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Residual stress and strain in the lamellar unit of the porcine aorta: experiment and analysis
ARTICLE IN PRESS
Journal of Biomechanics 37 (2004) 807–815
ASB-Microstrain Award 2002
Residual stress and strain in the lamellar unit of the porcine aorta: experiment and analysis Takeo Matsumotoa,b,*, Taisuke Gotob, Takao Furukawaa, Masaaki Satob a
Biomechanics Laboratory, Department of Mechanical Engineering, Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya 466-8555, Japan b Biomechanics Laboratory, Graduate School of Mechanical Engineering, Tohoku University, Aramaki-Aoba 01, Sendai, Miyagi 980-8579. Japan Accepted 2 August 2003
Abstract The opened-up configuration of the artery wall has long been assumed to be stress-free. This is questionable in a microscopic level. The aortic media is made of concentric layers whose unit is called a lamellar unit, a pair of elastic lamina (EL) and a smooth musclerich layer (SML). Recently, we found that the EL was about 2.5 times stiffer than the SML. If the circumferential stress in the in vivo condition is the same between the two layers, residual stress of each layer should be different because the stress–strain relationships differ. Such residual stress is not released fully by radial cutting, but is released in the area close to the cut surface, causing hills and valleys on the surface due to residual stress. To check this hypothesis, we have developed a scanning micro indentation tester, a scaled-up version of the atomic force microscope, and measured the topography and the stiffness distribution of the cut surface. The surface of the section of porcine thoracic aortas shows hill and valley pattern corresponding with their histology. The hills were more than three times stiffer than the valleys, indicating that the hills are the ELs and the valleys the SMLs, and the ELs are compressed and the SMLs stretched in the lamellar unit. A finite element analysis showed that the residual stress in the EL and the SML is much higher than those estimated in the unloaded ring-like segments. Fairly large stress may still reside in the opened-up aortic wall. r 2003 Elsevier Ltd. All rights reserved. Keywords: Lamellar unit; Indentation test; Heterogeneity; Stress analysis; Surface topography; Finite element analysis
1. Introduction Since Vaishnav and Vossoughi (1983) and Fung (1984) independently pointed out the existence of residual stress in the artery wall, many studies have been carried out on this subject. A ring-like segment of an aorta springs open to form an arc when it is cut radially. This indicates that there existed compressive residual stress near the inner wall and tensile near the outer before the radial cut. Such residual stress reduces stress concentration which would appear in the inner wall of a pressurized vessel, and many studies suggest that circumferential stress in the artery wall distributes uniformly in the radial direction in vivo. In most of these studies, the opened-up configuration has been assumed to be stress-free. This assumption would be correct as long as the wall is homogeneous. *Corresponding author. Tel./fax: +81-52-735-5049. E-mail address: [email protected] (T. Matsumoto). 0021-9290/$ - see front matter r 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.jbiomech.2003.08.014
However, the wall is not homogeneous in a microscopic level. For example, the aortic media is made of concentric layers whose unit is called a lamellar unit (Wolinsky and Glagov, 1967), a pair of elastic lamina (EL) and a smooth muscle-rich layer (SML). Elastic modulus of elastin is about 0.6 MPa (Fung, 1981), while that of the smooth muscle is in an order of 0.01 MPa in the relaxed state (Matsumoto et al., 2000). Thus, the EL is supposed to be much stiffer than the SML. To know the stiffness difference, we measured the change in thickness of the two layers during radial compression, and found that the EL was about 2.5 times stiffer than the SML (Matsumoto et al., 2003). If the stress–strain relationship is different between the EL and SML, residual stress of the two layers should be different. For example, if the circumferential stress in the in vivo condition is the same between the soft and stiff layers, compressive residual stress will reside in the stiff layer and tensile in the soft layer (Fig. 1). Such residual stress is not released fully by the radial cutting,
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Fig. 1. Residual stress caused by material heterogeneity. If the tensile stresses in the layers A and B consisting a body are equal to each other in a loaded state, the unloaded body has tensile stress in the soft layer (A) and compressive stress in the stiff layer (B) because the body collapses until the sum of the forces in layers A and B becomes zero. The body was assumed to have only two layers with equal thickness for simplicity.
but is released in the area close to the cut surface, causing hills and valleys on the surface due to compressive and tensile stresses, respectively. To check this hypothesis, we have developed a scanning micro-indentation tester (SMIT), a scaled-up version of an atomic force microscope (AFM), and measured the surface topography and the stiffness distribution of the cut surface of the porcine thoracic aortas, and estimated residual stress and strain in the lamellar unit with a finite element (FE) analysis.
2. SMIT Schematic diagram of the SMIT is shown in Fig. 2. Basic principle of its operation is the same as that of the AFM. Surface topography and stiffness distribution of a specimen surface are measured by pressing a cantilever tip against the specimen surface while scanning it like the AFM in the contact mode. In conventional AFMs, it is not possible to measure the surface topography with large height difference because the length of the cantilever is B200 mm and the tip height is B3 mm. The measurement area is not large, either. To measure the surface topography with height difference from 1 to 500 mm in the area of 10 mm 10 mm, we developed our scaled-up version of the AFM. The cantilever was made
Fig. 2. Schematic diagrams of the scanning micro indentation tester.
