MATHEMATICAL DESIGN OF THE VOLAR SURFACE OF THE RADIUS A Thesis Presented to The Graduate Faculty of The University of Akron In Partial Fulfillment of the Requirements for the Degree Master of Science Prashant Singh August 2006
MATHEMATICAL DESIGN OF THE VOLAR SURFACE OF THE RADIUS Prashant Singh Thesis Approved: Accepted: Advisor Dr. Glen Njus Department Chair Dr. Daniel Sheffer Committee Member Dr. Dale Mugler Dean of the College Dr. George Haritos Committee Member Dr. Stanley Rittgers Dean of the Graduate School Dr. George Newkome Date ii
ABSTRACT The distal radius is one of the more common fracture sites of the human long bones. No one-treatment modality is applicable to all the distal radius fractures. Due to various fixation and anatomic issues, the volar surface can be considered an appropriate site for the palmar locking plate for the treatment of unstable dislocated distal radius fractures. In case of a wrist injury involving distal radius, a distal volar radius plate used may not provide the optimum buttress effect due to its inefficiency to lie in close proximity with the distal volar surface. This project geometrically analyzed the distal volar surfaces of 9 randomly chosen radii. A family of polynomial equations representing the mid saggital deviations of the volar surface were obtained. This study will aid in the design of distal volar implants and will provide a more meaningful approach to distal fracture fixation techniques. A family of polynomial rational equations was obtained that defined the geometry of inter mid saggital volar surface of the given radii. The diaphysial region of the radius was more predictable with the residual dimensions, between the y values obtained from the equations and the volar surface, being in the clinically acceptable range of 0.5 to 1.5 mm. At the metaphysial region, in and around the centroidal plane, the equations iii
predicted the surface in the clinically significant range. As we approached the medial and lateral end of the metaphysis, the residual quantity surpassed the clinical significant range. The variance of the three lower order pertinent constants in the equations across the size distribution of the radii, were statistically analyzed and regressed to obtain pertinent relationships. The results obtained define the variation in the volar surface but fail to provide an applicable solution to the problem of obtaining a surface equation that will aid in manufacturing distal volar radius plates. A future study is recommended based on the protocol designed and the results obtained. iv
ACKNOWLEDGEMENTS There are a lot of people I would like to extend my deepest and sincere gratitude, without whose direct or indirect help, this project would not have been a successful journey. First and foremost, I would like to thank my advisor, Dr Glen O. Njus, who has been a terrific source of knowledge and inspiration. Without his insight and constant backing, this work would not have been possible and complete. A terrific balance of concentration and dedication, I am honored to have worked under his guidance. I would like to thank Dr Stanley Rittgers and Dr Dale Mugler for accepting to be in my committee and helping me out with details whenever I needed them. My heartfelt thanks to my parents, who have been a constant source of inspiration and determination, without whose blessings and prayers, this dream could not have been achieved. v
TABLE OF CONTENTS LIST OF TABLES... viii LIST OF FIGURES....ix CHAPTER I: INTRODUCTION...1 1.1Overview 1 Page 1.2 Objectives of the study..3 1.3 Test of Hypothesis.4 II: LITERATURE REVIEW.. 5 2.1 Anatomy 5 2.2 Distal Radius Fractures.6 2.3 Treatment Options 8 2.4 Distal Volar Radius Plates 9 III: MATERIALS AND METHODS.14 3.1 Materials..14 3.2 Methodology...14 3.21 Obtaining the boundaries of the radius CT slices...14 3.22 Determining the transformation factor 17 3.23 Determining the area of each slice..17 vi
3.24 Determining the thickness of each slice..18 3.25 Obtaining a size variance among the given samples..19 3.26 Determining the centroid for each slice..20 3.27 Determining the angle of rotation in x, y and z coordinates...21 3.28 Calculating a body fixed coordinate system...23 3.29 Positioning each CT slice along the 3 coordinate axes..25 3.30 Plotting all data points of each CT slices in the body fixed x and y..26 coordinate system. 3.31 Isolating the volar surface of the distal end of the radius...26 3.32 Obtaining a family of equations those functionally describe the...27 volar surface. 3.33 Inflexion Point 28 3.34 Obtaining the Y values from the boundary data points for the..28 medial and lateral deviations. 3.35 Calculating Y values from the family of polynomial equations.30 3.35 Sample size determination..30 3.36 Statistical Analysis..31 IV: RESULTS...33 4.1 Overview...33 4.2 Polynomial rational equations..34 4.3 Residual distance..34 4.4 Variance of lower order pertinent constants across the cross sectional..48 area distribution. 4.5 Relationship among the lower order constants..63 vii
V: DISCUSSION..66 REFERENCES..69 viii
LIST OF TABLES Table Page 3.1: Decoding the CT images 15 3.2: Determining the transformation factor...17 3.3: Y real values..29 3.4: Y calculated values.30 4.1-4.27: Residual.33 4.28-4.51: Variance of lower order constants across the cross sectional area... 48 ix
LIST OF FIGURES Figures Page 2.1: Anatomy of the joint of the wrist.6 2.2: Plate application sites...10 2.3: Conventional plates...11 2.4: Fixed angle plates...12 2.5: Placement of distal volar plates...13 3.1: CT orientation of the radius slice...15 3.2: Boundary representing the cortical surface of the CT slice...16 3.3: Calculating the area using Trapezoidal rule...17 3.4:Slice cross sectional area vs. slice number.. 19 3.5: Area distribution plot...20 3.6: Calculating centroids...20 3.7: Plot of X centroids vs. slice number...22 3.8: Plot of Y centroids vs. slice number...23 3.9: Body fixed coordinate system.24 3.10: Rotation along Y, X and Z axis respectively 25 3. 11: Graphical representation of the distal radius in x-y plane...26 3.12: Graphical representation of the volar surface of the radius in x-y plane..26 3.13: A curve representing the volar surface of the radius in y-z plane 27 3.14: Inflexion point...28 x
