Technical Brief. Journal of Biomechanical Engineering. Christof Hurschler 1. Paolo P. Provenzano. Ray Vanderby, Jr. 2.

Transcription

1 Journal of Biomechanical Engineering Technical Brief Application of a Probabilistic Microstructural Model to Determine Reference Length and Toe-to- Linear Region Transition in Fibrous Connective Tissue Christof Hurschler 1 Paolo P. Provenzano Ray Vanderby, Jr. 2 Department of Orthopedics and Rehabilitation and Department of Biomedical Engineering, University of Wisconsin, Madison, WI This study shows how a probabilistic microstructural model for fibrous connective tissue behavior can be used to objectively describe soft tissue low-load behavior. More specifically, methods to determine tissue reference length and the transition from the strain-stiffening toe-region to the more linear region of the stress-strain curve of fibrous connective tissues are presented. According to a microstructural model for uniaxially loaded collagenous tissues, increasingly more fibers are recruited and bear load with increased tissue elongation. Fiber recruitment is represented statistically according to a Weibull probability density function (PDF). The Weibull PDF location parameter in this formulation corresponds to the stretch at which the first fibers begin to bear load and provides a convenient method of determining reference length. The toe-to-linear region transition is defined by utilizing the Weibull cumulative distribution function (CDF) which relates the fraction of loaded fibers to the tissue elongation. These techniques are illustrated using representative tendon and ligament data from the literature, and are shown to be applicable retrospectively to data from specimens that are not heavily preloaded. The reference length resulting from this technique provides an objective datum from which to calculate stretch, strain, and tangent modulus, while the Weibull CDF provides an objective parameter with which to characterize the limits of lowload behavior. DOI: / Current address: Orthopadische Klinik, Medizinische Hochschule Hannover, Anna-von-Borries-Str. 1 7, D Hannover, Germany. 2 Ray Vanderby Jr., Orthopedic Research Laboratories, K4/738, 600 Highland Avenue, University of Wisconsin, Madison, Madison, WI vanderby@surgery.wisc.edu. Contributed by the Bioengineering Division for publication in the JOURNAL OF BIOMECHANICAL ENGINEERING. Manuscript received by the Bioengineering Division December 4, 2001; revised manuscript received November 19, Associate Editor: M. S. Sacks. Introduction The first goal of this study is to develop an objective method to define the reference length for soft fibrous connective tissues. Reference length is the basis for biomechanical descriptions, yet it is not selected in a consistent way throughout the literature. Reference length of a tendon or ligament specimen is often determined empirically by increasing deformation until a measurable load is detected, 1,2 a small arbitrarily assigned preload is applied, 3 5 or an inflection is observed in the load-deformation curve, 6,7. Tendon and ligament specimens, however, are compliant at the onset of load bearing such that small changes in load result in relatively large variations in reference length. In addition, relatively large preloads are sometimes applied that mask much of the toe-in region of the load-deformation response. Since stretch-ratio or strain is a normalized quantity, based on the reference length of the specimen, different choices of preload can introduce bias into biomechanical measures such as stretch, strain, strain to failure, or tangent modulus. The second goal of this study is to develop an objective method to identify the transition from the strain-stiffening toe-region to the linear region commonly observed in the stress-strain diagrams of fiber recruiting soft tissues. The definition of such a point will aid the study of low-load behavior in normal and pathologic connective tissues. Constitutive models, 8 14, have been developed using the assumption that the strain stiffening behavior of the toe-region results from gradual recruitment of collagen fibers during tissue elongation, 15, and that the fiber recruitment can be described by a probability density function PDF 8,9,11 13,16. Straightening of collagen fibers after the application of low loads has also been observed using scanning electron microscopy, 17. It should be noted that in this study low-load refers to the stresses present in the toe-region. Once a substantial percentage of fibers are recruited, tissue behavior transitions from the toe-region to a more linear region of the stress-stretch or strain curve. Although, many models fit the nonlinear behavior of fibrous connective tissue very well, the specific transition from the toe-region to the more linear portion of the curve has undergone less scrutiny. Most biomechanical studies of soft tissues, such