Introduction
A great challenge in biomechanics, and its related disciplines such as mechanobiology, is the determination of mechanical constitutive behaviors for tissues that undergo high strain. Even for homogeneous, single phase, rubber-like materials with behavior that is nearly isotropic and hyperelastic, the determination of constitutive behaviors has been ambiguous and problematic 1. For multi-phase, heterogeneous tissues with behavior that is anisotropic and visco-elastic, much theoretical and experimental work remains 2, and this work is a small step toward improving experimental methods to delineate tissue behavior.
There are, of course, approximate constitutive relations for most tissue types that have been validated with fits to experimental data and that have enabled modeling applications in biomechanics. Rather than integrate tests from multiple specimens to yield a generalized relation, this work is focused on a methodology to study individual specimen variations. Our reason for doing so is to exploit the natural variation in tissue structure to better understand the structure-function relationship. Each specimen has differing amounts of elastin, collagen, water, etc.; and with experimental methods that delineate rather than aggregate behavior, we seek to relate ultrastructure parameters to tissue response parameters. The ultimate goal of this work, albeit decades away, is to determine how ultrastructure maps to tissue behavior.
Toward this end, we developed a random stretch and random stretch-rate protocol for a one-dimensional sample (longitudinal strip from a porcine pulmonary artery). The average stress at a particular stretch is defined as the elastic or hyperelastic part of the response and the deviation from this average is defined as the inelastic part of the response. The inelastic response is assumed to be a function of three variables: stretch rate related, short tem stretch history Ht2; longer term stretch history Ht1; and time T since the start of the test protocol. At a particular stretch value, the inelastic response is assumed to be a linear function of these variables and dependence is found by Multivariable Linear Regression Analysis (MLRA). A randomized testing protocol eliminates any potential correlation among Ht2, Ht1, and T that conventional cyclic testing cannot. This randomness enables the use of MLRA.
Hoffman and Grigg have introduced the advantage of application of random stress stimuli to the soft tissue 3. The difference here is that stretch herein, rather than stress therein, is the independent variable that is randomized. Moreover, we seek to define and determine the elastic and inelastic parts of the response at particular chosen values of strain rather than fit behavior for the range tested. Though, there are many time-dependent theories that can capture or fit inelastic behaviors, such as, linear/nonlinear viscoelastic 45678910, poroviscoelastic 1112, and biphasic 13141516., this study is focused on experimental methodology for hyperelastic constitutive behaviors.
Remark on Hyperelasticity
We often mix the terms elastic and hyperelastic when referring to material behavior and certain aspects of the stress response. Conceptually, one can think about elasticity (i.e., stress as a function of stretch or strain) as different than hyperelasticity (i.e., strain energy as a function of stretch or strain); yet in fact, elastic behavior must be hyperelastic – or else it violates the first law of thermodynamics. This is easily shown by considering a closed, cyclic process involving an elastic material. An elastic material does not have viscous losses, and the work of the reverse path (or cycle) is the opposite of the forward path (or cycle). This is a direct result of the stress-power law. If the work done is zero for all closed paths, then there exists a state function (strain energy) that represents the storage, in a conservative manner, of work done on the material. This is hyperelasticity wherewith the constitutive law can be defined by its strain energy function and the stress is given by the derivative of the strain energy with respect to the strain. If the work done is not zero for all closed paths (i.e., non-hyperelastic), then choose one such path and go around the cycle in the direction that produces work, and then build a perpetual motion machine of the first kind. It is quite easy (in two and three dimensions) to fit an elastic model that violates the first law 17; and yet it is impossible for a hyperelastic model to violate the first law. To be explicit: 1) if an elastic model satisfies the first law then it can be expressed in a hyperelastic form, and 2) if an elastic model cannot be expressed in a hyperelastic form, then it violates the first law. When we refer to elastic behavior as hyperelastic, we are only excluding behavior that violates the first law of thermodynamics.