3.15: Plot of Y (mm) vs. X (mm) for a given Z value...29 4.1-4.27: Residual Quantities.35 4.28-4.45: Variance of lower order constants across the cross sectional area...49 4.46-4.49: Relationship among the lower order constants 63 xi
CHAPTER I INTRODUCTION 1.1 Overview The distal radius is one of the more common fracture sites of the human long bones. Colles [1] optimism that the limb will at some remote period again enjoy perfect freedom in all its motion and again be completely free from pain set a stage for generations of surgeons to apply a minimalist approach to the management of fractures at the distal end of the radius. However, studies that have monitored long term results and outcomes through the years have demonstrated the problems of long-term disability associated with sub optimal results from the fractures of distal radius [2,3]. Jupiter and Knirk [2] correlated the association of problems with failure to restore articular congruity while Taleisnik and Watson [3] assessed thirteen patients with malunited fractures of the distal radius that developed symptoms of pain and instability of the midcarpal joint. Furthermore, the incidence is increasing due to the general aging of the population, as these fractures are frequently associated with osteoporosis [4]. The difficulty of obtaining reliable fixation in osteoporotic bone presents a challenge to the surgeon that has been partially addressed by newer implants with screws that directly engage the plate, creating fixed angles that somewhat control fixation in osteoporotic bone [4]. Relative recent reports have shown that the use of advanced techniques can restore useful function [5,6,7]. 1
There are a wide variety of fracture patterns that occur in the distal end of the human radius [8,9]. The variation is based on the force and direction of the load creating the fracture, the position of the wrist at the time of the impact, and the quality of the bone and its surrounding tissue supportive structures. No one treatment modality is applicable to all the distal radius fractures. Although the external fixation is reliable in maintaining the reduction in displaced comminuted intra-articular fractures, it is inadequate in restoring articular congruity in many cases. The complications of external fixation are frequent and may be potentially serious in nature [10]. Open reduction and internal fixation represents distinct advantages over external fixation or pins including direct reduction of specific articular fragments with stable fixation as well as early range of motion [11]. Due to various fixation and anatomic issues, the volar surface can be considered an appropriate site for the palmar locking plate for the treatment of unstable dislocated distal radius fractures [12]. "Wolf's Law" states that the bone or tissues that are stressed remodel themselves to become stronger so that they can withstand even more stress [13]. Depending upon an individual s size, genes and the stresses involved in their daily physical activities, the radius size and shape may vary when compared to another individual s radius. In case of a wrist injury involving distal radius, a distal volar radius plate used may not provide the optimum buttress effect due to its inefficiency to lie in close proximity with the distal volar surface. This project undertook the study of the variation in the distal volar surface of 9 radii. A family of polynomial equations representing the proximal to distal 2
deviations as well as medial and lateral deviations of the volar surface were obtained. This study will aid in the manufacture of distal volar implants keeping in mind the volar surface variation and will provide a more meaningful approach to distal fracture fixation technique. 1.2 Objectives of the study A. To obtain the boundaries of the radius CT slices. B. To determine the transformation factor, area and the thickness of each slice. C. To obtain the size variance among the given samples. D. To determine the centroid for each slice. E. To determine the angle of rotation in x, y and z coordinates for inter-specimen analysis. F. To calculate a body fixed coordinate system. G. To position each CT slice along the 3 coordinate axis. H. To calculate the new profile area and the centroid of each CT slice. I. To graph all data points of each CT slice in the body fixed x and y coordinate system. J. To isolate the volar surface of the distal end of the radius. K. To obtain a family of equations those functionally describe the geometry of the mid saggital volar surface of the distal radius across a subset of human population. 3
1.3 Test of Hypothesis 1.3.1Null Hypothesis: - a. H 0 : A family of exponential functions with polynomial rational exponents having varying order cannot define the geometry of inter mid saggital volar surface of the distal radius across a subset of human population with the residual dimension being in the range of 0.5 to + 1.5 mm. b. H 0 : The variance of pertinent lower three constants in the polynomial rational equations with respect to inter medial and lateral deviation among the given samples is not less than an order of magnitude. 1.3.2Alternate Hypothesis: - a. H 1 : A family of exponential functions with polynomial rational exponents having varying order can define the geometry of inter mid saggital volar surface of the distal radius across a subset of human population with the residual dimension being in the range of 0.5 to + 1.5 mm. b. H 1 : The variance of pertinent lower three constants in the polynomial equations with respect to inter medial and lateral deviation among the given samples is less than an order of magnitude. 4
