Mechanical characterization of arteries affected by fetal growth restriction in guinea pigs (Cavia porcellus)

Author’s Accepted Manuscript Mechanical characterization of arteries affected by fetal growth restriction in guinea pigs (Cavia porcellus) Daniel Cañas, Claudio García-Herrera, Emilio A. Herrera, Diego Celentano, Bernardo J. Krause www.elsevier.com/locate/jmbbm

PII: DOI: Reference:

S1751-6161(18)30511-3 https://doi.org/10.1016/j.jmbbm.2018.08.010 JMBBM2921

To appear in: Journal of the Mechanical Behavior of Biomedical Materials Received date: 17 April 2018 Revised date: 9 August 2018 Accepted date: 10 August 2018 Cite this article as: Daniel Cañas, Claudio García-Herrera, Emilio A. Herrera, Diego Celentano and Bernardo J. Krause, Mechanical characterization of arteries affected by fetal growth restriction in guinea pigs (Cavia porcellus), Journal of the Mechanical Behavior of Biomedical Materials, https://doi.org/10.1016/j.jmbbm.2018.08.010 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.

Mechanical characterization of arteries affected by fetal growth restriction in guinea pigs (Cavia porcellus) Daniel Cañas1, Claudio García-Herrera1*, Emilio A. Herrera2, Diego Celentano3, Bernardo J. Krause4 1

Department of Mechanical Engineering, Faculty of Engineering, Universidad de

Santiago de Chile. Santiago, Chile. 2

Laboratory of Vascular Function and Reactivity, Pathophysiology Program, ICBM,

Faculty of Medicine, Universidad de Chile. Santiago, Chile. 3

Department of Mechanical and Metallurgical Engineering, Institute of Biological and

Medical Engineering, Pontificia Universidad Católica de Chile. Santiago, Chile. 4

Departament of Neonatology, Division of Pediatrics, Faculty of Medicine, Pontificia

Universidad Católica de Chile. Santiago, Chile. *Corresponding author: e-mail: [email protected] (CG)

1

Abstract Fetal growth restriction (FGR) is a perinatal condition associated with a low birth weight that results mainly from maternal and placental constrains. Newborns affected by this condition are more likely to develop in the long term cardiovascular diseases whose origins would be in an altered vascular structure and function defined during fetal development. Thus, this study presents the modeling and numerical simulation of systemic vessels from guinea pig fetuses affected by FGR. We aimed to characterize the biomechanical properties of the arterial wall of FGR-derived the aorta, carotid, and femoral arteries by performing ring tensile and ring opening tests and, based on these data, to simulate the biomechanical behavior of FGR vessels under physiological conditions. The material parameters were first obtained from the experimental data of the ring tensile test. Then, the residual stresses were determined from the ring opening test and taken as initial stresses in the simulation of the ring tensile test. These two coupled steps are iteratively considered in a nonlinear least-squares algorithm to obtain the final material parameters. Then, the stress distribution changes along the arterial wall under physiological pressure were quantified using the adjusted material parameters. Overall, the obtained results provide a realistic approximation of the residual stresses and the changes in the mechanical behavior under physiological conditions.

Keywords: Fetal growth restriction, systemic vessels, biomechanical characterization.

2

1.

