Am J Physiol Heart Circ Physiol 294: H66-H73, 2008.
First published October 12, 2007; doi:10.1152/ajpheart.00703.2007
0363-6135/08 $8.00
The validation of a generalized Hooke's law for coronary arteries
Chong Wang,1
Wei Zhang,2 and
Ghassan S. Kassab2,3,4
1Department of Mechanical and Aerospace Engineering, University of California, Irvine, California; and Departments of 2Biomedical Engineering, 3Surgery, and 4Cellular and Integrative Physiology and Indiana Center for Vascular Biology and Medicine, Indiana University-Purdue University Indianapolis, Indianapolis, Indiana
Submitted 16 June 2007
; accepted in final form 11 October 2007
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ABSTRACT
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The exponential form of constitutive model is widely used in biomechanical studies of blood vessels. There are two main issues, however, with this model: 1) the curve fits of experimental data are not always satisfactory, and 2) the material parameters may be oversensitive. A new type of strain measure in a generalized Hooke's law for blood vessels was recently proposed by our group to address these issues. The new model has one nonlinear parameter and six linear parameters. In this study, the stress-strain equation is validated by fitting the model to experimental data of porcine coronary arteries. Material constants of left anterior descending artery and right coronary artery for the Hooke's law were computed with a separable nonlinear least-squares method with an excellent goodness of fit. A parameter sensitivity analysis shows that the stability of material constants is improved compared with the exponential model and a biphasic model. A boundary value problem was solved to demonstrate that the model prediction can match the measured arterial deformation under experimental loading conditions. The validated constitutive relation will serve as a basis for the solution of various boundary value problems of cardiovascular biomechanics.
constitutive relation; material constants; nonlinearity
THE VASCULAR STRESS-STRAIN RELATION is fundamental for biomechanical studies of blood vessels. Various types of constitutive (stress-strain relation) models have been proposed, e.g., the polynomial (12), logarithmic (15), exponential (3, 4), and biphasic (6, 7, 9) functions. Reviews of the various constitutive equations can be found in papers by Sacks and Sun (14), Vito and Dixon (16) and Holzapfel et al. (7). The most widely used constitutive relation is the Fung-type pseudoelastic model (3, 4), which describes the strain energy of blood vessels as an exponential function of a quadratic sum of the Green strain components. The biphasic strain energy function proposed by Holzapfel et al. (8), which decouples the isotropic part from the anisotropic part, has been used recently (13, 18).
The stress-strain relation of blood vessels is highly nonlinear, and there is strong coupling between the circumferential and axial directions. A good constitutive model is expected to describe these mechanical behaviors with good fits to the experimental data and to have stable parameters, i.e., variations between material parameters are relatively small for a group of mechanically similar data. The logarithmic form has a limited ability to describe this anisotropic behavior of blood vessels and may generate "physically unrealistic predictions" (10). The polynomial form is less capable of distinguishing differences between various data sets, since it has larger variations in its material constants compared with the exponential form (3). Compared with these forms, Fung's exponential model captures the anisotropic behavior of blood vessels relatively well with seven parameters (1 linear parameter and 6 nonlinear parameters). The fit can still be inadequate, however, when the data include multiple loading curves from different axial stretch ratios. An additional issue is the oversensitivity of material constants, i.e., large variations of material constants can still be deduced for mechanically similar blood vessels. This sensitivity to the small change in experimental data is due to the nonuniqueness of the nonlinear optimization. The biphasic model (6) addresses this issue by reducing the number of nonlinear parameters. The parameters are argued to have a microstructure-based physical meaning, albeit this has not been validated.
We recently proposed a generalized Hooke's law for blood vessels (20) in which the stress is linearly dependent on a new strain tensor that is a nonlinear function of stretch ratios. In an inflation-stretch test without shear deformation, there are seven material constants. One is used in the definition of strains and represents the nonlinearity of the material, and the other six constants are linear parameters that can be interpreted as elastic constants with respect to the new strain measure. This is comparable to seven parameters for the exponential model and the polynomial model. In comparison, there are six nonlinear parameters and one linear parameter in Fung's exponential model (1, 3), and there are two linear parameters and three nonlinear parameters in the biphasic model (9).
The objective of this study was to compare the generalized Hooke's law with the exponential and biphasic constitutive relations based on a set of experimental data of porcine coronary arteries from a recent study (17). Comparisons were made on the basis of the goodness of fit. The sensitivity of material constants for these three models was also investigated. The linear model was used to predict the vessel pressure-diameter relations under experimental loading conditions. The limitations, implications, and applications of the novel constitutive relation are enumerated.