CHAPTER II LITERATURE REVIEW 2.1 Anatomy The radius extends from the elbow to the wrist and forms the skeletal part of the forearm along with ulna. The joint of elbow is a synovial hinge consisting of humeroradial, humeroulnar and proximal radioulnar articulations. The proximal end of the radius consists of the head, neck and radial tuberosity. The ulna and the radius in the middle region are joined by interosseous membrane. The distal end of the radius flares through a metaphyseal expanse to form an articular surface which seats the proximal carpal row. Two facets are found on this distal articular surface: one for the scaphoid and one for the lunate. The sigmoid notch provides a stable articular surface into which the head of the ulna is seated. Important radiocarpal ligamentous structures are oriented dorsally, radially and palmarly, providing stability to the carpus [14]. Palmar and dorsal radioulnar ligaments and the triangular fibrocartilage complex provide stability at the distal radioulnar joint as well as support and stability for the ulnar side of the radius. On the dorsal end of the radius, arranged in a fan like fashion are the six fibroosseous extensor compartments containing the extensor tendons of the fingers and wrist [14]. On the palmer side, the distal radius forms the floor of the carpal tunnel containing nine 5
finger flexors and the median nerve. The pronator quadratus provides both coverage of the diaphysial/metaphysical junction of the distal radius while maintaining a dynamic coupling with the distal ulna [14]. Figure 2.1: Anatomy of the joint of the wrist [14]. 2.2 Distal radius fractures Distal radius fractures are perhaps the most common fracture site in the upper extremity. In the turn of the 20 th century, residual deformity and shortening was of little concern as long as the arm was functional but in the past decade or so emphasis is being 6
laid upon the functionality as well as radiographic appearance [5]. In the light of various studies outcomes [2, 15, 16], we have become aware of the fact that articular incongruity and radial shortening should be unacceptable. Although restoration of palmar tilt to preoperative values is not critical, restoration to neutral axis is advised for a number of reasons. Firstly these will more adequately restore the radial length and avoid ulnar impaction or distal radioulnar joint incongruity and secondly it will preserve subsequent problems with midcarpal instability [3]. Similarly, correction of radial tilt, although not absolutely critical, will restore more normal joint mechanics [17]. The normal palmar inclination has been reported to be in the range of 0 0 to 22 0 ( μ: 14.5, σ: 4.3) [15]. According to a study conducted [18], restoration of the volar tilt angle during fracture reduction appears to be important in order to eliminate detrimental transverse loading of the distal radius articular cartilage. Numerous methods have been proposed to classify distal radius fractures, including those described by Frykman [19], Melone [20] and the AO group [9]. Because of the limitations of these classification systems in guiding treatment, Cooney et al. suggested that fractures be defined as combinations of: extraarticular or intraarticular; misplaced or non displaced; and, reducible or non reducible [21]. It follows therefore, that although undisplaced extraarticular and intraarticular fractures can be treated without surgical intervention, displaced and irreducible fractures, both extraarticular and intraarticular, require surgery. 7
2.3 Treatment options Treatment options for distal radius fractures range from cast immobilization to open reduction and internal fixation. Management is based on fracture pattern, degree of displacement of fracture fragments, and the individual patient's needs and demands. Open reduction and internal fixation is an option for displaced or unstable intra- or extraarticular fractures. There exists a wide range of plating techniques for distal radius fractures. New implant designs and plating strategies have made certain treatment options more attractive. Volar plating, dorsal plating, dual-column plating, and fragment-specific fracture fixations are options available to the treating surgeon, with multiple implants available for each form of plating. Although many of these decisions remain controversial, and are often based on surgeon preference, a few generalizations can be made [22]. Volar intraarticular lip fractures (Barton fractures) are best treated with a volar buttress plate. Once volar stability is restored, the dorsal metaphyseal fragments can be reduced against the stable volar buttress. Dorsal fixation with one plate frequently causes extensor tendon irritation or rupture. Dual plating of the distal radius is more stable than single plating, and early reports demonstrate less tendon problems. Fragmentspecific techniques address the concept of the radial, intermediate, and ulnar columns. Volar plating for distal radius fractures is a method that has gained support through good results with decreased tendon irritation [22]. 8
There is extensive experience with dorsal plating of distal radius fractures [6,7,23,24]. The dorsal metaphysis of the distal radius is subject to tensile and compressive forces during routine forearm activities. The typical mechanism of a distal radius fracture is a fall on an outstretched hand. This type of injury results in tensile forces across the volar surface, compressive forces on the dorsal surface, and supination of the distal fracture fragment. In the young adult, distal radius fractures are often caused by high-energy trauma. In the elderly patient, low-energy trauma, such as a fall from a standing height, can result in this injury. Compression and torsion across the articular surface can cause various patterns of intra-articular displacement. Dorsal and palmar transverse fractures of the medial complex are examples of compression applied to specific locations [25]. 2.4 Distal volar radius plates The great majority of these injuries are treated using conventional plates that are more effective when used in the buttress mode [26]. Dorsal plating may cause extensor tendon complications that include tenosynovitis and ruptures that occur because the tendons are in direct contact with dorsal plates [27]. On the other hand, there is more space on the volar side of the distal radius. The volar radius flares as it approaches the joint line and has a convex surface in the sagital plane. Relative to the volar surface the pronator quadratus muscle occupies this concavity. This structure widely separates the flexor tendons from the bone surface and allows the safe application of internal fixation 9