from a micro-glass plate of 65 mm 1 mm 0.15 mm (Asahi Techno Glass Corp., Japan) by pulling it with a pipette puller (PP-83, Narishige Co., Ltd., Japan) to make its tip diameter 3–5 mm and bending it with a microforge (MF-90, Narishige Co., Ltd., Japan) at right angle at the point 4 mm from the tip. The spring constant of the cantilever was less than 0.3 N/mm. The cantilever was mounted to an arm driven by a PZT actuator (AE505D1, Tokin Corp., Japan) and the displacement of the cantilever tip d was measured with a confocal laser displacement meter (LT8100, Keyence Corp., Japan) with the resolution of 0.2 mm. The measurement was done in a bath filled with a physiological saline solution at room temperature. The specimen bath was set on a motor-driven XY stage (MINI60X, Sigma Koki Co., Ltd., Japan) to scan the point of measurement with the resolution of 0.5 mm. The whole setup was mounted on a vibration isolation table (PB-5AH, Showa Science, Japan). The cantilever deflection D was obtained by subtracting the measured displacement d from the ordered displacement d0 obtained from the voltage applied to the PZT actuator. The D d curve is called a force curve and its initial slope after the tip comes into contact with the specimen surface was used as a stiffness index a
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(Fig. 3a). The contact point was determined at the point where the cantilever began deflecting, i.e., the displacement at the contact point dc was determined as the displacement at which the second derivative of the deflection D reached its peak (Fig. 3b). The index a was converted to elastic modulus E with the following formula obtained by calibrating the tester with silicone elastomers with known elastic moduli: EðkPaÞ ¼1:88 exp ð2:54aÞ: The PZT actuator and the XY stage were controlled with a PC (DynaBook Satellite 2210 SA50C/4C8, Toshiba, Japan) equipped with an I/O board (DAQ Card-1200, National Instruments, USA). Data were measured and analyzed on the same PC. Control of the system and the data measurement were performed with a measurement and automation software LabVIEW (version 5.0; National Instruments, USA).
3. Surface topography and stiffness distribution 3.1. Experimental procedure Tubular segments of porcine thoracic aortas between the end of the aortic arch and the sixth intercostal artery were obtained at a local slaughterhouse. After adventitia was carefully removed, each of the segment was immersed in a physiological saline solution and stored frozen at 20 C until measurement to remove the contractility of the smooth muscle cells completely. Just before the measurement, it was thawed and rectangular specimens of 5 10 mm (2–3 mm thick) were cut out with their major edge aligned to circumferential or axial directions of the aorta. Macroscopic residual stresses are removed during excision. Each of the specimens was then embedded in an agar gel having low gelling temperature (30–31 C, Nacalai Tesque, Japan) with its major edge upwards, and sliced with a tissue sectioner (Microslicer DTK1500, Dosaka EM Co., Ltd., Japan) in the saline to obtain a section in the middle of the specimen
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perpendicular to the circumferential or axial directions. The microslicer slices specimens horizontally with a razor blade vibrating in the direction of the blade edge like a saw. The frequency and the amplitude of the vibration were set to 40–50 Hz and 2 mm, respectively. The specimens were cut in the direction parallel to the lamellar unit at a cutting speed of 2 mm/min. The specimen with its bath was then mounted on the XY stage of the SMIT for the measurement. The indentation test was performed in the area of 100 mm 100 mm at 2 mm-intervals to obtain the surface topography and the distribution of the stiffness index a: In addition to the specimens mentioned above, formalin-fixed specimens were used to confirm that flat surface could be obtained in this setup and to estimate the depth of the residual stress release. The specimens were thawed and fixed with 10% formalin overnight, and embedded and sliced similarly to obtain a surface perpendicular to the circumferential direction (first section). Some of them were scanned immediately for the surface topography of the first section. The rest was sliced again to obtain the second section by removing a thin slice with various thickness from the first section, and scanned for the topography.
3.2. Experimental results Examples of the surface topography and the stiffness distribution of the non-fixed section perpendicular to the circumferential and the axial directions are shown in Figs. 4 and 5, respectively. The surfaces of the sections show hill and valley pattern perpendicular to the radial direction. The stiffness was higher in the hills than in the valleys. Similar topography and the stiffness distribution were obtained for all specimens tested (20 specimens for the section perpendicular to the circumferential direction and four for the axial direction). No significant difference was found between the sections with different directions. The average distance between the peaks and the peak height were B30 and B8 mm , respectively. The elastic modulus estimated from a was B180 kPa at the
Fig. 3. An example of the force curve (a) and its second derivative (b). The displacement at the contact point dc and the stiffness index a were obtained from the curves.
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Fig. 4. An example of the surface topography (a) and the distribution of the stiffness index (b) measured on a section perpendicular to the circumferential direction of a porcine thoracic aorta. Measurement area, 100 mm 100 mm.
peak and B50 kPa at the bottom. On a separate study, we observed the change in the thickness of the EL and the SML in response to radial compression of the whole wall and found that the elastic modulus of the EL is about 2.5 times higher than that of the SML (Matsumoto et al., 2003). Taking together, the hill was supposed to be the EL and the valley the SML, i.e., the EL may be compressed and the SML stretched in the unloaded lamellar unit. Fig. 6 shows examples of the surface topography of the formalin-fixed specimens. Fig. 6a is the topography of the first section. Figs. 6b and c are of the second sections after the removal of a thin slice whose thickness was 80 and 70 mm, respectively. The surface topography of Figs. 6a and b were essentially similar to that of nonfixed specimens, while that of Fig. 6c was almost flat.
Fig. 5. An example of the surface topography (a) and the distribution of the stiffness index (b) measured on a section perpendicular to the axial direction of a porcine thoracic aorta. Measurement area, 100 mm 100 mm.
Similar results were obtained for all formalin-fixed specimens whose topography was measured (seven specimens for the first section, three for 80 mm, five for 70 mm). These results indicate that the residual stress did not disappear in the formalin-fixed specimens and was not released at 80 mm from the surface but was released to the depth of 70 mm, i.e., the depth of the residual stress release is estimated to be 70 mm.