as ligament and tendon, focus on the middle linear and upper failure portions of the stress-stretch curve, while the equally important lower toe portion of the curve is less described. The ability to characterize the low-load region would increase a researcher s ability to understand and compare tissue behavior and the effect of treatment regimes. Panjabi et al. 18 characterized normal and subfailure damaged rabbit anterior cruciate ligament load-deformation behavior by measuring the tissue deformation at 5, 10, 25, and 50% of the previously determined failure force. Results of their study indicate a change in the load-deformation behavior manifested by significantly larger deformations in the lower half of the load-deformation curve. Nonetheless, more detailed descriptions of these changes in the lower portion of the structural or material property curve are largely unrealized due to the lack of objective parameters by which they can be characterized. Journal of Biomechanical Engineering Copyright 2003 by ASME JUNE 2003, Vol. 125 Õ 415

2 Several structural and microstructural models have been used to study the mechanical properties of various tissues 8,9,11 13,16. Hurschler et al. 11 previously developed a model that assumes increasing collagen fiber straightening and load bearing with increasing elongation i.e., fiber recruitment from a crimped or slack to a load bearing according to a Weibull 19 probability density function. Because it is an asymmetric one-tailed function, a parameter represented by a Weibull PDF has a zero probability of occurrence below a certain value. The Weibull PDF was chosen to represent fiber straightening in ligaments and tendons since in a slack configuration, no fibers have yet straightened to bear loads. Thus, fiber straightening is well represented by the one-tailed nature of the Weibull PDF. In contrast to the Gaussian PDF which is described by two parameters i.e., the mean and the standard deviation, the Weibull distribution is described by three parameters: shape, scale, and location, 20. In the context of the structural model, the shape and scale parameters are related to the shape of the toe-region where the fibers are being recruited; the location parameter is related to the stretch at which the first fibers begin to bear load. This location parameter, described in detail later in this manuscript, provides a parameter to objectively determine reference length of the tissue. An objective reference length provides a consistent basis for describing the material properties in the high load and low load regions of fibrous tissues. Hence, the goals of this study were: 1 to introduce the location parameter as a preload independent parameter to establish a reference length in biomechanical evaluations of fibrous connective tissues, such as ligaments and tendons, and 2 to establish a method for quantifying the toe-to-linear region transition-point in strain-stiffening soft fibrous tissues. In order to examine the first goal, representative data from the literature, 21, are used to illustrate how may be applied to data gathered from either initially slack or preloaded specimens. Also shown is how the experimental choice of reference length affects tangent modulus. In order to examine the second goal, the model is applied to previously published data for normal and damaged tissue, 22, to illustrate how it can be used to quantify microstructural changes which occur in the altered toe-to-linear region transition. Methods Microstructural Model. The probabilistic microstructural model 3 applied in this study is described in detail by Hurschler et al. 11 and will therefore only be briefly described here. The unloaded tissue is assumed to be dominantly composed of crimped collagen fibers that need to be extended various lengths before they begin to bear tensile load. Fibers are assumed to be oriented predominately along the longitudinal axis of the ligament or tendon. The straightening stretch ratio SRR, defined as the tissue stretch at which an individual fiber begins to bear load, is described as s l s /l o fiber length divided by tissue reference length. The load borne by the fiber is assumed to behave according to a constitutive law based on collagen fibril microstructure. The stress in the tissue is determined by the overall tissue stretch stretch ratio t l/l 0 deformed tissue length divided by tissue reference length and the deformation state, and hence load bearing contribution, of the fiber population. Thus, the longitudinal normal tissue stress ( t ) can be computed by integrating the contribution of all fibers over all possible SSRs ( s ): t t tpw s 33 t / s d s, t. (1) In Eq. 1, P w is the Weibull PDF as a function of SSR, 33 is the longitudinal normal stress in a fiber, and is the location parameter of the Weibull distribution and defines the onset of fiber loading. Hence, t ( t )0 when t, since none