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"Vl 3' . nrbl Ill? . , yiubll . . . x . lllJlHHllUlllzll““lelllllJII‘lUl“IIIHIUHIIUIIHHI z *c “m 300786 4311 F LIBRARY Michigan State ‘ University L___ ‘— This is to certify that the dissertation entitled A MICROSTRUCI'URAL MODEL OF SKIN: AN EVALUATION OF MATURING RAT DERMIS presented by Stephen Mark Belkof f has been accepted towards fulfillment of the requirements for Doctoral degree in Mechanics I ng Major rrp rofes Date April 27, 1990 MS U i: an Affirmative Action/Equal Opportunity Institution O~ 12771 ————-———~ _ _ __V / PLACE ll RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or betore due due. DATE DUE DATE DUE DATE DUE MSU I. An Affirmdlve Mort/Emil Opportunity lnetltulon C. A MICROSTRUCTURAL MODEL OF SKIN: AN EVALUATION OF MATURING RAT DERMIS By Stephen Mark Belkoff A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Metallurgy, Mechanics, and Material Science 1990 ABSTRACT A microstructural-based model for soft biological tissues was adapted to analyze the non-linear response of the collagen network in skin. Quasi-static uniaxial tensile tests were conducted on narrow and wide dorsal skin specimens of rats aged 1-4 months. Specimens were taken longitudinally and transversely to the spine and tested at 1.5 %/s. A simplified analysis in which the fibers for the narrow specimens were assumed aligned showed the effective stiffness to increase during maturation and to be greater laterally than longitudinally. It also indicated the migration of the "heel" portion of the response curve toward the origin during' maturation to be the result of less crimping in the fibers. These results were supported by microscopical observations. Based on the assumption that the difference in stiffness with specimen direction was caused by a preferential fiber orientation rather than different fiber material properties, a spatial distribution function to describe fiber orientation was added to the model. The fiber distribution was assumed elliptical with the major axis of the ellipse coinciding with the direction of greater stiffness. A collagen fiber stiffness and the major and minor radii values of the ellipse were determined from the effective values for the average narrow transverse and longitudinal specimens. These specimens were assumed to give the approximate response of a specimen for which all the fibers were oriented along the respective axes of the ellipse. The spatial distribution of fibers was found to change from a uniform (circular) distribution to a more elliptical distribution for ‘which. the ,preferential fiber orientation was in the transverse direction. The model incorporating the spatial distribution function was used to predict the wide specimen response. The model predicted the actual response better than the continuum-based prediction. When fiber stiffness was normalized based on a total content of collagen the resulting fiber modulus was shown to decrease during a 1 to 2 month age period. The model was found to be suited for response curves exhibiting a well-defined "heel" region. For curves exhibiting a more flat featureless "heel" region (typical of 1 month longitudinal specimens) the model was not adequate. The results of this study indicate that this simple five- parameter model is a: useful tool in analyzing the microstructure of collagenous tissue. It also provides a means of comparing the mechanical and geometrical properties of the microstructure of specimens of different size, shape and orientation. Such a means has been unavailable. To Lisa for her love, patience and support. ACKNOWLEDGEMENTS I would like to express my deepest appreciation to the following people for making the completion of this Doctoral Dissertation possible: To Dr. Roger Haut, my major professor and friend, for his insight, guidance and invaluable assistance. His dedication to the pursuit of the truth in science and his personal integrity serve as an example and an inspiration. To my committee members, Dr. Thomas Adams, Dr. Nicholas Altiero, Dr. Gary Cloud and Dr. Thomas Pence for serving on my committee, but more importantly, for appropriately preparing me for my dissertation research through their skillful instruction in the classroom. To the State of Michigan Research Excellence Fund for funding the research. To my mom and dad for their support and encouragement, for a childhood filled with love and discipline which gave me the skills necessary to complete this journey. To my grandparents on whose farm the hours spent hunting and visiting were necessary in maintaining a perspective on the research. ii To Jane Walsh, for her skillful assistance with the project and for her warm friendship. To Dr. Richard Hallgren for the use of his computer. To Cliff Beckett, for keeping things running in the lab. To Aimee Farquhar, for performing the biochemistry. And to Brenda Robinson, for preparing this document. iii II. TABLE OF CONTENTS Introduction.. ................... . ............ 1.1 The project........ ........ . ............. 1.2 Outline of the dissertation .............. Skin... .......... ......... ....... . ............ 2.1 Skin and its Function. ...... . ............ 2.1.1 Epidermis.. ...... .............. 2.1.2 Dermis ......................... 2.1.2.1 Collagen fibers ...... 2.1.2.2 Elastin fibers ....... 2.1.2.3 Reticulin fibers ..... 2.1.2.4 Ground substance ..... 2.2 Mechanical Behavior of Skin .............. 2.2.1 General ........................ 2.2.2 Langer's lines ................. 2.2.3 Elastic response ............... 2.2.4 Viscoelastic response .......... 2.3 Methods of Testing Skin .................. 2.3.1 General... ..................... 2.3.2 In Vivo Methods ................ 2.3.2.1 Uniaxial ............. 2.3.2.2 Torsion .............. 2.3.2.3 Diaphragm - Membrane. iv 10 11 11 ll 13 18 20 20 20 21 23 23 III. IV. 2.3.3 In Vitro.... ................... 2.3.3.1 Uniaxial ............. 2.3.3.2 Biaxial .............. 2.3.3.3 Membrane... .......... 2.3.3.4 Pure shear. .......... 2.4 Skin Models........ .................. ’.... 2.4.1 Continuum ...................... 2.4.2 Phenomonological ............... 2.4.3 Structural.... ................. Materials and Methods ........... . ............. 3.1 The model........... ..................... 3.1.1 Assumptions........ ............ 3.1.2 Aligned fiber model ............ 3.2 Experimental methods ..................... 3.3 Numerical Analysis ....... .... ............ 3.4 Spatial Distribution Function ............ 3.5 Wide Specimen Prediction ................. 3.6 Biochemical .............................. 3.7 Morphological .......... - .................. Results..... .. ........................... 4.1 Aligned fiber model ...................... 4.1.1 Model fit.......; .............. 4.1.2 Model parameters ............... 4.1.3 Parameter uniqueness ........... 4.1.4 Model fit to experimental data. 4.1.5 Morphological .................. 24 25 26 26 27 27 28 32 35 46 46 46 48 52 56 57 62 63 67 7o 70 7o 70 77 83 83 4.2 General model....... ..................... 4.2.1 Distribution function .......... 4.2.2 Wide specimen prediction ....... Discussion .................................... References .................................... Appendix A .................................... Appendix B........... ......................... Appendix C. .......... . ........................ vi 91 91 95 101 112 120 123 131 2.2 2.3 2.5 2.6 2.10 2.11 2.12 2.13 2.14 2.15 LIST OF FIGURES Schematic of cross section of skin (from Grays)... ...... ......... ........... Scanning electron micrograph of collagen fibers................. ......... Langer’s lines (from Ridge and wright)0.00.00.........OOOOOOOOO 000000000 Correlation of Langer's lines to preferred fiber orientation (from Gibson, et al.).. Three region explanation of mechanical response of skin (from Daly) ............. Direction dependent response of skin in vivo (from Ridge and Wright).... ...... Age related changes in the response of skin in viva (from Daly and Odland) ...... Strain rate dependence of skin (Haut).... The tensile response of four consecutive cycles during preconditioning (from Gibson and Kenedi) .............. . ........ Pure shear (from Trelear)... ............. ‘ Comparison of experimental response and power law fit (from Glaser, et a1.) ...... Collagen fiber model of Diamant, et al... Predicted response of zig-zag model by Diamant, et a1 ........................... Sinusoidal collagen model by Comminou and Yannas.... ........................... Predicted response of the model by Commonou and Yannas ...................... vii Page 12 13 14 16 17 19 20 31 35 37 37 38 38 2.16 2.17 2.18 2.19 2.20 2.21 2.22 Sequential straightening and loading model by Kastelic, et al.......... ....... Fit of model to experimental data by Kastelic, et al...... ...... . ............. Pinned sequence of unit collagen fibers by Markenscoff and Yannas. ....... . ....... Collagen-elastin fiber combination by Lanir Schematic planar views of the collagen and elastin network structure in flat tissues: (A) Tissues with high density of cross- links (HDCL) and elastin induced collagen undulation. (B) Tissues with low density of cross-links (LDCL) and inherent collagen undulation. Thick lines - collagen; thin lines - elastin; broken lines (in B) - overall collagen fiber direction (from Lanir)................................... Schematic of model by Decraemer, et a1... Sample fit of model by Decraemer, et al. to human tympanic membrane.. ............. Schematic of sequential fiber recruitment Uniaxial specimen stamp dimensions ....... Specimen in pneumatic grips.... .......... Tensile test of skin specimen ............ Elliptical distribution function ......... Wide specimen dimensions ................. Age average experimental response of transverse uniaxial skin samples ......... Age average experimental response of longitudinal uniaxial skin samples ....... Age related changes in the effective and tangent stiffnesses .................. Age related changes in model parameterpL Age related changes in model parameter 3. Sensitivity coefficient ratios for average 1.0 month transverse parameters.. viii 39 4O 40 42 43 44 45 49 54 55 55 58 63 71 71 75 75 76 78 4.8 4.9 4.20 4.21 Sensitivity coefficient ratios for average 1.5 month transverse parameters.. Sensitivity coefficient ratios for average 2.0 month transverse parameters.. Sensitivity coefficient ratios for average 3.0 month transverse parameters.. Sensitivity coefficient ratios for average 4.0 month transverse parameters.. Sensitivity coefficient ratios for average 1.0 month longitudinal parameters Sensitivity coefficient ratios for average 1.5 month longitudinal parameters Sensitivity coefficient ratios for average 2.0 month longitudinal parameters Sensitivity coefficient ratios for average 3.0 month longitudinal parameters Sensitivity coefficient ratios for average 4.0 month longitudinal parameters Average experimental response and average model fit and of 1.0 month uniaxial transverse specimens..... Average experimental response and average model fit and of 1.5 month uniaxial transverse specimens..... Average experimental response and average model fit and of 2.0 month uniaxial transverse specimens..... Average experimental response and average model fit and of 3.0 month uniaxial transverse specimens..... Average experimental response and average model fit and of 4.0 month uniaxial transverse specimens....... Average experimental response and average model fit of 1.0 month uniaxial longitudinal specimens .......... ix 0000000 78 79 79 80 80 81 81 82 82 84 84 85 85 86 86 4.22 4.23 4.24 4.25 4.26 4.27 4.28 4.29 4.30 4.31 4.32 4.33 4.34 4.35 4.36 4.37 Average experimental response and average model fit of 1.5 month uniaxial longitudinal specimens ......... Average experimental response and average model fit of 2.0 month uniaxial longitudinal specimens. ........ Average experimental response and average model fit of 3.0 month uniaxial longitudinal specimens.. ....... Average experimental response and average model fit of 4.0 month uniaxial longitudinal specimens ......... Typical model fit of experimental data.. Scanning electron micrographs ........... Age related changes in spatial distribution of fibers......... ......... Age related changes in fiber stiffness K Age related changes in dorsal skin thickness Age related changes in total and insoluble collagen content ...... . ..... . ........... Age related changes in the normalized fiber stiffness ..... ........ ............ Comparison of wide specimen, 5.5 times uniaxial and model predicted responses for 1.0 month transverse........ ........ Comparison of wide specimen, 5.5 times uniaxial and model predicted responses for 1.0 month longitudinal........ ...... Comparison of wide specimen, 5.5 times uniaxial and model predicted responses for 1.5 month transverse...... .......... Comparison of wide specimen, 5.5 times uniaxial and model predicted responses for 1.5 month longitudinal..... ......... Comparison of wide specimen, 5.5 times uniaxial and model predicted responses for 2.0 month transverse ................ 87 87 88 88 89 9O 92 93 94 94 95 96 96 97 97 98 4.38 4.39 4.41 Comparison of wide specimen, 5.5 times uniaxial and model predicted responses for 2.0 month longitudinal....... Comparison of wide specimen, 5.5 times uniaxial and model predicted responses for 3.0 month transverse................ Comparison of wide specimen, 5.5 times uniaxial and model predicted responses for 3.0 month longitudinal.... .......... Comparison of wide specimen, 5.5 times uniaxial and model predicted responses for 4.0 month transverse.......... ...... Comparison of wide specimen, 5.5 times uniaxial and model predicted responses for 4.0 month longitudinal........ xi 98 99 99 100 100 LIST OF TABLES Page Table 2.18 Continuum-based models. ............ 33 Table 4.1 Model parameters................... 72 xii xL XT LIST OF SYMBOLS area Major radius of elliptical spatial distribution function Minor radius of elliptical spatial distribution function Spatial distribution function Experimentally measured force at deformation x e 1 Model predicted force value at deformation Xi Initial specimen thickness Longitudinal effective fiber stiffness (total) Transverse effective fiber stiffness (total) Longitudinal effective fiber stiffness (per fiber) Transverse effective fiber stiffness (per fiber) Initial specimen length Sensitivity coefficient ratios Uncertainty of experimentally measured force at x- 1 Initial specimen width Number of fibers per unit thickness Deformation Longitudinal axis Transverse axis Incremental area xiii P(9)d9 Fiber orientation function Q Normalized fiber stiffness K Fiber stiffness (total) K’ Fiber stiffness (per fiber) “L Longitudinal mean of recruitment function “T Transverse mean of recruitment function 0L Longitudinal standard deviation of recruitment function 0T . Transverse standard deviation of recruitment function x2 Chi squared residual 0 Angle with respect to xTaxis in the XTXL plane n 3.14 Axi Incremental deformation C Dummy variable of integration n Dummy variable of integration subscripts i,j,m Counting index T Transverse L Longitudinal xiv I. INTRODUCTION 1.1 ID3.E£QJ§QL The intent of this study was to develop a method of analyzing the mechanical and geometrical properties of the collagen fibers in skin from the macromechanical response obtained. from tensile tests. The method. needed to be sensitive to changes in material properties and geometrical organization which occur with age, specimen location and direction. The model parameters had to have physical significance and be based on microstructure. In order to facilitate numerical analysis and to be useable in a clinical setting, the number of parameters needed to be kept to a minimum. The model also needed to be structurally based because it was felt that the response of the skin structure was as much a result of its constituent fiber stiffness as of the fiber geometry and spatial arrangement. The influence of fiber geometry could not be investigated with continuum models. The inability of continuum models to include fiber geometry is likely the reason continuum analyses have been unable to compare the responses from different specimen sizes and test configurations (simple extension, pure shear, torsion, etc.). 