4. Estimation of residual stress and strain 4.1. FE model Residual stress and strain in the EL and the SML were estimated from the surface topography and the
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Fig. 6. Examples of the surface topography of the sections of the formalin-fixed specimens obtained under various conditions: (a) a section obtained by slicing a specimen at its middle (first section); (b) a section obtained after removing a 80 mm thick slice from the first section; (c) similar to (b) but the thickness of the slice removed from the first section was 70 mm. All sections are perpendicular to the circumferential direction. Measurement area, 100 mm 100 mm.
elastic modulus measured in the previous section. Deformation of the cut surface following the sectioning was analyzed with a 2D FE model as schematically shown in Fig. 7. Circumferential–radial ðy rÞ section of a lamellar unit was modeled under the plane strain condition. The number of elements was 720 for the EL and 2400 for the SML. Following assumptions were made: (1) each layer is homogeneous and has uniform thickness before cut; (2) the EL is 6 mm thick, 150 mm high and uniformly compressed with stress s0EL ; while the SML is 22 mm thick, 150 mm high and uniformly stretched with stress s0SML before cut; (3) the EL and the SML are linearly elastic with the elastic modulus of 180 and 50 kPa, respectively; (4) layers are connected to each other and nearly incompressible with the Poisson’s ratio
of 0.49; (5) the model is constrained in the circumferential (vertical in the figure) direction at its bottom and in radial (horizontal in the figure) direction at both sides. Deformation and the stress and strain distributions following the slicing, i.e., upon the release of constraints on the top surface, were obtained for four different levels of initial residual stress. Residual stresses estimated in the simplified 2D model neglecting the connection between the layers, i.e., s0EL ¼ 12:6 kPa and s0SML ¼ 3:5 kPa, were used for Case 1 (Matsumoto et al., 2003). In Case n, the residual stresses in the two layers were multiplied by n ðn ¼ 124Þ: Analyses were performed with ABAQUS ver.5.8 (HKS, Inc.) at Center for Information and Media Studies, Nagoya Institute of Technology.
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4.2. Analytical result
Fig. 7. Diagrammatic representation of the finite element model of a lamellar unit and its constraints. Circumferential–radial ðy rÞ section of the smooth muscle-rich layer (SML) and two halves of the elastic lamina (EL) were modeled under the plane strain condition.
Deformation of the model following cut and resultant distribution of circumferential stress are graphically summarized in Fig. 8 for each case. As observed in Figs. 4 and 5, the EL became hill and the SML valley. However, the height difference between the hills and valleys Dh was smaller than the measured height difference (8 mm) in all cases. The height difference Dh was plotted against the amount of initial residual stress s0 for each case in Fig. 9. The relationship between the two parameters was almost linear. The stress which cause the measured height difference (8 mm) was considered to be the microscopic residual stress, and was estimated to be 70 kPa for the EL and 20 kPa for the SML from the linear regression line drawn in the figure. The depthwise distributions of the residual stress in the EL and the SML are summarized in Fig. 10.
Fig. 8. Distributions of the circumferential stress in the FE model shown in Fig. 7. Initial residual stresses s0EL and s0SML (kPa) are 12.6 and 3.5 for Case 1, 25.2 and 7.0 for Case 2, 37.8 and 10.5 for Case 3, 50.4 and 14.0 for Case 4, respectively.
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Circumferential stress on the centerline of the EL and the SML were normalized with the initial residual stress s0EL and s0SML, respectively, in the four cases. The normalized distributions in each layer were similar among the cases. Residual stress was released only in the region adjacent to the surface: more than 50% of the stress before slicing resides in the region deeper than 8 mm from the surface in the EL and 14 mm for the SML.
Fig. 9. Relationship between the amount of residual stress and the height difference between the hill and the valley Dh obtained in the FE analysis.
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5. Discussion 5.1. Residual stress analysis from a microscopic viewpoint Residual stress in soft biological tissues has been analyzed from a macroscopic viewpoint. Global changes in the specimen shape following cutting were used to obtain the measure of residual stress. Such approach would be enough if the specimen is considered to be homogeneous. Soft biological tissues are, however, not homogeneous, but made of materials with various elastic properties, i.e., cells, fibrous proteins like collagen and elastin, and other matrices such as proteoglycan. Therefore, residual stress in the soft tissues can be very much complicated in the spacial resolution of 1–10 mm. Without such knowledge on microscopic residual stress, we cannot estimate the stresses applied to the cells embedded in the tissue. It is indispensable to know the microscopic distribution of residual stress and strain while studying the mechanical responses of the soft biological tissues to applied load. When you cut a specimen having microscopic residual stress, the stress component perpendicular to the section is released in the region adjacent to the section, causing bumps and dents on the surface. Such component can be estimated as the stress necessary to restore the surface flat, i.e., it can be obtained from the surface topography and the stiffness distribution of the section by combining finite element analysis. For this purpose, we
Fig. 10. Depthwise distributions of the residual stress in the EL and the SML. Circumferential stress on the centerline of the EL and the SML were normalized by the initial stress s0EL and s0SML ; respectively in the four cases.
