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Cartilage is poroelastic but not biphasic
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LETTERS
CARTILAGE
Joumdlr
Ltd
TO THE EDITOR
IS POROELASTIC
BUT NOT BIPHASIC
TIMOTHY P. HARRIGAN Orthopedic
Biomechanics
Laboratory,
Massachusetts
Webster’s dictionary defines a phase as ‘a solid, liquid or gaseous homogeneous form existing as a distinct part of a heterogeneous system’. The keys to this definition are the terms ‘homogeneous form’ and ‘distinct part’. The precise scientific interpretation of this definition is the requirement that, on a microstructural scale, a continuum can be defined within the phase. The purpose of this note is to clarify the discussion of phases. especially as regards cartilage poroelasticity Cartilage and other biological tissues are unarguably poroelastic, but poroelasticitydoes not imply that multiple phasescan be defined in the tissue. Thisdistinction is most important when porosity or volume fraction is discussed or used in analysis of cartilage. In order to define a phase within a heterogeneous material, a continuum must be definable on either side of a phase boundary. Figure 1 illustrates this situation. Consider, for example. the distinction between a solid solution such as brass and a eutectic alloy such as AI-CuAl, (Kraft er al., 1962). A volume fraction of the Al phase in the eutectic can be measured, but discussion of a volume fraction of copper in brass makes no sense. Since a continuum must include at least several molecules of a given component, the phase boundary can only be considered to exist over several molecular dimensions. Thus, a porosity or volume fraction measurement requires the identification of a phase boundary and the accuracy of a volume fraction measurement is limited by the relative accuracy with which the phase boundary can be identified. If one applies the requirements above to cartilage, the clear conclusion is that no phase boundaries exist, since the components intermingle on a molecular scale. Thus it makes no more sense to define a volume fraction of water in cartilage than it does to define a volume fraction of carbon in steel, or of glucose in solution. Figure 2 illustrates the relative mixing in cartilage. Large proteoglycan aggregates attract cations in solution and imbibe water, and collagen fibrils have water at interstitial sites. The osmotic preswelling which allows cartilage to survive compressive joint loads depends on the molecular-level interactions of solid and fluid components. This molecular-level mixing makes the definition of phases within the tissue meaningless, and the only reasonable phase to define is a single phase in the cartilage as a whole. The relevance of this distinction to cartilage modelling occurs when a conservation of mass equation is written for mutually incompressible components (Mow and Lai. 1979). A further complication occurs in the generation of constitutive relations using mixture theory (Mow and Lai. 1979). A critical step in the formulation of those equations is the postulated minimization ofentropy generation at equilibrium. The form of the constitutive equations used in a mixture approach to poroelasticity is arrived at by using an equation for entropy generation. In order to model incompressible mixtures using the theory, the incompressible conservation of mass equation, multiplied by a Lagrange multiplier (which is later interpreted as pressure) is added to the entropy generation expression. 827
General
Hospital.
Boston,
MA 02 114, USA
Fig. I. A microstructure with definable and volume fractions.
Fig. 2. A schematic
phase
of cartilage.
boundaries
828
Letters to the Editor
This resulting equation is minimized and taken equal to zero. Thus the incompressible conservation of mass equation motivates use of porosity in the solid and fluid constitutive equations. Since porosity cannot be defined, however, the terms involving porosity in both the constitutive equations and the conservation of mass equation lose physical meaning. Thus the connection between the mixture formulation and the microstructure implied by the porosity terms does not exist. and the already complicated connection between theory and experiment becomes still more abstruse, Theory and experiment can be made to coincide using mixture theory in cartilage, through curve fitting the data in an experiment to a family of functions generated by the theory (Mow et al., 1980).These results can be interpreted as follows: Given a poroelastic response, one can almost always identify an imaginary solid-fluid two phase material which will have the same temporal response to loading, given material properties which are chosen to match that response. The correspondence between the imaginary material with a definable porosity and cartilage is based solely on the curve fitting done. Thus a truly predictive study using mixture theory appears impossible. Cartilage is not the only biomaterial in which phases are postulated; tendon, bone and skin are currently also being studied using mixture theories. Cartilage was chosen for discussion here because the theoretical basis is most widely published, and much of the theory on bone, tendon and skin uses that material as a starting point. The physical effects modelled in poroeiastic mixture theories are identical to those modelled in the now-classical Biot theory (Biot, 1941).Thus theadded theoretical complexity and arbitrariness in phase definition appear to make
REPLY
TO HARRIGAN’S
mixture theory less useful than the standard poroelastic approach. Also, in biomechanics, the solution of a boundary value problem is only a springboard to physiological study. The availability of theoretical solutions (Clearv. 1978)and of finite element methods (Ghabourry and.Wilson, 1972) based on the Biot theory seems to make standard poroelastic modelling clearly superior to re-deriving field equations and boundary value solutions using mixture theory.
