Modeling of Neutral Solute Transport in a Dynamically Loaded Porous Permeable Gel: Implications for Articular Cartilage Biosynthesis and Tissue Engineering

Journal of Biomechanical Engineering

Volumn 125, Issue 5, 2003, Pages 602-614

  Mauck, Robert L ^a   Hung, Clark T ^a   Ateshian, Gerard A ^a  

^a Columbia University ^*  (United States)

Author keywords

[No Author keywords available]

Indexed keywords

DIFFUSION; FRICTION; GELS; HYDRATION; MECHANICAL PERMEABILITY; POROUS MATERIALS; STRESSES; TISSUE;

SOLUTES;

BIOMECHANICS;

ANIMAL TISSUE; ARTICLE; ARTICULAR CARTILAGE; BIOSYNTHESIS; DIFFUSION; DYNAMICS; EXTRACELLULAR MATRIX; GEL; HUMAN; HUMAN TISSUE; HYDRATION; HYPOTHESIS; MODEL; NONHUMAN; PARTICLE SIZE; PERMEABILITY; SOLUTE; THEORY; TISSUE ENGINEERING;

ANIMALS; BIOLOGICAL TRANSPORT; CARTILAGE, ARTICULAR; COMPUTER SIMULATION; DIFFUSION; GELS; HUMANS; MECHANOTRANSDUCTION, CELLULAR; MODELS, BIOLOGICAL; MODELS, CHEMICAL; PERMEABILITY; POROSITY; RHEOLOGY; SOLUBILITY; SOLUTIONS; TISSUE ENGINEERING; WATER; WEIGHT-BEARING;

ANIMALIA;

EID: 0142200806     PISSN: 01480731     EISSN: None     Source Type: Journal    
DOI: 10.1115/1.1611512     Document Type: Article

References (93)