2 Although the ultimate goal of any biomechanical research on the skin is to describe the tissue accurately in its natural environment, experimental techniques are not well-suited for investigations of this type due to their complexity and their inability to isolate the actual tissue being tested from the mechanical influence of surrounding tissues. It is also more expedient to determine the validity of a model using in vitro experiments as the experimental protocol is less complicated. 1.2 Qutling 9f the Dissertation In the second chapter, background on the function and structure of the skin is presented, followed by an explanation on the mechanical behavior of the skin, especially’ in relation. to 'the :microstructural components which are assumed to govern the response. Past methods of mechanically testing skin and subsequent models proposed to explain the mechanical behavior are also presented. Finally, alterations in the microstructure and mechanical response of skin are considered in terms of aging and skin pathologies. Chapter III presents the research performed. The experimental protocol is outlined in this chapter. The microstructural model is developed and the numerical methods used to determine the model’s parameters are presented with respect to the experimental data. Methods are also presented for determining biochemical content and morphology. 3 Chapter IV presents the results of the experiments and the parameters obtained from the model. Biochemical and morphological results are also presented. Discussion of the results is presented in Chapter V. Of particular interest is the discussion on the physical significance of the parameters obtained from the numerical reduction of the experimental data via the model with respect to tissue maturation and specimen direction. Future research ideas and clinical impact of this study are presented. II . SKIN 2.1 Skin and Its Function In order to understand the mechanics of the skin it is essential to be familiar with its composition and function. Unless otherwise specified, the information in this section comes from Montagna (1962). The skin is the largest single organ in the body. In the adult human, it accounts for about 16% of the total body weight and has a surface area of 1.5-2.0 m? (Whitaker, 1972). The skin serves as a covering for' the internal organs which. protects against external mechanical, thermal, chemical, radiological and biological stimuli. It varies in thickness from 0.2 mm on the eyelid to 6.0 mm on the sole of the foot. The skin is divided into two layers; the superficial layer of epidermis which functions mainLy as a barrier and the deeper layer of dermis which is primarily responsible for the skin's structural integrity (Figure 2.1). Below the dermis is a layer of fat called the panniculus adiposus and below it, the panniculus carnosus. In most mammals, with the exception of humans, the panniculus carnosus is a well developed, flat sheet of muscle. It is this layer of muscle which controls "hackles" on dogs and gives the appearance of "goose bumps" in man. ./ F”, [f_- I: pulrrmu —— \tml. r n.rn ,"‘ “121*. SIN“. I/rrnl. ' _—_Drrmu ._ Surat gland Bulb ‘ Ilarr qul , Pa plua , Raul Selma. adipou I new aponcurotu‘a Figure 2.1 Schematic of cross section of skin (from Grays Anatomy) 2.1.1 E i rmi The epidermis is the cellular layer of the skin and is organized in several layers. The deepest layer, the stratum malpighian, is composed of the stratum basale which is in contact with the dermis; the stratum spinosum above it and the stratum granulosum. As the cells migrate from the stratum basale to the surface, they increase in their degree of keratanization. The most superficial layer is composed of dead keratanized cells which allows the epidermis to perform its barrier function. The stratum corneum is considered to contribute negligibly to the overall mechanical response of the skin (Schellander and Headington, 1974). 2.1.2 Dermis The dermis contains the fibrous connective tissue of the skin” Here elastic fibers (elastin), collagen fibers and reticulin fibers are found embedded in an amorphous gel— like matrix referred to as ground substance. Also found within the dermis are various glands and their ducts, hair shafts and hair follicles, nerve endings, blood vessels and lymph vessels. Although these other components are important to the physiological function of the skin, they contribute little to the mechanical behavior (Mendoza and Milch, 1964/65) and will be ignored. In the upper (papillary) layer of the dermis are found fine, randomly oriented collagen fibers (Brown, 1973). This layer' is :mainly concerned. with. connecting the reticular dermis with the epidermis. Because of the small fiber size and fine organization in this layer, its mechanical contribution is questionable. The deep layer of the dermis, the reticular dermis, is the layer which receives the most attention when the mechanical‘ response of the skin is investigated. The collagen fibers in this layer are thick and densely packed in jplanes running' parallel to the surface of the skin (Finlay, 1969; Lavker, et al., 1987) with some fibers running between the planes, presumably to prevent interplanar shearing (Millington et al., 1971). The fibers within the planes also appear to be oriented in a preferred direction (Ridge and Wright, 1966). This preferential 7 direction is believed to account for the skin’s anisotropic behavior and for the phenomenon known as Langer Lines, which will be discussed later. 2.1.2.1 C 11 n i r Collagen is the major component in the dermis and is considered responsible for skin’s structural integrity. It accounts for 50-80 percent of the dry weight of skin, (Bottoms, 1963) depending on age, sex and location (Dombi and Haut, 1985; Vogel, H.G., 1976). Collagen fibers, also referred to as collagen fiber bundles, range from 1—15 microns in diameter and are composed of finer collagen fibrils. The fibril ranges in diameter from 700 to 1400 °A (Craig and Parry, 1981) and gives the fiber bundle a rope- like appearance (Figure 2.2). The elastic modulus of ISKU XI’SBB Figure 2.2 Scanning electron micrograph of collagen fibers a collagen fiber is approximately 600MPa (Haut, 1983). During maturation the fibers increase in diameter, become more densely packed and lose some of their waviness. In aged skin, both in humans and in rats, the collagen fiber bundles change from cylindrical rope-like structures to flat ribbon-like fibers. These fibers were also reported to become increasingly more unravelled with age (Lavker, et al., 1987). It is interesting to note that collagen does not occur naturally in a pure state but is always associated with proteoglycans of the matrix (Curran, et al., 1965). The matrix at the fibril level is assumed to lubricate fibril interactions, and at the fiber level it may serve as a lubricant between fibers (Gibson, et al., 1965). It has also been reported that the crimping of the fibrils within the fibers (fiber bundles) may be caused by some fibril— matrix interactions (Dale and Baer, 1974). As the collagen matures, the crimping changes from a smooth sinusoidal crimp to a sharp zig-zag pattern (Kastelic, et al., 1978). Five types of collagen have been identified (Oxlund 1988). For the mechanical properties of skin, only type I collagen will be considered as it is the most ubiquitous, comprising 90% of the dermal collagen fibers (Smith, et al., 1962). The remaining collagen fibers are of type III. The content of collagen shifts from a large proportion of type III during fetal development to increasing amounts of type I 9 (Epstein, 1974). For a more extensive understanding of the collgen fibers, the reader is referred to a review by Elden (1968). 2.1.2.2 W Elastin makes up only 2-9 percent of the dry weight of human skin (Hult and Goltz, 1965; Weinstein, et al., 1960). It is found in fiber form, intertwining about collagen fibers. Elastin fibers are rubber-like protein polymers (Gotte, 1971) considered to be purely linear elastic (Carton, et al., 1962). Although the terms elastin and elastic fibers are often used synonomously, elastic fibers are essentially composed of an elastin core reported to be made of interwoven thread-like filaments (Gotte, 1971). The modulus of elasticity of elastin is reported to be between 500 kPa (Jenkins and Little, 1974) and 3MPa (Hass, 1942). Because of the low modulus of elasticity compared to collagen, and elastin’s low content in the skin, the mechanical contribution of elastin is usually ignored, except at very low strains. .2.1.2.3 R i i ibers Reticulin fibers account for only 0.4% of the dry weight of the skin (Montagna, 1962). They are found close to the epidermis and around blood vessels (Tregear, 1966). Although they are similar to collagen in composition, their 10 effect on the mechanical response of skin is usually neglected, on account of their minute quantity. 2.1.2.4 9W Ground substance is a gel-like matrix material consisting of water, glycosaminoglycans (GAG), glyco- proteins, and enzymes. The gel accounts for 70-90 percent of the volume of skin (Tregear, 1966). When the ground substance is digested in tendon, aorta and ligamentum nuchae, a reported. reduction in stress and Viscoelastic response is reported (Minns, et al., 1973) indicating some association between collagen and the matrix. This association is further strengthened by the report that in rats (Haut, 1989) the structural stiffness of skin increases as the strain rate is increased. The matrix flowing through the fiber network was also found to be responsible for the response of skin under' a compressive load (Tregear and Dirnhuber, 1965). Fessler (1960) further reports that this interaction of hyaluronic acid in the ground substance with the collagen fiber network inhibits the free flow of water within the matrix. Interfibilar interactions due to GAG components may be electrostatical (Egan, 1987). 11 2.2 Meehagieal Behavier ef Skin 2.2.1 General The skin demonstrates the well-documented three-phase non-linear response typical of soft biological tissues. Many factors such as fiber stiffness, matrix organization and composition, and fiber organization and geometry contribute to this response. These factors will be discussed in this section. 2.202 W, . Since microstructural fiber organization appears to be the primary factor in the composure of skin in vivo and plays a major role in the mechanical response to deformation, it is best to start here. In vivo the skin is under a state of tension. This was first noticed by Dupuytren (1835) and was later investigated in depth by Langer (1861) and Cox (1942). These investigators noticed that when they used an awl to punch circular holes in the skin of cadavers, instead of developing the expected circular hole, there was formed an elliptical one. This indicated that the skin was under some in vivo pretension and that this pretension had a preferred orientation, which corresponded to the direction of the major axis of the ellipse. The lines of maximum tension are referred to as cleavage, or more commonly, as Langer Lines. Langer published maps of the lines on the human body (Figure 2.3). Figure 2.3 Langer lines (from Ridge and Wright) These maps have been used extensively by surgeons to plan skin incisions. It is felt that cutting along the lines of maximum tensions, and not across them, aids in wound healing and reduces scarring (Flint, 1973). Cox later showed that wrinkle lines and creases are also closely related to Langer lines. Recent studies have shown that microstructurally, Langer’s lines seem to parallel a preferred orientation of the collagen fibers in the dermis (Gibson and Kenedi, 1965; Gibson, et al., 1969, Ridge and Wright, 1965; Flint, 1973), however, the pretension indicated by the retraction noticed after incision or puncture is likely due to elastin (Flint, 1973) and may also be affected by posture (Pierard and Lapiere, 1987). Gibson, et al. (1969) mapped the lines of maximum skin extension on the human chest (Figure 2.4). The direction of minimum skin extension, indicating maximum Figure. 2.4 Correlation of in. vivo sxin tension with Langer’s lines (from Gibson, et al.) stiffness, was found to correlate well with Langer’s lines. Langer’s lines, therefore, become important in mathematical modeling of the skin as an indicator of the material anisotropy, which must be considered before any finite elasticity approach is attempted (Alexander and Cook, 1977). 2.2.3 W The elastic response of skin is obtained from quasi static (slow strain rate) tensile experiments. With quasi static experiments the investigator is able to neglect the viscous contribution to the load response and focus on the purely elastic behavior. The Viscoelastic response will be discussed briefly in the next section. The classic elastic response of biological tissue is a non-linear curve of increasing stiffness with increasing deformation. The load response is traditionally divided 14 into three regions (Daly, 1982) for the purpose of explanation (Figure 2.5). The first region ("toe region) displays a low modulus response to deformation. There are two schools of thought as to which component of the skin is responsible for this compliant response. Some investigators suggest that in Region I elastin is being deformed and its low modulus accounts for the compliant response (Dick, 1951; Alexander and Cook, 1976; Lanir, 1979). Others propose that in this region the response is due to a very few collagen fibers already straightened and bearing load (Daly, 1979; Daly and Odlund, 1982). Oxlund (1988) digested elastin from skin samples, tested them, and found that the loading response changed slightly in the toe region, but that the contribution of elastin is mainly to restoring the stretched collagen fibers to their original undeformed configuration. 1 l A 3 c STRESS STRAIN Figure 2.5 Typical force deformation response of skin (from Daly) 15 The second region of the response curve demonstrates a rapid increase in stiffness. It is felt that this results from the wavy collagen fibers being sequentially straightened. Once straight, they are able to resist further elongation and generate load. In the non- straightened or wavy configuration, they are assumed to be like slack cables, unable to resist elongation until they are pulled taut. This microstructural explanation is based on morphological observations made under light and scanning electron microscopes (Brown, 1973; Craik and MeNeil, 1965; Gibson, et al., 1965; Finlay, 1969). These investigations subjected skin to sequential deformations, fixed the samples and observed then under the microscope. As the deformation continues, more and more fibers are straightened until all of the fibers are essentially straight and resisting elongation. This is the onset of Region III. Here it is assumed that all of the fibers are recruited and response is the sum of the linear response (collagen is assumed to be linear elastic) of all the collagen fibers. Dunn and Silver (1983) reported that once the collagen fibers are aligned in skin, the elastic behavior approaches that of tendon. Kenedi, et al., (1965) noticed a similar result which led to the claim that the "stress" encountered in the skin appears to be caused by the structural arrangement of the fibers (their crimping and spatial arrangement) and not solely due to their material properties. 16 Although the response curve qualitatively remains essentially the same for all biological tissues, there are some major quantitative changes which occur that depend on specimen location, direction and age. As is expected from the anisotropy noticed in skin under normal in vivo tension (as evidenced by Langer’s lines), there is a difference in response curves obtained from specimens taken from different sites and different orientations on the body. Investigators have noticed that specimens which are taken along Langer’s lines demonstrate a higher structural stiffness when elongated (Gibson, et al., 1965; Ridge and Wright, 1966a; Stark, 1977; Jansen and Rottier, 1957). l ‘ I l 350° ” Poet -'rnortern skin from anterior '“‘ l 1 «Indochina! ml! at 43 years eldmde [: ‘ . I r 1 1 . , . . 3000 {e 4 . J —._ Axial ‘ ; —o—-Circumtermm| i ‘ ~ 25°C % f .5 . g zooe ' 1 a i la .2. l o J :rwe é~ T ' . 3 i I . g 5... . ' / i I l f 5 l i " ; 5°C 1 L / : i . f /.' l l . ' 14‘0’1. * l o o: o 2 6-3 04 0.5 0-6 0 r Active strain», 1. Figure 2.6 Direction dependent response of skin in vivo (after Ridge and Wright) 17 This is explained by the preferred orientation of the collagen fibers. It is also noticed that the onset of Region II occurs earlier in specimens along Langer’s lines than perpendicular to them. This indicates the fibers are being recruited more quickly in that direction. There are also age-dependent changes in the response of skin. 3500 T Post - mortem stun, all samples ' anterior abdominal region 3000 I I I i Mole age 74 yrs U32-SXIO‘t‘2 - Mole age 43 yrs .