of the fibers are 3 The model by Hurschler et al. 11 has been implemented in Microsoft Excel and is available upon request. Please contact the corresponding author to obtain a copy of the model for research and academic purposes. loaded below the threshold. It should be noted however, that can be less than or greater than one depending on whether the tissue is initially in a preloaded or slack configuration, respectively. The form of the Weibull PDF used in Eq. 1 is P w s s 1 exp s, s P w s 0, s. (2) In Eq. 2, is the location parameter as stated above, and and are the shape and scale parameters, respectively. The shape, scale, and location parameters must all be greater than zero i.e., 0, 0, and 0 since stretch ratio is defined to be greater than zero. Shape and scale parameters are related to the shape of the toe-region while the location parameter is related to the stretch at which fibers begin to bear load. The Weibull PDF parameters,, and for a given set of stress-stretch data are determined by a nonlinear least-squares minimization, 11. Determining Tissue Reference Length. Consider a segment uniaxially loaded fibrous tissue such as tendon or ligament consisting of a population of collagen fibers. We define the true axial reference length l 0 of this segment of tissue as the length at which the first fibers straighten and just begin to bear load. Since the fibers comprising the tissue generally do not all straighten at once with elongation, there is in general no sharp discontinuity of the load-displacement curve signaling the attainment of l 0. The distance between the endpoints of the specimen determines the length l of the segment, which in an arbitrary state of deformation may be greater than, or less than, or equal to l 0. Let l r represent the experimentally assigned reference length based on specimen preload upon which measurement of stretch or strain are typically based. Then, generally, l r l 0, in which case the specimen is under some preload. If, on the other hand, l r l 0, the specimen is slack at that reference length and bears no load. Tissue stretch-ratio defined in terms of l 0 is t l/l 0. Stretchratio with respect to l r is primed, t l/l r. The stretch-ratio defined with respect to l r at which fibers just begin to bear load is t0 l 0 /l r. When t0 is known, stretch-ratio relative to l 0 can be obtained by t l l l r t l 0 l r l 0 t0. (3) Thus, if t0 is known, it is possible to convert from stretch based on an arbitrarily assigned reference length l r to stretch based on l 0. A technique by which t0, and hence l 0, can be determined independent of arbitrary load thresholds is now described. As stated in the previous subsection, the Weibull shape and scale parameters and determine the shape of the PDF for fiber straightening stretch and hence the shape of the toe-in region. Within the assumptions of the model, the location parameter determines the stretch at which fibers are first recruited and bear load by definition of the Weibull PDF, the probability that s is zero. Since, by definition, the stretch at which fibers first bear load being t0 l 0 /l r, thus t0. By fitting the model to the experimental stress-stretch data based on l r, t0, and it is possible to compute the stretch-ratio based on l 0 using Eq. 3 from stretch normalized with respect to the experimental reference length l r. It is important to note that the location parameter and hence t0 ) may take values greater or less than one. Thus, if 1 the specimen was initially slack before any load was applied, if 1 the specimen was initially preloaded which is the prevailing experimental practice. Typically, stress is zeroed at the preloaded reference length; in such cases it is still possible to determine the stress offset ( 0 ) associated with the specimen preload by evaluating Eq. 1 at t 1. This stress offset is automatically incorporated into the least-squares algorithm when becomes less than one and adds no additional parameters to the minimization. 416 Õ Vol. 125, JUNE 2003 Transactions of the ASME

3 Fig. 1 Example tendon stress-stretch data taken from Abrahams 21, retaining the units of the original paper. The open symbols represent data points, the dashed line represents the fit of the model to the data. Thus, by obtaining the Weibull PDF parameters from the experimental data based on l r, it is possible to adjust the stress-stretch data to include the stress offset ( 0 ) from the objective reference length l 0. As an example application, the proposed technique of determining t0 is applied to representative tendon data from the literature, 21, retaining the units used in the original paper. The original data ( t ) refers to stress and stretch based reference length l r, determined by some arbitrary preload, and adjusted data ( t ) refers to stress and stretch based on t0 determined by the model Eqs. 1 and 3. For purposes of illustration of the technique, load versus length data is computed by assuming an original specimen gauge length l r of 10 length units Fig. 1. Fitting the model to the original data results in t which indicates that a very small preload was applied to the specimen since 1) and that the chosen reference length for this specimen was Fig. 2 Simulated initially slack tendon stress-stretch data for a reference length of l r Ä9.9 length units open circles. Adjusted stress-stretch data and corresponding model fit filled circles, solid line agree with the fit of the original data l r Ä10 length units, dashed line. Journal of Biomechanical Engineering JUNE 2003, Vol. 125 Õ 417