Introduction

Fetal growth restriction (FGR) corresponds to one of the main complications that can be present during pregnancy. Affected neonates present increased risk of mortality and morbidity (Ashworth, 1998), impaired intellectual development (Gorman and Pollitt, 1992; Miller et al., 2016), and they are prone to develop cardiovascular diseases, obesity, and insulin resistance in adulthood (Alexander et al., 2015; Barker et al., 1993; Ong and Dunger, 2002). Notably, FGR neonates show a pro-atherogenic vascular structure, characterized by increased aortic intima-media thickness and stiffness (Cañas et al., 2017; Dodson et al., 2013a; Skilton et al., 2005), which would prime the risk of cardiovascular dysfunction later in life (Bradley et al., 2010; Thompson et al., 2011; Zanardo et al., 2013). In fact, low birth weight neonates show an increased media thickness of the aorta (Skilton et al., 2015), as well as an enhanced aortic stiffness later in life (Bradley et al., 2010). These suggest that FGR-associated vascular dysfunction could be mediated by an early onset of vascular remodeling. Therefore, it is of interest to unveil what functional biomechanical changes occur in the vascular tree of these subjects that lead to higher risk of cardiovascular diseases. A few studies have characterized the biomechanical behavior of FGR subjects, where pressure-diameter measurements on sheep (Dodson et al., 2014, 2013b) and ring tensile test measurements on guinea pigs (Cañas et al., 2017) can be found. Over the years, several tests have been developed to characterize the mechanical response of vascular tissues. Generally, inflation, uniaxial and multiaxial testing methods can be found in the study of different pathologies (García-Herrera and 3

Celentano, 2013). Vessels extracted from guinea pig fetuses have an inner diameter too small (Aorta ≈1.13mm, Carotid ≈ 0.52mm, Femoral ≈0.36mm (Cañas et al., 2017)) for the reliable obtainment of dogbone shaped samples for the longitudinal and circumferential directions. Moreover, the inflation test cannot be carried out due to the small length sample (≈4mm) and an inappropriate length/diameter ratio (Bersi et al., 2016. Consequently, the tensile test applied to rings was chosen for characterizing the tissue. This test presents some problems since the stress distribution along the sample during the test is strongly conditioned by the geometry, particularly its diameter, making its interpretation difficult. In the past, tensile tests applied to rings have been used to characterize arteries and determine the effects of exercise on the biomechanical properties of the rat aorta (Matsuda et al., 1993, 1989). However, only a few have addressed the numerical simulation of this test (Kowalik et al., 2016; Shazly et al., 2015) and none have considered residual stresses. The results of this test have been compared with those of a traditional inflation/uniaxial tensile test, and a good representation of the mechanical response was achieved (Bustos et al., 2016). Conversely, to obtain more information about the biomechanical changes, measurements of the opening angle have been used to determine the stress-free state of vessels (Han et al., 2006; Liu and Fung, 1988). Interestingly, opening angles of pathological and healthy guinea pig systemic vessels have been reported before (Cañas et al., 2017). Only few articles have modelled the mechanical behavior of vessels affected by FGR, where the assumption of a thin-walled tube was taken for the simulation of the common carotid artery on FGR fetuses, using the data obtained from the inflation test and an 4

anisotropic hyperelastic model (Dodson et al., 2013b) that assumes the presence of a symmetric fiber arrangement and orientation in the material (Gasser et al., 2006; Holzapfel et al., 2000). Regarding the residual stresses, we are unaware if they were incorporated in the simulation and could lead to differences in the simulation. Our aim in the present work consists in modeling the mechanical response of the aorta, the common carotid, and the femoral arteries for understanding the FGR-related cardiovascular diseases. For this purpose, we describe the materials, methods, and the constitutive model used in the analysis of the ring tensile and ring opening tests. Further, geometries, boundary conditions and the simulation method for these two tests are given. To this end, the residual stresses were obtained from the simulation and incorporated into the model; this is an important point in this work and a novel procedure in the understanding of the test. The modeling was performed via an isotropic constitutive model, where the initial material parameters were obtained from the ring tensile test measurements and then improved via numerical fitting considering the effect of residual stress estimated from the ring opening test. Finally, the simulation of the pressurization test under physiological pressure is subsequently carried out using the previously characterized material properties.

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2. Materials and methods 2.1.