\ J U'le05¢."/ 20°C I I | L . I Fernole age —1 I8 days 0,4“03‘65 . \»L/ . l . 500 *l l/ / l J e . I e 0 0‘2 04 0-6 08 Active strum, c. 2500‘——— \. \\ v I I Norninol stress c, lit/in2 I000 -— - .p..._~o.n—-——-._- Figure 2.7 Age related changes in the response of skin in vivo (from Daly and Odland) In young skin the modulus is very low in comparison with older samples. This difference is probably caused by the low collagen content, loose packing and fine fibers in young skin. Old skin, on the other hand, has a higher collagen content (ngel, 1983) and the fibers are thicker and more densely packed. Barenberg, et al. (1978) suggest that the 18 changes in mechanical properties associated with age result from increased fibril size and interaction with the matrix and not from increased fibrillar crosslinking. 2.2.4 Vi l i R onse Skin is not a purely elastic tissue but has Viscoelastic qualities as well. This means that stress in the tissue is not merely a function of strain, but also of time. This strain rate dependence can be seen in Figure 2.8 for rat skin strained at two different rates. The response of the specimen strained at a higher rate is much greater than for the specimen strained at the low strain rate. The viscous effect is greatest in the second region, which would indicate that the hydrated matrix material is being squeezed out as the collagen fibers align in the direction of load or deformation. The interaction of the fibers and the matrix during this process is considered to be the major contributor (Daly, 1982). The rate of flow of the matrix is considered 'to be very slow due to its high. viscosity, bonding to the fibers and the high density of fibers (Day, 1952; Tregaer and Dirnhuber, 1965). Since skin is Viscoelastic, its mechanical response is dependent on the strain history of the material. For this reason, an argument has raged in the biomechanics community on whether or not to precondition specimens. If skin is treated as a Viscoelastic solid, preconditioning the 19 . High Strain Rate l + Specimen 5ai|ure Slims, MP.) (5 t l ‘\ \ Low Strain hate /I ‘3 l ,1 l : // H. . . LI . 0 0.40 (180 1.20 1.60 Strain,mmlmm Figure 2.8 Strain rate dependence of skin (from Haut) specimen with successive cycle deformations obtains the equilibrium, elastic response of 'the ‘viscoelastic solid. While preconditioning may ensure a repeatable elastic response of the tissue, as shown in the third and fourth cycles of deformation in Figure 2.9, valuable information in the preconditioning cycles may be lost. Many investigators believe that the viscoelastic response is caused by the geometrical reorganization of the fibers and the interaction of the fibers with the matrix. Since the model proposed in this study is meant to describe much of the reorganization phenomena, the specimens were not preconditioned for fear of losing that information. 20 o l I I I”iL ‘XCI‘CD SKIN N TINEHON / 8 <. 5 — 3 f 3 .. 4 g /] J ( 2 2 3 // 9 ° 01 oz as. «we 05 51'th Figure 2.9 The tensile response of four consecutive cycles during preconditioning (from Gibson and Kenedi) 2.3 W 2.3.1 General There are two major divisions in mechanical testing methods, 1) In Viva - tests which occur in the living body. 2) In Vitro - tests in which the samples are excised from the body and tested (literally in a glass) outside the body. 2.3.2 I V'v The in viva test methods have the advantage of maintaining a viable normal environment for the test specimen. Since the ultimate goal of mechanical investigations of skin is to develop a data base and method of diagnosing skin dysfunction and pathologies in a non- 21 invasive and non-injurious manner, research into in viva methods is of vital importance. Unfortunately, measuring the mechanical properties of skin in viva is very difficult as several problematic circumstances are present. The task of attaching a loading or stretching mechanism to the skin is one such problem. A common method is to glue the grips of the device to the skin. This method, however, attaches the grips to the skin’s surface only and it is unknown how loads are transmitted from the epidermis to the dermis. Isolating the specimen from its mechanical interactions with the rest of the body without isolating it from its physiologic functional interaction is also not possible. The skin surrounding the specimen, and fascia and panniculus carnosus below the test site all have an influence on the mechanical response. Another problem is measuring skin thickness accurately in viva. Despite the limitations of in vivo studies, several investigations have been undertaken and have contributed to the understanding of the function and construction of the skin. A brief synopsis of these investigations follows. 2.3.2.1 Uniaxial Uniaxial tests are the most common type of in viva tests (Allaire, et al., 1977; Gibson, et al., 1969; Manschot and Brakkee, 1986a; Wijn, et al., 1976) likely due to the relative ease of conducting experiments in comparison with other in viva methods. Typically two pads are glued or 22 taped to the epidermis with a suitable distance between them to approximate the length-to-width aspect ratio typical of uniaxial tests. The two pads are then connected to each other via some mechanism which drives them. apart at a controlled force or deformation rate. The gauge length is taken as the distance between the pads and the width is considered to be the width of the pads. Typical results are plotted in Figure 2.6 and 2.7. Although this method makes analysis simple by assuming the specimen is in uniaxial tension, the test is clearly not uniaxial as the lateral boundary is not traction free, nor is the dermal-subdermal interface. Manschot (1985) recently accounted for the influence of subdermal interaction by approximating the skin’s effective thickness. In a similar manner he considered the influence of the lateral attachment of skin adjoining the test site. He reported that the effective width of the skin specimen is equal to the width of the test pads, provided the length-to— width ratio of the specimen is not too great. Manschot determined this effective specimen width by testing the skin using pads of widths ranging from 5 mm to 20 mm with the gauge length between the pads being constant. He found the force generated at a given strain was linearly proportional to the pad width. Knowing this information, he was able to account for the effective width of the specimen. 23 2.3.2.2 Taggigg Dissatisfied with the limitations of uniaxial testing, some investigators have turned to performing torsional tests on skin (Vlasblom 1967; Finlay, 1970). The apparatus usually consists of a disk which is taped or glued to the skin. A torsional load is then applied and the amount of twist is monitored. Vlasblom and Finlay controlled the amount of torque input whereas Duggan simply recorded the applied torque and resulting deformation. To isolate the test specimen from interacting with the surrounding tissue, Finlay glued a guard ring around the device’s torsional load disk. In this manner, only an annulus of skin was being tested. Sanders (1973) also tested human forearm skin in torsion and estimated the modulus of elasticity for collagen to decrease from 10 MPa to 2 MPa as the skin matured and aged from 6 to 61 years. Wijn, et a1. (1976) performed both torsion and uniaxial tests in an effort to correlate the two methods. Their results indicate no comparison- of material constants as would be allowed by the theory of elasticity for isotropic homogeneous media. They concluded that the skin must be treated as anisotropic. 2.3.2.3 Diaphragm - Membrane Like 'those who investigated 'the skin using torsion tests, Grahame and Holt (1969), Grahame (1970), and Jansen (1958) were dissatisfied with the assumptions and 24 approximations made for uniaxial testing. These investigators applied a method of testing skin in vitro (Dick, 1951) to testing skin in viva. Using a suction cup device to create a vacuum above the skin, they caused the skin to pull away from the subcutaneous tissue. The vacuum pressure was controlled and deformation (the height of the resulting dome of skin) was measured. The investigators assumed the tissue was isotropic and that the deformation field is homogeneous. By measuring the "dome" height and using equations for linear elastic membranes, they evaluated the skin and found an increase in elastic modulus with age. Because of the nonlinear response of the skin, the tangent modulus changed with deformation as well. The above method was modified by retracting the skin about the test site prior to testing in order to start from a stress free state (Alexander and Cook, 1977; Cook, et al., 1977). The investigators also printed a grid on the skin and monitored the deformation pattern and found it to be non-homogeneous. The diaphragm method has the advantage of testing the skin two-dimensionally. The influence of subcutaneous tissue on deformation was also investigated. 2.3.3 In Vitro In vitra tests, although unable to match the natural environment during testing, are often chosen because of the relative ease in preforming tests. Before complex models or test methods can be tested in viva, it is often necessary to 25 apply them to in vitro tests in order to establish feasibility of the models. It is also necessary to isolate the skin from the mechanical interaction of surrounding skin and subdermal tissue in order to evaluate the true material properties of the test sample. 2.3.3.1 ni x' 1 Most of the early tensile tests on skin were performed on strips of skin tissue (Elden, 1968; Rollhauser, 1950; Dirnhuber and Traeger, 1966). These strips were of non- standard dimensions which made comparison of material properties difficult. The skin strips were assumed to be continuous isotropic media" .Although. they exhibit non- linear material behavior, the modulus of the specimen is considered to be the tangent stiffness, the slope of the linear portion of the response curve, divided by the original specimen cross-sectional area. Inn this manner, a "Youngs" or elastic modulus is reported for skin. Glaser, et al. (1956) proposed a standardized method for testing uniaxial specimens. They also proposed. a standard for sample dimensions for uniaxial specimens which required the specimen width to be equal to its thickness. The standard required a special adjustable coupon stamp whose width had to be changed according to the thickness of each specimen tested. The standard was not widely accepted, but there has since been an almost universal use of dumbell-shaped specimens with dimensions equal or close to the ASTM 26 Standard D1708 coupon stamp (Haut, 1989; Vogel, 1976; Ridge and Wright, 1965). The modulus of rat skin is reported to range from 10 to 40 MPa (Vogel, 1976). 2.3.3.2 Biaxial Lanir' and. Fung (1974) investigated. the response of rabbit skin to biaxial deformations. They subjected the skin specimens to a series of tests that consisted of: 1. biaxial quasi-static stretching in which the rectangular specimen was subjected to a constant deformation rate (0.02, 0.2 and 2.0 mm/sec) in one direction, whiLe in the other direction the dimension was held constant, 2. uniaxial test as above, but where the other direction is traction free and allowed to retract. Their results indicated that biaxial specimens must be preconditioned biaxially and that maintaining the non- stretched dimension constant causes the stretched direction to generate a much higher load than the uniaxial specimen for a given deformation state. They also found that the considerable complexity of the experimental set up limits its practical use in determining the quantitative response of skin to biaxial deformations. 2.3.3.3 Membrane A simpler way of inducing a two-dimensional loading was reported by Dick (1951). He was the first to test skin in 27 vitra as a membrane. The experiment consisted of clamping a sample of Skin onto a ring which rested on a fluid-filled cylinder. The apparatus appeared much like that of a drum. The space below the skin was then pressurized, forcing the skin to deform outward. The amount of this deflection was measured as was the driving pressure. 2.3.3.4 Page Shear Snyder and Lee (1975) subjected frog skin to pure shear experiments in accordance with the experimental requirements outlined by Green and .Adkins (1960) for experimentally evaluating the properties of rubber. They also tested skin uniaxially but made no comments on comparing the response of the two methods or comparing the strain energy function constants . 2-4 W Mathematical models used to analyze the data obtained from the various experimental methods mentioned previously can be divided into three categories. There are the continuum. Ibased. models, phenomenological models and structural models. The continuum models normally idealize the skin as a homogeneous continuous isotropic medium subjected .to large deformations. The phenomenological models use power forms or exponential functions to describe the response. Like the continuum-based models, phenomenological models lack physical significance and are 28 only useful in curve fitting data. Structural models, however, describe the mechanical response of the skin in terms of the mechanical properties and geometrical organization of its microstructural components. These models offer the greatest hope in illuminating how the microstructure governs the mechanical behavior of skin. 2.4.1 mums There are only two methods reported for treating skin as a linear elastic material undergoing small deformations. The first method was mentioned earlier as being used to obtain an elastic modulus E from the uniaxial "strip" tensile tests on skin (Vogel, 1976; Ridge and Wright, 1965). Here the elastic modulus is given by E = o/e where o is the engineering stress defined by force per unit cross-sectional area (F/A) and the strain e is the change in length divided by original length (Al/lo). The elastic modulus is only determined in the linear portion of the response curve. The other method of analyzing skin using small deformation theory was presented by Grahame and Holt (1969). They model skin specimens in vivo as a membrane. They evaluated the equation for stress and strain proposed by Tregear (1966) at 500 different pressure inputs on a single piece of pig skin. ' Plotting the resulting stresses vs. strains they reported the modulus of the skin as the gradient of the stress-strain curve. Pressures were monitored such that strains did not exceed 8%. 29 Skin is normally strained in viva beyond values consistent with small deformation theory, and it is capable of undergoing large deformations up to 100-150% before failure. For this reason, most investigators proposed describing the mechancial response of skin and other biological tissues by means of a strain energy function consistent with the finite deformation theory of elasticity (Gou, 1970; Tang and Fung, 1976; Lanir and Fung, 1974; Soong and Huang, 1973; Demiray, 1972; Danielson, 1973; Alexander and Cook, 1977; Allaire, et al., 1977; Elden, 1968; DeHoff, 1981; Danielson and Natarajan, 1975) in which W is usually written as a function of the strain invariants W = W(I1,I2,I3). (2.1) The strain invariants are given by I1 = 112 + A22 + 132 I2 = AIZAZZ + A22A32 + 112A32 (2.2) I3 = 112A22132; and the stretch ratios are 11 = l/lo, A2 = w/wo, A3 = t/to; (2.3) where 1 is the deformed length and la is the original length and we and to are the original widths and thicknesses, respectively, as are w and t the deformed widths and thicknesses. 30 In the case of an incompressible material 11A2A3 = l, (2.4) which gives a convenient relation for the stretch ratios and simplifies experimental procedures, since one of the stretch ratios need not be measured as long as the other two are known. If the skin is modeled as a thin sheet undergoing pure homogeneous strain and has its major surfaces (not the edges) traction free, the stress in the specimen can be expressed (Green and Adkins, 1960)) as 2(112 - l/A12A22) (aW/aIl + A22 aW/aIz) T11 T22 = 2(122 - 1/112122) (BW/arl + 112 BW/BIZ) (2.5) T33 0 Tij = 0 (i E j): where, because of incompressibility, the strain invariants simplify to Iz = A12A22 + l/Alz + l/AZZ (2.6) I3 = 1. For the case of simple elongation, which is the case for which most uniaxial tests are modeled, T22 = T33 = O. From equation 2.5 it is shown that A1 = l/Azz T11 = 20-12 " 1/11) (OW/all + 1/11 aW/3I2) (2.7) 31 and 11 = 112 + 2/11, I2 = 211 + 1/112. (2.8) For this configuration only the axial load and elongation need be recorded in order to evaluate the unknown constants in the strain energy function. Another simple means of experimentally evaluating a specimen is to subject it to pure shear. This is approximated by testing a specimen that has a large width to length ratio such that the change in width is negligible (Figure 2.10). Figure 2.10 Pure shear (from Treloar) This means that k2 = 1. For this case, equation 2.5 becomes T11 = 2(112 - 1/112) (BW/dll + BW/BIZ) (2.9) Again, the strain energy function constants can be evaluated by knowing the axial force and deformation. 