1. McKibbin, B., and Maroudas, A., 1974, "Nutrition and Metabolism," Adult articular cartilage. M. A. R. Freeman, ed., Grune & Stratton, New York, pp. 461-486.
2. Maroudas, A., 1968, "Physicochemical Properties of Cartilage in the Light of Ion Exchange Theory," Biophys. J., 8(5), pp. 575-595.
3. Maroudas, A., 1975, "Biophysical Chemistry of Cartilaginous Tissues With Special Reference to Solute and Fluid Transport," Biorheology, 12(3-4), pp. 233-248.
4. Torzilli, P. A., Arduino, J. M., Gregory, J. D., and Bansal, M., 1997, "Effect of Proteoglycan Removal on Solute Mobility in Articular Cartilage." J. Biomech., 30(9), pp. 895-902.
5. Torzilli, P. A., Adams, T. C., and Mis, R. J., 1987, "Transient Solute Diffusion in Articular Cartilage," J. Biomech., 20(2), pp. 203-214.
6. Torzilli, P. A., 1993, "Effects of Temperature, Concentration and Articular Surface Removal on Transient Solute Diffusion in Articular Cartilage," Med. Biol. Eng. Comput., 31(Suppl), pp. S93-98.
7. Burstein, D., Gray, M. L., Hartman, A. L., Gipe, R., and Foy, B. D., 1993, "Diffusion of Small Solutes in Cartilage as Measured by Nuclear Magnetic Resonance (NMR) Spectroscopy and Imaging," J. Orthop. Res., 11(4), pp. 465-478.
8. Schneiderman, R., Snir, E., Popper, O., Hiss, J., Stein, H., and Maroudas, A., 1995, "Insulin-Like Growth Factor-I and Its Complexes in Normal Human Articular Cartilage: Studies of Partition and Diffusion," Arch. Biochem. Biophys., 324(1), pp. 159-172.
9. Potter, K., Spencer, R. G., and McFarland, E. W., 1997, "Magnetic Resonance Microscopy Studies of Cation Diffusion in Cartilage," Biochim. Biophys. Acta, 1334(2-3), pp. 129-139.
10. Quinn, T. M., Kocian, P., and Meister, J. J., 2000, "Static Compression is Associated With Decreased Diffusivity of Dextrans in Cartilage Explants," Arch. Biochem. Biophys., 384(2), pp. 327-334.
11. Foy, B. D., and Blake, J., 2001, "Diffusion of Paramagnetically Labeled Proteins in Cartilage: Enhancement of the 1-D NMR Imaging Technique," J. Magn. Reson., 148(1), pp. 126-134.
12. O'Hara, B. P., Urban, J. P., and Maroudas, A., 1990, "Influence of Cyclic Loading on the Nutrition of Articular Cartilage," Ann. Rheum. Dis., 49(7), pp. 536-539.
13. Garcia, A. M., Frank, E. H., Grimshaw, P. E., and Grodzinsky, A. J., 1996, "Contributions of Fluid Convection and Electrical Migration to Transport in Cartilage: Relevance to Loading," Arch. Biochem. Biophys., 333(2), pp. 317-325.
14. Urban, J. P., Holm, S., Maroudas, A., and Nachemson, A., 1982, "Nutrition of the Intervertebral Disc: Effect of Fluid Flow on Solute Transport," Clin. Orthop., 170, pp. 296-302.
15. Katz, M. M., Hargens, A. R., and Garfin, S. R., 1986, "Intervertebral Disc Nutrition. Diffusion Versus Convection," Clin. Orthop., 210, pp. 243-245.
16. Garcia, A. M., Lark, M. W., Trippel, S. B., and Grodzinsky, A. J., 1998, "Transport of Tissue Inhibitor of Metalloproteinases-1 Through Cartilage: Contributions of Fluid Flow and Electrical Migration," J. Orthop. Res., 16(6), pp. 734-742.
17. Fatt, I., and Goldstick, T. K., 1965, "Dynamics of Water Transport in Swelling Membranes," J. Colloid Sci., 20, pp. 962-989.
18. Sah, R. L., Kim, Y. J., Doong, J. Y., Grodzinsky, A. J., Plaas, A. H., and Sandy, J. D., 1989, "Biosynthetic Response of Cartilage Explants to Dynamic Compression," J. Orthop. Res., 7(5), pp. 619-636.
19. Kim, Y. J., Sah, R. L., Grodzinsky, A. J., Plaas, A. H., and Sandy, J. D., 1994, "Mechanical Regulation of Cartilage Biosynthetic Behavior: Physical Stimuli," Arch. Biochem. Biophys., 311(1), pp. 1-12.
20. Palmoski, M. J., and Brandt, K. D., 1984, "Effects of Static and Cyclic Compressive Loading on Articular Cartilage Plugs In Vitro," Arthritis Rheum., 27(6), pp. 675-681.