4 Fig. 3 Simulated initially preloaded tendon stress-stretch data for a reference length of l r Ä10.1 representing an initially preloaded specimen open circles. Adjusted stress-stretch data and corresponding model fit filled circles, solid line agree well with the fit of the adjusted original data set l r Ä10 length units, dashed line. Note that since some information in the toe-in region is lost, agreement is not as good as with the initially slack specimen Fig. 2. very close to l 0. Thus, no correction of stretch-ratio to l 0 for these data is necessary. The data is next modified to simulate two experimental conditions, first a test of the specimen from an initially slack configuration, and second a test of the specimen from an initially significantly preloaded configuration. To simulate a specimen that is initially slack and bears no load in the reference configuration, the original data are recomputed based on a reference length of l r 9.9 length units. This is accomplished by adding a zero-load data point at l9.9 to the original load-length data, and results in a shift to the right and slight decrease in the slope of the stressstretch curve open circles, Fig. 2. Furthermore, to simulate an experiment in which a significant amount of preload is applied to the specimen, the first load-length datum point is omitted. This results in a reference length of l r length units and a shift to the left and slight increase in slope of the stress-stretch curve open circles, Fig. 3. Note that by testing from this level of preload, some of the toe-in region of the stress-stretch curve is lost. Finally, to illustrate the effect of l r on tangent modulus, the modulus is computed from the last four data points of the example data for different values of reference length ranging from l r 9.9 to l r These moduli are then recomputed based upon example data adjusted to l 0. Determining the Toe Region to Linear Region Transition Point. Once the nonlinear load-deformation or stress-stretch curve is fit with the above model, the three Weibull parameters are obtained. The point of transition from the toe region to the linear region is determined by the analytical method described below. The cumulative distribution function CDF, F(), for the Weibull distribution can be obtained by integrating the Weibull PDF, F t tps d s t The resulting CDF is s 1 exp s d s. (4) F t 1exp t (5) where F is the percentile (0F1) of area under the PDF corresponding to the magnitude of fibers in tension and carrying load. Manipulating the equation algebraically leads to the inverse CDF, t ln1f 1/. (6) Equation 6 defines the stretch ratio at which any fraction of the fibers F have been recruited. Hence, by setting F to a particular value, say 0.75 at which 75% of fibers have been recruited, one can objectively define the stretch ratio and hence strain at which a particular percentage of collagen fibers are recruited, and at which most of the fibers are recruited and the tissue exits the nonlinear strain-stiffening toe-region and enters a more linear portion of the curve. After the stretch ratio for a specific fraction of loaded fibers are determined Eq. 6, the corresponding stress may be obtained Eq. 1. Probability density functions by definition have an area under their curves equal to unity. The PDF curves in the graphical representations presented herein are scaled to the maximum attained stress in for purposes of data visualization only. No parameters are distorted during scaling, i.e., scaling does not adversely effect the Weibull parameters. The above model was applied to rat medial collateral ligament MCL stress-strain or stretch ratio data obtained from a separate 418 Õ Vol. 125, JUNE 2003 Transactions of the ASME