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MATERIALS AND METHODS
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Constitutive model.
A new constitutive equation was recently proposed by our group (20). Only a brief description of the model is provided in this article. When the vessel is loaded with internal pressure and axial stretch, the deformation is described by the principal logarithmic-exponential (log-exp) strains Dii, namely,
 | (1) |
where 
, z, and r are principal stretch ratios along the circumferential, axial, and radial directions, respectively, n is a constant characterizing the material nonlinearity, and J1 is the first invariant of the right Cauchy-Green deformation tensor:
 | (2) |
A nominal strain energy function Wn (20) is written in the form of
 | (3) |
Analogous to conventional elasticity theory, it is assumed that the second Piola-Kirchhoff stress (
ii) can be derived by differentiating Wn in Eq. 3 with respect to the strain Dii in Eq. 1, i.e.,
 | (4) |
where p is an arbitrary scalar, Vn(Dii) = 0 is an internal constraint (i.e., incompressibility), and Sii = 
Wn/Dii is the stress with no constraints. No summation is intended in Eq. 4.
Based on Eqs. 3 and 4, the relation between the second Piola-Kirchhoff stresses without the constraint term and the log-exp strains is expressed by
 | (5) |
which, including n in Eq. 1, requires seven model parameters. The Cauchy stress can be computed from the second Piola-Kirchhoff stress as 
ii = 
Sii (7, 11). When the arterial wall is assumed to be incompressible, we have
 | (6) |
where –p has the significance of a hydrostatic pressure due to the incompressibility constraint.
For the purpose of comparison, a brief description of strain energy functions of the exponential model and the biphasic model is provided. The detailed derivations of Cauchy stresses from these functions can be found in Refs. 1 and 17 for the exponential model and in the report by Holzapfel et al. (9) for the biphasic model. The form of exponential strain energy function as proposed by Fung (3) is as follows:
 | (7) |
where
 | (8) |
W represents the pseudo-strain energy per unit volume, C has the units of stress (force/area), and b1, b2, b3, b4, b5, and b6 are dimensionless constants.
The strain energy function of the biphasic model was proposed by Holzapfel et al. (9) and can be written as follows:
 | (9) |
where I1 = 
+ 
and I4 > 1 are invariants, µ > 0 and k1 > 0 have the units of stress, and k2 > 0 and 
[0, 1] are dimensionless parameters. µ is associated with the noncollagenous matrix of the material, which describes the isotropic part of the overall response of the tissue (7). The constants k1 and k2 are associated with the anisotropic contribution of collagen to the overall response (7). Since there were no shear loadings in the experiments (17) and the assumption is that all the fibers are embedded in the tangential surface of the tissue (no components in the radial direction), I4 in Eq. 9 can be expressed as
 | (10) |
The collagen fibers in this model are assumed to be arranged in helical structures, and 
is the angle of the fibers with respect to circumferential direction. The second term in Eq. 9 contributes to W only when I4 > 1 (6, 9).
Determination of material constants.
Experimental data were provided by a previous study on the passive mechanical properties of porcine coronary arteries (17). Briefly, a series of inflation tests were done on cannulated vessels under different axial stretch ratios. Outer radius (ro), internal pressure (pi), and axial force (F) were measured. Vessel rings were taken from the specimen, and a radial cut was made to the vessel ring to reveal the zero-stress state. Inner circumference (Ci) outer circumference (Co), and wall area (A) were recorded.
There are two steps in determining the material constants for the Hooke's law. The first step is to derive the equations that express the external loadings (pi and F) as functions of strains and material constants. The second step is to use a separable nonlinear least-squares method to determine the material constants by minimizing the differences between theoretical and measured values of external loadings. Both steps are outlined in detail in the APPENDIX. This is the first time, to our best knowledge, that the separable nonlinear least-squares method has been used in determining material constants for blood vessels. Material constants for the exponential model were determined in the study (17). Material constants for the biphasic model as defined in Ref. 9 were determined using the standard nonlinear Levenberg-Marquardt method.
Statistical analysis.
To quantify the "goodness" of the fit, comparisons were made between experimental values and theoretical predictions with best-fit material constants; i.e., the correlation coefficient (R2) and the root mean square (RMS) error, expressed as a percentage of the mean value, were calculated for the fittings of pi and total axial force FT (sum of the force applied by the internal pressure and the external axial force). The sensitivities of the models are evaluated based on the variations of their corresponding material constants calculated from the same group of data sets.
