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Characterization of mechanical properties of lamellar structure of the aortic wall: Effect of aging
Author’s Accepted Manuscript Characterization of mechanical properties of lamellar structure of the aortic wall: Effect of aging Hadi Taghizadeh, Mohammad Tafazzoli-Shadpour
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S1751-6161(16)30271-5 http://dx.doi.org/10.1016/j.jmbbm.2016.08.011 JMBBM2030
To appear in: Journal of the Mechanical Behavior of Biomedical Materials Received date: 11 April 2016 Revised date: 18 July 2016 Accepted date: 3 August 2016 Cite this article as: Hadi Taghizadeh and Mohammad Tafazzoli-Shadpour, Characterization of mechanical properties of lamellar structure of the aortic wall: Effect of aging, Journal of the Mechanical Behavior of Biomedical Materials, http://dx.doi.org/10.1016/j.jmbbm.2016.08.011 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Characterization of mechanical properties of lamellar structure of the aortic wall: Effect of aging Hadi Taghizadeh§, Mohammad Tafazzoli-Shadpour¥* §
Division of Biomechanics, Mechanical Engineering Department, Sahand University of Technology, Tabriz 51335-1996, Iran; Email: [email protected]
¥
Cardiovascular Engineering Laboratory, Faculty of Biomedical Engineering, Amirkabir University of Technology, 424 Hafez Ave., Tehran 15875-4413, Iran; Email: [email protected]
*
Corresponding Author: Mohammad Tafazzoli-Shadpour, Cardiovascular Engineering Lab., Faculty of Biomedical Engineering, Amirkabir University of Technology, Tehran, Iran Email: [email protected]
Abstract: Arterial wall tissues are sensitive to their mechanical surroundings and remodel their structure and mechanical properties when subjected to mechanical stimuli such as increased arterial pressure. Such remodeling is evident in hypertension and aging. Aging is characterized by stiffening of the artery wall which is assigned to disturbed elastin function and increased collagen content. To better understand and provide new insight on microstructural changes induced by aging, the lamellar model of the aortic media was utilized to characterize and compare wall structure and mechanical behavior of the young and old human thoracic aortic samples. Such model regards arterial media as two sets of alternating concentric layers, namely sheets of elastin and interlamellar layers. Histological and biaxial tests were performed and microstructural features and stress-strain curves of media were evaluated in young and old age groups. Then using optimization algorithms and hyperelastic constitutive equations the stress-strain curves of layers were evaluated for both age groups. Results indicated slight elevation in the volume fraction of interlamellar layer among old subjects most probably due to age related collagen deposition. Aging indicated substantial stiffening of interlamellar layers accompanied by noticeable softening of elastic lamellae. The general significant stiffening of old samples were attributed to both increase of volume fraction of interlamellar layers and earlier recruitment of collagen fibers during load bearing due to functional loss of elastin within wall lamellae. Mechanical characterization of lamellar structure of wall media is beneficial in study of arterial remodeling in response to alternated mechanical environment in aging and clinical conditions through coupling of wall microstructure and mechanical behavior.
Keywords: Aging, Arterial wall, remodeling, lamellar structure, structure- mechanics coupling, constitutive modeling
Introduction The biological functions of cardiovascular tissues are directly correlated with their mechanical properties, i.e., proper function of the large arteries depends on their elasticity [1] and their elasticity is altered during physiologic and pathologic processes such as aging and hypertension [2]. Aging and hypertension are Associated with elevated blood pressure which deploys excessive circumferential tensile stress on the arterial wall. Cellular content of the wall, mainly smooth muscle cells (SMCs) and endothelial cells sense the extra tension through mechanotransduction pathways and synthesize fibrous content in the wall to restore their mechanical environment back to the normal conditions [3]. Such modifications alter the microstructure and hence, mechanical properties of the wall. Arterial walls possess a composite like lamellar structure of semi-concentric fibrous layers [4]. The lamellar structure can be regarded as repeated structural units through the thickness of the media [5]. With further focus on these structural units, two sets of sublayers are distinguished, i.e. highly elastic layers containing sheets of elastin, and interlamellar layers containing collagen bundles, fine elastic fibers, smooth muscle cells (SMCs) and ground substances such as proteoglycans [6, 7]. These two types of sublayers are mechanically different according to their structure and constituents. Elastic sheets contain homogenous elastin tissue while collagen fibers are arranged in preferred directions and these fibers are mechanically dominant among components of interlamellar layers. Such arrangement leads to anisotropic mechanical behavior of the arterial wall. A welldeveloped model of the arterial wall should be able to address these characteristics. Lamellar units of the tunica media is shown to provide a firm base toward constitutive modeling of the arteries in health and reflect disease-induced alterations [8-10]. There are some reported changes in the structure of the arterial wall with aging; The thickness of arterial media increases [11] and the amount of arterial elastin is reduced while the amount of collagen increases [12]. On the other hand, some suggest that aging does not necessarily decrease the physical content of elastin, however, fragmentation and loss of functionality of them are inevitable [13, 14]. Since collagen fibers are much stiffer than elastin fibers, age-related changes in collagen to elastin ratio lead to the well-established dilated and stiffened arterial wall [15]. Altered fibrillar content affects the coupling of the mechanical and biological environments of the artery and despite age-related remodeling of the arterial wall, aging is correlated with cardiovascular diseases [16-18]. Current research investigates the effects of aging on the mechanical behavior of the aortic microstructure utilizing the previously proposed lamellar model [9]. In the light of mentioned modeling framework, new insight on the aging process and alterations in mechanical performance of the lamellar structure in young and old aortic media will be provided. Materials and methods Biaxial tensile tests were carried out on human aortic samples of young and old subjects to characterize hyperelastic behavior of aortic tissue. Histological observations were concurrently performed on stained samples to observe structural layers of the wall media among young and old groups and provide needed microstructural data. Constitutive equations and optimization algorithms were recruited to characterize hyperelastic