Animals

All animal care and experimental procedures were approved by the Ethics Committees of the Faculty of Medicine of the Pontificia Universidad Católica de Chile (1130801) and the Universidad de Chile (protocol CBA N◦ 0694 FMUCH), and they were conducted according to the Guide for the Care and Use of Laboratory Animals published by the US National Institutes of Health (NIH Publication No. 85–23, revised 1996). Further, these procedures

were

reported

in

accordance

with

the

ARRIVE

guidelines

(https://www.nc3rs.org.uk/arriveguidelines). Four to five-month-old virgin sows in estrus were paired with a fertile male for 2 days. After the mating period, the females were individually housed with daily monitoring of body weight, food intake, and water consumption. Pregnancy was confirmed by ultrasonography at day 20-25 (Herrera et al., 2016). According to the work of Karatza and Varvarigou (2013) guinea pigs have displayed qualitative similarities when compared to human data. In this context, in a recent report we showed that structural and biomechanical properties of umbilical arteries in guinea pigs affected by fetal growth restriction (Cañas et al., 2017) have a comparable pattern as suggest previous studies in human umbilical cords (Krause et al., 2012).. 2.2.

FGR Induction

At day 35, all pregnant sows were subjected to aseptic surgery under general anesthesia (isoflurane 2% in O2), randomly assigned to either sham-operated (control) or progressive uterine artery occlusion (FGR) as described elsewhere (Cañas et al., 6

2017; Herrera et al., 2016). 2.3.

Euthanasia at Near-Term

At 61-63 days of gestation, the guinea pigs and their fetuses were euthanized with a maternal anesthetic overdose (Sodium Thiopentone 200mg kg-1, IP, Opet, Laboratorio Chile). Once the cardio-respiratory arrest was confirmed, the fetuses were extracted and their thoracic aorta (descending aorta), carotid and femoral vessels were dissected. 2.4.

Experimental tests

The experimental data used in this work was previously presented and details about the analyzed arteries of the control and FGR animals, together with the corresponding histological analysis can be found elsewhere (Cañas et al., 2017). An overview of the tests is given below. 2.4.1.

Ring tensile test

The test is used to obtain the stress-strain relationship, and it consists in subjecting a cylindrical sample of controlled length, approximately 2 mm, to a radial elongation performed by pins. For the simplified analytical interpretation of this test, the assumptions of negligible initial bending and frictionless condition are considered. All tests were performed immediately after dissection and carried out with the sample immersed in physiological serum at 37 ± 0.5 °C. During the test, the load (F) and the displacement of the pins (Δ) are recorded (Bustos et al., 2016). The variables used were the initial thickness (eo), the initial width (ao), the diameter of the pin ( φ ) and the mean diameter of the artery (d). The separation between the center pins and the initial length are represented by Δ and Δo, respectively. The initial length (Δo) is determined 7

by: 𝜋

∆0 = 2 [𝑑 − (𝜑 + 𝑒0 )]

(1)

Thus, it is possible to define the elongation considering the semiperimeter of the artery and the increase in separation of the pins: 𝜆 = 1+2

(Δ−Δ0 ) 𝜋𝑑

(2)

Finally, the expression of the Cauchy stress in the arterial wall is computed as: 𝐹

𝜎𝑒𝑥𝑝 = 2𝑎

0 𝑒0

2.4.2.

𝜆

(3)

Ring opening test

For the determination of residual stress, the opening angle (Han et al., 2006; Liu and Fung, 1988; García-Herrera et al., 2016) is used to measure the angle (α) subtended by the ends after making a radial cut of a circular segment of the artery (Fig 1). Briefly, the ring was immersed in physiological serum at 37 ± 0.5 °C for about 10 min, and it was then cut radially and the sample was photographed after 20 min (García-Herrera et al., 2016). After this, the internal diameter is obtained from the circumference that best fits the open ring and the mean thickness was considered as the average of 5 measurements. The specimen diameter and thickness were measured on each sample by means of an optical thickness gage (Keyence LS 7070M) with 5 μm accuracy.

8

Figure 1: Opening angle schematic measurement. a) initial and b) open configurations. R and t respectively denote the radius and thickness, αexp represents the opening angle. 2.5.