REFERENCES Biot. M. A. (1941) General theory of three dimensional consolidation. J. appl. Phys. 12, 155-164. Cleary. M. P. (1978) Moving singularities in elasto-diffusive solids with applications to fracture propagation. Inc. J. Solids Strucrures
Ann. Reo. Fluid
Mech.
II,
247-288.
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. Enyng
LETTER: CARTILAGE NOT BIPHASIC VAN
14, 81-97.
Ghaboussy, J. and Wilson, E. L. (1972) Variational formulation of dynamics of fluid saturated porous elastic solids. proc. AXE 98 (EM4), 941-963. Kraft, R. W., Lemkey, F. D. and George, F. D. (1962) Quantitative metallographic analysis of linear features in anisotropic structures. Substructure of lamellar eutectic alloy. Transactions OJ the Metallurgical Sociefy of Al ME Vol. 224, pp. 1037-1046. Mow, V. C. and Lai, W. M. (1979) Mechanics ofanimal joints.
102, 73-84.
IS POROELASTIC
BUT
C. Mow
Department of Mechanical Engineering and Qrthopaedic Bioengineering, Columbia University
Wedo not fully understand thecontent nor do we understand the real intent of Dr Timothy P. Harrigan’s letter. First, apparently, Dr Harrigan does not like the name ‘biphasic’ which we have chosen to call the binary mixture theory we used to describe the deformational behaviour of articular cartilage (Mow er al., 1980, 1985; Holmes, 1986). This theory was derived from modern mixture theory (Truesdell, 1965; Bowen, 1976;Green and Naghdi, 1967).Second, Dr Harrigan would like us to use the ‘now-classical’ Biot’s poroelasticity theorv (Biot. 1941: Freudenthal and Soillers. 1962:Schiffman, 1970jto describe articular cartilage and other soft hydrated biological tissues because he feels that the parameter ‘porosity’used in the biphasicequationscannot bedefined:‘Since porosity cannot be defined, however, the terms involving porosity in both the constitutive equations and the conservation of mass equation lose physical meaning’. Third. Dr Harrigan does not like the manner with which we curve-fit our experimental data because he believes that a truly predictive study using mixture theory appears impossible’. To answer the points raised by Dr. Harrigan, first, we have chosen the name ‘biphasic’ because there are significant ditTerences between our theory and the ‘now-classical’ Biot’s poroelasticity theory (Biot, 1941; Freudenthal and Spillers, 1962;Schiffman, 1970).These differences may be readily seen by referring to references (Mow er al.. 1980, 1985; Holmes, 1986; Biot, 1941; Freudenthal and Spillers, 1962; Schiffman,
1970). Second, in Biot’s poroelasticity theory, a volumetric ratio defining porosity ( = volume of voids to the total volume) is used! This is identical to that used in the biphasic theory. It is indeed a fact that this porosity factor is used in both the biphasic theory and the poroelasticity theory. Thus we do not understand the intent of this second point. The third point raised by Dr Harrigan raises a philosophical question: When does a physical theory become a valid physical theory? We use a number of simple and self evident criteria to guide us in constructing our theory: (1) the theory must be based upon valid and consistent physical principles, e.g. conservation of mass, conservation of energy, entropy inequality, principle of material objectivity, etc., (2) the mathematics used in the theory must be correct and the computations must be accurately achieved, and (3) the theory and mathematics must be able to predict the response of the material in a karge number oj independent experiments. We believe these criteria have been satisfied in our investigations. In closina. we would like to quote William Shakespeare concerning-names from Romeo bnd Julie& Act 11, Scene 1, lines 4345: What’s in a name? that which we call a rose, By any other name would smell as sweet; So Romeo would were he not Romeo call’d.