parameters of arterial lamellae and interlamellar layers. Utilizing this approach, we explored agerelated changes of the aortic wall based on lamellar modeling. This model couples microstructural information with mechanical behavior of the aortic media to better describe its mechanical performance. Human subjects under the age of 40 were categorized in young group and subjects older than 55 years were assigned to the old group. Such categorization originates from reports on the alteration in microstructure of the arterial walls with age [19]. The samples in the range of 40-55 years old were excluded from the comparison to address the age-induced alterations more clearly. Among each group, the average volume fractions of the major media layers and their respective mechanical properties were obtained and effects of aging were discussed. Specimen preparation Samples from thoracic aortas of 19 subjects, 10 young (31.4±8.1) and 9 old (61.3±3.1) samples, were provided from brain-dead patients that underwent organ donation surgery considering the ethical approval including consent of relatives. All steps of human tissue handling from extraction to experiments were conducted
according to the instructions of the Ethics Committee, Masih Daneshvari Hospital, the main center of organ donation in Iran. Healthy samples were harvested from descending region of the aorta above the diaphragm, since in this region branching and diameter changes are minimal and lesion formation occurs in fewer subjects. Tissue samples were preserved in Phosphate-Buffered Saline prior to experiments and the required tests are inquired in less than 48 hours post mortem. A thin ring of the tissue was separated for histological staining and the rest of the tissue was utilized for biaxial tensile tests. Data acquisition The proposed lamellar model employs microstructural and mechanical data to better approximate mechanical behavior of the aortic media. Histological staining of elastic fibers was performed to distinguish wall lamellae and interlamellar layers (Figure 1). Verhoeff Van Gieson (VVG) stain was utilized to distinguish the elastic lamellae in dark color within the bright interlamellar space. Microscopic images of tissue rings were captured and the volume fractions of two major layers were computed by developing an image-processing code in MATLAB. Planar biaxial tensile tests were carried out on at least two specimens from each subject. We employed the previously described protocol in biaxial testing of the samples [9]. Force-displacement data of the young and old samples were extracted and then converted to Second Piola stress-Green strain, the common stress-strain couple for mechanical description of hyperelastic materials [20, 21]. Continuum Preliminaries The non-linear elastic behavior of highly deformable soft tissues indicated these tissues are hyperelastic. Mechanical description of such materials necessitates adoption of large deformation continuum framework. In this context, deformation gradient relates reference and deformed configurations (F=∂x/∂X). The deformation gradient is composed of rotation (R) and deformation (U) parts (F=RU=VR). Rotation component is an orthogonal matrix with the property RRT=1 and main deformation tensors are defined not to depend on rotation component:
C FT F U 2 B FF T V 2 The Green-Lagrange strain tensor is defined by E
(1) 1 C 1 . 2
The first Piola stress is defined by force per initial section of the sample, which is related to the Cauchy stress, force per current section of the sample, by:
P J FT
(2)
Since the first Piola stress is not symmetric, the second Piola stress is defined as a symmetric tensor:
T F 1P JF 1 FT
(3)
For hyperelastic materials strain energy depends only on deformation field (deformation gradient, left or right Cauchy tensors and their invariants or Green strain):
ˆ E F C
(4)
Then the stress-strain relation of the material is obtained:
S 2
(C ) (E ) C E
(5)
The stress computation in eq. (5) is subjected to physical and computational constraints. For example, since arterial wall is considered to be incompressible [22], a Lagrange multiplier term is added to this relation to enforce incompressibility:
S 2
(C ) pC 1 C
(6)
The unknown parameter p is obtained by initial or boundary conditions of the problem. Additionally, arterial wall is anisotropic and appropriate selection of the strain energy density function (SEDF) is necessary for its mechanical characterization. Description of the Lamellar Model Data from histological and biaxial tests were fed into the lamellar model to evaluate mechanical properties of the aortic media layers. The lamellar modeling approach which is adopted in this study is described in our previous paper [9]. Briefly, we regard lamellar structure of the media, as a set of structural units. Additionally each structural unit was considered as a set of two sublayers: sheet of elastin (Layer I) described by the dark stained layers in Figure 1, and interstitial layer consisting of collagen bundles, fine elastin fibers and SMCs (Layer II). It is assumed that all of these lamellar units across the media thickness are identical in terms of the mechanical behavior of layer I and layer II respectively. Neo-Hookean and four parameter SEDFs were adopted for layer I and layer II, respectively (Eqs. 7 and 8 respectively) [9]. Neo-Hookean SEDF is suitable for capturing mechanical behavior of layer I regarding isotropic nature of elastic tissue [23, 24]. On the other hand, considering non-homogenous distribution and gradual uncrimp of collagen fibers in layer II, exponential SEDF is utilized to capture anisotropic and gradual stiffening of this layer. Then the overall mechanical behavior of the media is obtained by combining SEDFs of sublayers proportional to their fractional volumes.
WI c1 ( I1 3)
(7) (8)
W II c2 [exp(a1E12 a2E 22 2a3E1E 2 ) 1] W media f IW I f IIW II
(9)
In these equations, subscripts I and II denote layer I and layer II, respectively. Parameter W indicates SEDF, I1, E1 and E2 describe the deformation field of the material and finally f denotes volume fraction. The SEDF of the media shown in eq. (9) depends on five unknown material parameters, namely c1 for layer I, c2, a1, a2 and a3 for layer II. Then the SEDF in eq. (9) is substituted in eq. (6) to calculate stresses:
1 W p 2 1 1 1 S 0 0
0 1 W p 2 2 2 2 0
0 W 12 p 122 2 3 (1 , 2 ) 0
(10)
Where diagonal elements denote normal stress components in axial, circumferential and radial directions. It is shown that these directions coincide with principal directions of the arterial tissue and no shear stress will arise when test directions correspond to principal directions [25]. Hence in our formulation the shear deformations are not included and the respective components in stress tensor (off-diagonal elements) are zero. Since we carried out planar tests in axial-circumferential plane, the out of plane component of the stress will be zero (plane-stress assumption). Then the Lagrange multiplier term is obtained (
p
W I ) and axial 12 3 (1 , 2 ) 1
(Sa) and circumferential (Sc) stresses are determined accordingly.
