A model for passive elastic properties of rat vena cava

A model for passive elastic properties of rat vena cava

ARTICLE IN PRESS Journal of Biomechanics 40 (2007) 3130–3145 www.elsevier.com/locate/jbiomech www.JBiomech.com A model for passive elastic propertie...

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ARTICLE IN PRESS

Journal of Biomechanics 40 (2007) 3130–3145 www.elsevier.com/locate/jbiomech www.JBiomech.com

A model for passive elastic properties of rat vena cava Georg Wolfgang Descha,, Hans Werner Weizsa¨ckerb a

Institut fu¨r Mathematik und Wissenschaftliches Rechnen, Karl-Franzens-Universita¨t Graz, HeinrichstraX e 36, 8010 Graz, Austria Zentrum fu¨r Physiologische Medizin, Institut fu¨r Physiologie, Medizinische Universita¨t Graz, Harrachgasse 21/V, 8010 Graz, Austria

b

Accepted 19 March 2007

Abstract A two-dimensional model for the elastic properties of vena cava abdominalis under orthotropic deformation is introduced and tested against the experimental data obtained from six specimen of rat venae cavae by pressurization experiments. The model is based on membrane approximation and suited for vessels where most of the elastic elements are oriented axially, while circumferential contraction is exerted by redirecting axial stress by some network of oblique fibers. For the experimental data considered in this paper, the ratio between axial and circumferential stress depends almost exclusively on the circumferential extension ratio. As a consequence, the mechanical system can be formally decomposed in a kinematic system reacting by axial contraction on circumferential extension without any loss or storage of energy, serially connected to a hyperelastic system acting only in axial direction. Both systems are modeled separately by equations obtained by a purely phenomenological approach with two parameters for each system. This leads to reasonable reproduction of the experimental data. Introducing a correction parameter, which takes into account that the model assumption on the decomposition does not hold exactly, we get better reproduction of data. However, this is paid for by loss of physical rigor and in particular by departing from the assumption of hyperelasticity. r 2007 Elsevier Ltd. All rights reserved. Keywords: Rat vena cava; Elasticity; Stress–strain constitutive law; In vitro

1. Introduction The veins are more than simple conduits for the return of blood to the heart. With their active and passive wall properties veins contribute to the distribution of blood between periphery and heart influencing cardiac filling and regulating cardiac output. Veins participate in complex cardiovascular reflexes affecting the blood circulation both in health and in disease (Rothe, 1997). Although the incidence of vein disease is very high (Monos et al., 1995), investigation of the biomechanical properties of the venous wall is rather scant and mainly focused on distal veins and on their remodeling when used as arterial grafts (Monos et al., 1995; Wesly et al., 1975; Dobrin et al., 1988; Han et al., 1998). Surprisingly little information can be found in the literature on the mechanical Corresponding author. Tel.: +43 316 380 5177; fax: +43 316 380 9815.

E-mail addresses: [email protected] (G.W. Desch), hans. [email protected] (H.W. Weizsa¨cker). 0021-9290/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.jbiomech.2007.03.028

properties of proximal veins, particularly on those of the vena cava. In a seminal paper Moreno et al. (1970) have drawn attention to the complex aspects of vein collapsibility. They measured the cross-sectional shape and area of the inferior vena cava of dogs in the negative to positive domain of transmural pressure and modeled the cava’s deformational behavior using the classical theory of elasticity. Azuma and Hasegawa (1973) used results from uniaxial test on circumferential and axial strips of veins, along with histological evidence, to propose a qualitative two-dimensional histomechanical model of the venous wall. Their examination included the superior and inferior vena cava from dogs. Hasegawa (1983) complemented the aforementioned investigation by observing that circumferential strips display much more stress relaxation than longitudinal strips, a finding in agreement with axially oriented elastic fibers postulated by his model. Using circumferential rings from three different locations along the canine superior vena cava, Minten et al. (1986) found that both

ARTICLE IN PRESS G.W. Desch, H.W. Weizsa¨cker / Journal of Biomechanics 40 (2007) 3130–3145