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developed the SMIT with wide ranges of measurement area (10 mm 10 mm) and surface roughness ðDh ¼ 12500 mmÞ; both of which the conventional AFMs cannot meet, and measured the surface topography and the stiffness distribution of the specimen surface sectioned with a commercially available tissue sectioner. This system can be applicable to the microscopic residual stress analysis of other soft tissues. 5.2. Factors affecting the surface topography In this study, we froze and thawed the aortic specimens before measurement to deprive the smooth muscle cells of their contractility which would cause an artifact to the surface topography. However, the freezing may also cause physical damage to the cells and their intracellular contents may dissolve into the saline solution when they are cut. This may cause the valley in the SML. If the arterial wall deforms so as to increase the wall thickness after the sectioning, it would also cause the valley in the SML, because the softer SML become wider than the stiffer EL. To check these hypotheses, we measured the surface topography of the formalin-fixed specimens and obtained the similar hill and valley pattern (Fig. 6a). Elastin is not fixed firmly with formalin, while smooth muscle and collagen are (Fung, 1981). According to our radial compression test, elastic modulus is almost the same between the SML and the EL in formalin-fixed aortas (unpublished data). Thus, the dissolving of the intracellular components and the selective widening of the SML would be minor in the formalin-fixed specimens. And also, the hill and valley topography disappeared when the thin layer with the thickness of 70 mm was removed from the first section (Fig. 6c). This indicates that the irregular topography of the first section was caused by the release of the microscopic residual stress, and the irregularity was then removed by ‘planing’ the thin layer free from residual stress. Although the effects of the dissolving of the intracellular components and the selective widening of the SML cannot be fully excluded, we believe that the hill and valley topography was caused by the release of the microscopic residual stresses. 5.3. Physiological significance of the microscopic residual stresses This study indicated that the EL is in compression and the SML tension in the excised specimen. This can be confirmed from the histology of the aortic wall: the EL is usually wavy in the histological sections of the unloaded aorta. It is hard to assume that the stressfree configuration of the EL is such. Thus, the ELs are supposed to be compressed and buckled in the unloaded aortic wall and in return, the SMLs are stretched. Why such residual stresses appear in the aortic wall? One of the possible explanations would be that, as
schematically shown in Fig. 1, the stress in the EL are almost equal to that in the SML in a loaded state. It would be advantageous from the mechanical viewpoint if the two layers bear tensile stress equally. Are the stresses in the EL and the SML equal in a physiological state? This would be a very interesting question to solve. The heterogeneous residual stresses in the two layers were observed not only in the circumferential direction but also in the radial direction with similar magnitude. Generally speaking, in vivo strain in the artery wall is larger in the circumferential direction than in the axial. For example, it is 85.673.2% (mean7SEM, n ¼ 5) in the circumferential direction and 37.271.7% in the axial in the rat thoracic aortas (Matsumoto and Hayashi, 1996). Thus, the microscopic residual stress is expected to be larger in the circumferential direction than in the axial. The reason for this discrepancy should also be examined. 5.4. Residual stress estimated from the FE analysis The residual stress was estimated to be 70 kPa for the EL and 20 kPa for the SML. These values are relatively large. With regard to the circumferential direction, the macroscopic residual stress reported in various literature is in the range of 1–10 kPa and the physiological stress is about 300 kPa for the rat thoracic aortas (Matsumoto and Hayashi, 1994). The values obtained in the present analysis is considered to be the upper limit of the estimation for the following reasons. First, Dh might be smaller than the measured value. As stated in Section 5.2, the loss of the intracellular components and the selective widening of the SML cannot be fully excluded, although it is difficult to evaluate these effects quantitatively at this moment. Second and most probable one is that the EL and the SML were assumed to be isotropic in the analysis. Most of the components including smooth muscle cells and bundles of elastin and collagen are aligned in the circumferential direction both in the EL and the SML (Clark and Glagov, 1985). Cytoskeletal fibers in the smooth muscle cells also aligned in the same direction. Thus, tensile modulus is supposed to be much lower in the radial direction, i.e., across the fibers, than the circumferential direction, i.e., along the fibers, and so is the shear modulus. If this is the case, the height difference between the EL and the SML may increase because the constraint between the layers become smaller. It is necessary to measure the shear properties of artery wall as the next step. 5.5. Difference in the depth of stress release between the experiment and the analysis The depth of the residual stress release was estimated to be 70 mm in the measurement, while in the analysis, it was 10–20 mm. We used the formalin-fixed specimens for this purpose because it was not possible to slice
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non-fixed specimens with the thickness smaller than 100 mm. As noted in Section 5.2, the SML stiffens much more than the EL when fixed with formalin. Thus, the depth of the stress release should become smaller in the fixed specimens. Even though, the depth was larger in the experiment with the formalin-fixed specimens than the analysis, the reason for this discrepancy would again be that the layers were assumed to be isotropic in the analysis. If the shear modulus is smaller than the assumption, the depth may increase because the constraint between the layers become smaller. The discrepancy would also be attributable to heterogeneity in the SML. If its whole volume become stress-free when a cell is cut transversely, the depth of the stress release may correspond to the depth from the cut surface to the edge of the cell embedded in the EL. If this is the case, a half of the cell length would be an expected value of the depth of the stress release. The length of the smooth muscle cells in the wall is estimated to be 50–100 mm. The expected depth would be 50 mm and is close to the measured value (70 mm). This may also explain the drastic change in the surface topography between the slice thickness of 70 and 80 mm. In summary, we have developed the scanning micro indentation tester to measure surface topography and stiffness distribution on the section of soft biological tissues, proposed a novel method to estimate microscopic residual stress in soft biological tissues by combining the measurements with the SMIT and a FE analysis, and applied this method to micro-stress analysis in the porcine thoracic aortas. Fairly large stress may still reside in the opened-up segment of the aorta, which had been believed to be stress-free. Microscopic viewpoint is necessary to reveal the mechanical environment of the smooth muscle cells in the aortic media.
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Acknowledgements We thank Mr. Yoshiki Ogawara for his superb technical assistance. This work was supported in part by Grant-in-Aid from the Ministry of Education, Science and Culture in Japan (T. Matsumoto, Nos. 12558101 and 13450040).