LETTERS
CARTILAGE
Joumdlr
Ltd
TO THE EDITOR
IS POROELASTIC
BUT NOT BIPHASIC
TIMOTHY P. HARRIGAN Orthopedic
Biomechanics
Laboratory,
Massachusetts
Webster’s dictionary defines a phase as ‘a solid, liquid or gaseous homogeneous form existing as a distinct part of a heterogeneous system’. The keys to this definition are the terms ‘homogeneous form’ and ‘distinct part’. The precise scientific interpretation of this definition is the requirement that, on a microstructural scale, a continuum can be defined within the phase. The purpose of this note is to clarify the discussion of phases. especially as regards cartilage poroelasticity Cartilage and other biological tissues are unarguably poroelastic, but poroelasticitydoes not imply that multiple phasescan be defined in the tissue. Thisdistinction is most important when porosity or volume fraction is discussed or used in analysis of cartilage. In order to define a phase within a heterogeneous material, a continuum must be definable on either side of a phase boundary. Figure 1 illustrates this situation. Consider, for example. the distinction between a solid solution such as brass and a eutectic alloy such as AI-CuAl, (Kraft er al., 1962). A volume fraction of the Al phase in the eutectic can be measured, but discussion of a volume fraction of copper in brass makes no sense. Since a continuum must include at least several molecules of a given component, the phase boundary can only be considered to exist over several molecular dimensions. Thus, a porosity or volume fraction measurement requires the identification of a phase boundary and the accuracy of a volume fraction measurement is limited by the relative accuracy with which the phase boundary can be identified. If one applies the requirements above to cartilage, the clear conclusion is that no phase boundaries exist, since the components intermingle on a molecular scale. Thus it makes no more sense to define a volume fraction of water in cartilage than it does to define a volume fraction of carbon in steel, or of glucose in solution. Figure 2 illustrates the relative mixing in cartilage. Large proteoglycan aggregates attract cations in solution and imbibe water, and collagen fibrils have water at interstitial sites. The osmotic preswelling which allows cartilage to survive compressive joint loads depends on the molecular-level interactions of solid and fluid components. This molecular-level mixing makes the definition of phases within the tissue meaningless, and the only reasonable phase to define is a single phase in the cartilage as a whole. The relevance of this distinction to cartilage modelling occurs when a conservation of mass equation is written for mutually incompressible components (Mow and Lai. 1979). A further complication occurs in the generation of constitutive relations using mixture theory (Mow and Lai. 1979). A critical step in the formulation of those equations is the postulated minimization ofentropy generation at equilibrium. The form of the constitutive equations used in a mixture approach to poroelasticity is arrived at by using an equation for entropy generation. In order to model incompressible mixtures using the theory, the incompressible conservation of mass equation, multiplied by a Lagrange multiplier (which is later interpreted as pressure) is added to the entropy generation expression. 827
General
Hospital.
Boston,
MA 02 114, USA
Fig. I. A microstructure with definable and volume fractions.
Fig. 2. A schematic
phase
of cartilage.
boundaries
828
Letters to the Editor
This resulting equation is minimized and taken equal to zero. Thus the incompressible conservation of mass equation motivates use of porosity in the solid and fluid constitutive equations. Since porosity cannot be defined, however, the terms involving porosity in both the constitutive equations and the conservation of mass equation lose physical meaning. Thus the connection between the mixture formulation and the microstructure implied by the porosity terms does not exist. and the already complicated connection between theory and experiment becomes still more abstruse, Theory and experiment can be made to coincide using mixture theory in cartilage, through curve fitting the data in an experiment to a family of functions generated by the theory (Mow et al., 1980).These results can be interpreted as follows: Given a poroelastic response, one can almost always identify an imaginary solid-fluid two phase material which will have the same temporal response to loading, given material properties which are chosen to match that response. The correspondence between the imaginary material with a definable porosity and cartilage is based solely on the curve fitting done. Thus a truly predictive study using mixture theory appears impossible. Cartilage is not the only biomaterial in which phases are postulated; tendon, bone and skin are currently also being studied using mixture theories. Cartilage was chosen for discussion here because the theoretical basis is most widely published, and much of the theory on bone, tendon and skin uses that material as a starting point. The physical effects modelled in poroeiastic mixture theories are identical to those modelled in the now-classical Biot theory (Biot, 1941).Thus theadded theoretical complexity and arbitrariness in phase definition appear to make
REPLY
TO HARRIGAN’S
mixture theory less useful than the standard poroelastic approach. Also, in biomechanics, the solution of a boundary value problem is only a springboard to physiological study. The availability of theoretical solutions (Clearv. 1978)and of finite element methods (Ghabourry and.Wilson, 1972) based on the Biot theory seems to make standard poroelastic modelling clearly superior to re-deriving field equations and boundary value solutions using mixture theory.