21. Guilak, F., Sah, R. L., and Setton, L. A., 1997, "Physical Regulation of Cartilage Metabolism," Basic Orthopaedic Biomechanics, V. C. Mow and W. C. Hayes, eds., Lippincott-Raven, Philadelphia, pp. 179-207.
22. Guilak, F., Meyer, B. C., Ratcliffe, A., and Mow, V. C., 1994, "The Effects of Matrix Compression on Proteoglycan Metabolism in Articular Cartilage Explants," Osteoarthritis Cartilage, 2(2), pp. 91-101.
23. Quinn, T. M., Morel, V., and Meister, J. J., 2001, "Static Compression of Articular Cartilage Can Reduce Solute Diffusivity and Partitioning: Implications for the Chondrocyte Biological Response," J. Biomech., 34(11), pp. 1463-1469.
24. Bonassar, L. J., Grodzinsky, A. J., Frank, E. H., Davila, S. G., Bhaktav, N. R., and Trippel, S. B., 2001, "The Effect of Dynamic Compression on the Response of Articular Cartilage to Insulin-Like Growth Factor-I," J. Orthop. Res., 19(1), pp. 11-17.
25. Kuettner, K. E., 1992, "Biochemistry of Articular Cartilage in Health and Disease," Clin. Biochem., 25(3), pp. 155-163.
26. DiMicco, M. A., and Sah, R. L., 2003, "Dependence of Cartilage Matrix Composition on Biosynthesis, Diffusion, and Reaction," Transp. Porous Media, 50(1-2), pp. 57-73.
27. Freed, L. E., Vunjak-Novakovic, G., and Langer, R., 1993, "Cultivation of Cell-Polymer Cartilage Implants in Bioreactors," J. Cell. Biochem., 51(3), pp. 257-264.
28. Buschmann, M. D., Gluzband, Y. A., Grodzinsky, A. J., and Hunziker, E. B., 1995, "Mechanical Compression Modulates Matrix Biosynthesis in Chondrocyte/Agarose Culture," J. Cell. Sci., 108(Pt 4), pp. 1497-1508.
29. Mauck, R. L., Soltz, M. A., Wang, C. C., Wong, D. D., Chao, P. H., Valhmu, W. B., Hung, C. T., and Ateshian, G. A., 2000, "Functional Tissue Engineering of Articular Cartilage Through Dynamic Loading of Chondrocyte-Seeded Agarose Gels," J. Biomech. Eng., 122(3), pp. 252-260.
30. Mauck, R. L., Nicoll, S. B., Seyhan, S. L., Ateshian, G. A., and Hung, C. T., 2003, "Synergistic Action of Growth Factors and Dynamic Loading for Articular Cartilage Tissue Engineering," Tissue Eng., 9(4), pp. 597-612.
31. Mills, N., 1966, "Incompressible Mixture of Newtonian Fluids," Int. J. Eng. Sci., 4, pp. 97-112.
32. Craine, R. E., 1971, "Oscillations of a Plate in a Binary Mixture of Incompressible Newtonian Fluids," Int. J. Eng. Sci., 9(12), pp. 1177-1192.
33. Bowen, R. M., 1980, "Incompressible Porous Media Models by Use of the Theory of Mixtures," Int. J. Eng. Sci., 18(9), pp. 1129-1148.
34. Mow, V. C., Kuei, S. C., Lai, W. M., and Armstrong, C. G., 1980, "Biphasic Creep and Stress Relaxation of Articular Cartilage in Compression? Theory and Experiments," J. Biomech. Eng., 102(1), pp. 73-84.
35. Frank, E. H., and Grodzinsky, A. J., 1987, "Cartilage Electromechanics-II. A Continuum Model of Cartilage Electrokinetics and Correlation With Experiments," J. Biomech., 20(6), pp. 629-639.
36. Lai, W. M., Hou, J. S., and Mow, V. C., 1991, "A Triphasic Theory for the Swelling and Deformation Behaviors of Articular Cartilage," J. Biomech. Eng., 113(3), pp. 245-258.
37. Huyghe, J. M., and Janssen, J. D., 1997, "Quadriphasic Mechanics of Swelling Incompressible Porous Media," Int. J. Eng. Sci., 35(8), pp. 793-802.
38. Gu, W. Y., Lai, W. M., and Mow, V. C., 1998. "A Mixture Theory for Charged-Hydrated Soft Tissues Containing Multi-Electrolytes: Passive Transport and Swelling Behaviors," J. Biomech. Eng., 120(2), pp. 169-180.
39. Armstrong, C. G., Lai, W. M., and Mow, V. C., 1984, "An Analysis of the Unconfined Compression of Articular Cartilage," J. Biomech. Eng., 106(2), pp. 165-173.
