Center for Limb Loss and MoBility

Introduction

It is essential to accurately quantify the compressive properties of plantar tissue and changes that occur with disease that compromise these tissues to develop solutions that can compensate for these changes. For example, many people with diabetes suffer aberrant plantar pressures and are, hence, prone to plantar ulceration and, ultimately, amputation of the affected foot.^1,2 Knowledge of diabetic tissue properties could be used to develop impedance-matched orthoses that could potentially alleviate high plantar pressures under critical regions. Stress relaxation tests are routinely used for quantifying the viscoelastic properties inherent in most soft tissues, including the plantar subcutaneous tissue.^3,4 Further, the quasi-linear viscoelastic (QLV) theory⁵ is extensively used to quantify the relaxation response since it provides meaningful coefficients that can be applied to a wide range of tissues. Our research group is currently in the process of quantifying the differences between healthy and diabetic soft tissues by using stress relaxation tests and the QLV theory.

A limitation of the QLV theory is that its accuracy is dependent on the assumption that the applied strain includes an instantaneous ramp to a prescribed strain level. Since instantaneous ramps are experimentally unachievable, this assumption leads to errors,⁶ hence, amendments to the QLV theory were made to incorporate fast finite ramps.⁷ These changes, however, do not take into account the experimental target strain error (i.e., overshoot and undershoot) resulting from fast ramps. A few studies address target strain error,^8,9,10 but none examine this issue for plantar soft tissue in compression. Previous solutions to target strain errors in other tissues included using a very slow ramp time to minimize overshoot experimentally.¹⁰ It is unclear whether this approach is suited for plantar tissue since it requires extremely slow ramp rates that preclude the characteristic initial relaxation observed at high, physiologic ramp rates. Funk et al.⁸ replaced the erroneous part of the relaxation curve (i.e., the experimentally obtained force peak) by back-extrapolating from an accurate section of the force curve. This method is limited as it relies on a model (i.e., back extrapolation) to predict how the tissue would behave if the target error had not been present. Gimbel et al.⁹ used a direct fit approach to account for overshoot in the strain history, but this method still had a nearly 20% mean square error for a 7.5% overshoot. They also explored the effect of varying overshoot, but this was conducted as a simulated experiment that assumed the tissue behaved as an ideal QLV system.

Our current approach³ is to aggressively tune our testing control system to a fast ramp and hold test. By using this approach, we can successfully minimize our target strain error (i.e., overshoot) to approximately 0.3%. Although this error is considerably smaller than the 7.5% reported for supraspinatus tendons,⁹ even this small strain error might contribute to significant load errors for plantar tissue. In fact, we previously observed that a strain difference of 0.8% can lead to a considerable artifact in the observed stress (on the order of at least 10%).¹¹ It would be extremely useful to know to what extent small strain errors can affect stress values to minimize any unnecessary confounding of future results.

Thus, the objective of this study was to determine empirically how peak stress is affected by small target strain errors for the subcutaneous plantar soft tissue. While we experimentally cannot reduce this error any further for a ramp and hold test, we are capable of achieving 0% error (1–2 µm) by using a compensation feature of our testing software for cyclic tests. In our current relaxation testing protocol, we keep the ramp time constant for all 0.1 s so as to prevent relaxation from occurring during the ramp, i.e., < τ1, the short-term QLV time constant determined from previous plantar tissue tests. This ramp time of 0.1 s is equivalent to the ramp portion of a 5 Hz triangle wave. Hence, by using 5 Hz triangle waves in conjunction with compensation, we can accurately achieve the desired target strain error.

Methods

Experimental Protocol. Five specimens were obtained from four fresh frozen cadaveric feet, which were purchased from the National Disease Research Interchange. The Institutional Review Board's approval was obtained for this study from the Human Subjects Division at the University of Washington.

The plantar soft tissue was dissected free from the lateral midfoot, cut into cylindrical specimens by using a 1.905 cm diameter punch, and further dissected from the skin by using a scalpel (Figure 1). Each specimen was then placed in an environmental chamber between two platens covered with 220-grit sandpaper (Figure 1(d)). The chamber was designed to heat a water bath below the platen and create a moist environment near 100% humidity and at 35° C to approximate conditions in vivo. This setup was attached to an ElectroForce 3200 materials testing machine. The bottom platen was raised to apply a 0.1 N compressive load, and the specimen initial thickness was measured.