Sa fI(
W I 1 W I 1 1 W II 3 ) f II 1 1 1 2 3 (1 , 2 ) 1 1
W I 1 W I 1 1 W II S fI( ) f II 3 2 2 12 3 (1 , 2 ) 2 2 c
On the other hand, the layer stresses are related to the overall stress of the media using eq. (9):
(11)
S a f I S Ia f II S IIa c c c S f I S I f II S II
(12)
Comparing eq. (11) with eq. (12), layer stresses are obtained as:
S Ia
W I 1 W I 1 3 1 1 1 2 3 (1 , 2 ) S IIa
1 W II 1 1
(13)
W I 1 W I 1 S 3 2 2 12 3 (1 , 2 ) c I
S IIc
1 W II 2 2
Using these relations, the stresses in layer I and layer II are explored. For equibiaxial tests ( 1 2 ), the axial a
and circumferential stresses on layer I ( S I and
S Ic
respectively) are equal because of Isotropic SEDF used
for this sublayer. Determining unknown hyperelastic materials The explained modeling approach was applied to experimental data from histological and biaxial tests. The volume fractions, fI and fII, were obtained from images of stained samples and together with biaxial test data were used to evaluate hyperelastic parameters described in previous section. For each test specimen the set of hyperelastic parameters that best interpolate experimental data were determined by nonlinear optimization. We developed a computer code in MATLAB environment and used “fmincon” function to best determine these material parameters (adopted error function is explained in [9]). Averaging mechanical behavior of young and old age groups After obtaining experimental data from histological and mechanical tests and applying nonlinear regression to allocate material parameters, we averaged volume fractions and the set of hyperelastic parameters among samples of each group to obtain the representative volume fractions and material parameters for the young and old age groups. It should be noted that reporting the averaged set of constitutive parameters, is the popular method of representing the overall behavior of biological tissues, however it can lead to unrealistic results [26]. These issues will not considerably affect our comparison since the same constitutive model is adopted for young and old groups, causing similar errors in each group. However, careful attention should be given to mentioned discrepancy while the extracted material parameters are intended for computational models. The volume fractions and mechanical properties of the young and old groups, are then compared via two-tailed T test [27] and p-values smaller than 0.05 are considered as significant. Results A typical experimental data of a young subject (25 years old) and the respective fit of the lamellar model are provided in Figure 2 and corresponding material parameters are presented in Table 1. Such data were obtained for each test specimen. Averaged material parameters representing young and old age groups are provided in Table 2. Besides, the average volume fractions obtained from histological tests for young and old groups are presented in the same Table. Comparison of the volume fractions revealed an average 4.6 percent increased volume fraction of interlamellar layer among old subjects. The averaged hyperelastic parameters obtained for old subjects were different from those of young subjects. Parameter c1 representing isotropic SEDF of layer I (elastic lamellae) were substantially lower in the old group which might be due to loss of function of elastin. The parameters representing SEDF of layer II show a dual difference, c2 is decreased while the individual exponent parameters
(a1, a2 and a3) and their summation are elevated in the old group. From a mathematical point of view, the SEDF of layer II indicates milder slope at the beginning and then faster increase later in the stress-strain curve for old subjects compared to young subjects. The correspondent averaged stress-strain curves for two age groups are plotted in Figure 3 for axial and circumferential directions. The contribution of layer I and layer II on the overall mechanical behavior of the aortic media in young and old samples is also presented in Figure 3. The stresses generated on each sublayer (layer I and layer II) are regarded as their mechanical performance and contribution to the overall mechanical behavior of the media and these performances are compared in young and old groups. There are three functional regions in stress-strain curve of the healthy young aortic samples as shown in top panel of Figure 3. In lower strains, the stress- strain curve of the media is nearly linear. With elevation of the strain, the linear curve is transitioning toward a gradually stiffening mechanical behavior. In higher strains the stiffening behavior is noticeable and an incremental increase in the strain leads to substantial increase in the stress. Mechanical performance, of layers I and II in young and old aortas are compared in panels of Figure 4. Both panels reveal marked changes by aging. Elastic tissue has markedly smaller share on mechanical performance in the old aorta compared to young aortas. On the other hand, mechanical properties of layer II in old group indicate significant stiffening compared to young group and the altered mechanism of collagen recruitment is implied in this panel. The latter effect is more effective in alterations of mechanical properties in aging because corresponding volume fraction of layer II is more than three times that of layer I (elastic sheets). Discussion According to our model, mechanical properties of the aortic media were assigned through two layers, namely layer I and layer II as defined by their constituents. The proposed lamellar model is efficient in describing mechanical behavior of the aortic media for all ranges of the tissue deformation including physiologic and supraphysiologic strains (Figure 2) [9]. Such modeling approach couples the geometrical changes of the arterial wall directly with the changes in mechanical properties. Hence, it may be employed in study of aging and pathological conditions such as hypertension and the response of wall media to disturbed environment through alterations in micro-structural elements and subsequently changes in mechanical properties and stress distribution [10]. Such response, known as arterial wall remodeling is triggered after sustained changes in stress distribution across the wall media sensed by media and intima cells and results in extra synthesis of fibrous content, and usually acts by a trend reversing the stress pattern of the arterial wall to normal conditions [28, 29]. In aging, the increase in level of tensile stresses is due to gradual increase of blood pressure, while in systemic hypertension the elevation of luminal pressure due to clinical conditions is faster. Alterations in the microstructure of the artery influence macro mechanical behavior of the artery by incorporation of volume fractions in the proposed lamellar model. Such multiscale model is beneficial to establish both micro and macro scale geometry- mechanics couplings. The increased volume fraction of the interlamellar layer (layer II) due to aging is most likely a result of collagen deposition within this sublayer by SMCs which are activated through altered tension that affects their contractility [30, 31]. Our results indicate that among old group, Layer I has lower stress compared to that of young subjects (as seen in the left panel of Fig. 4), most probably due to the fragmentation of the elastic fibers with age. However the mentioned stiffening is caused by alterations in mechanical properties of layer II (right panel of Fig. 4) through stiffening and thickening of these layers. This is compatible with the idea of collagen synthesis with age in arterial wall [19]. Since the media of the old group is stiffened compared to the media of young subjects, one can conclude that stiffening of the layer II in the old group is the dominant factor in wall stiffening. It has been suggested that aging causes fragmentation of elastin fibers and reduced cross-link of the elastic fibers due to applied repeated load through pulsatile luminal pressure and consequently disruption of the organized structure of elastin sheets in wall lamellae and fine elastin meshes within interlamellar zones [16, 32]. This phenomenon seems to be similar to effect of mechanical fatigue in materials caused by repeated loading. In lower strains of the arterial wall, collagen fibers are crimped and wavy and hence inactive. In this range, elastic tissue plays the dominant role in overall stress-bearing of the arterial media. With elevation of the strain, collagen fibers start uncrimping and gradually engage, and mechanical role of layer II becomes more prominent