Methods
A uniaxial testing protocol wherein the stretch and stretch rate were randomized was developed to better delineate the average stress vs. stretch response of soft tissue. The randomized testing protocol was achieved by defining 71 time points or nodes and then randomly choosing (at the nodes) nominal stretch values of λ = 1.1, 1.3, 1.5, 1.7, or 1.9 and stretch-rate values in the range of [-5.26 s-1, 5.33 s-1]. With 71 time nodes there are 70 intervals, and in each interval we defined 50 points. The stretch value at points between nodes was obtained from the nodal stretches and stretch-rates by use of cubic Hermite interpolation in time, so to achieve C1 continuity across intervals 18. Figure 1 displays the stretch versus time of the stretching protocol. Upon inputting the reference length of the sample, the stretch versus time protocol was converted to displacement versus time and fed to opposing linear actuators (both ends of the sample were moved in opposite directions, so to double the stretch-rate range).
<p>Figure 1</p>
Stretch-controlled protocol showing the stretch and velocity of one of the actuators versus time
Stretch-controlled protocol showing the stretch and velocity of one of the actuators versus time.
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This stretching protocol was accomplished with custom LabVIEW (National Instruments, Inc) codes. The stretch values were chosen to cover the linear and non-linear portions of the stress-stretch response of the tissue, without damaging the tissue. The stretch-rate range was determined by the upper and lower bound for the speed of the actuators (CMA 25CCCL DC servo motor, max speed = 0.4 mm/s, Newport, Inc). The actuators were controlled by the Universal Motion Controller/Driver (ESP7000, Newport, Inc). The total testing time was 5600 seconds.
Fourier spectral analysis was performed on both randomized stretch and randomized stretch rate profiles to validate their randomness (Figure 2). The power spectrum is not ideally uniform across the spectrum, but it has power distributed across the spectrum with noise dominating any systematic trends.
<p>Figure 2</p>
Power spectra for stretch (left) and stretch rate (right)
Power spectra for stretch (left) and stretch rate (right). Although not uniform and flat, there is broad spectral distribution with noise dominating any systematic trends.
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The pulmonary artery from an adult swine was obtained from a local slaughter house immediately after slaughter. Two identical strips (25.4 mm × 2.5 mm × 2 mm) were cut from the porcine pulmonary artery (one was used for the stretch protocol and one for the relaxation analysis), preserved in chilled phosphate buffered saline (PBS) and tested at room temperature (approximately 23°C). We mounted a specimen by using custom-built alligator type grips and no slip was detected during the test. The stretch ratios were calculated based on a gauge length which is a grip-to-grip displacement. (In general, grip-to-grip strain is greater than average strain measured by optical strain measurement because of edge effects and local deformation at the tip of the grip. Since the length of our specimen is relatively long relative to its width, 10:1; and since this work is focused on archiving and discussing the methodology as opposed to archiving the behavior of a particular, now decomposed, sample, we neglected edge effects.) The average force at a particular stretch was defined as the hyperelastic response because it would be the best guess for the response if the stretch rate and stretch history were unknown or neglected – as per the assumption of elasticity. Inelastic deviation was defined and measured as the deviation from the average force at each nominal stretch ratio.
To characterize the inelastic deviation D at a particular stretch value, we assumed that it is a function of the following form.
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To quantify inelastic behavior, we assumed the functional form shown in equation (2) where FM and FW are measured and average force, respectively at a particular stretch.
FM = FW ± D(Ht2, Ht1, T)
To explicitly represent the limits of experimental methodologies, we included an unknown factor De that represents additional inelastic deviation (or random error) beyond that which is represented by DHt2, DHt1, and DT or deviation due to Ht2, Ht1, and T, respectively. Hence,
D = DHt2 + DHt1 + DT + De
To determine relative uncertainty, we normalized D (i.e., standard deviation of measured forces at each stretch ratio) with respect to the average force at a particular stretch and defined the fractional uncertainty FU.
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Stretch history
Stretch history is, generally, the deformation versus time for the entire life of the sample up to the current configuration. Such a concept is too all-inclusive to be practical, and yet the mechanical properties of tissues are highly sensitive to prior deformation – i.e., resting, preconditioning, or over-stretching of a sample will significantly perturb the stress-strain response. There are many ways to quantify a stretch history function. The function that we used for stretch history is shown in equation (5). It represents the average stretch for preceding time interval th, where λ(t) is stretch as a function of time, tN is a current time, and th is a preceding time interval.