40. Tinoco, I., 2002, Physical Chemistry: Principles and Applications in Biological Sciences, Prentice-Hall, Upper Saddle River, NJ.
41. Robinson, R. A., and Stokes, R. H., 1955, Electrolyte Solutions: The Measurement and Interpretation of Conductance, Chemical Potential and Diffusion in Solutions of Simple Electrolytes, Academic Press, New York.
42. Torzilli, P. A., Askari, E., and Jenkins, J. T., 1990. "Water Content and Solute Diffusion Properties in Articular Cartilage," Biomechanics of Diarthrodial Joints, V. C. Mow, A. Ratcliffe, and S. L. Y. Woo, eds., Springer-Verlag, New York, pp. 363-390.
43. Fournier, R. L., 1999, "Solute Diffusion Within Heterogenous Media," Basic Transport Phenomena in Biomedical Engineering, Taylor & Francis, Philadelphia, pp. 28-32.
44. Katchalsky, A., and Curran, P. F., 1975, "Isothermal Diffusion and Sedimentation," Nonequilibrium Thermodynamics in Biophysics, Harvard University Press, Cambridge, pp. 98-102.
45. Lai, W. M., and Mow, V. C., 1980, "Drag-Induced Compression of Articular Cartilage During a Permeation Experiment," Biorheology, 17(1-2), pp. 111-123.
46. Hou, J. S., Holmes, M. H., Lai, W. M., and Mow, V. C., 1989, "Boundary Conditions at the Cartilage-Synovial Fluid Interface for Joint Lubrication and Theoretical Verifications," J. Biomech. Eng., 111(1), pp. 78-87.
47. Sun, D. N., Gu, W. Y., Guo, X. E., Lai, W. M., and Mow, V. C., 1999, "A Mixed Finite Element Formulation of Triphasic Mechano-Electrochemical Theory for Charged, Hydrated Soft Tissues," Int. J. Numer. Methods Eng., 45(10), pp. 1375-1402.
48. Ateshian, G. A., Soltz, M. A., Mauck, R. L., Basalo, I. M., Hung, C. T., and Lai, W. M., 2003, "The Role of Osmotic Pressure and Tension-Compression Nonlinearity in the Frictional Response of Articular Cartilage," Transp. Porous Media, 50, pp. 5-33.
49. Van Holde, K. E., Johnson, W. C., and Ho, P. S., 1998, "Thermodynamics of Transport Processes," Principles of Physical Biochemistry, Prentice-Hall, Upper Saddle River, NJ, pp. 574.
50. Deen, W. M., 1987. "Hindered Transport of Large Molecules in Liquid-Filled Pores," AIChE J., 33, pp. 1409-1425.
51. Holmes, M. H., Lai, W. M., and Mow, V. C., 1985, "Singular Perturbation Analysis of the Nonlinear, Flow-Dependent Compressive Stress Relaxation Behavior of Articular Cartilage," J. Biomech. Eng., 107(3), pp. 206-218.
52. Lee, R. C., Frank, E. H., Grodzinsky, A. J., and Roylance, D. K., 1981, "Oscillatory Compressional Behavior of Articular Cartilage and Its Associated Electromechanical Properties." J. Biomech. Eng., 103(4), pp. 280-292.
53. Soltz, M. A., and Ateshian, G. A., 2000, "Interstitial Fluid Pressurization During Confined Compression Cyclical Loading of Articular Cartilage," Ann. Biomed. Eng., 28(2), pp. 150-159.
54. Grodzinsky, A. J., Levenston, M. E., Jin, M., and Frank, E. H., 2000, "Cartilage Tissue Remodeling in Response to Mechanical Forces," Annu. Rev. Biomed. Eng., 2, pp. 691-713.
55. Lee, D. A., and Bader, D. L., 1997, "Compressive Strains at Physiological Frequencies Influence the Metabolism of Chondrocytes Seeded in Agarose," J. Orthop. Res., 15(2), pp. 181-188.
56. Mow, V. C., and Lai, W. M., 1980, "Recent Developments in Synovial Joint Biomechanics," SIAM Rev., 22, pp. 275-317.
57. Lai, W. M., and Mow, V. C., 1979, "Flow Fields in a Single-Layer Model of Articular Cartilage Created by a Sliding Load," ASME Adv. Bioeng., pp. 101-104.
58. Ateshian, G. A., and Wang, H., 1995, "A Theoretical Solution for the Frictionless Rolling Contact of Cylindrical Biphasic Articular Cartilage Layers," J. Biomech., 28(11), pp. 1341-1355.
59. Ateshian, G. A., and Wang, X., 1998, "Sliding Tractions on a Deformable Porous Layer," J. Tribol., 120(1), pp. 89-96.