and reaches to that of layer I. This region is regarded as a transition zone between low and high strain ranges. Mechanical outcome of microstructural sublayers are nearly the same in this range. It has been suggested that physiologic strains reside in this transition zone [33]. Finally in the third region which corresponds to higher strains, collagen fibers increasingly uncrimp and become engaged in mechanical response of the artery which is evident in Figure 1 by hardening stress-strain curve of interlamellar sublayer. Such stiffening continues till most of the collagen fibers are involved in mechanical behavior. However, these functional regions are not distinguishable in the representative curve of the old group. The mechanism of gradual uncrimp of collagen fibers is disturbed and fibers recruitment starts in lower strains and leads to higher stress levels in old group compared to the young. This finding is in agreement with recent findings on more dispersed and irregular collagen fibers in aged aortas [19]. Unlike young group, initial dominancy of the elastic sheet is not observed in old subjects and the curve representing mechanical performance of layer II is above that of layer I from the beginning (Figure 4). Results confirm that aged aortic wall is stiffer than young aorta in both axial and circumferential directions, indicating that the increased mechanical contribution of layer II is dominant and compensates for the age related decrease of the stress of layer I. It is in agreement with previous findings of increased cross-link of collagen fiber due to aging [34]. The volume fraction of layer II is approximately three times that of layer I (Table 2), therefore, the overall mechanical behavior of the media will rather pursue the curve of the layer II, especially in higher strains. Comparing the overall mechanical behavior of these two groups, aged aortas exhibit higher stiffness which is a consequence of increased volume fraction of layer II. Such elevation is mainly due to deposition of collagen in this sublayer, however the small increase in volume fraction of layer II (4.6% according to Table 2) is not corresponding to the markedly higher stresses in old subjects. The earlier engagement and increased cross link of collagen fibers are most likely the main causes of higher performance of layers II, while the increased volume fraction plays smaller role. Altogether, loss of function of elastin accompanied by increased cross link and earlier recruitment of collagen fibers are main features of age-related stiffening of the aortic wall. Since the provided alterations in the fibrous content of the wall was judged based on the volume fraction of each sublayer instead of measuring fiber content directly, further investigations are necessary to quantify changes in elastin and collagen contents (fragmentation, fiber direction, waviness and cross link density). These data will lead to more accurate interpretation of collagen and elastin contribution in arterial mechanics. It should be noted that in the current study, age-related alterations in these microstructural features are reflected by the change of mechanical properties and altered stress-strain behaviors of layer I and layer II. In present study, age-related changes in the volume fractions and mechanical properties of lamellar structure of the arterial media is provided and led to new insights on altered shares of layer I and layer II on overall stressstrain behavior of the arterial tissue by comparing these features in young and old age groups. Acknowledgement Authors would like express their sincere gratitude towards relatives of brain-dead patients who dedicated aortic tissues of their beloved ones to scientific research.
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Figures
20 μm
Figure 1- stained section of the aortic media with VVG stain of young (left panel) and old (right panel) aortic samples. Wall lamellae and interlamellar layers are seen in dark and light colors respectively. These images were handled with an image-processing code to obtain volume fractions of microstructural sublayers for young and old samples. Elastic sheets (dark regions) seem more arranged and continues in the young aorta while these sheets are fragmented and disturbed in the old one.
Axial Stress 200 180
200 180 160
140
140
120
120
S(KPa)
S(KPa)
160
Circumferential Stress
Experimental data Layer I Layer II Lamellar Model
100
100
80
80
60
60
40
40
20
20
0 0
0.1
0.2
0.3 E
0.4
0.5
Experimental data Layer I Layer II Lamellar Model
0 0
0.1
0.2
0.3
0.4
0.5
E
Figure 2- experimental data of the typical sample (25 years old) in the axial and circumferential directions. The stresses in layer I and layer II is shown as well in both of the mentioned directions. Additionally the model prediction is illustrated against the experimental data. The proposed model can follow the stress-strain behavior of the arterial tissue in its different functional phases.
Figure 3-Average stress-strain curve of the young (top panel) and old (bottom panel) aortic media in axial (left) and circumferential (right) directions. The mechanical behaviors of the layer I and layer II are as well included in these panels. Comparing the respective stress-strain curve of the media in young and old groups reveals stiffer old media in both directions. In young group, the curve representing mechanical behavior of the layer I is above that of layer II in the range of low strains. Then the stress in the layer II rises gradually toward a stiffening behavior and exceeds the stress of the layer I for higher strains. Such behavior is not inspected in the old group and the curve of mechanical behavior of layer II lays above the stress-strain curve of layer I for all strains.
Layer II
Layer I Young Old
100
100
80
80
60
60
40
40
20
20
0
0
0.05
0.1
Young Old
120
S(KPa)
S(KPa)
120
0.15
0.2
0.25 E
0.3
0.35
0.4
0.45
0
0
0.05
0.1
0.15
0.2
0.25 E
0.3
0.35
0.4
0.45
Figure 4- Comparison of mechanical contribution of microstructural layers in young and old aorta samples. The loadbearing function of layer I, elastic sheets, is significantly lower in old samples compared to the healthy young group as seen in left panel. However, for the circumferential direction, average stress of the old group departs from the curve of young group and lays above that for the respective higher strain ranges.
Table 1- The obtained material parameters for a typical sample (25 years old). The Table contains results of histological (volume fractions) and mechanical (hyperelastic parameters) for this sample. The last column shows the goodness of the fit provided by the lamellar model.
Layer I
Layer II
c1 (KPa)
fI (–)
c2 (KPa)
a1 (–)
a2 (–)
a3 (–)
fII (–)
R2
36.387
0.2572
33.012
0.825
1.28
0.825
0.7428
0.9988
Table 2-Averaged data for young and old age groups. These data include hyperelastic material parameters and volume fractions of the lamellar structure. Increased volume fraction of layer II and respective decrease in volume fraction of the layer I is noticeable. The difference of the hyperelastic material parameters are also observed. The only material parameter of the layer I, c 1 is substantially decreased in old group. In the case of layer II, the parameter c2 is decreased however the individual exponent parameters (a 1, a2 and a3) and their sum are higher in the old group which mathematically corresponds to faster raising curve indicating that the required energy to apply a specific deformation field to the layer II is higher in old group. All of averaged material parameters of the old group were significantly (p<0.05) altered while changes of the volume fractions were not significant (p>0.1).
