A New Approach to C 2 Continuous Piecewise Bicubic Representation of the Articular Surfaces of Diarthrodial Joints

Abstract

Based on the force-deflection equation for a beam subjected to lateral point loads, a C² continuous piecewise bicubic mathematical representation was proposed to model complicated geometrical surfaces, e.g. the articular surfaces of human joints. The method was then extended so that it could be used for mathematical modelling of incomplete nets of data points, as well as smoothing of noisy and/or filtering of erroneous data points. Mathematical techniques were also developed to calculate the required unknown parameters explicitly, with no need to solve the system of equations simultaneously. The performance of the proposed method was evaluated on a number of surface modelling problems, including two known analytical surfaces and the human femoral and patellar articular surfaces. The results indicate that the proposed method is precise, flexible, and easy to apply and has several advantages over the conventional smoothing methods, i.e. the B-spline approach.

Keywords

surface modelling smoothing B-spline articular surface

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References

Huiskes

Kremers

de Lange

Woltring

H. J.

Selvic

van Rens

T. J. G.

Analytical stereophotogrammetric determination of three-dimensional knee-joint geometry

J. Biomechanics, 1985, 18, 159–170.

Ateshian

G. A.

Soslowsky

L. J.

Mov

V. C.

Quantization of articular surface topography and cartilage thickness in knee joints using stereo-photogrammetry.J. Biomechanics, 1991, 24, 761–776.

Hefzy

M. S.

Yang

A three-dimensional anatomical model of the human patellofemoral joint for the determination of patellofemoral motions and contact characteristics

J. Biomed. Engng, 1993, 15, 235–264.

Soslowsky

L. J.

Flatow

E. L.

Bigliani

L. U.

Mow

V. C.

Articular geometry of the glenohumeral joint

Clin. Orthop., 1992, 288, 181–190.

Hirokawa

Three-dimensional mathematical model analysis of the patellofemoral joint

J. Biomechanics, 1991, 24, 659–671.

Hefzy

M. S.

Hsieh

Y. F.

Raftopoulos

Jackson

W. T.

Saddemi

Determination of the patellofemoral joint contact: Effect of tibial rotations

Proceedings of the ASME Biomechanics Symp., 1991, 133–136.

Spilker

R. L.

Almeida

E. S.

Clutz

Shephard

M. S.

Ateshian

G. A.

Donzelli

P. S.

Three dimensional automated biphasic finite element analysis of soft tissues from stereophotogrammetric data. ASME Advances in Bioengineering Conference, 1993, ASME BED-26, pp. 15–18.

O'Bara

R. M.

Shephard

M. S.

Ateshian

G. A.

Geometric model construction and mesh generation for soft tissues in joints. ASME Advances in Bioengineering Conference, 1995, ASME BED-29, pp. 215–216.

Rushfeldt

P. D.

Mann

R. W.

Hariss

W. H.

Improved techniques for measuring in vitro the geometry and pressure distribution in the human acetabulum. I: ultrasonic measurement of acetabu-lar surfaces, sphericity and cartilage thickness

J. Biomechanics, 1981, 14, 253–260.

10.

Blankevoort

Kuiper

J. H.

Huiskes

Grootenboer

H. J.

Articular contact in a three-dimensional model of the knee

J. Biomechanics, 1991, 24, 1019–1031.

11.

Ateshian

G. A.

A least-squares B-spline surface-fitting method for articular surfaces of diarthrodial joints

J. Biomech. Engng, 1993, 115, 366–373.

12.

Lee

J. H.

Modeling generalized cylinders using direction map representation

J. Computer Aided Des., 2005, 37, 837–846.

13.

Jutller

Maurer

Cubic Pythagorean hodo-graph spline curves and applications to sweep surface modeling

J. Computer Aided Des., 1999, 31, 73–83.

14.

Jutller

Wagner

M. G.

Rational motion-based surface generation.J. Computer Aided Des., 1999, 31, 203–213.

15.

Zhang

Cheng

Triangular patch modeling using combination method

J. Computer Aided Geometric Des., 2002, 19, 645–662.

16.

Huang

Bajaj

C1 modeling with A-patches from rational trivariate functions

J. Computer Aided Geometric Des., 2001, 18, 221–243.

17.

Navau

J. C.

Garcia

N. P.

Modeling surfaces from meshes of arbitrary topology

J. Computer Aided Geometric Des., 2000, 17, 643–671.

18.

Navau

J. C.

Garcia

N. P.

Modeling surfaces from planar irregular meshes

J. Computer Aided Geometric Des., 2000, 17, 1–15.

19.

Wang

J. H. C.

Ryu

Han

J. S.

Rowen

A new method for the representation of articular surfaces using the influence surface theory of plates

J. Biomechanics, 2000, 33, 629–633.

20.

Dhaher

Y. Y.

Delp

S. L.

Ryner

W. Z.

The use of basic functions in modeling joint articular surfaces: Application to the knee joint

J. Biomechanics, 2000, 33, 901–907.

21.

Hirokawa

Takashi

Ohtsuki

A new approach for fitting method of articular joint surfaces

J. Biomechanics, 2004, 37, 1551–1559.

22.

Wahaba

Multivariate thin plate spline smoothing with positivity and other linear inequality constraints In Statistical Image Processing and Graphics (Eds Wegmanand

E. J.

DePriest

D. J.

), 1986, pp. 235–264 (Marcel Decker, New York).