60. Bonassar, L. J., Grodzinsky, A. J., Srinivasan, A., Davila, S. G., and Trippel, S. B., 2000, "Mechanical and Physicochemical Regulation of the Action of Insulin-Like Growth Factor-I on Articular Cartilage," Arch. Biochem. Biophys., 379(1), pp. 57-63.
61. Buschmann, M. D., Gluzband, Y. A., Grodzinsky, A. J., Kimura, J. H., and Hunziker, E. B., 1992, "Chondrocytes in Agarose Culture Synthesize a Mechanically Functional Extracellular Matrix," J. Orthop. Res., 10(6), pp. 745-758.
62. Soltz, M. A., Stankiewicz, A., Mauck, R. L., Hung, C. T., and Ateshian, G. A., 1999, "Direct Hydraulic Permeability Measurements of Agarose Hydrogels Used as Cell Scaffolds," ASME Adv. Bioeng., 43. pp. 229-230.
63. Andarawis, N. A., Seyhan, S. L., Mauck, R. L., Soltz, M. A., Ateshian, G. A., and Hung, C. T., 2001, "A Novel Permeation Device for Hydrogels and Soft Tissues," Proc. ASME IMECE, paper no. 23149.
64. Perdue, J. F., 1984, "Chemistry, Structure, and Function of Insulin-Like Growth Factors and Their Receptors: A Review," Can. J. Biochem. Cell Biol., 62(11), pp. 1237-1245.
65. Enberg, G., Carlquist, M., Jornvall, H., and Hall, K., 1984. "The Characterization of Somatomedin A, Isolated by Microcomputer-Controlled Chromatography, Reveals an Apparent Identity to Insulin-Like Growth Factor 1," Eur. J. Biochem., 143(1), pp. 117-124.
66. Kuffer, A. D., and Herington, A. C., 1984, "Proteolytic Conversion of Insulin-Like Growth Factors to an Acidic Form(s)," Biochem. J., 223(1), pp. 97-103.
67. Herington, A. C., and Kuffer, A. D., 1984, "Insulin-Like Growth Factor Characteristics of an Acidic Non-Suppressible Insulin-Like Activity," Biochem. J., 223(1), pp. 89-96.
68. Herington, A. C., Cornell, H. J., and Kuffer, A. D., 1983, "Recent Advances in the Biochemistry and Physiology of the Insulin-Like Growth Factor/Somatomedin Family," Int. J. Biochem., 15(10), pp. 1201-1210.
69. Kim, M. K., Warren, T. C., and Kimball, E. S., 1985, "Purification and Characterization of a Low Molecular Weight Transforming Growth Factor From the Urine of Melanoma Patients," J. Biol. Chem., 260(16), pp. 9237-9243.
70. Yamaoka, K., Hirai, R., Tsugita, A., and Mitsui, H., 1984. "The Purification of an Acid- and Heat-Labile Transforming Growth Factor From an Avian Sarcoma Virus-Transformed Rat Cell Line," J. Cell Physiol., 119(3), pp. 307-314.
71. Wang, C. C.-B., Soltz, M. A., Mauck, R. L., Valhmu, W. B., Ateshian, G. A., and Hung, C. T., 2000, "Comparison of the Equilibrium Axial Strain Distribution in Articular Cartilage Explants and Cell-Seeded Alginate Disks Under Unconfined Compression," Trans. Orthop. Res. Soc., 25, p. 131.
72. Maroudas, A., 1979, "Physicochemical Properties of Articular Cartilage," Adult Articular Cartilage, M. A. R. Freeman, ed., Pitman Medical, Kent, pp. 215-290.
73. Johnson, E. M., Berk, D. A., Jain, R. K., and Deen, W. M., 1995. "Diffusion and Partitioning of Proteins in Charged Agarose Gels," Biophys. J., 68(4), pp. 1561-1568.
74. Lai, W. M., Mow, V. C., Sun, D. D., and Ateshian, G. A., 2000, "On the Electric Potentials Inside a Charged Soft Hydrated Biological Tissue: Streaming Potential Versus Diffusion Potential," J. Biomech. Eng., 122(4), pp. 336-346.
75. Lai, W. M., Sun, D. D., Ateshian, G. A., Guo, X. E., and Mow, V. C., 2002, "Electrical Signals for Chondrocytes in Cartilage," Biorheology, 39(1-2), pp. 39-45.
76. Mow, V. C., Ateshian, G. A., Lai, W. M., and Gu, W. Y., 1998, "Effects of Fixed Charges on the Stress-Relaxation Behavior of Hydrated Soft Tissues in a Confined Compression Problem," Int. J. Solids Struct., 35(34-35), pp. 4945-4962.