Layer I
Layer II
c1 (KPa)
fI (–)
c2 (KPa)
a1 (–)
a2 (–)
a3 (–)
fII (–)
Young
25.863±8.011
0.262±0.008
30.198±8.526
1.239±0.169
1.704±0.432
0.270±0.289
0.738±0.008
Old
6.342±2.982
0.229±0.025
10.113±3.832
3.714±2.095
5.743±2.197
1.412±0.827
0.771±0.025
p-value
<0.001
>0.1
<0.001
0.008
0.001
0.003
>0.1
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PII: DOI: Reference:
S1751-6161(16)30271-5 http://dx.doi.org/10.1016/j.jmbbm.2016.08.011 JMBBM2030
To appear in: Journal of the Mechanical Behavior of Biomedical Materials Received date: 11 April 2016 Revised date: 18 July 2016 Accepted date: 3 August 2016 Cite this article as: Hadi Taghizadeh and Mohammad Tafazzoli-Shadpour, Characterization of mechanical properties of lamellar structure of the aortic wall: Effect of aging, Journal of the Mechanical Behavior of Biomedical Materials, http://dx.doi.org/10.1016/j.jmbbm.2016.08.011 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Characterization of mechanical properties of lamellar structure of the aortic wall: Effect of aging Hadi Taghizadeh§, Mohammad Tafazzoli-Shadpour¥* §
Division of Biomechanics, Mechanical Engineering Department, Sahand University of Technology, Tabriz 51335-1996, Iran; Email: [email protected]
¥
Cardiovascular Engineering Laboratory, Faculty of Biomedical Engineering, Amirkabir University of Technology, 424 Hafez Ave., Tehran 15875-4413, Iran; Email: [email protected]
*
Corresponding Author: Mohammad Tafazzoli-Shadpour, Cardiovascular Engineering Lab., Faculty of Biomedical Engineering, Amirkabir University of Technology, Tehran, Iran Email: [email protected]
Abstract: Arterial wall tissues are sensitive to their mechanical surroundings and remodel their structure and mechanical properties when subjected to mechanical stimuli such as increased arterial pressure. Such remodeling is evident in hypertension and aging. Aging is characterized by stiffening of the artery wall which is assigned to disturbed elastin function and increased collagen content. To better understand and provide new insight on microstructural changes induced by aging, the lamellar model of the aortic media was utilized to characterize and compare wall structure and mechanical behavior of the young and old human thoracic aortic samples. Such model regards arterial media as two sets of alternating concentric layers, namely sheets of elastin and interlamellar layers. Histological and biaxial tests were performed and microstructural features and stress-strain curves of media were evaluated in young and old age groups. Then using optimization algorithms and hyperelastic constitutive equations the stress-strain curves of layers were evaluated for both age groups. Results indicated slight elevation in the volume fraction of interlamellar layer among old subjects most probably due to age related collagen deposition. Aging indicated substantial stiffening of interlamellar layers accompanied by noticeable softening of elastic lamellae. The general significant stiffening of old samples were attributed to both increase of volume fraction of interlamellar layers and earlier recruitment of collagen fibers during load bearing due to functional loss of elastin within wall lamellae. Mechanical characterization of lamellar structure of wall media is beneficial in study of arterial remodeling in response to alternated mechanical environment in aging and clinical conditions through coupling of wall microstructure and mechanical behavior.
Keywords: Aging, Arterial wall, remodeling, lamellar structure, structure- mechanics coupling, constitutive modeling
Introduction The biological functions of cardiovascular tissues are directly correlated with their mechanical properties, i.e., proper function of the large arteries depends on their elasticity [1] and their elasticity is altered during physiologic and pathologic processes such as aging and hypertension [2]. Aging and hypertension are Associated with elevated blood pressure which deploys excessive circumferential tensile stress on the arterial wall. Cellular content of the wall, mainly smooth muscle cells (SMCs) and endothelial cells sense the extra tension through mechanotransduction pathways and synthesize fibrous content in the wall to restore their mechanical environment back to the normal conditions [3]. Such modifications alter the microstructure and hence, mechanical properties of the wall. Arterial walls possess a composite like lamellar structure of semi-concentric fibrous layers [4]. The lamellar structure can be regarded as repeated structural units through the thickness of the media [5]. With further focus on these structural units, two sets of sublayers are distinguished, i.e. highly elastic layers containing sheets of elastin, and interlamellar layers containing collagen bundles, fine elastic fibers, smooth muscle cells (SMCs) and ground substances such as proteoglycans [6, 7]. These two types of sublayers are mechanically different according to their structure and constituents. Elastic sheets contain homogenous elastin tissue while collagen fibers are arranged in preferred directions and these fibers are mechanically dominant among components of interlamellar layers. Such arrangement leads to anisotropic mechanical behavior of the arterial wall. A welldeveloped model of the arterial wall should be able to address these characteristics. Lamellar units of the tunica media is shown to provide a firm base toward constitutive modeling of the arteries in health and reflect disease-induced alterations [8-10]. There are some reported changes in the structure of the arterial wall with aging; The thickness of arterial media increases [11] and the amount of arterial elastin is reduced while the amount of collagen increases [12]. On the other hand, some suggest that aging does not necessarily decrease the physical content of elastin, however, fragmentation and loss of functionality of them are inevitable [13, 14]. Since collagen fibers are much stiffer than elastin fibers, age-related changes in collagen to elastin ratio lead to the well-established dilated and stiffened arterial wall [15]. Altered fibrillar content affects the coupling of the mechanical and biological environments of the artery and despite age-related remodeling of the arterial wall, aging is correlated with cardiovascular diseases [16-18]. Current research investigates the effects of aging on the mechanical behavior of the aortic microstructure utilizing the previously proposed lamellar model [9]. In the light of mentioned modeling framework, new insight on the aging process and alterations in mechanical performance of the lamellar structure in young and old aortic media will be provided. Materials and methods Biaxial tensile tests were carried out on human aortic samples of young and old subjects to characterize hyperelastic behavior of aortic tissue. Histological observations were concurrently performed on stained samples to observe structural layers of the wall media among young and old groups and provide needed microstructural data. Constitutive equations and optimization algorithms were recruited to characterize hyperelastic