devices (Fig. 2.2). Hence, the success of volar buttress plating for the volar unstable fractures [28,29]. Volar plate fixation of unstable distal radius fractures may prevent injury to the extensor tendons and avoids disruption of the dorsal fibrous cuff, facilitating reduction. The volar approach also provides an opportunity to release the pronator quadratus muscle, which is often trapped in the fracture and can be a cause of pronation contracture [30]. Distal Volar Radius Figure 2.2: Plate application sites [32]. The volar side, lower surface of Figure 2.2, of the distal radius is more suitable for the application of internal fixation. Interrupted lines represent the plate application sites. On the dorsal side, there is direct contact between the tendons (*) and the bone surface, allowing little space for internal fixation hardware. On the volar side, the flexor tendons are well separated from the bone and protected by the pronator quadratus [32]. 10
In a study conducted by the Miami Hand Center, Florida, USA [31], 29 patients with 31 unstable dorsally displaced distal radius fractures were treated with distal volar radius plates and results analyzed for more than one year. At the time of final evaluation and assessment (average 66 weeks, range: 53 98 weeks), all fractures had healed and there was no measurable loss of reduction after plating, and there were no instances of plate breakage. A plate used in the buttress mode has to resist only moderate axial and bending loads. Its mechanical function is to strengthen a weakened area of cortex, as there is bone contact across the fracture [32]. A plate fixing a distal radius fracture from the volar side and having no bone contact across the opposite cortex is subject to much higher axial and bending loads. It has to resist all the forces across the fracture site. Conventional screw fixation of the weak distal fragment thus fails by screw pullout, toggling of the distal fragment and recurrence of the deformity (Fig. 2.3). Figure 2.3: Conventional plates [31]. 11
Conventional plates are unable to fix dorsally unstable fractures from the volar side. This is due to the absence of the buttress effect resulting in loss of interface integrity with bone screw and toggling of the distal fragment [31]. Stable volar fixation of dorsally displaced fractures is only possible with fixed-angle devices (Fig. 2.4). These function on a different mechanical principle than buttress plates. They obtain fixation of the distal metaphyseal fragment through pegs, tines, nails or blades rigidly attached to a plate [33]. These devices are intended to neutralize all bone displacement across a fracture site and do not depend on bone contact for stability. Figure 2.4: Fixed angle plates [31] Fixed angle devices are able to fix dorsally unstable fractures from the volar side. These devices are constructed as single structural units. Thus, the buttress effect and toggling are secondary variables [31]. The placement of the distal volar radius plates has been studied by David L Nelson [34]. The projection of the volar plates should not exceed the WS line which is a 12
theoretical line marking the most volar aspect of the radius, as it would interfere with the flexor tendons. The plate is more effective when it lies as close to PQ line as it would enable the suture repair of pronator quadratus and allow a secure closure. This in turn will allow the muscle to form a thick interposition between the distal edge of the plate and the tendons. Also shown in Fig. 2.5 is the volar radial tuberosity represented by X and the volar radial ridge marked as VR. The plate should not be placed too radial as it will overlie these regions and will not lie flat on the radius. Figure 2.5: Optimum placement site of distal volar plates [34]. 13
CHAPTER III MATERIALS AND METHODS 3.1 Materials a. CT slices for 17 radii courtesy Washington University in St. Louis. b. DRX Viewer DeJarnette Research Systems, Inc. c. Able Software Corp. 3-D Doctor. d. Microsoft Excel 2000. e. Table Curve Jandel Scientific, AISN Software. 3.2 Methodology 3.21 Obtaining boundaries of the radius CT slices. 17 radii CT images were provided to us in the form of compact discs courtesy of The Washington University in St. Louis. Each compact disc contained 2 or 3 radii, each radius consisting of approximately 250 CT images with unknown thickness. Each image had an 8-digit file name and there was a need to arrange them in a pattern starting from the proximal end of the radius and ending at the distal end. 14
a. Using the DRX Viewer software, each CT slice was viewed and the files were renamed. Shown below in Fig. 3.1 is an image of a CT slice that was more toward the proximal end. Also seen are four reference points, which were used for calibration purposes. P-8-1 represented the second CT slice in the 8 th radius and the date and time represented when the scans were taken (Table 3.1). Due to the number of CT slices representing each radius, approximately every 5 th CT slice starting from the appearance of the 1 st proximal CT slice were analyzed. The orientation as seen in Fig. 3.1 was the CT coordinate system. Table 3.1: Decoding the CT images Master File Image No. Scan Date Scan Time Authors Comment Transition from the proximal end to the distal end of the 43414536 270 3/31/2005 7:16 p.m. radius Copied File Name P-8-1 Right hand side Reference point CT slice Figure. 3.1: CT orientation of the radius slice Platform 15
b. Using Able Software 3-D Doctor, boundaries were constructed for each slice in a pattern. There were 4 reference points in a rectangular orientation in the CT image. Data points representing the center of each reference point were taken and straight lines connecting these points were drawn. As can be seen in Figure 3.2, the boundary seems distorted. This was due to the box, in which the radii were placed while scanning, was tilted when compared to the CT reference frame. A boundary for the CT slice was then constructed by taking the data points representing the surface and connecting them with straight lines (Fig. 3.2). The number of data points selected depended upon the slope gradient of the surface. As the CT slices progressed towards the distal end of the radius, the structure became more complicated and more data points were required to represent the surface. Figure. 3.2: Boundaries representing the cortical surface of the CT slice and the reference points c. These data points were saved as ASCII files. 16