Constitutive modeling

An elastic and rate-independent material response is considered for the arteries analyzed in the present work. Moreover, their behaviors are taken as incompressible due to a large amount of water present in them. To this end, hyperelastic constitutive models are used (Gasser et al., 2006; Holzapfel et al., 2005). In this context, a deformation energy function W, assumed to describe the isothermal material behavior under any loading conditions, can be defined in terms of the right Cauchy deformation tensor C = FT ·F, where F is the deformation gradient tensor and T is the transpose symbol (note that det(F)=1 in this case). Invoking classical arguments of continuum mechanics, the Cauchy stress tensor is defined as σ = 2F · ∂W/∂C · FT. The isotropic response of tissues can be described via the following energy function (Demiray, 1972): 𝑎

𝑏

𝑊 = 𝑏 (𝑒 2

(𝐼1 −3)

9

− 1)

(4)

where I1 is the first invariant of C (I1 = tr(C), and tr is the trace symbol). This constitutive model has been used in this work to assess the capabilities of the ring tensile test in the characterization of the mechanical behavior of tissues, which in this work corresponds to vessels of FGR fetuses. This constitutive model was implemented in an in-house finite element code extensively validated in many biomechanical applications (GarcíaHerrera and Celentano, 2013). 2.6.

Fitting of material parameters

The measurements reported in Cañas et al. (2017) were used to determine the initial set of material parameters “a” and “b” of Equation (4), supposing a homogeneous distribution of stresses across the material and neglecting the bending to which the sample is subjected during the stretching procedure. The resulting uniaxial stressstretch equation used to determine a first set of the materials parameters (see figure 2) is: 1

𝑏

2 +2−3)] 𝜆

𝜎 = 𝑎 (𝜆2 − 𝜆) 𝑒 [2(𝜆

(5)

where the curve-fitting was performed using the Levenberg-Marquardt algorithm. In this context, the improvement of these parameters must be accomplished via numerical

simulation

in

order

to

consider

the

residual

stresses

and

the

nonhomogeneous stress distribution. Hence, the resultant force in the stretching direction of the vessel was computed from the simulation, and in conjunction with equations (1, 2, 3) transformed into the Cauchy stress, enabling the comparison between the simulation and the experimental data. With these initial parameters, two simulations were carried out. The residual stresses 10

determined from the analysis of the ring opening test were then incorporated into the ring tensile test simulation. This procedure is iteratively performed until a good adjustment of the material parameters is achieved (Fig 2). The parameters were readjusted using the Levenberg-Marquardt algorithm (Marquardt, 1963). Additionally, the Jacobian matrix which holds all the first-order partial derivatives was computed using the numerical results obtained from the simulation and a finite difference method approach. The error is quantified by the normalized root mean square deviation (NRMSD) with the purpose of having a comparable value between numerical and experimental curves. The parameter is given by: 1

1

𝑁𝑅𝑀𝑆𝐷 = Δ √𝑛 ∑𝑛𝑖=1(𝑦̂𝑖 − 𝑦𝑖 )2

(6)

where n is the number of experimental data, yi is the experimental measurement, ŷi is the numerical fitted value, and Δ = |ymax-ymin|.

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Figure 2: Fitting of the parameters. Diagram showing the fitting sequence to determine material parameters of the constitutive model.

2.7. Numerical simulation 2.7.1. Ring opening test The first step of the simulation corresponds to the closure of the rings, in order to determine the magnitude of the residual stress present in the vessel. To this end, the free-stress configuration is drawn considering a symmetric plane. The procedure used for the closure of the ring consists in displacing the surface denoted as CD using a series of local coordinate systems, rotated with respect to a global coordinate system. 12

After achieving almost completely the closure of the ring, a horizontal displacement is imposed to the surface denoted as CD''', and simultaneously the motion of a point on the surface AB is restricted to make possible the closure of the ring (Fig 3). 1200 elements were used in the mesh composed of 8-noded hexahedral elements. It had 10 elements in the thickness, 3 in the longitudinal, and 40 in the circumferential directions.

Figure 3: Scheme of the ring’s closure method. Multiple rotational steps and one circumferential displacement are needed to close the ring. CD represents the surface where the displacements are imposed. 2.7.2.