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RESULTS
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Data from the previous study by Wang et al. (17) were used to fit the model. Results are listed in Tables 1 and 2 for the porcine right coronary artery (RCA) and left anterior descending (LAD) artery, respectively. When compared with previous fits of the same data using the exponential (17) and biphasic models, RMS errors are largely smaller than those of exponential and biphasic models. The mean RMS% (±SD) values for RCA axial force and pressure are 13.8 ± 2.7 and 15.2 ± 5.2, 14.6 ± 2.9 and 19.1 ± 3.0, and 14.6 ± 7.0 and 23.1 ± 7.3 for the generalized Hooke's law, exponential, and biphasic models, respectively. For the LAD, the mean RMS% (±SD) values for axial force and pressure are 17.0 ± 7.7 and 15.2 ± 5.2, 26.6 ± 7.8 and 19.1 ± 3.3, and 13.8 ± 5.6 and 21.1 ± 7.4 for the generalized Hooke's law, exponential, and biphasic models, respectively. The differences are statistically significant between the RMS errors of the linear model and those of the exponential model in the LAD group (P = 0.007 for FT, P = 0.05 for pi).
The geometric data of the vessel sectors in the zero-stress state are listed in Table 3, which include the wall area A, inner circumference Ci, and outer circumference Co. Both material constants and geometric data are needed as inputs for numerical simulations of wall stress and strain. In this case, one set of data from heart 12 was used to solve the boundary value problem of inflation tests of blood vessel as conducted in the experiments. Essentially, the vessel diameter is computed to satisfy the boundary conditions of internal pressure and axial force specified by Eqs. A2 and A4. In this calculation, material constants are from the fitting results and zero-stress geometric data, internal pressure, and axial force values are from the measurements to mimic the experiments. Figure 1A shows the computed pressure-diameter curves and experimental data points. The agreement is good, especially near the high in vivo pressure range. Figure 1B shows the comparison between computed and measured diameter values for all LAD arteries (hearts 7–12) under the axial stretch ratio 1.4. The results are excellent. Figure 2A shows the computed pressure-total axial force curves and experimental data points. The agreement is not as good as the agreement of the diameter values in Fig. 1A. This is due to the larger RMS error of the total axial force (22.0%) compared with that of the internal pressure (11.1%), and this particular RMS error is also larger than that of average axial force of LAD vessels (17.0%). Figure 2B shows the comparison between computed and measured total axial force values for all LAD arteries (hearts 7–12) under the axial stretch ratio of 1.4. Although the results are satisfactory, they are not as good as the results of Fig. 1B because of the large variation of axial force RMS errors. The RCA results were similar to those for LAD arteries and hence are not shown.
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Fig. 1. A: computed diameter-pressure curves and experimental data points for heart 12 are shown from 3 different axial stretch ratios: = 1.3, 1.35, and 1.4. Symbols represent experimental data; lines represent predicted diameter-pressure curves. B: differences between predicted and measured values of vessel diameter are shown for LADs in hearts 7–12. The percentage of relative difference [(Dexp – Dpre)/Dexp; the ratio of the difference to the experimental value] was calculated for each data point. All data points are from the same axial stretch ratio, 1.4. The middle line is the mean value (0.11) of all the points; the other lines mark the boundaries of 2 standard deviations (±2SD; SD = 2.71) from the mean.
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Fig. 2. A: computed total axial force-pressure curves and experimental data points for heart 12 are shown from 3 axial stretch ratios: = 1.3, 1.35, and 1.4. Symbols represent experimental data; lines represent predicted total axis force-pressure curves. B: differences between the predicted and measured values of total axial force are shown for LADs in hearts 7–12. The percentage of relative difference was calculated for each data point. All data points are from the same axial stretch ratio, 1.4. The middle line is the mean value (0.09) of all the points; the other lines mark the boundaries of ±2SD (SD = 7.10) from the mean.
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The circumferential second Piola-Kirchhoff stress with no constraints (S) was calculated for the experimental data of heart 12 and is shown in Fig. 3 to be confined to a linear surface defined by Eq. 5.