3.22 Determining the transformation factor. Transformation factor was the pixel to length conversion factor. This was determined by finding the distance between the reference points in pixels and then dividing them by a known parameter. In this case the length of the rectangle was 101.6 mm and the width was 50.8 mm. The mean of these quantities gave us a transformation factor that was 96.34-pixels/ inch with standard deviation being 0.1436-pixels/ inch. Table 3.2: Determining the transformation factor Points CT X CT Y X i - X i-1 Y i - Y i-1 Transformation Factor 1 60.728 172.210 2 446.122 177.485 385.39 5.27 96.36 3 443.436 369.965-2.69 192.48 96.25 4 57.579 364.464-385.86-5.50 96.47 1 60.728 172.210 3.15-192.25 96.14 Mean 96.34 3.23 Determining the area of each slice. Y Y1 Y2 X X1 X2 Figure 3.3: Calculating the area using Trapezoidal rule 17
The area under the boundary may be visualized as comprising of a trapezoid and its area can be calculated as shown below, Area of the CT slice in pixels = Abs ( triangle areas + rectangle areas) Triangle Area= ((X2-X1) (Y2-Y1))/2 Rectangle Area=(X2-X1) Y1 The two consecutive data points may be visualized as falling on a straight line. While calculating the area of the CT slice, the quadrants under which the data points lie, were also taken into consideration. 3.24 Determining the thickness of each slice. Three radii were considered and distal most points were physically marked on the specimen. Second sets of points were marked based on any distinct characteristic on their surface. The distance between these two points was measured to an accuracy of thousands of an inch. Using DRX Viewer software, the number of slices between these points was determined. Thickness of a slice (mm)= (Thickness (inch)/ Number of slices) 25.4 The thickness of each slice after two trials was calculated as 0.4 mm 18
3.25 Obtaining a size variance among the given samples. 8000 Slice Area vs. Slice Number 7000 Cross sectional area (pixels) 6000 5000 4000 3000 2000 1000 0 50 100 150 200 250 Slice Number Diaphysial region CT slices being analyzed Metaphysial region Figure 3.4: Slice cross sectional area vs. slice number The slice area in pixels versus the slice number was plotted for each radius as shown in Figure 3.4. The diaphysial and metaphysial junction was obtained by graphing straight lines running parallel to the slopes and considering their intersection. A line normal to the diaphysial axis was drawn starting from this intersection. Cross sectional areas of 5 slices, 1 cm or 5 slices after the diaphysial-metaphysial junction, were considered for analysis. The plot for the area distribution is shown in Fig 3.5. 19
Area Distribution 200 Area (mm 2 ) 180 160 140 120 100 0 10 20 30 40 Radius Figure 3.5: Cross sectional area distribution plot 3.26 Determining the centroid of each CT slice. Calculating the centroid of each CT slice was important as it was used to establish a body fixed coordinate system for inter-specimen comparisons. Centroids for a representative triangle and rectangle are illustrated below. h/2 h b b/2 Figure 3.6: Calculating centroids The equations below were used to calculate the centroids, 20
X Centroid= ( (X CT A T ) + (X CR A R )) / ( A T + A R ) Y Centroid= ( (Y CT A T ) + (Y CR A R )) / ( A T + A R ) Where: - X CT = X Centroid of the triangle = ((X1+X2)/2) + ((X2-X1)/ABS (X2-X1)) (ABS ((1/6) (X2- X1))) Y CT = Y Centroid of the triangle = ((Y1+Y2)/2)+((Y2-Y1)/ABS (Y2-Y1)) (ABS ((1/6) (Y2- Y1))) X CR =X centroid of the rectangle=(x1+x2)/2 Y CR =Y centroid of the rectangle=y1/2 A T = Area of triangle A R = Area of rectangle (X1, Y1) = first data point and (X2, Y2) = successive data point. 3.27 Determining the angle of rotation in x, y and z coordinates. During CT scanning the radii were somewhat arbitrarily oriented and it was decided to construct a body fixed coordinate system for each radius as follows: 1. The z- axis coincident with the diaphyseal centroidal axis; 2. The y-axis normal to the distal flat volar surface; and, 3. The x-axis the cross product of y and z. The flat volar surface was found approximately 20 mm proximal to the distal articular surface and extended at least 10 mm. This surface orientation and geometry was found by 21
linear regression of that region of the CT slice boundary. This process was conducted on a number of relevant CT slices and the over all surface was defined as the average of linear regressions. All the above was conducted such that there was a logical system in place for inter-specimen comparison. a. Linear regression of the X centroids. The Fig. 3.7 shows regression analysis performed on the X centroids through slice 100. The adjusted R 2 value was > 0.95 and the slope of the linearly regressed data points was 0.101 and the intercept was 269.322. 300 X Centroid - Global Coord. System X=269.32+0.101Z X centroids (pixels) 290 280 270 260 0 50 100 150 200 Slice number Figure 3.7: Plot of X centroids vs. slice number b. Regression of the Y centroids. The Fig. 3.8 shows regression analysis performed on the Y centroids through slice 100. The adjusted R 2 value was > 0.98 and the slope of the linearly regressed data points was 0.085 and the intercept was 395.611. 22
400 Y Centroid - Global Coord. System Y=395.61-0.085Z Y centroids (pixels) 390 380 370 360 0 50 100 150 200 Slice number Figure 3.8: Plot of Y centroids vs. slice number c. X, Y and Z rotation angles were calculated as follows, Y rotation angle= (ATAN (slope of the X centroid linear regression)) X rotation angle= (ATAN (slope of the Y centroid linear regression)) Z rotation angle= π /2- (ATAN (mean of the slopes)) 3.28 Calculating a body fixed coordinate system. a. X and Y centroids from Regressed X and Y centroids respectively. Centroids were calculated in order to align the slices in a body fixed coordinate system using the regression analysis performed in the previous step. Shown below are formulae involved in calculating the X and Y centroids. 23