5 Fig. 4 Tangent modulus computed from stress-stretch data based on different choices of reference length l r white bars, and tangent modulus computed from the corresponding adjusted data gray bars. The unadjusted modulus changes linearly with l r, the adjusted modulus remains relatively constant. study, 22. These stress-stretch data from ligament are used as a further example of the application of the model to fibrous connective tissue. The stress-stretch curves used were obtained from subfailure damaged as well as contralateral control ligaments, 21. Results Tissue Reference Length. In the case of the simulated slack specimen with l r 9.9 length units, applying the model results in t The data adjusted to l 0 closely match the original data filled circles, Fig. 2. Application of the model to adjusted data results in very close agreement with the fit to the original l r 10.0 data solid versus dashed lines, Fig. 2. In the case of the preloaded specimen, applying the model results in t with a stress offset of psi. The data adjusted to l 0 also closely match the original data filled circles, Fig. 3. Again, the model applied to the adjusted data closely matches the fit to the original l r 10.0 data solid versus dashed lines, Fig. 3. Agreement is not as good as with the previous example because increasing preload results in a loss of information from the toe-in region. When further preloading eliminates most of the toe-in region from the data set, the least-squares minimization converges to nonunique solutions for the Weibull scale and shape parameters, and a consistent adjustment of the data to l 0 is not possible data not shown. Thus, too much preload results in a loss of information from the specimen such that l 0 cannot be recovered. The data show that considerable experimental bias in tangent modulus can be avoided by determining t0 and adjusting for choice of reference length Fig. 4. While the unadjusted modulus varies linearly with l r, as would be expected, the adjusted modulus remains relatively constant as reference length is increased Fig. 4. An adjusted modulus is more consistent even at relatively large preloads such as at l r i.e., up to 12% elongation relative to l 0 ) where much of the data from the toe-in region have been lost. The Toe Region to Linear Region Transition. The model fit the MCL data examined in this study extremely well (R in all cases. The stress and stretch-ratio values at the transition from the toe to linear portions of the stress-stretch curve in control and subfailure stretched MCLs were computed for a range F values Table 1. Within the assumptions of the model, F represents the ratio of collagen fibers recruited compared to total number of fibers. The larger F is, the more fibers are recruited. The Weibull parameters for the Control ligament are: 0.956, 0.019, 0.997, and for the ligament damaged by a subfailure stretch, the parameters are: 1.245, 0.021, Inspection of the data suggests that fiber recruitment over the 85th percentile provides a good indication of the transition Fig. 5. Values of F below 0.85 appear to still be well into the strain-stiffening region and therefore underestimate the transition-point from toe to linear regions. Values above an F value of 0.85 appear to describe Table 1 Tissue stress ratio t in mmõmm and stress t in MPa at the toe-region to linear region transition in control and subfailure damaged MCLs for varying values of F from 0.75 to Within the assumptions of the model, the parameter F represents a measure of the amount of collagen fiber recruitment. A larger ratio indicates more fibers have been recruited. For the connective tissue data fit in this study, the model shows that subfailure damage increases the stretch ratio and decreases the stress at which the transition occurred F t t t t t t t t t t Control Stretched Journal of Biomechanical Engineering JUNE 2003, Vol. 125 Õ 419

6 Fig. 5 Model fits for control and subfailure stretched rat MCL data from 22. The structural model proposed by Hurschler et al. 11 fit each curve with R 2Ì0.99. Note, the Weibull PDF has been scaled to the corresponding stress-stretch curve, see Methods. The Control and Damaged markers below the Stretch Ratio axis correspond to the stretch ratio calculated for a specific value of F values appear in Table 1. The tick marks represent each of the five values of F examined, starting with 0.75 and moving to the right by increments of 0.05 until the value 0.95 is reached. the transition more accurately. Of course, one can always apply a very high fiber recruitment measure i.e., 95 99% to measure the point of entire fiber recruitment, but beyond a particular threshold, say 0.85, the additional recruitment may provide only a small contribution to the strain-stiffening effect. The model reveals that subfailure damaged tissue displays an increase in the stretch-ratio or strain and a decrease in stress required for the transition when compared to the contralateral control tissue. Discussion The goal of this study was to illustrate methods in which a probabilistic microstructural model can aid in quantifying low load behaviors. Reference length of initially compliant tissues, such as tendons or ligaments, is often based on an arbitrarily assigned load threshold. These thresholds are selected more upon experimental convenience rather than physiological considerations i.e., differences between in situ and ex vivo reference