77. Wang, C. C., Hung, C. T., and Mow, V. C., 2001, "An Analysis of the Effects of Depth-Dependent Aggregate Modulus on Articular Cartilage Stress-Relaxation Behavior in Compression," J. Biomech., 34(1), pp. 75-84.
78. Soltz, M. A., and Ateshian, G. A., 2000, "A Conewise Linear Elasticity Mixture Model for the Analysis of Tension-Compression Nonlinearity in Articular Cartilage," J. Biomech. Eng., 122(6), pp. 576-586.
79. Huang, C. Y., Mow, V. C., and Ateshian, G. A., 2001, "The Role of Flow-Independent Viscoelasticity in the Biphasic Tensile and Compressive Responses of Articular Cartilage," J. Biomech. Eng., 123(5), pp. 410-417.
80. Horner, H. A., and Urban, J. P., 2001, "2001 Volvo Award Winner in Basic Science Studies: Effect of Nutrient Supply on the Viability of Cells From the Nucleus Pulposus of the Intervertebral Disc," Spine, 26(23), pp. 2543-2549.
81. Bhakta, N. R., Garcia, A. M., Frank, E. H., Grodzinsky, A. J., and Morales, T. I., 2000, "The Insulin-Like Growth Factors (IGFs) I and II Bind to Articular Cartilage Via the IGF-Binding Proteins," J. Biol. Chem., 275(8), pp. 5860-5866.
82. Pedrozo, H. A., Schwartz, Z., Gomez, R., Ornoy, A., Xin-Sheng, W., Dallas, S. L., Bonewald, L. F., Dean, D. D., and Boyan, B. D., 1998, "Growth Plate Chondrocytes Store Latent Transforming Growth Factor (TGF)-Beta 1 in Their Matrix Through Latent TGF-Beta 1 Binding Protein-1," J. Cell Physiol., 177(2), pp. 343-354.
83. Chintala, S. K., Miller, R. R., and McDevitt, C. A., 1994, "Basic Fibroblast Growth Factor Binds to Heparan Sulfate in the Extracellular Matrix of Rat Growth Plate Chondrocytes," Arch. Biochem. Biophys., 310(1), pp. 180-186.
84. Lai, W. M., Mow, V. C., and Roth, V., 1981, "Effects of Nonlinear Strain-Dependent Permeability and Rate of Compression on the Stress Behavior of Articular Cartilage," J. Biomech. Eng., 103(2), pp. 61-66.
85. Gu, W. Y., Yao, H., Huang, C. Y., and Cheung, H. S., 2003, "New Insight Into Deformation-Dependent Hydraulic Permeability of Gels and Cartilage, and Dynamic Behavior of Agarose Gels in Confined Compression," J. Biomech., 36(4), pp. 593-598.
86. Luby-Phelps, K., Castle, P. E., Taylor, D. L., and Lanni, F., 1987, "Hindered Diffusion of Inert Tracer Particles in the Cytoplasm of Mouse 3T3 Cells," Proc. Natl. Acad. Sci. U.S.A., 84, pp. 4910-4913.
87. Allhands, R. V., Torzilli, P. A., and Kallfelz, F. A., 1984, "Measurement of Diffusion of Uncharged Molecules in Articular Cartilage," Cornell Vet., 74(2), pp. 111-123.
88. Roger, P., Mattisson, C., Axelsson, A., and Zacchi, G., 2000, "Use of Holographic Laser Interferometry to Study the Diffusion of Polymers in Gels," Biotechnol. Bioeng., 69(6), pp. 654-663.
89. Johnson, E. M., Berk, D. A., Jain, R. K., and Deen, W. M., 1996, "Hindered Diffusion in Agarose Gels: Test of Effective Medium Model," Biophys. J., 70(2), pp. 1017-1023.
90. Mow, V. C., Hou, J. S., Owens, J. M., and Ratcliffe, A., 1990, "Biphasic and Quasilinear Viscoelastic Theories for Hydrated Soft Tissues," Biomechanics of Diarthrodial Joints, V. C. Mow, A. Ratcliffe, and S. L. Y. Woo, eds., Springer-Verlag, New York, pp. 215-260.
91. Williamson, A. K., Chen, A. C., and Sah, R. L., 2001, "Compressive Properties and Function-Composition Relationships of Developing Bovine Articular Cartilage," J. Orthop. Res., 19(6), pp. 1113-1121.
92. Freed, L. E., Langer, R., Martin, I., Pellis, N. R., and Vunjak-Novakovic, G., 1997, "Tissue Engineering of Cartilage in Space," Proc. Natl. Acad. Sci. U.S.A., 94(25), pp. 13885-13890.
93. Chang, S. C., Rowley, J. A., Tobias, G., Genes, N. G., Roy, A. K., Mooney, D. J., Vacanti, C. A., and Bonassar, L. J., 2001, "Injection Molding of Chondrocyte/Alginate Constructs in the Shape of Facial Implants," J. Biomed. Mater. Res., 55(4), pp. 503-511.