parameters of arterial lamellae and interlamellar layers. Utilizing this approach, we explored agerelated changes of the aortic wall based on lamellar modeling. This model couples microstructural information with mechanical behavior of the aortic media to better describe its mechanical performance. Human subjects under the age of 40 were categorized in young group and subjects older than 55 years were assigned to the old group. Such categorization originates from reports on the alteration in microstructure of the arterial walls with age [19]. The samples in the range of 40-55 years old were excluded from the comparison to address the age-induced alterations more clearly. Among each group, the average volume fractions of the major media layers and their respective mechanical properties were obtained and effects of aging were discussed. Specimen preparation Samples from thoracic aortas of 19 subjects, 10 young (31.4±8.1) and 9 old (61.3±3.1) samples, were provided from brain-dead patients that underwent organ donation surgery considering the ethical approval including consent of relatives. All steps of human tissue handling from extraction to experiments were conducted
according to the instructions of the Ethics Committee, Masih Daneshvari Hospital, the main center of organ donation in Iran. Healthy samples were harvested from descending region of the aorta above the diaphragm, since in this region branching and diameter changes are minimal and lesion formation occurs in fewer subjects. Tissue samples were preserved in Phosphate-Buffered Saline prior to experiments and the required tests are inquired in less than 48 hours post mortem. A thin ring of the tissue was separated for histological staining and the rest of the tissue was utilized for biaxial tensile tests. Data acquisition The proposed lamellar model employs microstructural and mechanical data to better approximate mechanical behavior of the aortic media. Histological staining of elastic fibers was performed to distinguish wall lamellae and interlamellar layers (Figure 1). Verhoeff Van Gieson (VVG) stain was utilized to distinguish the elastic lamellae in dark color within the bright interlamellar space. Microscopic images of tissue rings were captured and the volume fractions of two major layers were computed by developing an image-processing code in MATLAB. Planar biaxial tensile tests were carried out on at least two specimens from each subject. We employed the previously described protocol in biaxial testing of the samples [9]. Force-displacement data of the young and old samples were extracted and then converted to Second Piola stress-Green strain, the common stress-strain couple for mechanical description of hyperelastic materials [20, 21]. Continuum Preliminaries The non-linear elastic behavior of highly deformable soft tissues indicated these tissues are hyperelastic. Mechanical description of such materials necessitates adoption of large deformation continuum framework. In this context, deformation gradient relates reference and deformed configurations (F=∂x/∂X). The deformation gradient is composed of rotation (R) and deformation (U) parts (F=RU=VR). Rotation component is an orthogonal matrix with the property RRT=1 and main deformation tensors are defined not to depend on rotation component:
C FT F U 2 B FF T V 2 The Green-Lagrange strain tensor is defined by E
(1) 1 C 1 . 2
The first Piola stress is defined by force per initial section of the sample, which is related to the Cauchy stress, force per current section of the sample, by:
P J FT
(2)
Since the first Piola stress is not symmetric, the second Piola stress is defined as a symmetric tensor:
T F 1P JF 1 FT
(3)
For hyperelastic materials strain energy depends only on deformation field (deformation gradient, left or right Cauchy tensors and their invariants or Green strain):
ˆ E F C
(4)
Then the stress-strain relation of the material is obtained:
S 2
(C ) (E ) C E
(5)
The stress computation in eq. (5) is subjected to physical and computational constraints. For example, since arterial wall is considered to be incompressible [22], a Lagrange multiplier term is added to this relation to enforce incompressibility:
S 2
(C ) pC 1 C
(6)
The unknown parameter p is obtained by initial or boundary conditions of the problem. Additionally, arterial wall is anisotropic and appropriate selection of the strain energy density function (SEDF) is necessary for its mechanical characterization. Description of the Lamellar Model Data from histological and biaxial tests were fed into the lamellar model to evaluate mechanical properties of the aortic media layers. The lamellar modeling approach which is adopted in this study is described in our previous paper [9]. Briefly, we regard lamellar structure of the media, as a set of structural units. Additionally each structural unit was considered as a set of two sublayers: sheet of elastin (Layer I) described by the dark stained layers in Figure 1, and interstitial layer consisting of collagen bundles, fine elastin fibers and SMCs (Layer II). It is assumed that all of these lamellar units across the media thickness are identical in terms of the mechanical behavior of layer I and layer II respectively. Neo-Hookean and four parameter SEDFs were adopted for layer I and layer II, respectively (Eqs. 7 and 8 respectively) [9]. Neo-Hookean SEDF is suitable for capturing mechanical behavior of layer I regarding isotropic nature of elastic tissue [23, 24]. On the other hand, considering non-homogenous distribution and gradual uncrimp of collagen fibers in layer II, exponential SEDF is utilized to capture anisotropic and gradual stiffening of this layer. Then the overall mechanical behavior of the media is obtained by combining SEDFs of sublayers proportional to their fractional volumes.
WI c1 ( I1 3)
(7) (8)
W II c2 [exp(a1E12 a2E 22 2a3E1E 2 ) 1] W media f IW I f IIW II
(9)
In these equations, subscripts I and II denote layer I and layer II, respectively. Parameter W indicates SEDF, I1, E1 and E2 describe the deformation field of the material and finally f denotes volume fraction. The SEDF of the media shown in eq. (9) depends on five unknown material parameters, namely c1 for layer I, c2, a1, a2 and a3 for layer II. Then the SEDF in eq. (9) is substituted in eq. (6) to calculate stresses:
1 W p 2 1 1 1 S 0 0
0 1 W p 2 2 2 2 0
0 W 12 p 122 2 3 (1 , 2 ) 0
(10)
Where diagonal elements denote normal stress components in axial, circumferential and radial directions. It is shown that these directions coincide with principal directions of the arterial tissue and no shear stress will arise when test directions correspond to principal directions [25]. Hence in our formulation the shear deformations are not included and the respective components in stress tensor (off-diagonal elements) are zero. Since we carried out planar tests in axial-circumferential plane, the out of plane component of the stress will be zero (plane-stress assumption). Then the Lagrange multiplier term is obtained (
p
W I ) and axial 12 3 (1 , 2 ) 1
(Sa) and circumferential (Sc) stresses are determined accordingly.