References Clark, J., Glagov, S., 1985. Transmural organization of the arterial media. The lamellar unit revisited. Arteriosclerosis, Thrombosis, and Vascular Biology 5, 19–34. Fung, Y.C., 1981. Biomechanics. Springer, New York, pp. 196–214. Fung, Y.C., 1984. Biodynamics: Circulation. Springer, New York, pp. 54–66. Matsumoto, T., Hayashi, K., 1994. Mechanical and dimensional adaptation of rat aorta to hypertension. ASME Journal of Biomechanical Engineering 116, 278–283. Matsumoto, T., Hayashi, K., 1996. Analysis of stress and strain distributions in hypertensive and normotensive rat aorta considering residual strain. Journal of Biomechanical Engineering 118, 62–73. Matsumoto, T., Sato, J., Yamamoto, M., Sato, M., 2000. Smooth muscle cells freshly isolated from rat thoracic aortas are much stiffer than cultured bovine cells: possible effect of phenotype. JSME International Journal, Series C 43, 867–874. Matsumoto, T., Goto, T., Sato, M., 2003. Residual stress and strain in the artery wall: from macroscopic to microscopic viewpoint. Proceedings of Advanced Technology in Experimental Mechanics 2003, (in CD-ROM), 0S07W0160. Vaishnav, R.N., Vossoughi, J., 1983. Estimation of residual strains in aortic segments. In: Hall, C.W. (Ed.), Biomedical Engineering II, Recent Developments. Pergamon Press, New York, pp. 330–333. Wolinsky, H., Glagov, S., 1967. A lamellar unit of aortic medial structure and function in mammals. Circlation Research 20, 99–111.
Journal of Biomechanics 37 (2004) 807–815
ASB-Microstrain Award 2002
Residual stress and strain in the lamellar unit of the porcine aorta: experiment and analysis Takeo Matsumotoa,b,*, Taisuke Gotob, Takao Furukawaa, Masaaki Satob a
Biomechanics Laboratory, Department of Mechanical Engineering, Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya 466-8555, Japan b Biomechanics Laboratory, Graduate School of Mechanical Engineering, Tohoku University, Aramaki-Aoba 01, Sendai, Miyagi 980-8579. Japan Accepted 2 August 2003
Abstract The opened-up configuration of the artery wall has long been assumed to be stress-free. This is questionable in a microscopic level. The aortic media is made of concentric layers whose unit is called a lamellar unit, a pair of elastic lamina (EL) and a smooth musclerich layer (SML). Recently, we found that the EL was about 2.5 times stiffer than the SML. If the circumferential stress in the in vivo condition is the same between the two layers, residual stress of each layer should be different because the stress–strain relationships differ. Such residual stress is not released fully by radial cutting, but is released in the area close to the cut surface, causing hills and valleys on the surface due to residual stress. To check this hypothesis, we have developed a scanning micro indentation tester, a scaled-up version of the atomic force microscope, and measured the topography and the stiffness distribution of the cut surface. The surface of the section of porcine thoracic aortas shows hill and valley pattern corresponding with their histology. The hills were more than three times stiffer than the valleys, indicating that the hills are the ELs and the valleys the SMLs, and the ELs are compressed and the SMLs stretched in the lamellar unit. A finite element analysis showed that the residual stress in the EL and the SML is much higher than those estimated in the unloaded ring-like segments. Fairly large stress may still reside in the opened-up aortic wall. r 2003 Elsevier Ltd. All rights reserved. Keywords: Lamellar unit; Indentation test; Heterogeneity; Stress analysis; Surface topography; Finite element analysis
1. Introduction Since Vaishnav and Vossoughi (1983) and Fung (1984) independently pointed out the existence of residual stress in the artery wall, many studies have been carried out on this subject. A ring-like segment of an aorta springs open to form an arc when it is cut radially. This indicates that there existed compressive residual stress near the inner wall and tensile near the outer before the radial cut. Such residual stress reduces stress concentration which would appear in the inner wall of a pressurized vessel, and many studies suggest that circumferential stress in the artery wall distributes uniformly in the radial direction in vivo. In most of these studies, the opened-up configuration has been assumed to be stress-free. This assumption would be correct as long as the wall is homogeneous. *Corresponding author. Tel./fax: +81-52-735-5049. E-mail address: [email protected] (T. Matsumoto). 0021-9290/$ - see front matter r 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.jbiomech.2003.08.014
However, the wall is not homogeneous in a microscopic level. For example, the aortic media is made of concentric layers whose unit is called a lamellar unit (Wolinsky and Glagov, 1967), a pair of elastic lamina (EL) and a smooth muscle-rich layer (SML). Elastic modulus of elastin is about 0.6 MPa (Fung, 1981), while that of the smooth muscle is in an order of 0.01 MPa in the relaxed state (Matsumoto et al., 2000). Thus, the EL is supposed to be much stiffer than the SML. To know the stiffness difference, we measured the change in thickness of the two layers during radial compression, and found that the EL was about 2.5 times stiffer than the SML (Matsumoto et al., 2003). If the stress–strain relationship is different between the EL and SML, residual stress of the two layers should be different. For example, if the circumferential stress in the in vivo condition is the same between the soft and stiff layers, compressive residual stress will reside in the stiff layer and tensile in the soft layer (Fig. 1). Such residual stress is not released fully by the radial cutting,
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Fig. 1. Residual stress caused by material heterogeneity. If the tensile stresses in the layers A and B consisting a body are equal to each other in a loaded state, the unloaded body has tensile stress in the soft layer (A) and compressive stress in the stiff layer (B) because the body collapses until the sum of the forces in layers A and B becomes zero. The body was assumed to have only two layers with equal thickness for simplicity.
but is released in the area close to the cut surface, causing hills and valleys on the surface due to compressive and tensile stresses, respectively. To check this hypothesis, we have developed a scanning micro-indentation tester (SMIT), a scaled-up version of an atomic force microscope (AFM), and measured the surface topography and the stiffness distribution of the cut surface of the porcine thoracic aortas, and estimated residual stress and strain in the lamellar unit with a finite element (FE) analysis.