length are not known due to complex geometry and loading patterns. The choice of this load threshold can bias biomechanical measures that depend on length. An objective technique for determining reference length, one that is independent of load thresholds, was therefore developed and illustrated. The technique relies on a mathematical model that assumes that fibers are recruited and bear load according to a Weibull PDF for fiber straightening as the tissue elongates, 11. Thus, it is implicitly assumed that enough low load data are present so that an accurate representation of the toe-in region is possible via the three parameters of the Weibull PDF. When insufficient toe-in data are available, it is not possible to uniquely determine the Weibull parameters, and specifically the location parameter t0, and adjustment is not possible. The technique presented herein can be applied retroactively to normalize and compare existing data. Optimally, it is applied prospectively to data that capture the toe-in region in its entirety. It eliminates entirely the need to use a load threshold i.e., 0.1 N, 1 N, or 5N to establish a reference length: the entire low-load behavior of the specimen can be captured because the specimen can be tested from an initially slack configuration. The applied microstructural model and accompanying analytical method proposed herein fit the experimental data well, and provide a parameter that can be used to consistently select a transition-point between the toe-region and the linear region in the stress-stretch curve. The parameter F defining the transition, as is the entire model, is based on the phenomenon of collagen fiber recruitment in the microstructural model. Hence, this model provides a useful tool for evaluation of experimental data, with parameters deriving from physical characteristics of the tissue. Evaluating parameters from the toe-region, as well as tangent modulus and failure parameters, allows for a more complete description of the tissue than is possible by limiting the analysis to the linear and failure regions alone. Quantifying this transitionpoint may for instance provide useful insight into the effect of experimental therapies such as mobilization and exercise, and tissue engineering methods on the recovery of normal low load strain-stiffening behavior in injured connective tissue. Furthermore, low-load behavior of connective tissues is important in daily activity and understanding and quantifying the low-load region of the stress-strain curve in connective tissue may provide insight into pathologic tissue laxity. Laxity in ligament, for example, may contribute to pathologic joint laxity, and pathologic joint laxity can lead to joint deterioration and arthritis. Quantifying the toe region to linear region transition allows for better understanding of normal tissue behavior and length and provides a baseline from which to quantify changes. The microstructure of fibrous connective tissues has a direct impact on the mechanical load-deformation and stress-stretch behaviors of the tissues. In ligament and tendon, for instance, collagen crimp drives the nonlinear strain-stiffening behavior in the low-load region when the tissue is loaded in the longitudinal direction. The stiffness of the tissue increases only as the fibers straighten and are they able to bear load. The microstructural model applied to data in this study assumes that this strainstiffening behavior is the result of collagen fiber recruitment during tissue stretch. To further examine this assumption the model was fit to the data of two existing studies, 15,23, that examine changes in collagen crimp as a function of tissue stretch Figs. 420 Õ Vol. 125, JUNE 2003 Transactions of the ASME

7 Fig. 6 Model fits to data from a Viidik 15 in which changes in collagen fiber crimp were examined with polarized light during tissue strain measured grip to grip and b Hanson et al. 23 in which changes in collagen fiber crimp were examined with optical coherence tomography during tissue strain measured grip to grip. In both cases the original units of the studies were maintained with the exception of strain being converted to stretch ratio. In the case of the Viidik data the model agrees very well, in the case of the Hanson et al. data the model slightly underpredicts the stretch at which complete fiber straightening occurs. 