Ring tensile test

The initial stresses computed from the closure of the ring are inserted in this simulation in order to achieve a better representation of the stress field obtained during the actual test. Therefore, an adjustment of the material parameters was made. Symmetry 13

conditions reduce the size of the problem, hence the geometry used for the simulation of the contact problem corresponds to the upper half of the previous vessels geometry (Fig 4), resulting in 1/8 of the original geometry used during the tensile test. The geometry composed of hexahedral elements considers 15 elements in the thickness, 30 in the longitudinal, and 60 in the circumferential directions, with a greater density in the zone of contact with the pins in order to guarantee the contact between geometries during the simulation. Appropriate boundary conditions were imposed according to the symmetry planes, restricting the displacement of some surfaces. In relation to the pin, it considers 50 elements in the longitudinal and 50 in the circumferential directions. There are a total of 35420 nodes and 33200 elements between both geometries. Moreover, it must be noted that the problem does not consider friction in the contact between the vessel and the pin. The contact model used for the simulation considers that the contact pressure (pn) increases quadratically with the penetration (gn) of the surfaces: 𝑃𝑛 = 𝐸𝑛0 𝑔𝑛 + 𝐸𝑛 𝑔𝑛2

(7)

where En0 and En are constants that control the penetration. This contact law was found to deal properly with the large stiffness variation developed by the material during the ring tensile test.

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Figure 4: Scheme of the ring tensile test simulation. Planes of symmetry were used to reduce the problem to 1/8 of the size and the sliders represent the boundary conditions on each surface. 2.7.3.

Pressurization test

The simulation consists of applying a pressure on the inner surface of the vessels and it considers the closure and pressurization of the ring. The geometry itself is the same used previously in the closure of the ring and the boundary conditions on the "AB" and "CD" surfaces were fixed in the x-direction. The mesh used in our problem was composed of hexahedral elements distributed in 10 elements in the thickness, 3 in the longitudinal, and 40 in the circumferential directions.

15

3. Results 3.1.

Ring tensile test

The experimental curves and simulations computed using the adjusted parameters in Table 1 are presented below (Fig 5). A stiffer response was observed in aorta (Fig 5a) and femoral arteries (Fig 5c) from FGR animals every stretch level, but no significant differences were seen in the rupture stress of both vessels. Conversely, no significant differences were found between FGR and control carotid arteries (Fig 5b).

16

(a)

(b)

(c) Figure 5: Average experimental and numerically fitted stress-stretch curves of the ring tensile test. (a) Aorta, (b) Carotid, and (c) Femoral arteries simulations for control (blue line) and FGR (red line). Experimental data from control (open circles) and FGR animals (solid circles).

17

Table 1: Material parameters for the Demiray model. Initial Parameters Control FGR Vessel a [kPa] b a [kPa] b Aorta 6.92 1.71 10.50 1.65 Carotid 10.08 1.96 7.77 1.93 Femoral 9.16 1.34 15.19 2.02

Final Parameters Control a [kPa] b 6.98 ± 0.64 1.99 ± 0.02 6.47 ± 0.25 2.90 ± 0.18 7.04 ± 0.77 1.91 ± 0.01

FGR a [kPa] b 9.16 ± 0.74 2.17 ± 0.10 6.23 ± 0.68 2.53 ± 0.18 13.03 ± 2.47 2.74 ± 0.03

Fig 6 shows the stress obtained from the simulation with initial values (residual stresses) divided by the experimental stress during the ring tensile tests. It can be observed the presence of a stress concentration near the inner face of the vessels during the test, which is 1.6 - 2.0 times higher than the experimental stress. Also, Fig 7 shows the evolution of the same ratio during the stretching of control aorta, where the same effects can be observed.

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(a)

(b)

(c) Figure 6: Computed stress profiles along the thickness at the end of the ring tensile test for each vessel. (a) Aorta, (b) Carotid, and (c) Femoral arteries simulation for control (blue line) and FGR (red line). The stress ratio was calculated from the data across the thickness on the central zone and the experimental stress, schematized by a horizontal line in the figure.

