condyles if three or more photographs are obtained and additional corresponding realignments are performed to relate information from one region of the surface to the other (Huiskes et al., 1985). This additional procedure was not performed in the current study. The entire surface area of the articular surfaces of the patella, tibia1 plateau, and menisci could be obtained (Table 6). The surface area of the distal femoral surface represents the patellar groove and anterior condylar areas only. Furthermore, the tibia1 plateau was divided into its medial and lateral compartments and their surface areas were tabulated as well (Table 6). The surface area of the medial compartment (11.8 cm2) was found to be statistically greater than that of the lateral compartment (10.7 cm2) using a Wilcoxon signed-rank test (p < 0.05). No statistical difference was found between the surface areas of the lateral (5.6 cm2) and medial (5.5 cm2) menisci. Surface areas of these joints (Table 6) were determined using other techniques which have been reported in the literature. For example, the areas of the articular surfaces of the patella, tibia, and menisci have been measured as 11.8,23.2 and 15.9 cm2 respectively (Ahmed and Burke, 1983; Ahmed et al., 1983). The small discrepancy between the surface area results of our study and those of other studies may be related to the measurement technique. While some studies have estimated surface areas from planar projections, our spatial modeling of the articular surface allows a more accurate and direct measurement of surface area. Conversely, our analytical SPG method slightly

G. A. ATESHIAN et al.

(b)

(4

Fig. 5. Three-dimensional geometric models of the cartilage surfaces of the distal femur (a), patella (b), tibia1 plateau (d), as well as the superior surface of the menisci (c) are displayed in schematics of the human knee illustrating the applications of the stereophotogrammetry method (P = proximal, M = medial).

Table 6. Anatomical data for the patella femur, tibia1 plateau, and meniscus (averagefstd

dev.)

Joint surface

n

Surface area (cm3

Mean cart. thick. (mm)

Min. cart. thick. (mm)

Max. cart. thick. (mm)

Coeff. of var. (%)

Patella Femur* Tibia1 plateau Lateral tibia1 plateau Medial tibia1 plateau Lateral meniscus Medial meniscus

12 3 12 12 12 12 12

10.7 + 1.6 29.5 + 2.5 22.5 + 2.4 10.7_+1.1 11.8k1.6 5.6k1.2 5.5+ 1.4

3.33 kO.39 1.99k0.12 2.92 k 0.52 3.51kO.87 2.42kO.61 NAt NA

0.89 +0.31 0.16*0.06 0.35 kO.27 0.95 -+ 1.20 0.56kO.43 NA NA

5.91 kO.87 3.61 kO.30 6.25 +0.85 6.25 _+0.85 4.46 +_1.04 NA NA

33*11 29&l 44+8 39*13 33+9 NA NA

*Excluding posterior aspect of the condyles. tNA: not applicable.

underestimates the total articular area of a joint by neglecting those patches which project across the periphery of the articular surface. The more significant difference in meniscal area results cannot be attributed entirely to the measurement techniques, but may result in part from the difficulty in clearly delineating the boundaries of the superior surfaces of the menisci. For each specimen, the cartilage thickness was evaluated at all the nodes of the bone surface, and the mean, minimum, and maximum values were recorded for that surface. Table 6 displays the average of those

quantities for the entire set of retropatellar, tibial, and femoral specimens. In theory, the minimum cartilage thickness should be zero, which is the value to be expected at the periphery of the cartilage layer. In practice however, due to the finite size of the projected grid, it is not possible to precisely capture the peripheral border of the cartilage layer, and consequently, the minimum recorded cartilage thickness will be nonzero. Results show that the mean cartilage thickness on the lateral tibia1 compartment (3.51 mm) is statistically greater than on the medial compartment

Quantitation

of articular

surface

topography

115

CONCLUSION (2.42 mm) (p
776

G. A. ATESHIANet al.

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