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Note that as th gets smaller, H(t) approaches λ(tN) and gives the near term history or instantaneous stretch ratio if th ≅ 0. On the contrary, if th gets larger, H(t) approaches the average stretch ratio (which is λ = 1.5 in this study because we set λ = 1.0 and λ = 2.0 as bounds on a random distribution). Figure 3 displays our stretch history concept. Although stretch functions λ(t)1, λ(t)2, and λ(t)3 give the same stretch and stretch rate at tN, the histories are different.
<p>Figure 3</p>
Illustration of various stretch histories with the same stretch and stretch rate at time tN
Illustration of various stretch histories with the same stretch and stretch rate at time tN. Although the stretch ratio functions λ(t)1, λ(t)2 and λ(t)3 give the same stretch and stretch rate at time tN, the average stretch ratios for the time duration th for each are distinctly different.
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Figures 4 and 5 show the stretch history obtained by using th = 10, 400, 1000, and 2000. Note that the stretch history obtained by th = 1000 and th = 2000 approaches the average of λ = 1.5. Thus, finding a reasonable th is important for stretch history to characterize the inelasticity due to history. Upon analyzing the relaxation spectrum for a sample held at a particular stretch (λ = 2, max stretch in a random distribution), it is evident that there is short term (fast) relaxation and longer term (slow) relaxation. We fit exponential decays to the stress relaxation curve, and it was fit well by the summation of four exponential functions. The stress relaxation curve is shown in Figure 6, and F(t) is the force to maintain a step increase in stretch and fit by the multi-exponential decay as follows:
<p>Figure 4</p>
Stretch history obtained by th = 10, th = 400, th = 1000, and th = 2000 (for 0–3000 sec)
Stretch history obtained by th = 10, th = 400, th = 1000, and th = 2000 (for 0–3000 sec). When th is small, the stretch history is nearly the same as the instantaneous stretch, and as th increases, stretch history reaches a plateau of λ = 1.5, the average stretch ratio of a random distribution in the range [1,2].
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<p>Figure 5</p>
Continuation of Figure 4
Continuation of Figure 4.
![]()
<p>Figure 6</p>
Stress relaxation curve for the tissue and the multi-exponential fit
Stress relaxation curve for the tissue and the multi-exponential fit.
![]()
F(t) = 6.1 + 1.5e-t/400 + 0.5e-t/30 + 1.3e-t/10 + 0.5e-t
We consider the two fastest as fast relaxation spectrums (th = 1, th = 10) and the two slowest as slow relaxation spectrums (th = 30, th = 400), and note that the weights of the exponential functions corresponding to th = 10 and th = 400 are greater than the others. Hence, we judiciously chose th = 10 and th = 400 for short-term history (Ht2) and long-term history (Ht1), respectively. Our system could not capture fast deformation histories within 1 second because our lead screw actuators are too slow to deliver a step function in stretch, ideally needed to analyze stress relaxation. Note that the stretch history obtained by th = 10 is of relatively short duration compared to our acceleration, and it depends, thus, on the direction of approach or stretch rate (see Figure 7) and is roughly the current stretch minus half of the stretch rate times 10 sec. Equation (7) shows the relationship between short term stretch history Ht2 and stretch rate λ˙(t)
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<p>Figure 7</p>
Relationship between short-term stretch history and stretch rate
Relationship between short-term stretch history and stretch rate. (a) When the stretch rate is positive, the stretch history for th = 10 sec is less than the current stretch ratio. (b) When the stretch rate is negative, the stretch history for th = 10 sec is greater than the current stretch ratio.
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Multiple linear regression analysis
To characterize inelastic behavior in terms of Ht2, Ht1, and T, multivariable linear regression analysis, goodness-of-fit test and t-test were used 19. Basically, we did a separate statistical analysis for each nominal stretch value. Because each nominal stretch value was retested many times (71 nodes and only 5 possible stretch values), there is a set of force measurements at the same stretch – but each with a different history. T, Ht2, and Ht1 are calculated from the stretching protocol, and FM is measured by the force transducer and captured in synchrony with the stretching protocol. T, FM, Ht2, and Ht1 were grouped together for each stretch ratio. The grouped data for λ = 1.1, for example, can be expressed in matrix form as in equation (8) where n is the number of data corresponding to stretch ratio λ = 1.1.