32 A summary of the strain energy equations used to evaluate skin is presented in Table 2.1. All the investigations assume that the tissue undergoes homogeneous deformation and ignore the structurally significant unit of skin, namely collagen. All of the investigators, except Tang and Fung, ignore the material’s well-documented anisotropy. When evaluating the strain energy function with respect to experimental data, all the investigators except Alexander and. Cook (1977) and Lanir and Fung (1974) start their experiments from a non-stress-free state, thus making comparisons between tests on the same sample, not to mention different samples from different sources, impossible. Inspection of the several strain energy forms presented in Table 2.1 make it clear that there is no unique description of skin. Nor is there any physical significance to the constants in the proposed strain energy functions. It is the lack of physically significant material parameters in the strain energy functions which has kept the finite deformation formulations from offering any insight into the microstructural organization of the skin. Also, the often analytically complex strain energy forms have made it difficult to evaluate the function experimentally. 2.4.2 Phenomenologieal In lieu of the mathematically complex continuum approach, other investigators have turned to a simple phenomenological description of the non-linear mechanical lnxsatlsatgr Alexander 5 Cook (1976) Allaire, et al. (1977) Danielson (1973) Dehoff a Key (1981) Demiray (1972) Con (1910) Snyder 6 Lee (1975) Tang 6 Fung (1976) Veranda s Westmann (1970) 133 Table 2.1 Continuum Based Models u - clitiz-ixxzxpztcxptx(11-3)21 K. C - constants W I C1(Il-3) * C2(Iz-3l c1,c2 - constants "an - Adam exp (3"51"; 4» WP?) A0919. ek3 - tensors of elastic constants w - 9.24 x 10-5 .xp(4.4111-3)-1t -2.03 x 10"(12-3t N - fl/lez-3l B - constant w - C(exp(k11(112-312ll-ll k, c - constants w - (k/azl exp(a(B-l)l -(k/alfl p - (116(113-312t1’2t/12 k, a- constants Po" ' 91°11 t “2'22 + 264 011 .22 * Cexptalell + 32°22 + a3e12 t 2a4011922 * 11°11 + 12922 + 14911222 * tseiiezzl a1..., 01.... 11... and c are constants. e11 - strain Po - material density W-Cllexp(3(11-3ll-ll +cztlz-3t + 9(13) C1, C; an - constants Membrane uniaxial lncompresslble Membrane Incompresslble used in finite element analysis Incompresslble uniaxial cylindrical bar not fit to data incompressible Incompres. pure shear. simple extension Orthotroplc Uniaxial, cat skin compressslble 34 response of skin. These models are usually exponential functions or power forms. For stress up to 15 g/mm2, Ridge and Wright (1965) found that skin can be described by E = a + b log L where a and b are constants and E and L are extension and load, respectively. For values greater than 15 g/mm2 up to about 1000 g/mm2 the response curve is better described by E = c + kLb where E, L and b are the same as above and c and k are two more constants. They showed the constants b and k to change with age. The first constant was shown related to the properties of the collagen, while k was suggested to be related to the meshwork of collagen, although no specific correlations were given. Kenedi, et al. (1965) used the relation 0 = A8n where A and n are constants and O'and S are the nominal stress and strain of the specimen, respectively. Like Ridge and Wright, they found changes in the model constants A and n occurring with age. Glaser, et al. (1965) used the same relation to compare wounded and unwounded guinea pig skin for strains less than 30% after which a Hooke’s (linear) law takes over (Figure 2.11). Although the models were able to track changes in the mechanical response of skin as reflected in the model constants, these models are really only exercises in curve fitting and offer little insight into what is happening microstructurally. 35 2400 l 1 l l ALMOST LINEAR Ponnou/ 2000 - .. .. m, M_EAN ugxmuu- 2L __ '; STRESS 4' .3 / new I 8, TYPICAL ,’ 3’, srarss-srnAm U 9‘ I200 — 0‘; o’ - Mr) N :3 l g / \ r- ,’ rowan LAW 400 L. APPROXIMATION ’1 or - 20.000 8-" FOR 05 r s 0.3 o l l l 0 0.: 0.2 0.3 0.4 0.5 TENSILE STRAIN. I (IN MN) Figure 2.11 Comparison of experimental response and power law fit (from Glaser, et al.) 2.4-3 Structural The structural model has several important advantages over either the continuum or the phenomonological model. Most important is that the parameters of the model all have physical meaning, whether describing the stiffness of the fibers, the content, crimp pattern or spatial distribution. Structural models are in contrast to continuum models in which the response of the tissue is described in terms of same strain energy function whose constants lack physical meaning, and in contrast to phenomenological models in which a mathematical function is selected to curve fit data from particular experiments. Because structural model parameters are in terms of the physical microstructure, they are able 36 to give some insight into why the whole structure responds the way it does to external mechanical stimuli. Since the component of skin thought to be responsible for its structural integrity is collagen, structural models for skin find their roots in structural models for collagen fibers. The biological tissue most often used to determine the characteristics of collagen is the rat tail tendon, as it is composed of almost pure collagen fibrils, which are aligned. Diamant, et a1. (1972) found that while the fibrils in this tendon are aligned, they have some inherent waviness. They hypothesized that the waviness might explain the low modulus response in the toe region of a stress- strain curve of tendons. They documented the decrease in the crimp angle of the periodic crimp or wave structure with age. To explain the changes in the mechanical response of tendon with respect to age, they developed a model of a collagen fiber as a series of planar zig-zag fibrils (Figure 2.12). Their model predicted the characteristic non-linear response of the fiber for older specimens but did not model young specimens well (Figure 2.13). Comninou and Yannas (1976) varied. the zig-zag' beam approach to a beam sinusidal in geometry (Figure 2.14). They did not experimentally verify their model, but did report some predicted responses and compared them to experimenally obtained data from the literature. While in general agreement with the experimental plots, the toe 37 Figure 2.12 Collagen fiber model of Diamant, et al. 2wk 10' x stress, o/dyno cm" % strain. ex 100 Figure 2.13 Predicted response of zig-zag model by Diamant, et al. Figure 2.14 region response was high 38 \Z, I I \/ I 90 - 60‘4" 20"( I l H——-—— 2b! =1] Sinusoidal collagen model by Comninou and Yannas (Figure 2.15). This was worsened when they considered the fiber to be bonded to the ground substance. Figure 2 . 15 A T U - ° t-attnto‘ £ - O O I c. N ‘ U . ~ (I O I ~ ‘ Reduced Stress. ‘ _ ~ )- l 25 5 7.5 Amrent Strom s' in IO'Z Predicted response of model by Comninou and Yannas. - 39 Unsatisfied that the low modulus response in the toe region of the rat tail tendon should be caused by a bending resistance in the fibrils, Kastelic, et al. (1980) proposed that the IOW' modulus response was due to a sequential uncrimping of the individual fibrils (Figure 2.16). Figure 2.16 Sequential straightening and loading model by Kastelic, et al. They noticed that the fibril crimp angle increased from 0° at the center of the fiber radially to a maximum crimp angle on the exterior. They proposed a crimp angle distribution function which is a function of the fiber radius and adjusted the parameters of the distribution function to fit experimental data from 3, 6 and 24 month rat tail tendon data (Figure 2.17). The results of their study indicate that the toe region response for the tendon could be described by a sequential recruitment of fibrils. 40 7 1 r f I 6 ' d 5 - , 3 month RTT 7 O - l8' ‘ "’ cl 0/E (:03) 3 - .. 2 - .. o I 0.825 , _ b . 0.25 - doto envelope 0 l l L 2 2.5 3 3.5 4 Elongation (96) Figure 2.17 Fit of model to experimental data by Kastelic, ' et a1. Unconvinced that collagen fibers in skin can posess some bending stiffness, Markenscoff and Yannas (1979) developed a model which consists of unit length fibers pinned together at their ends (Figure 2.18). Figure 2.18 Pinned sequence of unit collagen fibers by Markenscoff and Yannas. 41 The spatial angle (6) of each fiber unit is described by a random distribution ranging from 0 to 180° Their results predicted the onset of the linear region of the stress strain curve for skin to occur at about 57%, which is in general agreement with previous reports in the literature. Although the model successfully predicted the onset of the linear region, it could not model the characteristic shape of the curve. Recognizing that the nonlinear response in rat tail tendon was caused by the fibril crimping and that this was analogous to the nonlinear response of skin being caused by fiber undulations, Manschot (1985) used the sinusoidal beam model to analyze skin as if it were composed of a single collagen fiber. He fit the model to data obtained from in viva uniaxial tensile tests of normal and pathological human calf (leg) skin. This three parameter model - the parameters are for the fiber stiffness, undulation period and diameter - was shown to fit the experimental data well. Greater stiffness is reported transversely versus longitudinally, and fiber stiffness in males is shown to be greater than for females. Realizing that unlike in tendon, collagen fibers in skin are not aligned. but are arranged spatially, Lanir (1979) proposed a model for flat biological tissues incorporating a spatial arrangement of fibers. He assumed a collagen-elastin fiber unit considered to behave in one of two ways. First that the collagen fiber is wavy and this 42 waviness is caused by elastin cross linked to the ends of each collagen undulation (Figure 2.19). COLLAGEN(K.) L: ,._/\ /.._. fl... / \f“ ELASTIN (K.) Figure 2.19 Collagen-elastin fiber combination by Lanir The ratio of length between the undulated fiber unit and the distance between its ends (presumably the length of the elastin fiber) is given by the stretch ratio (743) under which the collagen fiber becomes straight. The distribution of 7‘3 along a fiber is governed by a normal distribution function P(ls). The fibers are further arranged spatially with respect to a set of principle axes (along and across Langer’s lines) according to another distribution function. This is the high density cross link (HDCL) model (Figure 2.20A). His other model, the low density cross link (LDCL) model (Figure 2.208) is similar to the previous model except the cross links between elastin and collagen are fewer and the undulation in the collagen fiber is not caused. by elastin but to some inherent crimp in the fiber. Despite the inherent crimp, the fiber’s resistence to straightening is assumed 'negligible. For the LDCL model a separate 43 '.—l t ' I \ t ‘ 0 I ‘ Figure 2.20 Schematic planar ‘views of 'the collagen and elastin network structure in flat tissues: (A) Tissues with high density of cross-links (HDCL) and elastin induced collagen undulation. (B) Tissues with low density of cross-links (LDCL) and inherent collagen undulation. Thick lines - collagen; thin lines - elastin; broken lines (in B) - overall collagen fiber direction (from Lanir). angular distribution function for elastin and collagen, Re(0) and RC(0) respectively, is necessary, but‘ the undulations in the collagen fiber are still described by Pas) where 2.3 is the stretch ratio at which the collagen fiber unit. becomes straight. For both the HDCL and LDCL cases, the fibers are assumed to be linear elastic. A strain energy function is developed for each fiber system and incorporated into finite deformation theory for thin sheets. Lanir presents some theoretical predictions which resemble experimental data previously reported. He did not fit the model to experimental data and, on account of the 44 complexity of the model, it is not practical to fit the model to experimental data. Further, from a parameter estimation point of view, because of the parallel arrangement of collagen and elastin fibers, it is not possible to determine their stiffnesses independently. Shortly after Lanir proposed his model, Decraemer, et al. (1980a) reported another model for soft biological tissues which uses a similar sequential recruitment of fibers but which is considerably simpler as it has only three model parameters. These parameters are b, a fiber stiffness term; u, the mean of the fiber slack lengths; and o, the standard deviation of the mean. In their model the undeformed. shape of’ the fibers is insignificant (Figure 2.21). A “- \N \N\‘\ ‘.‘\ ‘-’\. ‘mNAM\A.‘./~‘v\“ A Figure 2.21 Schematic of model by Decraemer, et al. The only concern is the deformation state at which the fibers become straightened. Like Lanir (1979) and Kastelic (1980), it is the sequential straightening of the fibers which gives the material its characteristic non-linear 45 that all the fibers are aligned in the direction of applied deformation and they do not include the effects of elastin or the matrix. They also did not verify the model with experiments but did show its feasibility by fitting some sample curves from the literature (Figure 2.22). Load “ (g) 0.00% lenotnle. ascending branch e. Jet 10" a. u 1.395 ‘6’!!! a .. s ; 1.3“10'3m 038.0800“ l 1 o l L l l LO 1.4 to 20 (in ends at n 2 we cm. Figure 2.22 Sample fit of model by Decraemer, et al. to human fascia III. MATERIALS & METHODS 3.1 The Model 3.1.1 Assumptions The following assumptions were made in developing the model; they are based on previous studies reported in the literature and observations made during pilot studies. a) The structural component responsible for the mechanical response of skin is assumed to be collagen. This idealization is justified by the overwhelming presence of collagen which is 40-80% of the dry weight of skin (Bottoms and Shuster, 1963; Dombi and Haut, 1985) in comparison to elastin, 2-4% of the dry weight (Hult and Goltz, 1965), and collagen's dominant material stiffness. Collagen is reported to have a modulus of 600 MPa - 2 GPa (Haut, 1983; Lanir, 1979), whereas elastin is about 600 kPa - 3MPa (Jenkins and Little, 1974; Hass, 1942). b) Collagen in tension is assumed to be linear elastic. c) Collagen fibers are perfectly flexible, having no bending stiffness and. will Ibuckle ‘under any compressive loads. d) According to scanning electron micrographs of skin, collagen fibers are wavy or crimped in the relaxed 46 47 (stress-free) state. These fibers are assumed to be unable to resist elongation until they become straightened. e) Fiber-fiber interactions are neglected. This assumption was made to simplify the development of the model, however, there is some evidence to indicate that there are cross-links between fibers (Pierard and Lapiere, 1987). Yet these interactions may be electrostatic and not offer much mechanical effect at low strain rate. f) The fibers are arranged essentially in a planar fashion with few fibers passing from plane to plane. The fibers which course from one plane to another are neglected for modelling simplicity. g) The spatial arrangement of fibers in the planes is assumed to be described by an elliptical distribution. The elliptical distribution is assumed constant through the thickness of the dermis. The major axis of the distribution ellipse coincides with the direction of maximum skin stiffness. h) Mechanical contribution of the epidermis is neglected (Schellander and Headington, 1974). . i) Since the tensile tests used to evaluate the method are run at a slow strain rate (1.5%/sec.), all viscous effects are neglected. j) Matrix interaction with the fibers is reportedly responsible for the viscous response (Minns, et al., 1973; Tregear and Dirnhuber, 1965; Egan, 1987). Since no viscous 48 effects are considered, neither are the matrix/fiber interactions. 