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Figure 7: Computed stress distribution in control aorta wall during ring tensile test. The stress ratio was calculated from the data across the thickness on the central zone and the experimental stress at different stretch levels (λ = 1.25 (black line); λ = 1.50 (blue line); λ = 1.75 (red line); λ = 2.00 (green line)).

3.2. Ring opening test An increase was seen in the opening angle of FGR subjects, aorta (+55.63%) and carotid artery (+17.65%). Conversely, a reduction of the opening angle (-25.16%) was seen in the femoral artery, shown in Table 3. σIN and σOUT represent the stress in the inner and outer face of the vessels after the ring closure. Table 2 summarizes the diameter and thickness of both situations, where "d" corresponds to the inner diameter and "t" is the thickness.

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Table 2: Comparison between experimental measures (without any pressure) and simulation of the ring dimension Experimental

Vessel Contol aorta FGR aorta Control carotid FGR carotid Control femoral FGR femoral

d [mm] 1.130 ± 0.116 0.996 ± 0.094 0.518 ± 0.060 0.432 ± 0.050 0.360 ± 0.074 0.221 ± 0.056

Simulation

t [mm] 0.164 ± 0.027 0.148 ± 0.011 0.108 ± 0.014 0.089 ± 0.011 0.072 ± 0.008 0.061 ± 0.011

d [mm] 1.128 0.968 0.490 0.416 0.369 0.236

t [mm] 0.166 0.144 0.104 0.086 0.071 0.060

Table 3: Residual stresses in control and FGR subjects. Vessel

Control Angle [°]

Aorta 55.74 ± 3.03 * Carotid 86.10 ± 4.01 * Femoral 135.10 ± 4.72 *

σIN[kPa]

σOUT[kPa]

Angle [°]

-1.090 -2.151 -3.297

0.955 2.044 3.317

86.75 ± 4.16 * 101.30 ± 6.86 * 101.10 ± 3.57 *

FGR σIN[kPa] -2.227 -2.431 -5.959

σOUT[kPa] 2.182 2.277 5.512

(*) Data obtained elsewhere [7]

3.3. Pressurization test The opened configuration of the ring is closed and residual stresses are determined for an axial stretch λz = 1.0, then a mean blood pressure of 40 mmHg was applied on the inner face of the ring. The procedure is performed in order to obtain the behavior under a larger range of pressures, knowing that average values of 28 mmHg are found in guinea pig fetuses (Kihlström, 1981). The circumferential stress in control and FGR aorta with residual stresses, as well as, pressure-circumferential stretch curves are presented in Figures 8 and 9, respectively.

21

(a)

(b)

Figure 8: Computed circumferential stress fields considering residual stresses [kPa] in the pressurization simulation at 28 mmHg and λz=1.0. (a) Control, (b) FGR aortas. A summary of the cases under control and FGR conditions is presented in Table 4, where the circumferential stress in the inner and outer face is denoted as σθin and σθout, respectively; the average circumferential stress as σθm, which is calculated as follows (σθm = (σθin + σθout)/2 ), and the difference between the inner and outer face, ∆σθ = |(σθin - σθout)|.

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Table 4: Circumferential stress comparison between control and FGR cases (28 mmHg). Artery

Condition Control Aorta FGR Control Carotid FGR Control Femoral FGR

σθin [kPa] σθout [kPa] 57.848 34.447 38.246 36.459 24.665 24.704 23.799 28.210 23.596 38.358 8.312 21.287

(a)

∆σθ [kPa] 23.401 1.787 0.039 4.411 14.762 12.975

σm [kPa] 46.147 37.352 24.684 26.004 30.977 14.799

(b)

(c) Figure 9: Pressure [mmHg] vs. circumferential stretch. (a) Aorta, (b) carotid, and (c) 23

femoral. Simulations were computed within a range of pressure, 0-40 mmHg, for control (open circle) and FGR (solid circle) vessels. The horizontal red line represents 28 mmHg.