2. SMIT Schematic diagram of the SMIT is shown in Fig. 2. Basic principle of its operation is the same as that of the AFM. Surface topography and stiffness distribution of a specimen surface are measured by pressing a cantilever tip against the specimen surface while scanning it like the AFM in the contact mode. In conventional AFMs, it is not possible to measure the surface topography with large height difference because the length of the cantilever is B200 mm and the tip height is B3 mm. The measurement area is not large, either. To measure the surface topography with height difference from 1 to 500 mm in the area of 10 mm 10 mm, we developed our scaled-up version of the AFM. The cantilever was made
Fig. 2. Schematic diagrams of the scanning micro indentation tester.
from a micro-glass plate of 65 mm 1 mm 0.15 mm (Asahi Techno Glass Corp., Japan) by pulling it with a pipette puller (PP-83, Narishige Co., Ltd., Japan) to make its tip diameter 3–5 mm and bending it with a microforge (MF-90, Narishige Co., Ltd., Japan) at right angle at the point 4 mm from the tip. The spring constant of the cantilever was less than 0.3 N/mm. The cantilever was mounted to an arm driven by a PZT actuator (AE505D1, Tokin Corp., Japan) and the displacement of the cantilever tip d was measured with a confocal laser displacement meter (LT8100, Keyence Corp., Japan) with the resolution of 0.2 mm. The measurement was done in a bath filled with a physiological saline solution at room temperature. The specimen bath was set on a motor-driven XY stage (MINI60X, Sigma Koki Co., Ltd., Japan) to scan the point of measurement with the resolution of 0.5 mm. The whole setup was mounted on a vibration isolation table (PB-5AH, Showa Science, Japan). The cantilever deflection D was obtained by subtracting the measured displacement d from the ordered displacement d0 obtained from the voltage applied to the PZT actuator. The D d curve is called a force curve and its initial slope after the tip comes into contact with the specimen surface was used as a stiffness index a
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(Fig. 3a). The contact point was determined at the point where the cantilever began deflecting, i.e., the displacement at the contact point dc was determined as the displacement at which the second derivative of the deflection D reached its peak (Fig. 3b). The index a was converted to elastic modulus E with the following formula obtained by calibrating the tester with silicone elastomers with known elastic moduli: EðkPaÞ ¼1:88 exp ð2:54aÞ: The PZT actuator and the XY stage were controlled with a PC (DynaBook Satellite 2210 SA50C/4C8, Toshiba, Japan) equipped with an I/O board (DAQ Card-1200, National Instruments, USA). Data were measured and analyzed on the same PC. Control of the system and the data measurement were performed with a measurement and automation software LabVIEW (version 5.0; National Instruments, USA).
3. Surface topography and stiffness distribution 3.1. Experimental procedure Tubular segments of porcine thoracic aortas between the end of the aortic arch and the sixth intercostal artery were obtained at a local slaughterhouse. After adventitia was carefully removed, each of the segment was immersed in a physiological saline solution and stored frozen at 20 C until measurement to remove the contractility of the smooth muscle cells completely. Just before the measurement, it was thawed and rectangular specimens of 5 10 mm (2–3 mm thick) were cut out with their major edge aligned to circumferential or axial directions of the aorta. Macroscopic residual stresses are removed during excision. Each of the specimens was then embedded in an agar gel having low gelling temperature (30–31 C, Nacalai Tesque, Japan) with its major edge upwards, and sliced with a tissue sectioner (Microslicer DTK1500, Dosaka EM Co., Ltd., Japan) in the saline to obtain a section in the middle of the specimen
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perpendicular to the circumferential or axial directions. The microslicer slices specimens horizontally with a razor blade vibrating in the direction of the blade edge like a saw. The frequency and the amplitude of the vibration were set to 40–50 Hz and 2 mm, respectively. The specimens were cut in the direction parallel to the lamellar unit at a cutting speed of 2 mm/min. The specimen with its bath was then mounted on the XY stage of the SMIT for the measurement. The indentation test was performed in the area of 100 mm 100 mm at 2 mm-intervals to obtain the surface topography and the distribution of the stiffness index a: In addition to the specimens mentioned above, formalin-fixed specimens were used to confirm that flat surface could be obtained in this setup and to estimate the depth of the residual stress release. The specimens were thawed and fixed with 10% formalin overnight, and embedded and sliced similarly to obtain a surface perpendicular to the circumferential direction (first section). Some of them were scanned immediately for the surface topography of the first section. The rest was sliced again to obtain the second section by removing a thin slice with various thickness from the first section, and scanned for the topography.
3.2. Experimental results Examples of the surface topography and the stiffness distribution of the non-fixed section perpendicular to the circumferential and the axial directions are shown in Figs. 4 and 5, respectively. The surfaces of the sections show hill and valley pattern perpendicular to the radial direction. The stiffness was higher in the hills than in the valleys. Similar topography and the stiffness distribution were obtained for all specimens tested (20 specimens for the section perpendicular to the circumferential direction and four for the axial direction). No significant difference was found between the sections with different directions. The average distance between the peaks and the peak height were B30 and B8 mm , respectively. The elastic modulus estimated from a was B180 kPa at the
Fig. 3. An example of the force curve (a) and its second derivative (b). The displacement at the contact point dc and the stiffness index a were obtained from the curves.
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Fig. 4. An example of the surface topography (a) and the distribution of the stiffness index (b) measured on a section perpendicular to the circumferential direction of a porcine thoracic aorta. Measurement area, 100 mm 100 mm.
peak and B50 kPa at the bottom. On a separate study, we observed the change in the thickness of the EL and the SML in response to radial compression of the whole wall and found that the elastic modulus of the EL is about 2.5 times higher than that of the SML (Matsumoto et al., 2003). Taking together, the hill was supposed to be the EL and the valley the SML, i.e., the EL may be compressed and the SML stretched in the unloaded lamellar unit. Fig. 6 shows examples of the surface topography of the formalin-fixed specimens. Fig. 6a is the topography of the first section. Figs. 6b and c are of the second sections after the removal of a thin slice whose thickness was 80 and 70 mm, respectively. The surface topography of Figs. 6a and b were essentially similar to that of nonfixed specimens, while that of Fig. 6c was almost flat.
Fig. 5. An example of the surface topography (a) and the distribution of the stiffness index (b) measured on a section perpendicular to the axial direction of a porcine thoracic aorta. Measurement area, 100 mm 100 mm.