A sensitivity analysis was done to evaluate the stability of material constants. For the heart 11 LAD data, the zero-stress inner circumference Ci and outer circumference Co are both hypothetically varied from 80 to 120% of their measured values in 10% steps. This simulates a softer to stiffer vessel, respectively, compared with actual data. The reference lengths are varied, because these have the most uncertainty experimentally. Five cases were calculated to determine their material constants. The new model, the exponential model, and the biphasic model were used, and the results are listed in Table 4. Compared with the exponential model and the biphasic model, the material constants of the new model have smaller coefficients of variation, i.e., the ratio of the standard deviation to the mean. Therefore, the new model may be more suitable to detect the changes in the blood vessels due to growth or remodeling. This is because when two groups are compared, small variations create less overlap. Therefore, the differences are more likely to be statistically significant. The goodness of fit is maintained throughout all data sets with the new model, whereas in the exponential model, the RMS errors of the internal pressure are increased with perturbations of data. For the biphasic model, the RMS errors are significantly affected when the lengths are elongated to 120% of the original.
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DISCUSSION
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Our group has recently introduced a generalized Hooke's law (2-dimensional in Ref. 19 and 3-dimensional in Ref. 20), which is equivalent to the Fung exponential model for a given set of material parameters. For a set of experimental data, however, it is difficult to obtain the exponential material parameters because of the nonlinear dependence of the stress on the model constants. The material parameters determination is simplified in the generalized linear Hooke's model, which yields better curve fits. In this study, it was demonstrated that the material constants in the new model fit experimental data generally better than those determined by the exponential and biphasic models (Tables 1 and 2). The material constants of the new model are also more stable than those of exponential and biphasic models (Table 4). Therefore, the new model has a number of advantages as outlined below.
First, the material constants in the proposed generalized Hooke's law can be obtained more easily. A separable least-squares method limits the nonlinear optimization to only one nonlinear model parameter. For the exponential model, a much more complicated fitting method is needed (17) for six nonlinear parameters. For the biphasic model, there are still three nonlinear parameters. Linear optimization is much more definitive than the nonlinear optimization, and the goodness of fit is improved in the new model by employing linear optimization to determine six linear material constants.
Second, it was shown that the material constants of the new model have a smaller coefficient of variance (CV = SD/mean) compared with those determined by the exponential and biphasic models. For example, the maximum CV value for RCA is 0.74, 0.97, and 1.17 (Table 1) in the linear, exponential, and biphasic models, respectively. For LAD, the maximum CV is 0.69, 1.09, and 0.88 (Table 2). It was shown that the material constants have a smaller CV compared with those determined by the exponential and biphasic models (Table 4). The maximum CV was 0.49, 1.36, and 0.94 in the linear, exponential, and biphasic models, respectively. By utilizing the constitutive equation with more reliable and stable parameters, computational models can predict the vascular mechanics more realistically.
For the constitutive relation to be used in a computational model, the convexity of the strain energy function has to be satisfied to ensure stability under loadings (11). In the new model, the nominal strain energy Wn should be positive definite. To ensure this condition, the eigenvalues of the matrix
have to be positive. This condition is equivalent of the following three expressions being positive:
 | (11) |
These conditions are not enforced in the linear optimization directly, because these constraints are nonlinear inequalities and are difficult to implement in linear optimization. Instead, we first add lower boundaries in the linear optimization to ensure positive values for all constants. If there are negative eigenvalues, we then increase the lower boundary of c33 to make the eigenvalues positive. The third inequality is typically the only negative one. Increase of c33 makes the third equation more positive given the second inequality is already positive. Although this is not a systematic mathematical method, we obtain good fits from this method, and the manipulation of c33 has a very minor effect on the goodness of the fit. Furthermore, c33 is the constant corresponding to the radial direction whose deformation is an indirect measurement from a biaxial test. Therefore, this method of determining linear constants was used, and all the constants were verified to satisfy the convexity condition as shown in Fig. 4.
The material constants in the generalized Hooke's law have clearer physical meanings than those in the exponential model; i.e., n represents the material nonlinearity, c values are elastic moduli with respect to the log-exp strain defined by n, and the anisotropy of the material is revealed by the moduli in various directions (20). In the biphasic model, the constants are thought to have a direct connection to the microstructure of the vessel wall as stated earlier. It was noted the angle in the biphasic model sometimes reaches 90° in the data curve fit (Table 2), which implies the collagen fibers are parallel along the axial direction. This contradicts, however, the biphasic model assumption that the collagen fibers in the vessel have helical structures. This may be because we did not use the biphasic model to fit individual layer data as it was originally intended (6). Regardless, this might be a potential problem for the physical meanings of material parameters in the biphasic model, which remains unvalidated microstructurally.