REFERENCES Biot. M. A. (1941) General theory of three dimensional consolidation. J. appl. Phys. 12, 155-164. Cleary. M. P. (1978) Moving singularities in elasto-diffusive solids with applications to fracture propagation. Inc. J. Solids Strucrures
Ann. Reo. Fluid
Mech.
II,
247-288.
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. Enyng
LETTER: CARTILAGE NOT BIPHASIC VAN
14, 81-97.
Ghaboussy, J. and Wilson, E. L. (1972) Variational formulation of dynamics of fluid saturated porous elastic solids. proc. AXE 98 (EM4), 941-963. Kraft, R. W., Lemkey, F. D. and George, F. D. (1962) Quantitative metallographic analysis of linear features in anisotropic structures. Substructure of lamellar eutectic alloy. Transactions OJ the Metallurgical Sociefy of Al ME Vol. 224, pp. 1037-1046. Mow, V. C. and Lai, W. M. (1979) Mechanics ofanimal joints.
102, 73-84.
IS POROELASTIC
BUT
C. Mow
Department of Mechanical Engineering and Qrthopaedic Bioengineering, Columbia University
Wedo not fully understand thecontent nor do we understand the real intent of Dr Timothy P. Harrigan’s letter. First, apparently, Dr Harrigan does not like the name ‘biphasic’ which we have chosen to call the binary mixture theory we used to describe the deformational behaviour of articular cartilage (Mow er al., 1980, 1985; Holmes, 1986). This theory was derived from modern mixture theory (Truesdell, 1965; Bowen, 1976;Green and Naghdi, 1967).Second, Dr Harrigan would like us to use the ‘now-classical’ Biot’s poroelasticity theorv (Biot. 1941: Freudenthal and Soillers. 1962:Schiffman, 1970jto describe articular cartilage and other soft hydrated biological tissues because he feels that the parameter ‘porosity’used in the biphasicequationscannot bedefined:‘Since porosity cannot be defined, however, the terms involving porosity in both the constitutive equations and the conservation of mass equation lose physical meaning’. Third. Dr Harrigan does not like the manner with which we curve-fit our experimental data because he believes that a truly predictive study using mixture theory appears impossible’. To answer the points raised by Dr. Harrigan, first, we have chosen the name ‘biphasic’ because there are significant ditTerences between our theory and the ‘now-classical’ Biot’s poroelasticity theory (Biot, 1941; Freudenthal and Spillers, 1962;Schiffman, 1970).These differences may be readily seen by referring to references (Mow er al.. 1980, 1985; Holmes, 1986; Biot, 1941; Freudenthal and Spillers, 1962; Schiffman,
1970). Second, in Biot’s poroelasticity theory, a volumetric ratio defining porosity ( = volume of voids to the total volume) is used! This is identical to that used in the biphasic theory. It is indeed a fact that this porosity factor is used in both the biphasic theory and the poroelasticity theory. Thus we do not understand the intent of this second point. The third point raised by Dr Harrigan raises a philosophical question: When does a physical theory become a valid physical theory? We use a number of simple and self evident criteria to guide us in constructing our theory: (1) the theory must be based upon valid and consistent physical principles, e.g. conservation of mass, conservation of energy, entropy inequality, principle of material objectivity, etc., (2) the mathematics used in the theory must be correct and the computations must be accurately achieved, and (3) the theory and mathematics must be able to predict the response of the material in a karge number oj independent experiments. We believe these criteria have been satisfied in our investigations. In closina. we would like to quote William Shakespeare concerning-names from Romeo bnd Julie& Act 11, Scene 1, lines 4345: What’s in a name? that which we call a rose, By any other name would smell as sweet; So Romeo would were he not Romeo call’d.