Sa fI(
W I 1 W I 1 1 W II 3 ) f II 1 1 1 2 3 (1 , 2 ) 1 1
W I 1 W I 1 1 W II S fI( ) f II 3 2 2 12 3 (1 , 2 ) 2 2 c
On the other hand, the layer stresses are related to the overall stress of the media using eq. (9):
(11)
S a f I S Ia f II S IIa c c c S f I S I f II S II
(12)
Comparing eq. (11) with eq. (12), layer stresses are obtained as:
S Ia
W I 1 W I 1 3 1 1 1 2 3 (1 , 2 ) S IIa
1 W II 1 1
(13)
W I 1 W I 1 S 3 2 2 12 3 (1 , 2 ) c I
S IIc
1 W II 2 2
Using these relations, the stresses in layer I and layer II are explored. For equibiaxial tests ( 1 2 ), the axial a
and circumferential stresses on layer I ( S I and
S Ic
respectively) are equal because of Isotropic SEDF used
for this sublayer. Determining unknown hyperelastic materials The explained modeling approach was applied to experimental data from histological and biaxial tests. The volume fractions, fI and fII, were obtained from images of stained samples and together with biaxial test data were used to evaluate hyperelastic parameters described in previous section. For each test specimen the set of hyperelastic parameters that best interpolate experimental data were determined by nonlinear optimization. We developed a computer code in MATLAB environment and used “fmincon” function to best determine these material parameters (adopted error function is explained in [9]). Averaging mechanical behavior of young and old age groups After obtaining experimental data from histological and mechanical tests and applying nonlinear regression to allocate material parameters, we averaged volume fractions and the set of hyperelastic parameters among samples of each group to obtain the representative volume fractions and material parameters for the young and old age groups. It should be noted that reporting the averaged set of constitutive parameters, is the popular method of representing the overall behavior of biological tissues, however it can lead to unrealistic results [26]. These issues will not considerably affect our comparison since the same constitutive model is adopted for young and old groups, causing similar errors in each group. However, careful attention should be given to mentioned discrepancy while the extracted material parameters are intended for computational models. The volume fractions and mechanical properties of the young and old groups, are then compared via two-tailed T test [27] and p-values smaller than 0.05 are considered as significant. Results A typical experimental data of a young subject (25 years old) and the respective fit of the lamellar model are provided in Figure 2 and corresponding material parameters are presented in Table 1. Such data were obtained for each test specimen. Averaged material parameters representing young and old age groups are provided in Table 2. Besides, the average volume fractions obtained from histological tests for young and old groups are presented in the same Table. Comparison of the volume fractions revealed an average 4.6 percent increased volume fraction of interlamellar layer among old subjects. The averaged hyperelastic parameters obtained for old subjects were different from those of young subjects. Parameter c1 representing isotropic SEDF of layer I (elastic lamellae) were substantially lower in the old group which might be due to loss of function of elastin. The parameters representing SEDF of layer II show a dual difference, c2 is decreased while the individual exponent parameters
(a1, a2 and a3) and their summation are elevated in the old group. From a mathematical point of view, the SEDF of layer II indicates milder slope at the beginning and then faster increase later in the stress-strain curve for old subjects compared to young subjects. The correspondent averaged stress-strain curves for two age groups are plotted in Figure 3 for axial and circumferential directions. The contribution of layer I and layer II on the overall mechanical behavior of the aortic media in young and old samples is also presented in Figure 3. The stresses generated on each sublayer (layer I and layer II) are regarded as their mechanical performance and contribution to the overall mechanical behavior of the media and these performances are compared in young and old groups. There are three functional regions in stress-strain curve of the healthy young aortic samples as shown in top panel of Figure 3. In lower strains, the stress- strain curve of the media is nearly linear. With elevation of the strain, the linear curve is transitioning toward a gradually stiffening mechanical behavior. In higher strains the stiffening behavior is noticeable and an incremental increase in the strain leads to substantial increase in the stress. Mechanical performance, of layers I and II in young and old aortas are compared in panels of Figure 4. Both panels reveal marked changes by aging. Elastic tissue has markedly smaller share on mechanical performance in the old aorta compared to young aortas. On the other hand, mechanical properties of layer II in old group indicate significant stiffening compared to young group and the altered mechanism of collagen recruitment is implied in this panel. The latter effect is more effective in alterations of mechanical properties in aging because corresponding volume fraction of layer II is more than three times that of layer I (elastic sheets). Discussion According to our model, mechanical properties of the aortic media were assigned through two layers, namely layer I and layer II as defined by their constituents. The proposed lamellar model is efficient in describing mechanical behavior of the aortic media for all ranges of the tissue deformation including physiologic and supraphysiologic strains (Figure 2) [9]. Such modeling approach couples the geometrical changes of the arterial wall directly with the changes in mechanical properties. Hence, it may be employed in study of aging and pathological conditions such as hypertension and the response of wall media to disturbed environment through alterations in micro-structural elements and subsequently changes in mechanical properties and stress distribution [10]. Such response, known as arterial wall remodeling is triggered after sustained changes in stress distribution across the wall media sensed by media and intima cells and results in extra synthesis of fibrous content, and usually acts by a trend reversing the stress pattern of the arterial wall to normal conditions [28, 29]. In aging, the increase in level of tensile stresses is due to gradual increase of blood pressure, while in systemic hypertension the elevation of luminal pressure due to clinical conditions is faster. Alterations in the microstructure of the artery influence macro mechanical behavior of the artery by incorporation of volume fractions in the proposed lamellar model. Such multiscale model is beneficial to establish both micro and macro scale geometry- mechanics couplings. The increased volume fraction of the interlamellar layer (layer II) due to aging is most likely a result of collagen deposition within this sublayer by SMCs which are activated through altered tension that affects their contractility [30, 31]. Our results indicate that among old group, Layer I has lower stress compared to that of young subjects (as seen in the left panel of Fig. 4), most probably due to the fragmentation of the elastic fibers with age. However the mentioned stiffening is caused by alterations in mechanical properties of layer II (right panel of Fig. 4) through stiffening and thickening of these layers. This is compatible with the idea of collagen synthesis with age in arterial wall [19]. Since the media of the old group is stiffened compared to the media of young subjects, one can conclude that stiffening of the layer II in the old group is the dominant factor in wall stiffening. It has been suggested that aging causes fragmentation of elastin fibers and reduced cross-link of the elastic fibers due to applied repeated load through pulsatile luminal pressure and consequently disruption of the organized structure of elastin sheets in wall lamellae and fine elastin meshes within interlamellar zones [16, 32]. This phenomenon seems to be similar to effect of mechanical fatigue in materials caused by repeated loading. In lower strains of the arterial wall, collagen fibers are crimped and wavy and hence inactive. In this range, elastic tissue plays the dominant role in overall stress-bearing of the arterial media. With elevation of the strain, collagen fibers start uncrimping and gradually engage, and mechanical role of layer II becomes more prominent