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For use in MLRA, the model for inelastic deviation from the mean or elastic part is:
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The partial regression coefficients β^j
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MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaacciGaf8xSdeMbaKaaaaa@2D89@ and β^j
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaacciGaf8NSdiMbaKaadaWgaaWcbaGaemOAaOgabeaaaaa@2F14@. To avoid scale-dependency in the partial regression coefficients, we normalized each data set by their magnitudes. Upon fitting the β^j
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaacciGaf8NSdiMbaKaadaWgaaWcbaGaemOAaOgabeaaaaa@2F14@, we checked R¯2
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGafmOuaiLbaebadaahaaWcbeqaaiabikdaYaaaaaa@2E37@, the adjusted coefficient of determination to observe how well the derived sample regression line describes the observed variable Di and then examined the significance of each β^j
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaacciGaf8NSdiMbaKaadaWgaaWcbaGaemOAaOgabeaaaaa@2F14@ using t-test. Successive iterations of t-test were performed whenever there was/were insignificant variable(s) for a particular stretch. Since we tested a single specimen, we set the level of significance α as 0.1 which represents the 90% probability that the partial regression coefficient β^j
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaacciGaf8NSdiMbaKaadaWgaaWcbaGaemOAaOgabeaaaaa@2F14@ is likely to be contained within the interval. In general, a level of significance α is preferably set as 5%.
Predictive capability
The predictive capability was examined by comparing the predicted inelastic deviations calculated by the derived regression model to the measured inelastic deviations from unused stretch ratios. Only the 71 nodal time points were used to regress the coefficients; yet there were many more times, within the interval between nodes, that the stretch protocol prescribed one of our nominal stretch values of interest (i.e., λ = 1.1, 1.3, 1.5, 1.7, or 1.9). Equation (10) was used for the interval prediction where D^fc
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGafmiraqKbaKaadaWgaaWcbaGaemOzayMaem4yamgabeaaaaa@2FC4@ is critical forecasting interval for the D^f
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGafmiraqKbaKaadaWgaaWcbaGaemOzaygabeaaaaa@2E75@ with a 100(1-α) confidence, D^f
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGafmiraqKbaKaadaWgaaWcbaGaemOzaygabeaaaaa@2E75@ is the predicted value of a certain point, tc(α/2;n-k-1) is a critical t value with a degree of freedom n-k-1 where n and k are the number of samples and the number of independent variables, respectively, sf is the standard deviation which consists of variance of ei and variance of D^i
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGafmiraqKbaKaadaWgaaWcbaGaemyAaKgabeaaaaa@2E7B@.
D
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MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGafmiraqKbaKaadaWgaaWcbaGaemOzayMaem4yamgabeaakiabg2da9iqbdseaezaajaWaaSbaaSqaaiabdAgaMbqabaGccqGHXcqScqWG0baDdaWgaaWcbaGaem4yamMaeiikaGccciGae8xSdeMaei4la8IaeGOmaiJaei4oaSJaemOBa4MaeyOeI0Iaem4AaSMaeyOeI0IaeGymaeJaeiykaKcabeaakiabgwSixlabdohaZnaaBaaaleaacqWGMbGzaeqaaaaa@49A8@
Lastly, to assess the goodness of fit and search for further systematic behavior, we calculated the deviation of the predication, Di - D^f
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGafmiraqKbaKaadaWgaaWcbaGaemOzaygabeaaaaa@2E75@, which is the difference between observed deviation of a data point Di and point forecasting value D^f
MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGafmiraqKbaKaadaWgaaWcbaGaemOzaygabeaaaaa@2E75@ obtained by the statistical model for the particular, nominal stretch value. If the deviation of the predication has systematic variation, then it can reasonably be said that there must be other factors (or perhaps higher orders of Ht2, Ht1, and T) that induce inelastic deviations from hyperelasticity.