3-1-2 Alighsd_Eiber_luniaxiall_Mgdel As mentioned previously, the model used was based heavily on those proposed by Lanir (1979) and Decraemer et al. (1980). Like their models, this model assumes that all fibers are recruited sequentially as their slack is removed and they become straightened. The slack length of the fibers is described by a normal Gaussian distribution. For the uniaxial specimen the fibers are assumed to be aligned in the direction of elongation. This assumption is, in part, justified by the length to width ratio of the uniaxial specimen. The specimen was narrow enough that the fibers running from grip to grip formed an angle with the long axis of the specimen of less than 10°. For the wider specimens, a distribution function describes their spatial orientation (assumption g). Unlike Lanir’s model, the mechanical contribution of elastin is not included. The exclusion of elastin is due to assumption (a) and from. a ;parameter estimation perspective, i.e., the stiffness parameters for collagen. and. elastin cannot. be determined. independently. This simplified form of Lanir’s model is essentially the aligned fiber model proposed by Decraemer, et al., with an important exception to be discussed shortly. Lastly, the stretch notation was dropped and engineering strain terms were used instead. 49 The fibers are assumed to be linear elastic and organized in a wavy fashion in the relaxed state. When a uniaxial specimen, cut transverse to the body axis, is stretched, fibers become straightened according to a recruitment function R(x), which is a normal Gaussian distribution. R(xp = exp [-(xi-HT)2/26T2]/(OTV§_1?) (3.1) The distribution is centered about a mean, uT, which is the deformation state for which half the fibers will have been recruited for a transverse-oriented specimen. The distribution has a standard deviation, CT. The parameter, OT, determines how quickly the fibers are recruited with respect to the mean, “T. A similar distribution exists for the longitudinal direction. H At. AT’QXZ 0' o\ o‘ Figure 3.1 Schematic of sequential fiber recruitment 50 Starting from a stress-free state, the specimen is elongated some increment Axl (Figure 3.1) along its long axis. A few fibers become straight. Once straight, they are able to resist stretch thus creating a resistive force. The percentage of the number of fibers (N) in a given thickness recruited due to a deformation increment Axl, ending at x1, is approximated by R(x1)Ax1. The force generated is the product of the percentage of fibers per thickness recruited, the strain in the fibers and the "effective" fiber stiffness, k’T, for a specimen of initial thickness hO and the number of fibers per unit thickness N. le = k'tIth'N-[Axl/(lo + X1)]-R(X1)AX1 (3.2) The strain in the fibers is Axl/(lo+x1), 10 is the gauge length of the specimen and lo + Xi is the original length of the fibers as they are recruited. Since the total number of fibers N is unknown, k’T-N is replaced with kT which is the stiffness of all the fibers, that is k’T-N=kT. Let the specimen be deformed some Ax2 further. The percentage of fibers newly recruited due to Ax2 at the deformation state x2 is R(x2)Ax2. The strain in the newly recruited fibers is Ax2/(10+x2). The force generated is now fT2 = kT°ho-[AX1/(lo + x1)]-R(x1)Ax1 + kT°ho'[AX2/(lo + x1)]-R(x1)Ax1 (3.3) + kT'hO°[AX2/ (10 + XZ) ]-R(X2)AX2 _ 51 The fibers already recruited generate a load le caused by Axl which is the first term in equation 3.3. These fibers are also stretched some sz further, thus generating the second term in the above equation. The third term is the force generated by the newly recruited fibers. This is the point where this proposed model departs from the models of Lanir and Decraemer, et al. Their models show the number of fibers recruited and stretched initially as being the same as the first term in equation (3.3) above. When the specimen is elongated further, new fibers are recruited and stretched yielding the third term. Unlike this model, the fibers which were previously recruited and stretched some Axl are not stretched the sz further. This further stretching of the previously recruited fibers is the second term in the above equation. Physically, it seems reasonable to assume that the previously recruited and stretched fibers will be stretched further when the specimen is also stretched further. The force generated at any deformation state Xn resulting from a deformation increment AXn from a previous state of deformation Xn-l (where n = 2,3, ... n) is fTi = kT-ho-[Axl/ (lo + x1)]-R(x1)Ax1 + kT'ho°[AX2/ (.10 4* X1) ]-R(X1)AX1 (3.4) + kT'ho'[Ax2/(lo + x2)]-R(x2)Ax2 + kT-hO-[Axn/(lo + xi)]-R(xi)Axi If the above expression is placed in summation form 52 n i fTi = kT-ho- XAxiZ [R(Xj)/(lo + XjHAXj (3.5) = :1 Admittedly, the initial length of the fibers recruited at xj is not actually 15 + xj. The initial lengths are off by the initial increment x1 = Axl. This amount, which is the deformation increment recorded experimentally, is small (llum) compared to the typical gauge length (27 mm) of the skin specimen. Therefore, some accuracy has been sacrificed for simplicity. Assuming there are a significantly large quantity of fibers and that the deformation increments are infinitesimal, equation (3.5) can be expressed in integral form as x n fT =kT°hof [ [R(C)/(lo + C) 1d; d“ (3.6a) X1 X1 Similarly for the longitudinal direction X n fL =kL-hof f [R(C)/(lo + CHdt; dn (3.6b) Xl x1 3.2 Experimental Meghegs To verify the model, force-deformation data were obtained from dorsal skin of 64 male Fischer 344 rats. A minimum of 6 samples (one per rat) per age group (1.0, 1.5, 53 2.0, 3.0, 4.0 months) per direction (lateral and longitudinal) to the spine were used. The rats were euthanized by a lethal dose of inhalant anesthetic Halothane*‘ (AVMA, 1986). Immediately post-mortem, the animal’s back was clipped and shaven clean. A 38.1 mm x 38.1 mm grid of 2.54 mm squares was stamped on the shaven area, cranial to the base of the rib cage, caudal to the neck, using a specially made rubber stamp with normal stamp pad ink. The resulting grid marked a segment of skin at normal in vivo tension. The gridded segment was excised and with the grid pattern facing up, was placed in a pool of 0.1M phosphate buffer on a cutting board. It is customary for the skin to retract when removed from its natural environment on account of pre-tensions, in vivo. In order to return the specimen to in viva dimensions, the skin was tacked down in one corner of the grid with a dissection pin and stretched such that one side of the grid was returned to its original 38.1 mm dimension. That side was pinned down. The opposite corners were then pulled to 38.1 mm and pinned, returning the marked skin sample to its original 38.1 mm square dimensions. An ASTM standard D1708 test coupon stamp was used to stamp out a "dumb-bell" shaped test specimen (Figure 3.2). Specimens were stamped out longitudinally *Fluothane, Ayerst Laboratories, Inc., New York, NY 10017 54 3.0‘ 7.9 I.- ‘C—W] .1 _J 15.9 ....)1 fl Figure 3.2 Uniaxial specimen stamp dimensions (along the spine) or laterally (transverse to the spine) depending on the desired orientation. The stamped out specimen was removed from the grid of skin and the paniculus carnosus, adipose tissue and fascia were cut away. Care was taken not to further stretch or damage the specimen. Thickness was measured using an electrical contact-sensing micrometer. The specimen was then placed in pneumatically activated grips at a clamping force of 81 N (Figure 3.3). Quasistatic tensile tests were performed on all samples using a servohydraulic testing machine (Figure 3.4). The tests were stroke controlled yielding a grip-to-grip strain rate of l.5%/sec. Load and deformation data were recorded every 50 msec. until the specimen failed or reached a load of 50 N. Gauge length was determined to be the grip-to—grip distance Figure 3.3 Specimen in pneumatic grips Figure 3.4 Tensile test of skin specimen 56 when the specimen generated a load of 0.1 N. The specimens were tested in nonphysiologic conditions, namely at room temperature with no bath. Testing in a bath would have complicated the test protocol. However, great care was taken to ensure that the specimens did not dry out as they were bathed in phosphate buffered saline during preparation and tested within 10 minutes post mortem. 3.3 W To determine the parameters for the aligned fiber model, the merit function x2, x2 =i§1(1/Si)2'[fiexp - fimodel(xi;aj)]2 (3.7) was minimized in a least squares sense using Marquardts method as outlined by Press, et al. (1988). Here fiexp is the experimentally measured force at deformation xi. The variable fimodel is the model value predicted force and aj (j=l,2,3) are the parameters k, p, and a, respectively. Model parameters were obtained for both directions and all 5 age groups. The uncertainty of the experimental force, Sir was assumed uniform and equal to .1 1L. .Although data was recorded at one data point every 50 msec, only every third data point (every 150 msec) was used during analysis to speed the minimization procedure. Minimization was suspended when x2 decreased by less than 0.01. Convergence to a global minimum was determined 57 subjectively by inputting several initial parameter guesses and comparing the experimental curve with the model curve and objectively by monitoring the x2 residuals. To ensure that the parameters can be determined uniquely, the ratio of the sensitivity coefficients Rm=(aj/ak)'(afi/aaj)'(aak/afi) (3,8) (ml jlk=112l 31°j¢k) were plotted versus deformation to check linearly independence of the parameters (Beck and Arnold, 1977). 3-4 W A spatial distribution function was added to the model in order to account for the anisotropic behavior of the skin which is reportedly caused by a preferred fiber alignment (Ridge and Wright, 1965). Most investigators agree that it is unlikely that the difference in stiffnesses with respect to specimen orientation is caused by stiffer fibers in that direction, but rather is a result of more fibers being aligned in that direction. It also seems physically unlikely that the spatial arrangement of fibers would be described by a discontinuous function. For these reasons, the spatial distribution of fibers in a given thickness of skin was assumed to be elliptical, where the major and minor axes coincide with the direction of maximum and minimum skin stiffness and coincide with Langer lines (Figure 3.5). 58 Figure 3.5 Elliptical diStribution function The percentage of total fibers at a given angle 8 from the xT axis is [ .321)2 ] 1/2 r(9) a ------------------- (3.9) azsinze + bzcosze The percentage of total fibers in an incremental area dA about 8 is dA = (r(9)2/2) d8 (3.10) Assuming d8 is sufficiently small, the incremental areas are integrated from 0 to 21: to yield 100 percent of the fibers in a given thickness. 100% = 1 = A = nab (3.11) This gives a relation for a and b. 59 a = l/fib (3.12) The contribution along the XT axis of fibers oriented at some angle 8 is the percentage of fibers in an incremental slice about 6 times its xT components of those fibers, as shown below. (3.13a) PT(8)d6==cos 9 [a2b2/2(b2cos2 6 + azsin2 8)] d6 Similarly, for the XL direction, (3.13b) PL(9)d9 = sin 8 [a2b2/2(b2cos2 8 + azsin2 0)] dB The force generated in ‘the xT direction by fibers oriented at 8 forga 5.6 $,§2 is given by :2 X 11 (3.14a) fT = K’-hoN f P1461 f [R(C)/(lo + C) JdCdOd'n §1 *X1 x1 Similarly, for the XL direction (3.14b) §2 X n fL =K’-hoN f PL(e)f I [Rm/(1O + §)]d§dedn E1 x1 x1 Where K’ is the fiber stiffness per unit thickness and hO is the original specimen thickness. Again let K = K’N where K is the stiffness value of all the fibers combined. Since the distribution function is symmetric about the xT axis, equation (3.13a) can be rewritten as 60 g x ’n (3.15) fT =2-k—hof PT (8) f f [Ran/(lo + C) 1dCd8dn 0 x1 x1 It is assumed that only the fibers which run from clamp to clamp are actively resisting elongation. With respect to the physical dimensions of the specimen shown in Figure 3.2 gl = arctan [we/10] (3.16) The uniaxial approximation of equation (3.15) is compared with equation (3.6a). The effective stiffnesses kT and kL obtained by curve fitting equation (3.6a) and (3.6b) to the transverse and longitudinal data, respectively, are really a combination of the spatial distribution function and the fiber stiffnessICin equation (3.15). Since the "effective" stiffnesses of a specimen are the product of the distribution function and fiber stiffness K, and as such are linear dependent, the fiber modulus and distribution radii, a and b, could not be solved for directly. Therefore, even if the model had been originally developed for the general case of spatially distributed fibers, the uniaxial approximation of equation (3.15) would still be a necessary intermediate step in determining K, a and b. Recall that the fibers in the uniaxial specimens were assumed aligned in the direction of deformation and the force in the xT direction was modeled as equation (3.6a). 61 If equation (3.15) is adjusted to reflect the uniaxial case, the bounds of integration with respect to 6 are 0 g 8 _<_ E, where E = arctan [wo/lo]. This assumes that the fibers remain at their original orientation and do not align during deformation. When the force responses of equation (3.6a) and equation (3.15) are set equal to each other, then x n who] I [R (C) / (10+C) JdCdn x1 x1 é x n = zx-hof PT(e)def I [R<§)/(1o+§) JdCdn 0 X1 X1 The bracketed terms are clearly equal to each other, as is the thickness ho. Thus the above equation reduces to i kT. =2K-J PT (9) d6 (3.17) 0 Similarly, for the uniaxial response in the XL direction §+n/2 kL~ =2x- PL(6)d9 (3.18) fi/Z 62 Yielding two equations to solve for the two unknowns K and a. Parameter a was determined by solving equations (3.17) and (3.18) for Kand setting them equal to each other to yield 5 §+u/2 kT/f PT(9)d9 = kL/j‘ PL(9)d9 0 n/2 rearranging terms to get § §+n/2 kL' PT (9) d9 - kT' PL (9) d9 0 ll 0 n/Z and solving for a using the bisection method outlined by Press, et al. (1988). Once a is determined, it is substituted back into equations (3.17) and (3.18) to find fiber stiffness parameter K. 3.5 Wi imnriin As a further check on the model, specimens 5.5 times wider (Figure 3.6) than the uniaxial specimens, but of the same gauge length, were tested according to the protocol outlined in section 3.3. A minimum of 6 specimens per age per direction were tested. The experimental response was compared with the response predicted by the model. For the wide specimens the bounds of integration on the distribution function increased to reflect the increase in width. Here é 30.1 Figure 3.6 Wide specimen dimensions = arctan [Wo/lo] where wo is now 5.5 times the uniaxial specimen width. Using the fiber stiffness parameter K, the recruitment parameters [.1 and c and the distribution parameter a and b obtained from the average uniaxial response curves from both directions and each age group, the response of the wide specimens was predicted for their respective age groups and directions. These predicted responses were compared with the wide specimen experimental response data. The predicted response was also compared to the uniaxial response magnified 5.5 times, which is the predicted response from a continuum analysis. 