and reaches to that of layer I. This region is regarded as a transition zone between low and high strain ranges. Mechanical outcome of microstructural sublayers are nearly the same in this range. It has been suggested that physiologic strains reside in this transition zone [33]. Finally in the third region which corresponds to higher strains, collagen fibers increasingly uncrimp and become engaged in mechanical response of the artery which is evident in Figure 1 by hardening stress-strain curve of interlamellar sublayer. Such stiffening continues till most of the collagen fibers are involved in mechanical behavior. However, these functional regions are not distinguishable in the representative curve of the old group. The mechanism of gradual uncrimp of collagen fibers is disturbed and fibers recruitment starts in lower strains and leads to higher stress levels in old group compared to the young. This finding is in agreement with recent findings on more dispersed and irregular collagen fibers in aged aortas [19]. Unlike young group, initial dominancy of the elastic sheet is not observed in old subjects and the curve representing mechanical performance of layer II is above that of layer I from the beginning (Figure 4). Results confirm that aged aortic wall is stiffer than young aorta in both axial and circumferential directions, indicating that the increased mechanical contribution of layer II is dominant and compensates for the age related decrease of the stress of layer I. It is in agreement with previous findings of increased cross-link of collagen fiber due to aging [34]. The volume fraction of layer II is approximately three times that of layer I (Table 2), therefore, the overall mechanical behavior of the media will rather pursue the curve of the layer II, especially in higher strains. Comparing the overall mechanical behavior of these two groups, aged aortas exhibit higher stiffness which is a consequence of increased volume fraction of layer II. Such elevation is mainly due to deposition of collagen in this sublayer, however the small increase in volume fraction of layer II (4.6% according to Table 2) is not corresponding to the markedly higher stresses in old subjects. The earlier engagement and increased cross link of collagen fibers are most likely the main causes of higher performance of layers II, while the increased volume fraction plays smaller role. Altogether, loss of function of elastin accompanied by increased cross link and earlier recruitment of collagen fibers are main features of age-related stiffening of the aortic wall. Since the provided alterations in the fibrous content of the wall was judged based on the volume fraction of each sublayer instead of measuring fiber content directly, further investigations are necessary to quantify changes in elastin and collagen contents (fragmentation, fiber direction, waviness and cross link density). These data will lead to more accurate interpretation of collagen and elastin contribution in arterial mechanics. It should be noted that in the current study, age-related alterations in these microstructural features are reflected by the change of mechanical properties and altered stress-strain behaviors of layer I and layer II. In present study, age-related changes in the volume fractions and mechanical properties of lamellar structure of the arterial media is provided and led to new insights on altered shares of layer I and layer II on overall stressstrain behavior of the arterial tissue by comparing these features in young and old age groups. Acknowledgement Authors would like express their sincere gratitude towards relatives of brain-dead patients who dedicated aortic tissues of their beloved ones to scientific research.
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Figures
20 μm
Figure 1- stained section of the aortic media with VVG stain of young (left panel) and old (right panel) aortic samples. Wall lamellae and interlamellar layers are seen in dark and light colors respectively. These images were handled with an image-processing code to obtain volume fractions of microstructural sublayers for young and old samples. Elastic sheets (dark regions) seem more arranged and continues in the young aorta while these sheets are fragmented and disturbed in the old one.
Axial Stress 200 180
200 180 160
140
140
120
120
S(KPa)
S(KPa)
160
Circumferential Stress
Experimental data Layer I Layer II Lamellar Model
100
100
80
80
60
60
40
40
20
20
0 0
0.1
0.2
0.3 E
0.4
0.5
Experimental data Layer I Layer II Lamellar Model
0 0
0.1
0.2
0.3
0.4
0.5
E
Figure 2- experimental data of the typical sample (25 years old) in the axial and circumferential directions. The stresses in layer I and layer II is shown as well in both of the mentioned directions. Additionally the model prediction is illustrated against the experimental data. The proposed model can follow the stress-strain behavior of the arterial tissue in its different functional phases.
Figure 3-Average stress-strain curve of the young (top panel) and old (bottom panel) aortic media in axial (left) and circumferential (right) directions. The mechanical behaviors of the layer I and layer II are as well included in these panels. Comparing the respective stress-strain curve of the media in young and old groups reveals stiffer old media in both directions. In young group, the curve representing mechanical behavior of the layer I is above that of layer II in the range of low strains. Then the stress in the layer II rises gradually toward a stiffening behavior and exceeds the stress of the layer I for higher strains. Such behavior is not inspected in the old group and the curve of mechanical behavior of layer II lays above the stress-strain curve of layer I for all strains.
Layer II
Layer I Young Old
100
100
80
80
60
60
40
40
20
20
0
0
0.05
0.1
Young Old
120
S(KPa)
S(KPa)
120
0.15
0.2
0.25 E
0.3
0.35
0.4
0.45
0
0
0.05
0.1
0.15
0.2
0.25 E
0.3
0.35
0.4
0.45
Figure 4- Comparison of mechanical contribution of microstructural layers in young and old aorta samples. The loadbearing function of layer I, elastic sheets, is significantly lower in old samples compared to the healthy young group as seen in left panel. However, for the circumferential direction, average stress of the old group departs from the curve of young group and lays above that for the respective higher strain ranges.
Table 1- The obtained material parameters for a typical sample (25 years old). The Table contains results of histological (volume fractions) and mechanical (hyperelastic parameters) for this sample. The last column shows the goodness of the fit provided by the lamellar model.
Layer I
Layer II
c1 (KPa)
fI (–)
c2 (KPa)
a1 (–)
a2 (–)
a3 (–)
fII (–)
R2
36.387
0.2572
33.012
0.825
1.28
0.825
0.7428
0.9988
Table 2-Averaged data for young and old age groups. These data include hyperelastic material parameters and volume fractions of the lamellar structure. Increased volume fraction of layer II and respective decrease in volume fraction of the layer I is noticeable. The difference of the hyperelastic material parameters are also observed. The only material parameter of the layer I, c 1 is substantially decreased in old group. In the case of layer II, the parameter c2 is decreased however the individual exponent parameters (a 1, a2 and a3) and their sum are higher in the old group which mathematically corresponds to faster raising curve indicating that the required energy to apply a specific deformation field to the layer II is higher in old group. All of averaged material parameters of the old group were significantly (p<0.05) altered while changes of the volume fractions were not significant (p>0.1).
Layer I
Layer II
c1 (KPa)
fI (–)
c2 (KPa)
a1 (–)
a2 (–)
a3 (–)
fII (–)
Young
25.863±8.011
0.262±0.008
30.198±8.526
1.239±0.169
1.704±0.432
0.270±0.289
0.738±0.008
Old
6.342±2.982
0.229±0.025
10.113±3.832
3.714±2.095
5.743±2.197
1.412±0.827
0.771±0.025
p-value
<0.001
>0.1
<0.001
0.008
0.001
0.003
>0.1