X centroid = (A M XC ) + B XC Y centroid =(A M YC ) + B YC where :- A- distance in mm from the proximal end M XC & M YC - slope of the X and Y centroid linear regression respectively B XC & B YC - intercept of the X and Y centroid linear regression respectively b. Moving the centroid to the body fixed coordinate system. In order for inter specimen comparison, each CT slice had to be positioned in a global coordinate system by shifting their centroids using the equations shown below, X centroid = (X1-calculated centroids from Reg. X centroids)/ inches to mm factor Y centroid = (Y1-calculated centroids from Reg. Y centroids)/ inches to mm factor 435 Radius 4 Slice Name P-4-1 10 Radius 4 Slice Name P-4-1 y (pixels) 420 405 390 375 y (mm) 5 0-5 360 230 250 270 290 x (pixels) Figure 3.9: Body fixed coordinate system. 24-10 -10-5 0 5 10 x (mm)
3.29 Positioning each CT slice along the 3 coordinate axes. Due to the difference in the orientation among the specimens, we calculated the angles of rotation in order to align the specimen in the body fixed coordinate system and obtained the volar surface of the distal radius on top half of the graph. A rotation transformation matrix was used to calculate rotations along each axis. X1 Cos θ -Sin θ X Y1 Sin θ Cos θ Y where (X1, Y1) are the new data points after the rotation by an angle θ along Z coordinate axis. Fig. 3.10 shows y- axis rotation of 4.3012 o, x-axis rotation of 4.7706 o and z-axis rotation of 191.39 o. 10 Radius 4 Slice Name P-4-1 Radius 4 Slice Name P-4-1 10 10 Radius 4 Slice Name P-4-1 5 5 5 y (mm) 0 y (mm) 0 y (mm) 0-5 -5-5 -10-10 -5 0 5 10 x (mm) -10-10 -5 0 5 10 x (mm) -10-10 -5 0 5 10 x (mm) Figure 3.10: Rotation along Y, X and Z axis respectively. 25
3.30 Plotting all data points of each CT slices in the body fixed x and y coordinate system. Y (mm) 20 15 10 5 0-5 -10-15 -20 Radius 4-15 -10-5 0 5 10 15 20 25 X (mm) Figure 3. 11: Graphical representation of the distal radius in x-y plane. 3.30 Isolating the volar surface of the distal end of the radius. Radius 4 Y (mm) 20 15 10 5 0-5 -10-15 -20-15 -10-5 0 5 10 15 20 X (mm) Figure 3.12: Graphical representation of the volar surface of the radius in x-y plane 26
We were only concerned with the flat surface of the distal radius and hence the dorsal side data was excluded (Fig. 3.12). 3.31 Obtain a family of equations which functionally describe the volar surface. We analyzed the data in 3-dimensions by keeping x constant at intervals 0, 3, -3, 6, - 6 and 10 mm and collecting the corresponding y and z values. Equations were then fitted using Table Curve Windows v1.12 AISN software as shown below in Figure 3.13. The intervals signify the lateral and medial deviations from the midline of the distal radius. As the specimens vary in structure, a more likely solution to this problem was to obtain a family of equations representing the volar surface. Fig. 3.13 shows the proximal to distal deviation in one of the intervals for radius 4. Figure 3.13: A curve representing the volar surface of the radius in y-z plane 27
3.32 Inflexion point A point on the curve at which the curvature changes signs is called an inflexion point. This point signifies the end of the metaphysic region and beginning of the epiphysis region (Fig 3.14). As discussed in the literature review, the placement of the distal volar plate should be such that it should not cover the volar lip of the radius in order to avoid coming in contact with the flexor tendons. Hence the results were obtained starting from the inflexion point and progressing towards the proximal end. 7000 Radius 5 Area vs. Slice number Area (pixels) 5000 3000 1000 0 50 100 150 200 250 Slice number Diaphysis Metaphysis inflexion point Figure 3.14: Inflexion point Epiphysis 3.34 Obtaining the Y values from the boundary data points for the medial and lateral deviations 28
In the 3-dimensional coordinate system, the medial deviations were represented by X= -3 mm and 6 mm; the lateral deviations was represented by X=3 mm, 6 mm and 10 mm where X=0 was the centroidal axis. Z-axis represented the proximal to distal distance of the radius. Starting from the inflexion point, the boundary data points at random Z intervals were plotted and regressed to obtain a polynomial third order equation (Fig. 3.15). The Z value corresponding to the closest X=0 value at each interval was noted. Y values at intervals X=-6, -3, 0, 3, 6 and 10 mm were predicted based on the regressed equation (Table 3.3). We denoted these Y values as Y real values. Y (mm) 7.0 6.8 6.6 6.4 6.2 6.0 5.8 5.6 5.4 5.2 5.0 Radius 5, Z=57.6 y = -0.0011x 3 + 0.0186x 2 + 0.0213x + 5.4309 R 2 = 0.9544-10 -5 0 5 10 15 X (mm) Figure 3.15: Plot of Y (mm) vs. X (mm) for a given Z value Table 3.3: Y real values X (mm) -6-3 0 3 6 10 Y real(mm) 6.210 5.564 5.431 5.633 5.991 6.404 29
3.35 Calculating Y values from the family of polynomial equations obtained from step 3.31. For the noted Z values from the previous step (3.34), Y values were calculated using the best fit corresponding polynomial rational equations at different Z intervals (Table 3.4). These calculated Y values were denoted as Y calculated values. Table 3.4: Y calculated values X R 2 y=f(z) a b c d e Z values Y cal. -6 0.999 lny=(a+cz+ez 2 )/(1+bz+dz 2 ) 1.3919-0.0286-0.0407 0.0002 0.0003 59.4 10.611-3 0.998 lny=(a+cz+ez 2 )/(1+bz+dz 2 ) 1.6385-0.0287-0.0474 0.0002 0.0004 59.4 8.951 0 0.999 lny=(a+cz+ez 2 )/(1+bz+dz 2 ) 1.7404-0.0204-0.0404 0.0001 0.0003 59.4 7.901 3 0.998 lny=(a+cz)/(1+bz+dz 2 ) 1.5582-0.0089-0.0175-0.0001 59.4 7.014 6 0.995 lny=(a+cz)/(1+bz) 1.2316-0.0129-0.0132 59.4 6.775 10 0.991 lny=(a+cz)/(1+bz) 1.1515-0.0137-0.0135 59.4 6.515 3.36 Sample Size Determination Power analysis was performed to determine the sample size of the distal radii using the following equation; n 2 ( σ / δ ) t α,[ ν ] ( + t ) 2 2 2 ( 1 P )[ ν ] 30
where, n : sample size σ : true standard deviation δ : smallest true difference to be detected ν : degrees of freedom α : significance level P : power of the test t : two-tailed t-table value For the present study, a true standard deviation of 20% was assumed, the smallest true difference to be detected was assumed to be 50%, the power of the test was assumed to be 95% and a significance level of 0.05 was assumed in order to determine the sample size. The number of samples was calculated to be 7 using 10 degrees of freedom. 3.37 Statistical Analysis The statistical model assumed for this study was a single factor analysis of variance (ANOVA) fixed effects model. The main effect in this project was the radius size. The model was given by the equation, Y i = μ +α i + ε i 31