24

4. Discussion A new approach to characterize the material response was taken and a good improvement of the material parameters was obtained in the presence of residual stresses. In the long run, this new way of approaching the problems could be coupled with non-invasive pressure, geometrical and resistance determination in order to obtain parameters from healthy and pathological subjects. In general, a good correlation was achieved between the constitutive model and the experimental response of the tissue material. The presence of a nonuniform stress field across the vessels makes difficult to determine a representative stress level to compare with the experimental data, hence the resultant force was calculated from the simulation for each stretch level, and using equations (1, 2, 3) the Cauchy stress was evaluated. The variation of the stress along the thickness in the area of interest, plotted in (Fig 6), is a consequence of the traction and bending to which the ring-shaped specimen is subjected during the stretching, (Fig 7). The stress concentrations in the inner face and the zone closest to the pins make the experimental stress lower than the actual maximum stress in the simulation. This has been reported before and depends on the diameter and thickness of the sample (Bustos et al., 2016). In addition, since the rupture of the sample occurs in this zone, it is not possible to obtain a rupture stress representative of the material. It should be noted that with the inclusion of the residual stresses, the parameters related to the initial stiffness of the material were smaller, Table 1. This trend was seen in all vessels except for the healthy aorta, and this can be explained by the relatively low residual stress compared 25

to the rest of the vessels. The final parameters show clear differences between healthy and pathological subjects in the aorta and femoral vessels, which are directly related to the stiffer pathological response of these vessels (Cañas et al., 2017). Conversely, carotid parameters were slightly lower in the pathological subject (p > 0.05), an effect that can be visualized in a smaller “a” of the model. Table 3 presented the residual stress values for the inner and outer face of the vessels. It is worth noting that residual stress values in this study were greater in the pathological subjects, differences that have been attributed to changes in the smooth muscle response and the distribution of elastin and collagen in the vessel (Matsuda et al., 1993; Cardamone et al., 2009, Cañas et al., 2017). Considering the opening angles of each vessel, it can be suggested that the residual stresses not only depend on the opening angle, which was greater in the aorta and carotid but a smaller in the femoral artery compared to their pathological counterparts, but also on the mechanical properties of the tissue, since looking only at the angle may be misleading. This trend of opening angles for the healthy vessels has been seen before (Li and Hayashi, 1996; Zhao et al., 2002), but this is not always found (Wang et al., 2014), where an increased alignment of collagen fibers in the media and a stiffer response of the vessel at physiological loads was found. In our simulations, a 55% increase in the aorta's opening angle led to an increase close to 104% in the residual stresses. For the common carotid, which underwent an increase of 17% in the opening angle, the residual stress was ~13% higher, while in the femoral artery a 25.16% reduction of the angle was observed, but since the pathologic tissue is stiffer, an increase in the residual stress of ~80% was 26

obtained in the simulation. It is known that the effects of residual stresses reduce the variation of the circumferential stress across the vessel walls (García-Herrera et al., 2016). In general, the stress variation from the inner to the outer face with respect to the average circumferential stress is not greater than 25%. The only exception corresponds to the femoral artery, where the difference was 43% on the average, a value that may be that high due to the absence of axial stretching in the simulation that exists in the actual vessels (Cardamone et al., 2009; Dobrin et al., 1990) which were not taken into account for this simulation. The selection of an anisotropic constitutive model would doubtlessly cause differences in the stress distribution across the wall when stretched in the axial direction for healthy and unhealthy subjects (Gizzi et al., 2014; Tricerri et al., 2016). Similar values can be found in healthy subjects with differences of 18% (García-Herrera et al. 2016) which could be up to ~38% in the case of pathological cases. The modeled pressurization retrieved an approximation of the potential behavior under a range of normal fetal arterial pressure (0-40 mmHg). Studies in healthy and restricted fetal sheep showed no differences in the mean pressure near term (Edwards et al., 1999; Louey et al., 2000), and also in human childhood and adolescence no significant differences have been established (Keijzer et al., 2005; Williams et al., 1992). Therefore, for the aims of this study, the blood pressure was assumed to be similar between healthy and restricted fetuses, using 28 ± 0.5 mmHg as reported for guinea pig fetuses (Kihlström, 1981). Fig 9 shows the internal pressure on the circumferential stretch, where it is seen that the effect of the pathology on the mechanical response of the vessel under pressure showed a stiffer behavior; the slope at physiological pressure 27