Similar results were obtained for all formalin-fixed specimens whose topography was measured (seven specimens for the first section, three for 80 mm, five for 70 mm). These results indicate that the residual stress did not disappear in the formalin-fixed specimens and was not released at 80 mm from the surface but was released to the depth of 70 mm, i.e., the depth of the residual stress release is estimated to be 70 mm.
4. Estimation of residual stress and strain 4.1. FE model Residual stress and strain in the EL and the SML were estimated from the surface topography and the
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Fig. 6. Examples of the surface topography of the sections of the formalin-fixed specimens obtained under various conditions: (a) a section obtained by slicing a specimen at its middle (first section); (b) a section obtained after removing a 80 mm thick slice from the first section; (c) similar to (b) but the thickness of the slice removed from the first section was 70 mm. All sections are perpendicular to the circumferential direction. Measurement area, 100 mm 100 mm.
elastic modulus measured in the previous section. Deformation of the cut surface following the sectioning was analyzed with a 2D FE model as schematically shown in Fig. 7. Circumferential–radial ðy rÞ section of a lamellar unit was modeled under the plane strain condition. The number of elements was 720 for the EL and 2400 for the SML. Following assumptions were made: (1) each layer is homogeneous and has uniform thickness before cut; (2) the EL is 6 mm thick, 150 mm high and uniformly compressed with stress s0EL ; while the SML is 22 mm thick, 150 mm high and uniformly stretched with stress s0SML before cut; (3) the EL and the SML are linearly elastic with the elastic modulus of 180 and 50 kPa, respectively; (4) layers are connected to each other and nearly incompressible with the Poisson’s ratio
of 0.49; (5) the model is constrained in the circumferential (vertical in the figure) direction at its bottom and in radial (horizontal in the figure) direction at both sides. Deformation and the stress and strain distributions following the slicing, i.e., upon the release of constraints on the top surface, were obtained for four different levels of initial residual stress. Residual stresses estimated in the simplified 2D model neglecting the connection between the layers, i.e., s0EL ¼ 12:6 kPa and s0SML ¼ 3:5 kPa, were used for Case 1 (Matsumoto et al., 2003). In Case n, the residual stresses in the two layers were multiplied by n ðn ¼ 124Þ: Analyses were performed with ABAQUS ver.5.8 (HKS, Inc.) at Center for Information and Media Studies, Nagoya Institute of Technology.
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4.2. Analytical result
Fig. 7. Diagrammatic representation of the finite element model of a lamellar unit and its constraints. Circumferential–radial ðy rÞ section of the smooth muscle-rich layer (SML) and two halves of the elastic lamina (EL) were modeled under the plane strain condition.
Deformation of the model following cut and resultant distribution of circumferential stress are graphically summarized in Fig. 8 for each case. As observed in Figs. 4 and 5, the EL became hill and the SML valley. However, the height difference between the hills and valleys Dh was smaller than the measured height difference (8 mm) in all cases. The height difference Dh was plotted against the amount of initial residual stress s0 for each case in Fig. 9. The relationship between the two parameters was almost linear. The stress which cause the measured height difference (8 mm) was considered to be the microscopic residual stress, and was estimated to be 70 kPa for the EL and 20 kPa for the SML from the linear regression line drawn in the figure. The depthwise distributions of the residual stress in the EL and the SML are summarized in Fig. 10.
Fig. 8. Distributions of the circumferential stress in the FE model shown in Fig. 7. Initial residual stresses s0EL and s0SML (kPa) are 12.6 and 3.5 for Case 1, 25.2 and 7.0 for Case 2, 37.8 and 10.5 for Case 3, 50.4 and 14.0 for Case 4, respectively.
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Circumferential stress on the centerline of the EL and the SML were normalized with the initial residual stress s0EL and s0SML, respectively, in the four cases. The normalized distributions in each layer were similar among the cases. Residual stress was released only in the region adjacent to the surface: more than 50% of the stress before slicing resides in the region deeper than 8 mm from the surface in the EL and 14 mm for the SML.
Fig. 9. Relationship between the amount of residual stress and the height difference between the hill and the valley Dh obtained in the FE analysis.
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5. Discussion 5.1. Residual stress analysis from a microscopic viewpoint Residual stress in soft biological tissues has been analyzed from a macroscopic viewpoint. Global changes in the specimen shape following cutting were used to obtain the measure of residual stress. Such approach would be enough if the specimen is considered to be homogeneous. Soft biological tissues are, however, not homogeneous, but made of materials with various elastic properties, i.e., cells, fibrous proteins like collagen and elastin, and other matrices such as proteoglycan. Therefore, residual stress in the soft tissues can be very much complicated in the spacial resolution of 1–10 mm. Without such knowledge on microscopic residual stress, we cannot estimate the stresses applied to the cells embedded in the tissue. It is indispensable to know the microscopic distribution of residual stress and strain while studying the mechanical responses of the soft biological tissues to applied load. When you cut a specimen having microscopic residual stress, the stress component perpendicular to the section is released in the region adjacent to the section, causing bumps and dents on the surface. Such component can be estimated as the stress necessary to restore the surface flat, i.e., it can be obtained from the surface topography and the stiffness distribution of the section by combining finite element analysis. For this purpose, we
Fig. 10. Depthwise distributions of the residual stress in the EL and the SML. Circumferential stress on the centerline of the EL and the SML were normalized by the initial stress s0EL and s0SML ; respectively in the four cases.