where: Y i : dependent variable (output) μ : parametric mean of the population α i : fixed treatment effect for the ith group ε i : error term 32
CHAPTER IV RESULTS 4.1 Overview A family of polynomial rational equations was obtained that defined the geometry of inter-mid-saggital volar surface of the given radius. The diaphysial region of the radius was more predictable with the residual dimensions being in the clinically allowable range of 0.5 to 1.5 mm. At the metaphysial region, in and around the centroidal plane, the equations predicted the surface in the clinically significant range. As we approached the medial and lateral end of the metaphysis, the residual dimensions surpassed the clinical significant range. The variance of the three lower order pertinent constants in the equations across the cross sectional area distribution of the radii were linearly regressed in order to obtain a relationship between the cross sectional area and the constants. An attempt was also made to establish a trend between the constants. 33
4.2 Polynomial rational equations A family of exponential functions with polynomial rational exponents with a degree of 3 or less was applied because the regression coefficients were greater than 0.9 and their frequency of occurrence was highest among all the other equations. 1. lny=(a+cz+ez 2 )/(1+bz+dz 2 +fz 3 ) 2. lny=(a+cz+ez 2 )/(1+bz+dz 2 ) 3. lny=(a+cz)/(1+bz+dz 2 ) 4. lny=(a+cz)/(1+bz) where:- y- normal to the volar surface z- distance along the diaphysial axis a, b, c, d, e and f- constants These equations predicted the mid-saggital volar surfaces of the distal radius across the subset of human population. 4.3 Residual distance The residual distance between the volar surface and the polynomial rational equations was established by determining the difference between the Y real values and the Y calculated values for Z values, starting from the inflexion point and plotting the results, as shown in the Tables 4.1-4.27 and Figures 4.1-4.27. 34
Radius 3 Z=65.4 X Table 4.1: Residual Y calc. Y real (mm) (mm) Residual -6 10.576 8.777 1.799-3 9.197 9.515-0.318 0 9.954 9.615 0.339 3 9.398 9.499-0.101 6 10.405 9.588 0.816 10 11.645 10.755 0.891 Y (mm) Y calc. & Y real vs. X (mm) 12 11 10 9 8-10 -5 0 5 10 15 X (mm) Figure 4.1: Residual Z=54.6 X Table 4.2: Residual Y calc. Y real (mm) (mm) Residual -6 4.788 4.594 0.194-3 4.980 4.980 0.000 0 4.792 5.024-0.232 3 5.130 5.002 0.128 Y (mm) 7 6 5 4 3 Y calculated & Y real vs. X (mm) 6 5.446 5.189 0.257-10 -5 0 5 10 15 X (mm) 10 5.624 6.238-0.614 Figure 4.2: Residual where: - Y calc. Y values from the equations Legend: - Y calc. Y real Y values from the surface - Y real 35
X Table 4.3: Residual Y calc. (mm) Y real (mm) Residual -6 N/A N/A N/A Z=15.3 7 6 Y calculated & Y real vs. X (mm) -3 3.959 4.223-0.263 0 4.494 4.513-0.019 3 4.004 3.817 0.188 6 N/A N/A N/A 10 N/A N/A N/A Y (mm) 5 4 3 2-4 -2 0 2 4 X (mm) Figure 4.3: Residual Radius 4 Z=65.2 X Table 4.4: Residual Y calc. Y real (mm) (mm) Residual -6 11.011 12.279-1.268-3 10.625 11.323-0.698 0 10.848 11.046-0.198 3 11.120 11.253-0.133 6 16.001 11.750 4.251 Y (mm) 20 18 16 14 12 10 Y calculated & Y real vs. X (mm) -10-5 0 5 10 15 X (mm) 10 18.431 12.527 5.904 Figure 4.4: Residual where: - Y calc. Y values from the equations Legend: - Y calc. Y real Y values from the surface - Y real 36
X Table 4.5: Residual Y calc. Y real (mm) (mm) Residual -6 6.286 7.003-0.716 9 8 Z=58.9 Y calculated & Y real vs. X (mm) -3 6.303 6.404-0.102 0 6.174 6.158 0.015 3 6.244 6.120 0.124 6 6.147 6.142 0.005 10 6.345 6.015 0.330 Y (mm) 7 6 5 4-10 -5 0 5 10 15 X (mm) Figure 4.5: Residual Z=48.3 X Table 4.6: Residual Y calc. (mm) Y real (mm) Residual -6 N/A N/A N/A -3 4.280 4.247 0.033 0 4.342 4.434-0.092 3 4.306 4.356-0.050 6 N/A N/A N/A 10 N/A N/A N/A Y (mm) 7 6 5 4 3 2 Y calculated & Y real vs. X (mm) -4-2 0 2 4 X (mm) Figure 4.6: Residual where: - Y calc. Y values from the equations Legend: - Y calc. Y real Y values from the surface - Y real 37
Radius 5 Z=64.2 X Table 4.7: Residual Y calc. Y real (mm) (mm) Residual -6 14.383 11.911 2.472-3 13.063 10.878 2.185 0 10.671 10.026 0.645 3 10.009 9.451 0.558 6 9.338 9.250 0.087 10 10.608 9.733 0.875 Y (mm) 16 14 12 10 8 Y calculated & Y real vs. X (mm) -10-5 0 5 10 15 X (mm) Figure 4.7: Residual X Table 4.8: Residual Y calc. (mm) Y real (mm) Residual Z=56.3 Y calculated & Y real vs. X (mm) -6 8.586 8.736-0.150-3 7.438 7.580-0.142 0 6.898 6.776 0.122 3 6.168 6.225-0.057 6 5.949 5.832 0.117 Y (mm) 10 9 8 7 6 5-10 -5 0 5 10 15 X (mm) 10 5.522 5.382 0.140 Figure 4.8: Residual where: - Y calc. Y values from the equations Legend: - Y calc. Y real Y values from the surface - Y real 38
X Table 4.9: Residual Y calc. Y real (mm) (mm) Residual -6 4.087 4.381-0.294 7 6 Z=29.1 Y calculated & Y real vs. X (mm) -3 5.147 5.162-0.015 0 5.069 5.148-0.080 3 4.546 4.601-0.054 6 N/A N/A N/A 10 N/A N/A N/A Y (mm) 5 4 3 2-8 -6-4 -2 0 2 4 X (mm) Figure 4.9: Residual Radius 7 X Table 4.10: Residual Y calc. Y real (mm) (mm) Residual -6 14.383 11.911 2.472-3 13.063 10.878 2.185 0 10.671 10.026 0.645 3 10.009 9.451 0.558 6 9.338 9.250 0.087 Y (mm) 16 14 12 10 8 39 Z=64.2 Y calculated & Y real vs. X (mm) -10-5 0 5 10 15 X (mm) 10 10.608 9.733 0.875 Figure 4.10: Residual where: - Y calc. Y values from the equations Legend: - Y calc. Y real Y values from the surface - Y real
X Table 4.11: Residual Y calc. Y real (mm) (mm) Residual -6 4.642 4.819-0.177 7 6 Z=45.2 Y calculated & Y real vs. X (mm) -3 5.052 4.885 0.167 0 5.060 4.931 0.129 3 4.982 4.860 0.123 6 4.651 4.574 0.076 10 N/A N/A N/A Y (mm) 5 4 3 2-10 -5 0 5 10 X (mm) Figure 4.11: Residual Z=25.2 X Table 4.12: Residual Y calc. Y real (mm) (mm) Residual -6 N/A N/A N/A -3 4.477 4.621-0.144 0 4.960 4.904 0.056 3 4.276 4.448-0.172 6 N/A N/A N/A Y (mm) 7 6 5 4 3 2 Y calculated & Y real vs. X (mm) -4-2 0 2 4 X (mm) 10 N/A N/A N/A Figure 4.12: Residual where: - Y calc. Y values from the equations Legend: - Y calc. Y real Y values from the surface - Y real 40