6a and 6b. Data were obtained from these studies, 15,23, by scanning the original figures, digitizing, and replotting the data. The model fits both data sets well (R ), and the model predicts fiber recruitment with good agreement to the experimental measures of collagen fiber straightening. In the second case Fig. 6b the measured strain at which crimp was observed to be extinguished somewhat exceeded the model prediction. It is possible that although crimp is was no longer observable using the technique in that study, that the fibers were nonetheless still not in a fully elongated state. In addition, neither study measured strain on the ligament locally, but rather grip to grip. This has been shown to increase the measured strain parameter and may therefore explain the somewhat larger toe-region strain for fiber straightening in these studies. Lastly, nonlinearities exceeding solely fiber straightening i.e., nonlinearity in the fiber themselves may be present in the tissues resulting in some small error between theory and experiment. However, the model assumes linear fiber stress-stretch which is supported by experiments performed on very small subunits of rat tail tendon, 24, and by X-ray diffractometry of bovine Achilles tendon fibril deformation, 25,26. Further study into the structure-function relationship is required. In this manuscript the proposed model was applied to tendon and ligament, although the technique could be analogously applied to other strain-stiffening fibrous connective tissue. In addition, this methodology is not restricted to the Weibull distribution, but may be implemented into other probabilistic microstructural models that utilize other PDFs. Both Gaussian and Gamma distri- Journal of Biomechanical Engineering JUNE 2003, Vol. 125 Õ 421

8 butions for fiber straightening may be capable of describing the behavior of various connective tissues with adequate fidelity, although continuous PDF s e.g., Gaussian do not yield a l 0. Once an appropriate distribution that describes tissue microstructure is selected to fit a particular data set and the data are fit, a CDF can be applied to determine the magnitude of fiber recruitment. For the data fit in this study, the 85th percentile appeared the most reasonable choice for the lower limit of transition linear region of the stress-strain curve. Values of F below 0.85 appear to be displaying substantial strain-stiffening behavior and therefore underestimate the transition-point while values above an F value of 0.85 appear to describe the transition more accurately. Above 0.85, additional fiber recruitment results in very little strainstiffening behavior, yet, one can always apply a very high fiber recruitment measure i.e., 95 99% to measure the point of entire fiber recruitment. However, the percentile used is not critical as long the value of F is fixed and reported. Once a value for F is chosen, it can be objectively applied to various groups and treatments for experimental comparison. In conclusion, the methods presented herein provide new tools with which to address some of the major difficulties associated with quantifying connective tissue low-load behavior. These tools, based on a probabilistic microstructural model for fiber straightening, now allow us objectively describe the events that govern low-load behavior, that is the initiation of load bearing and the transition from the toe to the linear region. They allow the capture of the entire low-load region of tissue behavior since the specimen can be tested from a slack configuration without the application of an arbitrary assigned preload. Acknowledgment The authors thank Dr. Dennis Heisey and Dr. Glen Leverson for their statistical insight, and Dr. David Corr for his helpful suggestions and critique. Funding for this work was provided by NASA grant # NAG and by a grant from the University of Wisconsin Surgical Associates. References 1 Lam, T. C., Shrive, N. G., and Frank, C. B., 1995, Variations in Rupture Site and Surface Strains at Failure in the Maturing Rabbit Medial Collateral Ligament, ASME J. Biomech. Eng., 1174, pp Chimich, D., Frank, C., Shrive, N., Dougall, H., and Bray, R., 1991, The Effects of Initial End Contact on Medial Collateral Ligament Healing: A Morphological and Biomechanical Study in a Rabbit Model, J. Orthop. Res., 91, pp Przybylski, G. J., Carlin, G. J., Patel, P. R., and Woo, S. L., 1996, Human Anterior and Posterior Cervical Longitudinal Ligaments Possess Similar Tensile Properties, J. Orthop. 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S., and Vanderby, Jr., R., 2002, Sub-Failure Damage in Ligament: A Structural and Cellular Evaluation, J. Appl. Physiol., 1, pp Hansen, K. A., Weiss, J. A., and Barton, J. K., 2002, Recruitment of Tendon Crimp With Applied Tensile Strain, ASME J. Biomech. Eng., 1241, pp Kato, Y. P., Christiansen, D. L., Hahn, R. A., Shieh, S.-J., Goldstein, J. D., and Silver, F. H., 1989, Mechanical Properties of Collagen Fibers: A Comparison of Reconstituted and Rat Tail Tendon Fibers, Biomaterials, 10, pp Sasaki, N., and Odajima, S., 1996, Elongation Mechanism of Collagen Fibrils and Force-Strain Relations of Tendon at Each Level of Structural Hierarchy, J. Biomech., 299, pp Sasaki, N., and Odajima, S., 1996, Stress-Strain Curve and Young s Modulus of a Collagen Molecule as Determined by The X-Ray Diffraction Technique, J. Biomech., 295, pp Õ Vol. 125, JUNE 2003 Transactions of the ASME