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developed the SMIT with wide ranges of measurement area (10 mm 10 mm) and surface roughness ðDh ¼ 12500 mmÞ; both of which the conventional AFMs cannot meet, and measured the surface topography and the stiffness distribution of the specimen surface sectioned with a commercially available tissue sectioner. This system can be applicable to the microscopic residual stress analysis of other soft tissues. 5.2. Factors affecting the surface topography In this study, we froze and thawed the aortic specimens before measurement to deprive the smooth muscle cells of their contractility which would cause an artifact to the surface topography. However, the freezing may also cause physical damage to the cells and their intracellular contents may dissolve into the saline solution when they are cut. This may cause the valley in the SML. If the arterial wall deforms so as to increase the wall thickness after the sectioning, it would also cause the valley in the SML, because the softer SML become wider than the stiffer EL. To check these hypotheses, we measured the surface topography of the formalin-fixed specimens and obtained the similar hill and valley pattern (Fig. 6a). Elastin is not fixed firmly with formalin, while smooth muscle and collagen are (Fung, 1981). According to our radial compression test, elastic modulus is almost the same between the SML and the EL in formalin-fixed aortas (unpublished data). Thus, the dissolving of the intracellular components and the selective widening of the SML would be minor in the formalin-fixed specimens. And also, the hill and valley topography disappeared when the thin layer with the thickness of 70 mm was removed from the first section (Fig. 6c). This indicates that the irregular topography of the first section was caused by the release of the microscopic residual stress, and the irregularity was then removed by ‘planing’ the thin layer free from residual stress. Although the effects of the dissolving of the intracellular components and the selective widening of the SML cannot be fully excluded, we believe that the hill and valley topography was caused by the release of the microscopic residual stresses. 5.3. Physiological significance of the microscopic residual stresses This study indicated that the EL is in compression and the SML tension in the excised specimen. This can be confirmed from the histology of the aortic wall: the EL is usually wavy in the histological sections of the unloaded aorta. It is hard to assume that the stressfree configuration of the EL is such. Thus, the ELs are supposed to be compressed and buckled in the unloaded aortic wall and in return, the SMLs are stretched. Why such residual stresses appear in the aortic wall? One of the possible explanations would be that, as
schematically shown in Fig. 1, the stress in the EL are almost equal to that in the SML in a loaded state. It would be advantageous from the mechanical viewpoint if the two layers bear tensile stress equally. Are the stresses in the EL and the SML equal in a physiological state? This would be a very interesting question to solve. The heterogeneous residual stresses in the two layers were observed not only in the circumferential direction but also in the radial direction with similar magnitude. Generally speaking, in vivo strain in the artery wall is larger in the circumferential direction than in the axial. For example, it is 85.673.2% (mean7SEM, n ¼ 5) in the circumferential direction and 37.271.7% in the axial in the rat thoracic aortas (Matsumoto and Hayashi, 1996). Thus, the microscopic residual stress is expected to be larger in the circumferential direction than in the axial. The reason for this discrepancy should also be examined. 5.4. Residual stress estimated from the FE analysis The residual stress was estimated to be 70 kPa for the EL and 20 kPa for the SML. These values are relatively large. With regard to the circumferential direction, the macroscopic residual stress reported in various literature is in the range of 1–10 kPa and the physiological stress is about 300 kPa for the rat thoracic aortas (Matsumoto and Hayashi, 1994). The values obtained in the present analysis is considered to be the upper limit of the estimation for the following reasons. First, Dh might be smaller than the measured value. As stated in Section 5.2, the loss of the intracellular components and the selective widening of the SML cannot be fully excluded, although it is difficult to evaluate these effects quantitatively at this moment. Second and most probable one is that the EL and the SML were assumed to be isotropic in the analysis. Most of the components including smooth muscle cells and bundles of elastin and collagen are aligned in the circumferential direction both in the EL and the SML (Clark and Glagov, 1985). Cytoskeletal fibers in the smooth muscle cells also aligned in the same direction. Thus, tensile modulus is supposed to be much lower in the radial direction, i.e., across the fibers, than the circumferential direction, i.e., along the fibers, and so is the shear modulus. If this is the case, the height difference between the EL and the SML may increase because the constraint between the layers become smaller. It is necessary to measure the shear properties of artery wall as the next step. 5.5. Difference in the depth of stress release between the experiment and the analysis The depth of the residual stress release was estimated to be 70 mm in the measurement, while in the analysis, it was 10–20 mm. We used the formalin-fixed specimens for this purpose because it was not possible to slice
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non-fixed specimens with the thickness smaller than 100 mm. As noted in Section 5.2, the SML stiffens much more than the EL when fixed with formalin. Thus, the depth of the stress release should become smaller in the fixed specimens. Even though, the depth was larger in the experiment with the formalin-fixed specimens than the analysis, the reason for this discrepancy would again be that the layers were assumed to be isotropic in the analysis. If the shear modulus is smaller than the assumption, the depth may increase because the constraint between the layers become smaller. The discrepancy would also be attributable to heterogeneity in the SML. If its whole volume become stress-free when a cell is cut transversely, the depth of the stress release may correspond to the depth from the cut surface to the edge of the cell embedded in the EL. If this is the case, a half of the cell length would be an expected value of the depth of the stress release. The length of the smooth muscle cells in the wall is estimated to be 50–100 mm. The expected depth would be 50 mm and is close to the measured value (70 mm). This may also explain the drastic change in the surface topography between the slice thickness of 70 and 80 mm. In summary, we have developed the scanning micro indentation tester to measure surface topography and stiffness distribution on the section of soft biological tissues, proposed a novel method to estimate microscopic residual stress in soft biological tissues by combining the measurements with the SMIT and a FE analysis, and applied this method to micro-stress analysis in the porcine thoracic aortas. Fairly large stress may still reside in the opened-up segment of the aorta, which had been believed to be stress-free. Microscopic viewpoint is necessary to reveal the mechanical environment of the smooth muscle cells in the aortic media.
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Acknowledgements We thank Mr. Yoshiki Ogawara for his superb technical assistance. This work was supported in part by Grant-in-Aid from the Ministry of Education, Science and Culture in Japan (T. Matsumoto, Nos. 12558101 and 13450040).
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