X Table 4.13: Residual Y calc. Y real (mm) (mm) Residual -6 9.147 9.673-0.525 18 16 Radius 8 Z=55.9 Y calculated & Y real vs. X (mm) -3 10.649 9.312 1.338 0 10.086 9.508 0.579 3 11.036 10.097 0.939 6 12.690 10.919 1.770 10 16.320 12.097 4.223 Y (mm) 14 12 10 8-10 -5 0 5 10 15 X (mm) Figure 4.13: Residual Z=40.7 X Table 4.14: Residual Y calc. Y real (mm) (mm) Residual -6 4.605 4.649-0.043-3 5.142 5.283-0.140 0 5.669 5.607 0.062 Y (mm) 7 6 5 4 Y calculated & real vs. X (mm) 3 5.594 5.574 0.020 3 2 6 5.211 5.133 0.077-10 -5 0 5 10 X (mm) 10 N/A N/A N/A Figure 4.14: Residual where: - Y calc. Y values from the equations Legend: - Y calc. Y real Y values from the surface - Y real 41
X Table 4.15: Residual Y calc. Y real (mm) (mm) Residual -6 N/A N/A N/A -3 5.171 5.044 0.127 0 5.438 5.546-0.108 Y (mm) 7 6 5 4 Z=21.2 Y calculated & real vs. X (mm) 3 4.784 4.862-0.079 6 N/A N/A N/A 10 N/A N/A N/A 3 2-4 -2 0 2 4 X (mm) Figure 4.15: Residual Radius 9 Z=60.4 X Table 4.16: Residual Y calc. Y real (mm) (mm) Residual -6 7.079 7.558-0.479-3 6.945 7.407-0.462 0 7.678 7.773-0.095 3 8.804 8.575 0.229 Y (mm) 12 11 10 9 8 7 Y calculated & real vs. X (mm) 6 9.320 9.731-0.411 10 10.474 11.684-1.210 6 42-10 -5 0 5 10 15 X (mm) Figure 4.16: Residual where: - Y calc. Y values from the equations Legend: - Y calc. Y real Y values from the surface - Y real
X Table 4.17: Residual Y calc. (mm) Y real (mm) Residual -6 4.788 5.256-0.469-3 4.846 4.919-0.073 0 4.770 4.710 0.060 Y (mm) 7 6 5 4 Z=49.8 Y calculated & real vs. X (mm) 3 4.465 4.563-0.098 6 4.752 4.415 0.337 10 4.622 4.103 0.519 3 2-10 -5 0 5 10 15 X (mm) Figure 4.17: Residual X Table 4.18: Residual Y calc. Y real (mm) (mm) Residual -6 N/A N/A N/A -3 4.380 4.282 0.098 0 4.303 4.303 0.000 Y (mm) 7 6 5 4 Z=40.7 Y calculated & real vs. X (mm) 3 4.074 4.031 0.043 3 2 6 N/A N/A N/A -4-2 0 2 4 X (mm) 10 N/A N/A N/A Figure 4.18: Residual where: - Y calc. Y values from the equations Legend: - Y calc. Y real Y values from the surface - Y real 43
X Table 4.19: Residual Y calc. (mm) Y real (mm) Residual Radius 10 Z=64.8 Y calculated & real vs. X (mm) -6 12.463 12.256 0.207-3 13.448 12.586 0.862 0 12.669 12.829-0.160 3 13.826 13.116 0.710 6 15.551 13.575 1.976 10 26.868 14.680 12.188 Y (mm) 30 25 20 15 10-10 -5 0 5 10 15 X (mm) Figure 4.19: Residual X Table 4.20: Residual Y calc. (mm) Y real (mm) Residual Z=37.6 Y calculated & real vs. X (mm) -6 4.633 5.118-0.485-3 5.733 5.260 0.473 0 5.554 5.591-0.037 Y (mm) 8 7 6 5 3 5.720 5.609 0.111 4 3 6 4.760 4.812-0.052-10 -5 0 5 10 X (mm) 10 N/A N/A N/A Figure 4.20: Residual where: - Y calc. Y values from the equations Legend: - Y calc. Y real Y values from the surface - Y real 44
X Table 4.21: Residual Y calc. Y real (mm) (mm) Residual Z=4.8 Y calculated & real vs. X (mm) -6 4.077 3.341 0.736-3 5.386 5.454-0.068 0 6.081 6.106-0.024 3 5.499 5.506-0.007 6 4.080 3.866 0.214 10 N/A N/A N/A Y (mm) 8 7 6 5 4 3-10 -5 0 5 10 X (mm) Figure 4.21: Residual Radius 21 Z=65.1 X Table 4.22: Residual Y calc. Y real (mm) (mm) Residual -6 7.325 7.727 0.402-3 6.778 7.647 0.869 0 7.421 7.544 0.123 Y (mm) 10 9 8 7 Y calculated & real vs. X (mm) 3 7.359 7.437 0.077 6 8.297 7.340-0.957-10 -5 0 5 10 15 10 7.520 7.255-0.265 X (mm) Figure 4.22: Residual where: - Y calc. Y values from the equations Legend: - Y calc. Y real Y values from the surface - Y real 6 5 45
X Table 4.23: Residual Y calc. Y real (mm) (mm) Residual -6 4.738 5.416-0.677-3 6.561 6.526 0.034 0 5.871 6.381-0.510 Z=30.5 Y (mm) 8 7 6 5 Y calculated & real vs. X (mm) 3 5.471 5.514-0.043 6 4.557 4.459 0.098 10 N/A N/A N/A 4 3-10 -5 0 5 10 X (mm) Figure 4.23: Residual X Table 4.24: Residual Y calc. Y real (mm) (mm) Residual -6 N/A N/A N/A -3 5.660 5.681-0.021 0 5.997 5.838 0.159 Y (mm) 8 7 6 5 Z=5 Y calculated & real vs. X (mm) 3 4.984 4.920 0.064 4 3 6 N/A N/A N/A -10-5 0 5 10 10 N/A N/A N/A X (mm) Figure 4.24: Residual where: - Y calc. Y values from the equations Legend: - Y calc. Y real Y values from the surface - Y real 46
X Table 4.25: Residual Y calc. (mm) Y real (mm) Residual -6 7.457 6.651 0.806-3 7.446 6.430 1.015 0 6.355 6.422-0.067 3 6.245 6.561-0.316 Y (mm) 9 8 7 6 5 Radius 31 Z=53 Y calculated & real vs. X (mm) 6 6.958 6.783 0.175 10 7.369 7.096 0.273 4-10 -5 0 5 10 15 X (mm) Figure 4.25: Residual X Table 4.26: Residual Y calc. (mm) Y real (mm) Residual -6 5.068 4.741 0.326-3 5.084 4.568 0.516 0 4.731 4.621 0.110 Y (mm) 7 6 5 4 Z=45.3 Y calculated & real vs. X (mm) 3 4.612 4.740-0.128 3 2 6 4.917 4.761 0.157-10 -5 0 5 10 15 10 5.001 4.357 0.644 X (mm) Figure 4.26: Residual where: - Y calc. Y values from the equations Legend: - Y calc. Y real Y values from the surface - Y real 47
X Table 4.27: Residual Y calc. Y real (mm) (mm) Residual -6 N/A N/A N/A -3 4.673 4.194 0.480 0 4.406 4.251 0.156 Y (mm) 7 6 5 4 Z=40.8 Y calculated & real vs. X (mm) 3 4.375 4.268 0.107 6 4.485 4.198 0.287 10 N/A N/A N/A 3 2-4 -2 0 2 4 6 8 X (mm) Figure 4.27: Residual 4.4 Variance of lower order pertinent constants across the cross sectional area distribution of the radii. The variance of the lower order constants belonging to the family of polynomial rational equations was plotted against the cross sectional area distribution of the radii, for the medial and lateral volar surface deviations. This was done in order to determine if the constants followed a pattern across the cross sectional areas of the specimen. A linear relationship between the cross sectional area distribution and constants was attempted. If the data points were randomly scattered and could not be linearly regressed, the means *12and standard deviations were determined (Tables 4.28-4.51 and Figures 4.28-4.45). 48
X=0 (centroidal axis) Table 4.28: Constants distribution across the population Cross sectional Radius area a b c 3 113.2786 1.5205-0.0287-0.0446 4 102.5215 1.5383-0.0235-0.0425 5 156.8984 1.7404-0.0204-0.0404 7 146.7242 1.6092-0.0293-0.0470 8 151.3497 1.7623-0.0121-0.0259 9 119.3474 1.6636-0.0250-0.0474 10 184.4886 1.8002-0.0321-0.0565 21 190.0246 1.7930 0.0010 0.0016 31 108.0613 1.4810-0.0100-0.0200 Constant a 2.0 Relation between cross sectional area and constant a at X=0 y = 0.0034x + 1.1726 R 2 = 0.8075 1.8 a values 1.6 1.4 1.2 1.0 100 120 140 160 180 200 size (mm 2 ) Figure 4.28: Linearly regressed plot of constant a vs. cross sectional area 49