3.6 W In the previous sections a method was described to determine the stiffness K of the collagen in skin. Recall, 64 however, that K was really the stiffness of all the collagen, that is K = K'-N where K’ is the stiffness of a fiber and N is the total number of fibers. Morphometric techniques are not yet available to determine the total number of fibers in a skin sample, therefore, the total number of fibers was unknown. The total quantity of collagen in a sample, however, can be determined. For this reason the stiffness was normalized on a per mg of collagen basis rather than on a per fiber basis. The normalized stiffness K where K = K/m where K is the stiffness for all the collagen and m was the total mass of the collagen. The collagen content in the rat skin was determined through a process of extractions and enzyme digestion. Three aliquots of tissue from three specimens at each age were placed in phosphate buffered saline and weighed. For each aliquot, two soluble fractions and one insoluble fraction collagen was determined by the method of Schnider and Kohn (1981). A separate assay was conducted to determine the total collagen content (Stegemann, 1958) in each tissue, and served as a check for losses during sequential extraction. For each assay approximately 15-20 mg. of skin was frozen in liquid nitrogen, mechanically smashed, lyophilized overnight, weighed, and placed into small test tubes. The lipids were removed by agitating the samples overnight in chloroform-methanol (2:1 V:V) at 4°C. 65 The following day the samples were centrifuged at 2000 rpm for 1.5 hours. The supernatant was removed and discarded. The above procedure was repeated two additional times for a total of 72 hours of extraction. After removal of the supernatant on the third day, the sample was dried with nitrogen gas. The total collagen samples were immediately hydrolyzed and neutralized by the procedure to follow. Once dried the aliquots used for solubility tests were suspended in two ml. of 0.5 M acetic acid, agitated overnight at 4°C, and centrifuged for 1.5 hours on the following day. The supernatant was removed and placed in a clean, dry test tube. The above procedure was repeated for two additional nights. The supernatants were collected and pooled. After removal of the supernatant on the third day the pellet was suspended in 2 ml. of acetic acid and digested in 2 ml. of pepsin enzyme at a concentration of 1 mg/ml for 18 hours at 4%:. After digestion a neutral salt solution of 1.0 M NaCl in 0.05 M Tris-HCl buffer (1:1 V:V) ‘with a pH of 7.4 was added to the aliquot. Enough salt solution was added to completely neutralize (pH 7.0) each sample. The samples were centrifuged for one hour at 4000 rpm. The supernatant was removed and transferred to a clean, dry test tube. The pellet was rewashed twice with the salt solution, centrifuged for 20 minutes, and the supernatants were combined. This was the pepsin soluble fraction. The remaining sample was the insoluble fraction. 66 The pepsin soluble fraction was dialyzed in 0.1 M acetic acid overnight using cellulose membranes (VWR Scientific, m.w. cutoff 12,000-14,000) to remove excess salt. All fractions were frozen and lyophilized overnight and then hydrolyzed. Two ml. of 6 M HCl was added to the test tube to dissolve the precipitate. The sample was incubated overnight at 105°C in a sealed test tube. The following day the sample was neutralized by the following procedure: 2-3 drops of methyl-red indicator was added to the hydrolyzed sample. 2400 ul. of 2.5 M NaOH was then added and vortexed. Additional NaOH was added until a pH of 7.0 was reached. The acid- and pepsin-soluble, and total collagen fractionS' were diluted to 10 ml. The insoluble fraction was diluted to 25 ml. An assay for collagen was conducted on all the samples. A set of ten standards (1-10 pg/ml) was prepared from a stock of 1.0 mg hydroxyproline/IOO ml. Two water samples were used for zero standards. The assay was conducted using 1 ml of sample. Three solutions were used during the assay preparation including: 3 M HClO4, 0.03 M chloramine-T solution (0.845 g Chloramine- T in 20 ml H20, 30 ml propanol, 50 ml citrate-acetate buffer, pH 6.0), and 5% solution of p—dimethylamino- benzaldehyde (p-Dab) in propanol (2.0 9/40 ml) dissolved under heat. One ml. of the Chloramine-T solution was added to each test tube, vortexed, and left to settle for 20 67 minutes. One ml. of 3 M HClO4 was then added to each test tube, vortexed, and left to settle for five minutes. One ml. of p-dab solution was then added, vortexed, and incubated at 60W3. for 18 minutes (Stegemann, 1958). Once the test tubes had cooled to room temperature, the absorbances of the standards and samples were measured on a Beckman Model DG-spectrophotometer. The water samples were prepared with reagents as above and served as reference. The absorbances of the samples were compared with the standard curve for the calculation of hydroxyproline per dry weight of tissue. 3.7 W To investigate if the changes in microstructure predicted by the model could be seen microscopically, transversely oriented skin were taken from three 1.0 and 4.0 month old rats and one 9 month old rat for electron microscopy. With the exception of the 9 month old rat, one sample from each age was allowed to relax in phosphate buffer for several minutes and was then clamped to a tongue depressor and submerged in 10% formalin. for 36 hours. Another specimen 'was excised and clamped at one end to a tongue depressor. The free end was pulled, stretching the specimen an amount typically sufficient to load it into the "linear" third region and was clamped to the tongue depressor. The specimen was submerged in formalin for 36 hours. The third specimen from each age group was placed in 68 the pneumatic grips and elongated at l.5%/s grip to grip strain rate. Unlike the regular tensile tests reported previously, this test was performed in a bath fixture, which was empty. The specimen was elongated until the load response showed the onset of the "heel" region. At that time the test program was suspended and the bath was filled with formalin. The specimen was allowed to fix for four hours. The specimen was then removed carefully, placed on a tongue depressor and both ends of the specimen were clamped down. It was submerged in formalin for another 32 hours. The 9 month old skin sample was prepared in the relaxed state like the previously mentioned relaxed samples. All samples were removed from the formalin and the paniculus carnosus and adipose tissue were removed with a scalpel until the midreticular dermis was exposed. The specimens were removed from the clamps, a piece of the midzone of the specimen suitable to be placed on a SEM stub was taken. The specimens were subjected to hyalaronidase (100-200 I.U./ml phosphate buffer at pH 6.5 at 20°C for 24-36 hours) to digest the ground substance. The digested fragments were rinsed several times in double distilled water and dehydrated using the critical point method. They were then glued to SEM stubs with silver colloidal paint, gold sputter coated and examined in a JEOL T-330 Scanning Electron Microscope operated at 15 kV. The micrographs of the samples were then studied for differences in fiber size, composition and organization with 69 respect to age and for differences in organization with respect to the three deformation states. IV. RESULTS 4.1 AW 4.1.1 ModeLEit The average experimental response of specimens from the five age groups for the transverse and longitudinal directions are shown in Figure 4.1 and 4.2, respectively. The curves for the transverse direction show a higher stiffness in the linear region compared with the longitudinal direction for each age group. There is also an apparent stiffening which occurs throughout maturation as well as a shift of the "heel" region toward the origin. 4.1.2 M l P m rs The model parameters describing the uniaxial responses and their averages and standard deviations per age group per direction are presented in Table 4.1. The average parameter values are presented in Figures 4.3, 4.4, 4.5. The average effective stiffness, kT, for the transverse direction is shown to increase during maturation, whereas the average longitudinal effective stiffness, kL, decreases. Both directions show a jump increase in effective stiffness from 1 to 1.5 months (Figure 4.3). The transverse stiffness is greater than longitudinal stiffness, except for the young 70 LOAD (N) 13 10 71 {#40/ a f I'_ I I I r I I I f ‘ c a 10 12 14 DEFORHATION (mm) Age average experimental response of transverse uniaxial skin samples Figure 4.1 30 2. 20 24 22 .. 20 5 u g to 3 :4 12 10 I O 4 2 0 Figure 4.2 Age Cum 0 x: we // O/D 0 ‘ D/D1mo .4/ D/ / / / . I Ifi I I I I I I I T T 2 4 0 average 8 IO 12 14 DEFORUATION (mm! experimental response longitudinal uniaxial skin samples Of 1.0 month 1.5 months Longitudinal Transverse Transverse Longitudinal ID SK18 SK38 SK41 SK51 SK52 SK53 SK134 Average S.D. SK17 SK19 SK39 SK40 SK49 SK50 Average S.D. SK42 SK44 SK46 SK48 SK54 SK55 Average S.D. SK43 SK45 SK47 SK56 SK57 SK58 Average S.D. k (N/mm%) 100 144 233 213 148 204 182 176 i 47 230 205 247 i117 191 313 111 243 269 1% 248 :t 89 72 Table 4.1 )1 (mm 0 (mm) 6.22 5.17 12.23 5.80 11.61 6.00 9.68 5.20 7.02 5.48 8.83 5.94 8.118. 3.3.7 9.10 5.29 $2.24 10.87 6.22 24.56 4.40 25.26 9.70 9.82 4.77 11.32 -1.70 5.84 ;O..Ql 1.5.6 3.89 13.39 i4.17 19.34 7.75 4.31 5.97 2.43 9.77 4.96 10.11 4.22 4.15 2.45 5.1.6. 2.91 7.15 3.56 $2.46 $1.08 13.65 9.81 12.76 9.06 4.56 5.27 10.59 8.39 6.33 4.76 8,58 9,48 9.41 7.80 i3.59 $2.21 10(mm) 27.76 23.75 25.33 25.65 26.38 24.65 26.76 $3.35 30.27 26.51 25.23 25.88 28.72 27.56 il.97 25.62 $0.76 ho(mm) .965 .991 .660 .610 .813 .787 .864 .813 i.143 .838 .635 1.14 1.09 .813 .610 .854 i.222 .12 .19 .02 .914 .965 HF‘H 1:06 .762 .838 .762 _._8_3§ .923 i.211 2.0 months 3.0 months ID SK21 SK35 SK37 SK59 SK60 SK61 SK64 Transverse Average S.D. 7; 51:20 c: SK22 '3 SK36 =3 SK62 f) SK63 g; sx135 0 Average ”3.0. <1) SK24 :3 SK32 a) SK65 if, SK66 g SK67 s.) SK68 5" Average S.D. F+SK23 co SK25 g sx33 'g SK69 .18K70 g, SK71 c: SK72 ..1 Average S.D. k (N/mm%) 145 237 351 140 240 161 1.11 207 i 75 44 42 174 182 158 28.6 148 1: 93 154 401 233 210 214 1Q: 253 :1: 88 115 184 126 223 155 106 2.02 160 21:46. 73 L63. 7 17 i1 28 3 46 2.14 12.07 9 18 6 19 7, 2 6 81 6.32 12.53 3.52 3.99 5.08 6.45 13.00 Table 4.1 continued 0(mm) 2.99 3.46 3.57 3.28 3.72 2.89 34.2.6 3.31 $0.30 4.08 3.72 9.93 7.93 6.08 Lfl 5.90 i2.59 10mm) 29.42 25.60 25.50 27.13 28.58 24.95 4.0 months ID a) SK27 m SK29 33x31 5, SK139 c. SK140 {3 SK141 E1Average S.D. SK26 SK28 SK30 SK136 SK137 SK138 Average S.D. Longitudinal k(N/mm%) 196 264 178 286 370 212 262 $ 69 90 117 107 176 142 235 145 $ 54 74 )1 (mm) 7.25 8.93 9.71 5.86 8.62 6.33 7.78 11.54 Table 4.1 continued 0(mm) 3.33 4.46 4.98 3.40 3.96 2.5.6 3.78 $0.87 4.06 4.08 4.59 3.77 4.98 3,31 4.13 $0.59 lo(mm) 26.54 27.58 26.29 23.71 24.80 30.51 26.57 $2.36 22.43 21.84 25.24 21.55 23.36 20.59 22.50 $1.63 ho(mm) 2.08 1.63 2.08 1.14 1.14 1,27 1.56 $.44 HHt—Ir—‘HH oo o 75 a EFFECTIVE TRANSVERSE + EFFECTIVE LONGITUDINAL snrmns (N/mm 5) - OTANGENT TRANSVERSE - ATANGENT LONGITUDINAL ‘ «3888888888888888 I l I I I I I 1 2 3 4 AGE (Io) Figure 4.3 Age related changes in effective and tangent stiffnesses 15- A mmsvense + LONGITUDINAL ‘ O J # V PARAMETER )1 (mm). In ..CI ‘1 AGE (mo) Figure 4.4 Age related changes in model parameter u 76 157 A TRANSVERSE + LONGITUDINAL SE PARAMETER 0’ (mm) ' a AGE (mo) Figure 4.5 Age related changes in model parameter 6 animals (1 and 1.5 months) when it is the same for both orientations. Tangent stiffnesses per unit thickness (obtained by measuring the slope of the linear region of the experimental response curve and dividing' by the initial thickness) are also shown in Figure 4.3. Note that the model-determined effective stiffnesses and the tangent stiffnesses show a jump increase at 1.5 months. There is a strong correlation between the transverse effective stiffness, kT, and transverse tangent stiffness (r = .94; p < 0.02) but a weak correlation between tangent stiffness and effective stiffness longitudinally (r = .3; p > .6). The location of the heel of the response curve represented by u is plotted in Figure 4.4. Transversely, there is a drop in the value of u from 1 to 1.5 months after 77 which it remains essentially constant. For the longitudinal direction, there is an increase from 1 to 1.5 months and then a steady decrease throughout maturation. This has the effect of moving the heel of the curve away from the origin between 1.0 and 1.5 months followed by a steady migration toward the origin. Microstructurally, this means the fibers become less wavy during maturation. The parameter which governs how rapidly the stiffness in Region II increases is model parameter 6. The average values plotted in Figure 4.5 show a decrease in 61 transversely during the period between 1 and 1.5 months after which the value remains constant at about 3.5 mm. Longitudinally, there is a more rapid decrease in 0 during -maturation. This parameter indicates that the waviness of the fibers in the longitudinal direction is more random during the first months of life. The waviness of the fibers then becomes more homogeneous, like those in the transverse direction. 4.1.3 We: It is not enough for a model to exhibit a good visual fit to the experimental data. The ability to determine the model parameters uniquely must also be guaranteed. To ensure the parameters of the uniaxial model could be determined uniquely, the ratio of sensitivity coefficients were plotted vs. deformation and are presented in Figures 4.6 through 4.15. These plots were generated using the 78 0.5 O.‘ -' 0.3 - 0.2 ‘ 0.1 -' o \ -O.1 ‘" '2’ -oz - é -o.3 - -o.4 - -o.s ~ -o.s - -O.7 "' -O.B " 2 :‘2 -o.0 « ° '3 -‘ I I r I I I I r r I I i I I O 2 ‘ U 8 ‘IO 12 14 Figure 4.6 Sensitivity coefficient ratios for average 1 month transverse parameters M7003 Figure 4.7 Sensitivity coefficient ratios for average 1.5 month transverse parameters 79 Illlllll RATIO. h 111111114 O N Figure 4.8 Sensitivity coefficient ratios for average 2 month transverse parameters 0.0 0.8 0.7 0.0 0.5 0.4 0.3 0.2 0. 1 -o.1 -o.z -o.3 -o.4 -o.s -o.o 41.7 -o.a -o.n -1 llllllll RATIOS lllllllll 9+0 333 q .4 —1 fi .1 q .1 -( q q 4- .1 :- Figure 4.9 Sensitivity coefficient ratios for average 3 month transverse parameters 80 RKUOS é. Figure 4.10 Sensitivity coefficient ratios for average 4 month transverse parameters -1 d _2-1 In 4'1 am + R: 0 R3 ‘5‘ 4‘ ‘7 I I I I I I I I r I I I 7T I 0 2 0 8 10 ‘2 1‘ Figure 4.11 Sensitivity coefficient ratios for average 1 month longitudinal parameters 81 RATIOS -1 I- -‘1 .1 0+1: 333 -1 .‘ O .... Q C D d 0 d N d O» Figure 4.12 Sensitivity coefficient ratios for average 1.5 month longitudinal parameters ' '7' VVVV'T' v‘fo" RATIOS 0+1: 033 Figure 4.13 Sensitivity coefficient ratios for average 2 month longitudinal parameters 82 RATIOS Figure 4.14 Sensitivity coefficient ratios for average 3 month longitudinal parameters -‘ ‘ _2- RAMS _3- _5... .7- Figure 4.15 Sensitivity coefficient ratios for average 4 month longitudinal parameters 83 average parameters (k, 11, and 0) obtained from uniaxial specimens for each age group and direction. For the well- defined three-phase response of the transverse specimens of each age group and the longitudinal specimens from the 1.5 through 4 month age groups, the sensitivity coefficients are linearly independent. For the "flat" more featureless 1 month old longitudinal response curve, (Figure 4.12) sensitivity coefficients approach (Figure 4.11), or are more linearly dependent. 4.1.4 M 1 fi rim n a The experimentally measured force data were averaged at intervals of 2 mm of deformation for the uniaxial transverse and longitudinal specimens for each of the five age groups and is shown plotted in Figures 4.16 - 4.25. This is compared to the model response generated by using the average model parameters for each age group and direction. In all cases, except for the 1 month longitudinal responses, the average model force is nearly the same as the average experimental force or lies within one standard deviation. This is not surprising since the model fit the experimental data very well. An example of a typical fit of the response of a single specimen is given in Figure 4.26. 4.1.5 Morghglggigal Typical scanning electron micrographs are presented in Figure 4.27. These micrographs are from one month old skin, 84 LOAD (N) Figure 4.16 Average experimental responses and average model fit of 1 month uniaxial transverse specimens LOAD (u) Figure 4.17 Average experimental responses and average model fit of 1.5 month uniaxial transverse specimens 85 LOAD (w) Figure 4.18 Average experimental responses and average model fit of 2 month uniaxial transverse specimens LOAD (N) W110! (nun) Figure 4.19 Average experimental responses and average model fit of 3 month uniaxial transverse specimens 86 LOAD (N) Figure 4.20 Average experimental responses and average model fit of 4 month uniaxial transverse specimens LOAD (a) 10- Figure 4.21 Average experimental responses and average model fit of 1 month uniaxial longitudinal specimens 87 50 40.- 30.. 3 9 3 20.. 