The target load, based on the specimen cross-sectional area, donor weight, and normative ground reaction force and contact area¹² was used to determine the target displacement. In load control, the specimen underwent ten 1 Hz sine waves from 10 N to the target load; the maximal absolute displacement was noted as the target displacement. Target strain was calculated as the target displacement divided by the initial thickness. Six additional "erroneous" target strains were calculated to undershoot and overshoot the target displacement in 0.3% increments, leading to seven test groups of −0.9%, −0.6%, −0.3%, 0.0%, 0.3%, 0.6%, and 0.9% strain error. A special compensation function of the testing software WinTest v4.0 enabled us to achieve 0% error (1–2 µm) for cyclic tests. We elected to use 5 Hz triangle waves to emulate the ramp portion of our typical relaxation test. The sample was allowed to recover in an unloaded state for 25 minutes after the load control test, followed by a brief tuning and recovery period, and then seven triangle tests at wave tests corresponding to each strain level. Each triangle wave test consisted of 30 cycles to the prescribed strain to ensure sufficient cycles for compensation and preconditioning, followed by 10 minutes of recovery before the next triangle wave test. The force and displacement data were acquired at a rate of 5,000 Hz. Stress (force divided by original specimen area) between the maximum undershoot and maximum overshoot (i.e., −0.9% and 0.9% strain error) was used to calculate the total stress variation for each specimen.

Figure 1: Dissection of specimen showing a) removal of plantar tissue flap at lateral midfoot location from which a b) cylindrical specimen was punched and the c) skin was removed. The specimen was then placed d) between two platens covered with sandpaper in a humidity chamber at 35° C and sealed with plastic wrap (not shown).

Statistical Analysis. Linear mixed effects regression was used to determine the relationship between the percent target strain error (the independent variable) and peak stress (the dependent variable). The specimen number was modeled as a random effect, with variability assessed for both the intercept and slope of the percent target strain error across specimens. Analyses were carried out using R 2.9.0.

Results

The peak stress for all specimens was highest for the largest strain error of 0.9% and lowest for the smallest strain error of 0.9%. This result is also seen in the stress versus strain response for a typical specimen for all seven target strain errors. The average total stress variation for all specimens was 25%. A strain overshoot of 0.3%, the target strain error observed in our typical stress relaxation experiments, corresponded to a stress overshoot of 2.2%, 5.1%, 1.2%, 2.7%, and 2.7% for specimens 1–5 respectively, i.e., an average stress overshoot of 3 ± 1%. Further examination of the tuning results of a representative specimen demonstrated that the peak stress is reached just prior to the maximal target strain error of 0.3% and just after the time at which the actuator first reaches target.

There was a significant linear relationship between the target strain error and stress (p = 0.0002). However, upon plotting the data, a nonlinear component in the association between the target strain error and stress was evident, whereby the difference in peak stress tended to decrease with increasing total displacement (−16% stress error at −0.9% strain error versus 8% stress error at 0.9% strain error). We carried out another regression with the target strain error modeled as an orthogonal polynomial and found a significant cubic relationship (p < 0.0001) that was stronger than the linear relationship.

Conclusion

The results of this study demonstrate that plantar tissue in compression is sensitive to small target strain errors that can result in stress errors that are several fold larger. Since these errors are difficult to completely eliminate experimentally while maintaining fast physiologic ramp rates, we recommend understanding the extent to which overshoot may affect the peak stress for a particular tissue and loading regime. For plantar tissue in compression, we are able to maintain a target strain error of 0.3% and, hence, a small peak stress error of 3%. We expect that these findings will differ in magnitude for other soft tissues. Future studies should examine strain versus stress errors for plantar tissue for other physiologic loading modes, e.g. shear and combined loading, where the tissue properties and associated overshoot error are likely to differ.

To read the full project description, please see:

Effect of Target Strain Error on Plantar Tissue Stress. Pai S, Ledoux WR. J Biomech Eng. 2010 Jul;132(7):071001. PMID: 20590279.

Acknowledgements

This study was supported by NIH under Grant No. 1R01 DK75633-03 and the Department of Veterans Affairs, RR&D Service.

Research Team

Shruti Pai, Ph.D.
William R. Ledoux, Ph.D.

Related

• Compressive Material Properties of the Plantar Soft Tissue
• Compressive Mechanical Properties of Diabetic and Non-Diabetic Plantar Soft Tissue
• Quasi-Linear, Viscoelastic, Structural Model of the Plantar Soft Tissue with Frequency-Sensitive Damping Properties

References