In this study, we have validated a new blood vessel constitutive model proposed recently (20). Experimental data on the porcine coronary arteries were used to fit the model. Material constants for the new model have generally better fit of data compared with those determined by the exponential and biphasic models. Material constants for the new model are also more stable than those determined by the exponential model and the biphasic model. The major cause of these improvements is that the new model absorbs the nonlinearity of stress-strain relation into one parameter and yields a linear dependence of stress on the other six parameters. Therefore, we make use of the separable nonlinear least-squares method, which divides the parameter determination into two steps: 1) a nonlinear least-squares optimization to determine the one nonlinear constant and 2) a linear optimization to determine the linear constants. Problems caused by the nonlinear least-squares fitting of multiple parameters are avoided, i.e., the instability of material constants and unsatisfactory fits. A simple boundary value problem is solved to mimic the experimental condition and verify the model predictions. In future studies, the present model should be extended to individual layers of the vessel wall to investigate various boundary problems in cardiology and cardiovascular physiology.
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APPENDIX
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Equations of internal pressure and axial force.
Derivations of the equations for internal pressure and axial force were provided in detail previously (2, 7). We provide a brief description below. For a cylindrical vessel under inflation and axial stretching, the equilibrium equation is
 | (A1) |
From this equation and the boundary condition that the pressure at the outer surface is zero, the internal pressure can be obtained in the form of
 | (A2) |
where ri is the inner radius and ro is the outer radius. Because one end of the tested vessels was closed, the total axial force (FT) has two components; one part is the force applied by the internal pressure (pi), and the other part is the measured axial force (F) from the force transducer (17), namely,
 | (A3) |
The use of Eqs. A2 and A3 (7) leads to
 | (A4) |
Both Eqs. A2 and A4 are used in the nonlinear least-squares method to determine the material constants.
To determine the values of the Eqs. A2 and A4, i (i = , z, r) must be known. The axial stretch ratio z was measured and assumed to be uniform in the vessel wall. Based on the incompressibility assumption and the measurements of ro, inner and outer circumferences Ci and Co, and wall area A of the vessel ring in zero-stress state, the values of i (i = , r) can be determined as follows. First, the inner radius of the blood vessel is calculated as
 | (A5) |
The circumferential stretch ratios at the inner and outer surface (
, 
) can be calculated as
 | (A6) |
Based on the incompressibility condition, the circumferential stretch ratio in the vessel wall is a function of the radius r:
 | (A7) |
The radial stretch ratio r is then determined by the incompressibility condition as
 | (A8) |
Given the stretch ratios i (i = , z, r) and the material constants, Eqs. 6, A2, and A4 can be used to calculate the theoretical values of pi and F.
Separable nonlinear least squares.
The parameters pi and F were measured in experiments. Theoretical values of both can be determined from Eqs. 6, A2, and A4. A nonlinear least-squares method was used to determine elastic constants in the constitutive model by minimizing the differences between theoretical and measured values.
If we substitute Eq. 6 into Eqs. A2 and A4, it can be shown that pi and F are both linear functions of elastic constants c11–c23:
 | (A9) |
where 
(n) and 
(n) are 1 x 6 vectors that nonlinearly depend on parameter n. Dii (i = , z, r) is defined in Eq. 1. To take advantage of this specific form, the separable nonlinear least-squares method (5) was used to determine the nonlinearity constant n and the six linear constants c11, c12, c13, c22, c23, and c33. First, a new nonlinear target function for the nonlinear least-squares method is generated based on functions in Eq. A9, which eliminates all the linear constants and only includes the nonlinear constant n. This new function T(n) is
 | (A10) |
where ||... || stands for the L2 vector norm. A nonlinear least-squares method is then used to determine n by minimizing T(n). Once n is determined, linear constants can be determined by solving a linear optimization problem as
 | (A11) |
This method reduces the number of parameter spaces to only one in the nonlinear least-squares method compared with the exponential model with six nonlinear parameters and hence greatly simplifies the fitting. The above fitting algorithm has been programmed with Matlab (www.mathworks.com.
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FOOTNOTES
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Address for reprint requests and other correspondence: G. S. Kassab, Depts. of Biomedical Engineering, Surgery, and Cellular and Integrative Physiology, Indiana Univ., Purdue Univ. at Indianapolis, 635 Barnhill Drive MS 2069, Indianapolis, IN 46202 (e-mail: gkassab{at}iupui.edu)
The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
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Copyright © 2008 by the American Physiological Society.
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ABSTRACT
MATERIALS AND METHODS