10a 0 mm» (m) Figure 4.22 Average experimental responses and average model fit of 1.5 month uniaxial longitudinal specimens LOAD (u) Figure 4.23 Average experimental responses and average model fit of 2 month uniaxial longitudinal specimens 88 mm (m) Figure 4.24 Average experimental responses and average model fit of 3 month uniaxial longitudinal specimens LOAD (N) Figure 4.25 Average experimental responses and average model fit of 4 month uniaxial longitudinal specimens 89 LOAD (N) mm (mm) Figure 4.26 Typical model fit of experimental data relaxed (Figure 4.27a), stretched into Region II (Figure 4.27b), stretched into the linear region (Figure 4.27c), and for the 4 month old skin, relaxed (Figure 4.27d), stretched into Region II (Figure 4.27e) and stretched into the linear region (Figure 4.27f). There appears to be more waviness in the fibers of the 1 month old rats than the 4 month old rats. This supports the model’s claim that the heel of the curve moves toward the origin during maturation. The collagen in Figure 4.27e and Figure 4.27f look very similar. This is in contrast to the corresponding micrographs (Figure 4.27b & c) from the 1 month old animals. It appears that for the 1 month animals the fibers are straightened gradually, whereas for the 4 month old animals the fibers are more homogeneous (small value for parameter 6) and are straightened more rapidly causing a more rapid change in 11'11'1)‘ [II 20000: .' I) 53 L" l 0 Figure 4.27 Scanning electron micrographs 91 slope of the response curve. There also appears to be an increase in fiber diameter from 1 to 4 months of age. This may account for the increase in stiffness reported by the model. There also appears an increased fiber packing density which occurs during the 4 month period. 4.2. W1 4.2.1 Won The elliptical distribution parameters a and b, the major and minor radii, respectively, are presented in Figure 4.28. The spatial distribution of fibers in the plane of the skin appears to go from a uniform distribution at 1.0 months of age to a more elliptical one as the animal matures. Note that the major axis of the ellipse coincides with the transverse direction, which is the direction of maximum stiffness. The fiber stiffness K values are presented in Figure 4.29. Recall that the fiber stiffness K is the stiffness per mm of sample thickness. Fer each age group with the' exception. of ‘the stiffness value for 'the 1.5 month old samples, the stiffness K remains essentially constant at about 3 kN/mm %). .The age related change in thickness of the skin samples is shown in Figure 4.30. There is a rapid increase in thickness from 1.5 months of age to 2 months with the overall thickness doubling from 1.0 months to maturation between 3-4 months. 92 1 MONTH t5 MONTH V j 2 MONTH 3 MONTH / \ 6\ b‘.560 6N 6\ 4 MONTH / Figure 4.28 Age related changes in spatial distribution of .fibers 6N 93 snrrnrss (EN/mm u) “.1 . AGE (Io) Figure 4.29 Age related changes in fiber stiffness K The total collagen and insoluble fraction per wet weight versus age are presented in Figure 4.31. Both contents increase rapidly between 1 and 2 months. Since it was not possible to evaluate the number N of collagen fibers in a given sample, a normalization of the fiber stiffness K in terms of the total amount of collagen was made. The resulting' normalized fiber' modulus ‘K is presented in Figure 4.32 as it changes with age. The units on Kare Newtons per mg of collagen. K is shown to decrease more than two-fold with age in the period between 1.0 and 2.0 months after which is remains essentially constant. 94 2 n. u... a. “.4 “1 ‘E "- E “- U u" j .3 "1 ‘ m I 5»- oin- 2 mu 0- ud- u- u. 0.3- u- u... 0 f 7 I I I I fl 1 2 3 4 AGE (mo) Figure 4.30 Age related changes in dorsal skin thickness 9’9” .3 . ? 0-. ? A: ? A TOTAL + INSOLUBLE ‘ a I l 'COLLAOEN CONTENT PEN WET WEIGHT (In I' T F T 1 2 3 I 4 AGE (mo) .1 Figure 4.31 Age related. changes in 'total and insoluble collagen content 95 NONMALIZED DTIFFNEOB (Nlmg M. 6883888555885838 l I u. 5 dc! N“ AGE (mo) Figure 4.32 Age related changes in the normalized fiber stiffness 4.2.2 W The model predicted responses for the wide specimens are plotted in Figures 4.33 through 4.42. Also plotted are the predicted responses of 5.5 times the uniaxial response, which is what would be expected for the wide specimens from a continuum analysis. Both predictions are compared to the actual response for the wide specimens in both directions and at all age groups. In all but the one month longitudinal responses, the model prediction is better than the continuum prediction. LOAD (N) 96 -X- CONTINUUM PREDICTION —- MODEL PREDICTION —0— EXPERIMENTAL Fix! Figure 4.33 Comparison of wide specimen, 5.5 times uniaxial and model predicted responses for 1.0 month transverse 1.0410 (w) 10 ‘ .0- EXPERIMENTAL -X- CONTINUUM PREDICTION -—- MODEL PREDICTION Figure 4.34 Comparison of wide specimen, 5.5 times uniaxial and model predicted responses for 1.0 month longitudinal 97 -X- CONTINUUM PREDICTION '00 . — MODEL PREDICTION -o- EXPERIMENTAL 1.011001) 88888388 Figure 4.35 Comparison of wide specimen, 5.5 times uniaxial and model predicted responses for 1.5 month transverse I” n _ -X- CONTINUUM PREDICTION '° ‘ — MODEL PREDICTION X -o- EXPERIMENTAL N -. co .2’: 9 so 3 40 30 20 to o Figure 4.36 Comparison of wide specimen, 5.5 times uniaxial and .model predicted responses for 1.5 month longitudinal 98 In I80 - “° ' -x— CONTINUUM PREDICTION x ‘3“ ‘ — MODEL PREDICTION m - -0- EXPERIMENTAL LOAD (N) EIIOISSIHE 88 Figure 4.37 Comparison of wide specimen, 5.5 times uniaxial and model predicted responses for 2.0 month transverse so 79 - -X- CONTINUUM PREDICTION — MODEL PREDICTION -o- EXPERIMENTAL LOAD (N) Figure 4.38 Comparison of wide specimen, 5.5 times uniaxial and model predicted responses for 2.0 month longitudinal mum) Figure 4.39 LOAD (N) Figure 4.40 99 -X- CONTINUUM PREDICTION —- MODEL PREDICTION -0- EXPERIMENTAL 0388888388 °Illllllllllllllllllll Comparison of wide specimen, 5.5 times uniaxial and model predicted responses for 3.0 month transverse 1” "° ‘ -x- CONTINUUM PREDICTION X 1m - —— MODEL PREDIOTION .0 _ -0- EXPERIMENTAL u.. 70 U 50 40 30 N 10 0 Comparison of wide specimen, 5.5 times uniaxial and model predicted responses for 3.0 month longitudinal 100 -X'- CONTINUUM PREDICTION '°° ‘ — MODEL PREDICTION -o- EXPERIMENTAL LDAD(N) 88888388 Figure 4.41 Comparison of wide specimen, 5.5 times uniaxial and model predicted responses for 4.0 month transverse -X- CONTINUUM PREDICTION - — MODEL PREDICTION -0- EXPERIMENTAL LOAD (N) 88888888 .0 O Figure 4.42 Comparison of wide specimen, 5.5 times uniaxial and model predicted responses for 4.0 month longitudinal V. DISCUSSION The average experimental responses for’ the uniaxial specimens show trends consistant with the literature. Transverse specimens demonstrate a1 stiffer structural response than longitudinal specimens. .Haut (1989) made similar observations in rats. Directional differences in skin stiffness have also been reported for humans (Manschot, et al., 1986b) and other animals (Lanir and Fung, 1974). As the rat matures, the structural response of the skin becomes stiffer. This stiffening with age has been noticed in humans (Daly and Odlund, 1979), as well as rats, (Vogel, 1976). Another age related trend is that the "heel" region of the curves tends toward the origin. This was reported to occur in humans (Ridge and Wright, 1966; Gibson and Kenedi, 1969) and in rats (Haut, 1989). The response of the four- month transverse uniaxial specimen is structurally less stiff than the three-month transverse response. Vogel (1976) and Flint, et al. (1984) report that the skin matures sometime between three and four months, and they also noticed a slight drop in the response of four month animals with respect to the three month old specimens. The collagen in rat tail tendon is also reported to mature around 3-4 months (Torp, et al., 1974). 101 102 The effective stiffness parameter, k, for the most part determines the linear region response. This parameter is analogous to and shows similar trends with age and direction as the tangent stiffness from a continuum analysis. Manschot and Brakkee (1986b) and others (Daly, 1979; Lanir and Fung, 1974) report higher modulus values in the transverse direction than in the longitudinal direction. Several investigators have reported an increase in stiffness occurring during maturation (Vogel, 1976) a finding which is supported by the model’s "effective" transverse stiffness. Longitudinally' the effective stiffness appears to remain essentially constant after a jump at 1.5 months. This jump in stiffness at 1.5 month is a consequence of the structural stiffness being normalized by skin thickness. The structural stiffness increases more rapidly than the skin thickness, so when the stiffness is normalized per unit thickness,. there is a dramatic increase in effective stiffness. During the next half‘ month, the structural stiffness increases further, but nowr the skin thickness increases more rapidly. The result is an apparent drop in effective stiffness from 1.5 to 2.0 months. That is to say that there is not a one-to-one relationship between skin thickness and structural stiffness. There is no information on rat skin stiffness at 1.5 months of age in the literature to which to compare this phenomenon. For comparison purposes, Vogel (1976) reports an increase in transverse tangent modulus from 17 MPa at 1 month to 38 MPa at 4 103 months. The lateral specimens in this study have tangent moduli of 14 MPa at 1 month and 36 MPa at 4 months from a continuum analysis. It should be noted that Vogel used Sprague Dawley rats. Vogel also reported a correlation between the insoluble collagen content and transverse tangent modulus. A similar correlation for the effective stiffness of transverse specimens was found; however, for longitudinal specimens, a correlation between tangent stiffness and model stiffness and insoluble collagen content was not apparent. If there is a pmeferred orientation of collagen fibers, then attempting to correlate the structural stiffness in a given direction with the per-wet-weight content of collagen without accounting for the amount of collagen oriented in that direction would yield no correlation. The desire to correlate structural stiffness with collagen content was the motivation for incorporating a spatial distribution function in the model. The hypothesis is that the difference in structural stiffness was not caused by the material properties being direction dependent, but rather that the fibers had a preferred orientation. The parameters 0 and a, which describe the waviness of the fibers, indicate that transversely the fibers become less wavy between 1 and 1.5 months after which time the average waviness remains constant. Not only is the waviness decreased between 1 and 1.5 months, but the standard deviation of this waviness becomes smaller making the fibers more homogeneous. Longitudinally, the fibers appear to 104 become more wavy during the period between 1 and 1.5 months, after which the waviness steadily decreases making the fiber waviness more homogeneous. Qualitatively these trends could be seen microscopically. The waviness of the collagen fibers does appear to decrease with age. Not much quantitative information is available from the literature on the distribution of waviness of collagen fibers. However, Craik (1966) reports a decrease in skin extensibility (which is analogous to model parameter (I) with age. Further, Manschot (1985) reports that skin extensibility decreases significantly across Langer lines (longitudinal) with age "but remains constant along them (transverse). The reported increase in waviness longitudinally from 1 to 1.5 months is probably because the model being unable to determine a unique set of parameters for the average l-month longitudinal response. Recall that for these specimens the sensitivity coefficients were linearly dependent (Figure 4.11). Similarly, the large standard deviation in average parameter 6 reported for the longitudinal specimen at 1.0 month is likely due to this problem as well. In order for a microstructural model to be useful it should have physically significant parameters, the parameters should be uniquely determinable, and the model should fit the experimental data. This study showed the model to satisfy all three of these considerations for all but a few of the response curves. '5'- 105 The sensitivity coefficients for the model evaluated with the average parameters from each age group and direction for the uniaxial specimens indicate that the model is well suited for all specimens except the one month longitudinal. One month longitudinal specimens often lack a well defined "heel" region (Region II) which allows the model to select large standard deviations, parameter 6, for the recruitment function. When 0’ is large, any combination of effective modulus k and recruitment function mean 0 allow the model to fit the experimental curves. That is, in this case k and u can not be determined uniquely. The sensitivity ratios for the other longitudinal specimens show a nearly linear dependence for R2. These ratios are not linear, but are very close to it. This does not cause a problem in determining the parameters uniquely, however, it does cause the minimization technique to converge very slowly. Adding a constraint to the distribution function would likely make the minimization converge more quickly and would force the sensitivity coefficients to be better behaved. The constraint should require the mean and standard deviation parameters to be values less than the deformation at failure. Further, the standard deviation should be constrained such that all of the fibers are recruited (straightened) before the specimen fails. This constraint would be supported physically, in that once the specimen fails, rupturing into two pieces, it would seem unlikely that there remains any fibers left to recruit. The problems 106 encountered with an unconstrained problem can be seen in the one-month longitudinal data. in“; model indicates that not until u (3.89mm) plus 0 (13.39mm) are the majority of the fibers recruited. The specimens generally failed at a deformation well under 17mm. The age-related changes in the distribution of collagen fibers in the plane of the skin indicate that the young skin has a uniform spatial distribution. As the animal matures, the collagen fibers become more preferentially oriented in the transverse direction. This apparent change in the distribution of fibers spatially allows for an explanation of the different structural response obtained when tests are performed in two different directions at the same location on the skin. That is, the distribution function is a means of describing the planar anisotropy of the skin. The only other investigator to include a spatial distribution function is ILaniru ILanir assumes ‘that the number of fibers oriented along a given angle with respect to the direction of deformation of the specimen is given according to a Gaussian distribution function, plus a uniform distribution. The mean of the distribution coincides with the axis of deformation. His distribution functions add three parameters to the model, which already consists of six parameters. The three additional parameters are the height of the uniform distributions and a standard deviation for both distribution functions. Whenever more parameters are added to a model, there exists the increased 107 risk of the parameters being linearly dependent. When parameters are linearly dependent, they can not be determined uniquely. Lanir's description of the fiber in terms of two distribution functions is also disadvantaged by their not being able to describe the distribution of fibers continuously as a function of their angular orientation in the skin. He also did not experimentally evaluate the distribution functions and there is no information available with which to compare these results. When the stiffness per unit thickness K was normalized on a per mg total collagen content basis, the modulus decreased during the period from 1 to 2 months of age. After two months, the modulus remained essentially constant. It seems unlikely that the collagen fibers become more compliant during that period because the stiffness of collagen fibers in tendons (where collagen is aligned) is reported to increase during maturation. Likely there is some as yet unaccounted secondary arrangement of the fibers. In the model the spatially distributed fibers were assumed to run from grip to grip at a given angle. From inspection of the scanning electron micrographs, it appears that this is a simplification of the actual morphology. The fibers appear to ”wander" in the skin, starting in one direction and then turning in another direction. This wandering not only occurs in the plane of the skin, but also