for the pathological subject is 6% and 18% higher than the control specimens for the aorta and femoral arteries, respectively. In contrast, the carotid artery showed a compliant behavior compared to the controls (8% lower), Wladimiroff et al. (1986) observed that some subjects affected by this pathology present offsetting effects on the peripheral vascular resistance in the fetal brain, in contrast with the higher vascular resistance of the body in order to foster the proper development of the brain. The determination of the intima-media thickness from the pressurization test was done in conjunction with the histological data presented previously (Cañas et al, 2017). An increase of the aorta and femoral intima-media thickness (IMT) can be computed from the simulation, and it is 1.9% and 13.1% thicker in the pathological subject’s aorta and femoral, respectively. Conversely, the calculated carotid IMT was 13.6% smaller in the affected animal, a result that can be misleading since no axial stretch is considered in the simulation. Studies have shown that there is a relationship between FGR and IMT, showing an increase of the aorta IMT (Skilton et al., 2005; Koklu et al., 2006) [9,43]. Regarding the carotid artery, there are no conclusive data, and there are cases in which an increase or no changes at all can be found (Crispi et al., 2012; Dratva et al., 2013; Oren et al., 2004). Further studies should focus on non-invasive pressure and resistance determinations (pressure cuff, ultrasound among others) to predict biomechanical behavior of the arteries in clinical settings. This information shall help and assess clinicians in their therapeutical decisions. Limitations of this work that will be addressed in further research are: the assumption of the isotropic behavior, which was basically made because of the difficulty of characterizing the longitudinal direction; the assumption of the longitudinal stretch used 28

in this work, which was not measured during the experiments; and the consideration of experimental tests in which the multiaxial effects are not so relevant. In particular, an anisotropic model could be used together with restricted bounds assigned to some parameters in an inverse approach, where the ill conditioned identification inherent of this technique can be improved by adding more measurements. Although a more complete instrumentation of the ring tensile test could be an option to address this, we are planning for feature works to carried out additional tests, e.g., uniaxial tensile test using longitudinal samples.

5. Conclusions The FGR was found to have a relevant effect on the mechanical behavior of the aorta and femoral arteries. This affects the arterial function, stiffening the artery and changing the residual stresses of these vessels. Moreover, a good representation of the initial geometry was achieved when closing the opened ring; that made it possible to determine residual stress in both subjects. The numerical simulation of the ring tensile test showed that there is a large difference in the stress across the wall during the test, and because of this concentration the actual rupture stress is higher than the stress obtained from the test. The pressurization of the vessels for the aorta and femoral arteries showed a stiffer behavior compared to the controls, that can be related to decreased damping capabilities. Conversely, FGR carotid arteries showed a completely different behavior, related to offsetting effects on the vascular resistance. Furthermore, we propose that this fitting procedure can be adapted to be used together with a non29

invasive

pressure,

characterize

human

resistance

and

pathologies

to

geometrical predict

determinations

different

conditions

techniques with

to

extreme

physiological loads (hypertension) and, therefore, to help physicians in clinical situations to diagnose patient outcomes.

Conflicts of Interest The authors have no conflicting interests regarding this paper.

Acknowledgments The supports provided by the National Council for Scientific and Technological Research CONICYT (FONDECYT Project No. 1170608) and “Proyecto Fortalecimiento Usach USA1799_GC131612 (Universidad de Santiago de Chile)”, are gratefully acknowledged.

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