from plane to plane. Perhaps the interplanar wandering of the fibers increases with age, as the thickness increases and may 108 account for the predicted model response for the wide specimens fitting the actual response more poorly as the tissue matures. A possible topic for future research might be to determine what effect a more complex spatial arrangement would have on the predicted tissue response. Another useful investigation would be a morphometric study to determine the number of fibers in a skin specimen. A morphometric study would allow for the calculation of the modulus of an individual collagen fiber which could. be compared with collagen fiber moduli for other tissues, such as the well researched rat tail tendon. When comparing the responses between uniaxial specimens and wide specimens for each age group and direction, based on a simple strength of materials prediction, one would expect the wide specimen to have a greater structural response over the uniaxial specimen. Although the wide specimen is 5.5 times wider than the uniaxial specimen, the wide specimen response is run: 5.5 times greater. It was felt that this discrepency was caused by the spatial distribution. of fibers and. was the motivation for incorporating a distribution function into the model. The predictions of the wide specimen response by the model incorporating the distribution function showed closer agreement with the actual response than did the pmedicted response based on a continuum analysis. The only exception to this was the model prediction for the 1 month old longitudinal response. For this case, the continuum 109 prediction was closer to the experimental curve than was the model prediction. As mentioned earlier, the model was not well-suited to describe responses typical of l-month longitudinal specimens. It is also not surprising that the model predicted a stiffer response for the 1-month longitudinal wide specimen because the model fit of the 1— month uniaxial longitudinal responses was stiffer than the actual experimental response. The results obtained from the model, including the spatial distribution function, indicate that the response of skin is not only dependent on the material properties of the individual fibers, but it also dependent on the spatial orientation of those fibers. Further, since the model-predicted response, in many cases lies outside the range of actual data, it is hypothesized that there may be yet another undetermined spatial parameter that governs the response as well. This secondary fiber arrangement was previosusly discussed. Another fiber arrangement phenomena that was not included in the model, but likely takes place in the tissue, is the aligning of fibers in the direction of deformation. In this model it was assumed that a fiber at a given angle 0 always remained at that angle throughout deformation. However, as the specimen is elongated, the length to width ratio of the specimen changes and the angle the fiber makes with the long axis of the specimen decreases. This could be modeled by forcing the distribution function to change shape with deformation, such that the percentage of fibers actively 110 deformed is the same but their orientation changes. The result of such a deformation-dependent distribution function would be a lesser value of Kreported for the fiber than was the case for the fibers whose orientation was assumed constant. The force generated. by the fiber of lesser stiffness would be the same, as the realignment would result in a larger force component per fiber to be in the direction of deformation. Despite the difference in model and actual response, the model does predict closer the wide specimen response based on uniaxial data. Such a prediction has been previously unavailable. Snyder and Lee (1975) attempted to compare the uniaxial response with the wide specimen response (pure shear) in frog skin using a large deformation theory strain energy function, but were unsuccessful. Wijn (1976) compared the response of skin in torsion with the uniaxial response and reported the lack of comparison was a result of the anisotropy of the skin. Because this structural model is based on the physical. architecture of the skin's microstructure, it is able to describe the mechanical response 'of skin not only in terms of the material properties of the component's fibers, but also in terms of their geometric configuration and their spatial orientation. It is the coupling of the spatial and geometric properties with the material properties which has made it difficult for investigators to determine the "real" material properties of collagen in skin. The ability to de- 111 couple the material properties from the geometric and spatial properties is now available through the model and gives investigators the tool to analyze and predict the response of specimens independent of their shape and size. Further, knowing the spatial distribution function of the fibers also allows for the prediction of specimens oriented in different directions. REFERENCE 3 Alexander, H. & Cook. T. (1976) "Variations with age in the mechanical properties of human skin in vivo". Bedsore Biomechanics, McMillan, London, pp. 109-117. Alexander and Cook, T.H. (1977) "Accounting for natural tension in the mechanical testing of human skin", J. Inves. Derm., 69:310-314. Allaire, P.E., Thacker, J.C., Edlich, R.F., et al. (1977) "Finite deformation theory in vivo human skin". J. of Bioeng., 1:239-249. American Veterinary Medical Association. (1986) 1986 Report of the AVMA Panel on Euthanasia. JAVMA 188, 252-268. Barenberg, S.A., Filisko, F.E. and Geil, P.H. 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(1965) "The structural components of the dermis and their mechanical characteristics". Gibson, T., Stark, H. and Evans, J.H. (1969) "Directional variation in extensibility of human skin in vivo". g; Biomechanics, 2:201-204. Glaser, A.A., Marangoni, R.D., Must, T.S., Beckwith, T.G. Brody, G.S., Waxler, G.R. and White, W.L. (1965) "Refinements in the method for the measurement of mechanical properties of unwounded and wounded skin". Med. Electron. Biol. Engng., 3:411-419. Gotte, L. (1971) "Recent observations on the structure and composition of elastin". Adv. Exp. Med. Biol., 79:105-115. Gou, P.-F. (1970) "Strain energy function for biological tissues". J. Biomechanics, 3:547-550. Grahame, R. and Holt, P.J.L. (1969) "The influence of aging on the in vivo elasticity of human skin". Gerontologia, 15:121-139. Green, A.E. and Adkins, J.E. (1960) "Large elastic deformations". Oxford, pp. 296-307. Hass, G.M. 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I uber die spaltbarkeit der cutis". S.B. der Acad. in Wien, 44:19-46. Lanir, Y. and Fung, Y.C. (1974) "Two-dimensional mechanical properties of rabbit skin - II. Experimental results". J. Biomechanics, 7:171-182. Lanir, Y. (1976) "Biaxial stress-relaxation in skin". Ann. Biomedical Eng., 4:250-270 Lanir, Y. (1979) "A structural theory for the homogeneous biaxial stress-strain relationships in flat collagenous tissues". J. Biomechanics, 12:423-436. Lavker, R.M., Zheng, P. and Dong, G. (1987) "Aged skin: a study by light transmission and scanning electron microscopy". J. Invest. Dermatol., 88:445-518. Manschot, J.F.M. (1985) The mechanical properties of the skin in vivo. Thesis, Katholicka, Universiteit Nymegen, The Netherlands. 117 Manschot, J.F.M. and Brakkee, A.J.M. (1986a) "The measurement and modeling of the mechanical properties of human skin in vivo - I, the measurement". J. Biomechanics, 19:511-515. Manschot, J.F.M. and Brakkee, A.J.M. 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(1958) "Mikrobestimmung von Hydroxyprolu mit Chloramin-T und p-Dimethyl-aminobenzaldehyd". Hoppe- Seyler's Z. Physiol. Chem., 311:41-45. Tong, P. and Fung, Y.C. (1976) "The stress strain relationship for the skin". J. Biomechanics, 9:649-657. Torp, S., Arridge, R.G.C., Armeniades, C.D. and Baer, E. (1974) "Structure-property relationships in tendon as a function of age". Structure of Fibrous Biopolymers (Edited by Atkins, E.D.T. and Keller, A.), pp. 197—221, Butterworths, London. Tregear, R.T. (1966) "Physical function of skin". Academic Press, London. Tregear, R.T. and Dirnhuber, P. (1965) "Viscous flow in compressed human and rat skin". J. Invest. Derm., 45:119- 125 O Treloar, L. (1958) "The physics of rubber elasticity". Clarendon Press, Oxford. 119 Veronda, D.R. and Westmann. (1970) "Mechanical characterization of skin". J. Biomech., 3:111-124. Vlasblom, D. (1967) "Skin elasticity". Ph.D. Thesis, Univ. of Utrecht, Utrecht, The Netherlands. Vogel, H.G. (1983) "Effects of age on the biomechanical and biochemical properties of rat and human skin". J. Soc. Cosmet. Chem., 34:453-463. Vogel, H.G. (1976) "Altersabhangige Veranderungen der Mechanischen und Biochemischen Eigenschaften der cutis bei ratten". Akt. Geront., 6:477-487. Weinstein, G.D. and Boucek, R.J. (1960) "Collagen and Elastin of Human Dermis". J. Invest. Dermat., 35:227-230. Whitaker, L.A. (1972) "Characteristics of skin of importance to the surgeon". Int. Surg., 57:877-879. Wijn, P.F.F., Brakkee, A.J.M., Stienen, G.J.M. and Vendrik, A.J.G. (1976) "Mechanical properties of human skin in vivo for small deformations: a comparison of uniaxial strain and torsion measurements". Bedsore Biomechanics, McMillan, Longon, pp. 103-108. APPENDICES APPENDIX A 120 ittt***************i*iiittiititAi*i********f*t************titittttiittitttttttt FILE CONVERSION STEPHEN BELKOFF 9 MAY 1989 t k t t t t * THIS PROGRAM CONVERTS NICOLET VOLTAGE VALUES TO LOAD AND STROKE * VALUES. A A program con real STRK(2000),LOAD(2000).X(2000),Y(2000),LCAL,LO character FNAME*20, ANSWER*20,FN*20,ANS*20 RttAttttttttttttttt*ttttttttttiiitttttttt * INPUT CALIBRATION VALUES * RRxA*ttttttttttt*ttttt*tttttttxtttttttttt print *, 'PLEASE LOCK CAPS' print *, 'USE Y FOR YES AND N FOR NO' 750 print *, 'ENTER STROKE CALIBRATION (M/V)’ read*,SCAL print *, 'ENTER LOAD CALIBRATION (N/V)’ read*,LCAL PRINT*,' ENTER TIME PER POINT SETTING (SECONDS PER POINT)’ read*,TPP PRINT*,’ENTER STARTING LENGTH (IN)' READ*,SLNTH RAt*Mtittttttitittitttttttttttttttttt*ttA * CONVERT To MILLIMETERS * titA*iittttt*****t***************A*****tA . SLNTH=SLNTH*25.4 700 print *, 'PLEASE ENTER FILENAME’ read*, FNAME print *, 'ENTER END TIME OF TEST (SECONDSI' read*,TEND print *, 'DATA FILE IS BEING LOADED’ print *, 'PLEASE WAIT’ K-TEND/TPP+1 *ttttttttttttittttttktt*fittttttttttttiitit*i************t*****AAAAtARt*tttttA * THIS LOOP READS IN STROKE AND LOAD VOLTAGE VALUES. TWO FORMAT STATEMENTS * * ARE INCLUDED DUE TO THE FILE STRUCTURE OF THE SYMPHONY GENERATED FILES. * * THE FORMAT STRUCTUE WILL BE OBVIOUS IF YOU INSPECT THE DESIRED * . DATA FILE AND NOTICE COLUMNS OF CHARACTERS FOLLOWED BY DATA. * *ttttt*tt*****tAtttttttt**t****Alt******t*fitt***************t********t*itttit open(4,file='A:'//FNAME) ALOAD=0 ASTRK-O do 40 I=1,14 read(4,800) STRK(I),LOAD(I) ALOAD=ALOAD+LOAD(I) ASTRK=ASTRK+STRK(I) 40 continue ASTRK=ASTRK/14 ALOADaALOAD/14 PRINT*,ASTRK,ALOAD do 10 I=15,K read(4,900) STRK(I),LOAD(I) 10 continue Close(4) *t******ti**t**Attttiit***A*t******k***tt****A********t*t*****tktttt****tttAt * ALOAD IS THE AVERAGE OF THE FIRST FOUR DATA POINTS. THIS VALUE IS USED * * AS ZERO AND IS SUBTRACTED FROM ALL SUBSEQUENT POINTS. BE SURE WHEN ENTERING * TEST START AND FINISH TIMES THAT YOU INCLUDE SEVERAL POINTS To BE USED . * AS ZERO. * *tttt**i*i***ik**ti*****titttitt*A***i*********ttitttittttkittttAtktit*tfitfiiA do 20 M=1,K X(M)-(STRK(M)-ASTRK)*SCAL Y(M)-(LOAD(M)-ALOAD)*LCAL PRINT*,M,X(M),Y(M) 20 continue VL=0.012*LCAL VS=0.012*SCAL L=l 50 CONTINUE ti************************t************ttfitttii************************t***tk t A t i A A A A A t 60 70 80 121 IF(Y(L).GT.VL)THEN ICOUNT-O CHECK-Y(L) DO 60 MM-1.9 CHECK-Y(L+MM)+CHECK CONTINUE CHECK-CHECK/lo IF(CHECK.GT.VL)THEN ICOUNTaICOUNT+1 ELSE ENDIF ELSE L-L+1 GO TO 50 ENDIF IF(ICOUNT.EO.1)THEN LHIT-L ELSE L-L+1 GO TO 50 ENDIF L31 CONTINUE IF(X(L).GT.VS)THEN ICOUNT-O CHECK-X(L) DO 80 MM-1,9 CHECK-X(L+MM)+CHECK CONTINUE CHECK-CHECK/lo IFICHECK.GT.VS)THEN ICOUNT-ICOUNT+1 ELSE ENDIF ELSE L=L+1 GO TO 70 ENDIF IF(ICOUNT.EQ.1)THEN IHIT-L ELSE L=L+1 GO TO 70 ENDIF TDEF-IHIT*TPP TLOD-LHIT*TPP GL=SLNTH+X(LHIT) FL-SLNTH+X(K) itkitiitikfiitiiiAittttttittt‘kttfiiiitttktt i PRINT OUTPUT TO THE SCREEN * AAAitt*t**t*titfit***tttt**t**ttt*tfitttt't* 30 PRINT‘,TLOD,TDEF PRINT*,'GAUGE LENGTH IS',GL PRINT*,’FINAL LENGTH IS',FL print*,'ENTER FILENAME OF CONVERTED FILE' read*,FN 'N-K-LHIT Open(l,file-’B:’//FN) write(1,200) N/3 do 30 KKaLHIT,K,3 write(1,100) Y(KK),X(KK)-X(LHIT) PRINT*,KK,X(KK) continue PRINT*,'DO YOU WANT TO CONTINUE' READ*,ANS IF(ANS.ne.'N')then go to 700 > else endif 122 it*ittttttittifiiifitfit*tititttttttitittttt * FORMAT STATEMENTS ~ ttthfitttttrttttttttttttitttittttttttttttt 100 format(2X,E20.10,2x,820.10) 200 format(2X,14) 800 format(16x,F6.3,2x,F6.3) 900 format(dx,F6.3,2x,F6.3) end tittttttttttitttttt*ttttttfittttttttittttttttit*ttkttttttttttttttttttittttttttta * END OF PROGRAM * itiiti**t**fiit****ttitt**t*fiiittittttiflittfiti*i******tt**t*kfifikflfittttttttfittttt APPENDIX B 123 tittttitttttitttfiiitifittt*tiififiitfitfitfitfit*fitfiititttfitfitifiit*****itifiiittttitt0t SKIN PROGRAM FEB 89 REVISED 1 AUG 89 MARQURDT METHOD * I t t t * fi t t t * THIS PROGRAM USES THE LEVENEERG-MAROUARDT METHOD WHICH Is A * * NON-LINEAR LEAST SQUARES MINIMIzATION TECHNIQUE TO FIND THE * * FARAMETERS OF THE SKIN MODEL. FOR MORE INFORMATION ABOUT * * THE METHOD AND THE SUBROUTINES USED HERE, YOU ARE ENCOURAGED * * TO CONSULT THE TEXT ”NUMERICAL RECIPES" BY FRESS,ET AL.. * t t t * it!tittiit*t***i*tttttfi***itti*****i**t*******ttittititt********tit*********t PROGRAM MRQRDT DIMENSION AP(3),COVP(3,3),ALPHP(3,3),BETAP(3),LREC(150) DIMENSION NSTART(10),NSTOP(10),M(200),IDEN(6,3,3,9),X1(8), $X2(8),X3(8),IFIL(40) CHARACTER*10 FNME(141),ANS,81,32,B3,84,85,86,AANS,YES,FFIL,EN REAL PI,LO INTEGER SH COMMON PI,X(1500),Y(1500)pSIG,NDATA,MFIT,MA,LISTA(3),OCHISQ,LI,LO PI-ACOS('1.) SIG=.1 it******t*itttfiiiit*fittfi*t*fi*****ti****** . LISTA(I) ORDERS THE PARAMETERS . * TO BE FIT * tittiitiflit*********k***t*tittifitt***t*** LISTA(1)=1 LISTA (2) '2 LISTA(3)=3 t***t*ittt**********tktti**fi*t*****i***fi* * INITIALIZES CHI SQUARED RESIDUAL“ *********i****fi**t**tktfittttit*********** OCHISQ'100 *fitttfiflfitfltflfitkttttiflfiti*ttttttitt****tti * STORAGE FILE SET UP * *t***t***t***tfi***ti*ttt*****i****fiflfi***t EI-'FILE' BZ-‘K’ BB-‘M’ B4-’S’ BS-'CHISQ’ BG-'ME' ID=1 WRITE(5,*)’PLEASE ENTER STORAGE FILENAME' READ(5,900)FFIL OPEN(60,FILE-'c:\SMOD\FNAM') DO 3 L-17,140 LJ=L IF(L.GE.34)LJ-L+1 READ(60,900)FNME(LJ) 3 CONTINUE 4 CONTINUE *******t*********tit*t******t*i***t*****t . INITIALIZE PARAMETERS * tttfiktttfittttt*ttfiititt*ttt**tittfiitt**** WRITE(5, *) 'ENTER INITIAL K' READ(5,201)AP(1) WRITE (5, *) ' ENTER INITIAL M' READ(5.201)AP(2) WRITE(5,*)’ENTER INITIAL S' READ(5,201)AP(3) Al-AP(1) A2=AP(2) A3-AP(3) WRITE(5,*) 'THESE ARE YOUR INITIAL PARAMETER INPUTS' WRITE(5,*)A1,A2,A3 tt*********t****fit*t**ttitttttttiitittitt * MA - TOTAL NUMBER OF PARAMETERS * * MFIT 8 NUMBER OF PARAMETERS TO * * BE FIT * ttitttittNiki!tttttttttflttfiittttttittitit 124 MA=3 MFIT=3 WRITE(5,*)'ENTER NUMBER OF FILES YOU WISH TO FIT' READ(5:121)IN DO I I-1,IN WRITE(5,*)'FILE NUMBER?’ READ(5,121)IFIL(I) l CONTINUE DO 2 I-1,IN LI=IFIL(I) WRITE(5,121)LI CALL DATA(FNME,VALUE) k**fi****t*****k******************fi******* * l INITIALIzE LAMDA * ******fi***********************t****t***** ALAMP--1. JD-z CALL MRQMIN(AP,COVP,ALAMP,ALPHP,BETAP) ***************t*t*********************** * WRITE RESULTS TO DATA FILE * ti*************t*****i**********t***fi**** HRITE(5,100)JD,FNME(LI),Ap(1),Ap(2),Ap(3),OCHIso,NDATA OPEN(65,ACCESS='DIRECT',filea'c:\SMOD\'//FFIL,FORM-'FORMATTED', $RECL=62) WRITE(65,100,REc-LI) JD,FNME(LI),AP(1),AP(2),AP(3),OCHISQ,NDATA AP(1)-A1 AP(2)-A2 AP(3)-A3 2 CONTINUE WRITE(5,*)’DO YOU WANT TO CONTINUE?’ READ(5,900)ANS IF(ANS.EQ.’Y’)GO TO 4 CLOSE(65) tit*********ii*titiitkttfifiiiitttt * FORMAT STATEMENTS * ***********t*******t************* 100 FORMAT(11,1x,A10,4 F10.2,I4) 111 FORMAT<11) 121 FORMAT(I$) 200 FORMAT(6 A10) 201 FORMAT= MFIT) ARE USED AS WORKING SPACE DURING MOST ITERATIONS. NORMALLY YOU WOULD SUPPLY A SUBROUTINE FUNCS(X, A,YFIT,DYDA,MA) THAT EVALUATES THE FITTING FUNCTION YFIT, AND ITS DERIVATIVES DYDA WITH RESPECT TO THE FITTING PARAMETERS A AT X, HOWEVER, THE FUNCTION SUBROUTINE HAS BEEN WRITTEN INTO THIS PROGRAM. ON THE FIRST CALL PROVIDE AN INITIAL GUESS FOR THE PARAMETERS A, AND SET ALAMDA