the composition and mechanical properties of the venous wall strongly depend on topography. Minten et al. (1987) studied the geometrical variations of the canine vena cava in vivo during open and closed chest experiments. The authors found that the cava underwent complex multiaxial deformations throughout the cardiac cycle, but that the cross-section was nearly circular during most phases of the cycle. To be able to cope with the strong anisotropy observed in caval veins, Sakanishi et al. (1988) used two different exponential formulae and a total of six material constants to separately fit stress– strain data stemming from circumferential and axial strips, respectively. Xie et al. (1991) presented the most detailed and systematic description of the geometry of rat’s veins in their in situ, load-free and stress-free states. Their study included the abdominal vena cava, the circumference of which was found to be quite round in vivo but very soft, flimsy and wrinkly in the no-load and stress-free states. The irregular geometry of the opened-up sectors allowed only a crude estimate of the opening angle of the cava abdominalis. Ohhashi et al. (1992) studied the regional differences of mechanical properties of veins and the contribution of tissue components to these properties by recording pressure–volume curves from tubular segments of veins in vitro. As for the inferior vena cava their results indicated an increase of distensibility in distal direction, within the pressure range 0–3.7 mmHg (0–0.5 kPa). Smooth muscle activation appeared to have little impact on the mechanical behavior of the lower abdominal cava. Not surprisingly, the interest of several authors in the biomechanics of veins has been clinically motivated (Attubato et al., 1994; Hayashi et al., 2003; Kitano et al., 1999; Macha et al., 1996; Wagenseil et al., 2004; Wesly et al., 1975). Garcia et al. (2002), for example, studied the effect of the cava’s compliance on the performance of a pulsating respiratory support catheter. Given the paucity of available data in the pressure range of interest (5–15 mmHg, i.e., 0.7–2 kPa) the authors measured the specific compliance of the bovine vena cava, fabricated an elastic tube with comparable compliance and analyzed its influence on the gas exchange performance. From the literature quoted above it is quite evident that research on the elasticity of the vena cava has hitherto been limited essentially to one-dimensional tests. These were either simple elongation experiments on vascular strips or pressurization tests on tubular segments of cava held at constant length. Therefore it does not surprise that, differently as is the case for the aorta, constitutive relations do not exist for the vena cava. Yet it is clear that these relations are a prerequisite for a rational approach to the vein’s performance under normal, pathological and clinical conditions. The purpose of the present study was to develop a constitutive model for the pseudoelastic properties of the vena cava, based on multiaxial experiments performed on the vena cava abdominalis of rats.

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2. Experimental methods and results The experiments used in this study consisted essentially of two sets of in vitro tests. The first set was aimed at establishing the deformational behavior of the vena cava when the vessel was subjected to transmural pressure and axial stretch; the second set complemented the first and focused on establishing the load-free and zero-stress state of the vessel. A total of 12 segments of abdominal vena cava from 12 adult Wistar rats were used, 6 for each set. Experimental design, setup and protocols have been described elsewhere (Weizsa¨cker, 1988); for the sake of completeness they are summarized here. In a typical pressurization test, a section of abdominal vena cava was surgically exposed and its axial retraction upon excision (or equivalently its in vivo pre-stretch) was determined using ink dots applied to the vessel wall. The tubular segment was then closed at one end, cannulated at the other, connected to a piston pump and suspended in vertical position between the clamps of a tensile testing machine. A cuvette containing Caþþ -free physiological salt solution (PSS) with EDTA at 37  C was then used to fully submerge the vein. After having preconditioned the vessel and repeatedly determined its load-free dimensions, the tubular segment was stretched to several levels of axial strain (including and past the in vivo one), and pressurized at each level with PSS from 0 to 20 mmHg (0–2.7 kPa). During the test the transmural pressure, p, and the axial force, f , were recorded by transducers while deformation at mid-portion of the specimen (i.e. the outer diameter, d, and the axial stretch, lz ) were measured using a Video System. Given the tremendous deformability of the caval wall at low transmural pressure, proper precautions were taken to exactly set and control transmural pressure and to avoid spurious torsions, end-effects and axial overstretch, as detailed in Weizsa¨cker (1988). After the mechanical test was completed, the wall thickness of the cava was measured under microscope on cross-sectional rings cut from the specimen. This latter measurement proved to be more difficult than one would expect, the reason being twofold. Firstly, the wall of the cava was extremely soft and deformable in the no-load state and easily ‘‘foldedover’’, which tended to hide the actual wall thickness. Secondly, no clear demarcation could be found between vessel wall and perivascular tissue. Hence the measured value of the wall thickness likely depended somewhat on both the dissection procedure and the personal judgment. To further address these problems and in order to assess the zero-stress state of the cava, a complementary set of tests was added, as mentioned earlier: 6 tubular segments were dissected from 6 rats, carefully freed from loose connective tissue, cut into rings and successively in ‘‘opened-up sectors’’, all while floating in the PSS at 37  C. Surprisingly, both rings and sectors showed an irregular, wavy and wrinkly geometry with varying thickness along the perimeter, as very well described in Xie et al. (1991). Thus while the perimeter of the ‘‘rings’’

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could be determined fairly well on enlarged photographs, an opening angle (as defined e.g. in Greenwald et al., 1997) could be defined reasonably well only for some 20–30% of the sectors. All these data (gathered from over 100 sections) and a reevaluation of corresponding ones from the first set of experiments yielded the following best estimates: loadfree outer circumference, C ¼ 7:64  0:61 (SD) mm; loadfree wall thickness, H ¼ 0:08  0:01 (SD) mm; opening angle, a  60 . From the value of C the load-free ‘‘reference diameter’’ was calculated as D ¼ C=p. Due to the selection mentioned above, the value of a is open to doubt and likely biased toward small values. Even more importantly, following the results by Vossoughi et al. (1993) and by Greenwald et al. (1997) it appears questionable whether the radially cut-open sectors were actually stress-free, as has been previously assumed (Xie et al., 1991).

-3

x 10 3.4

Outer Diameter [m]

3.2

A representative result out of the six pressurization experiments is shown in Fig. 1. The upper diagram gives the pressure–diameter relationship for various constant levels of axial stretch. The lower diagram shows the corresponding relation for the axial force. The inset arrows indicate the sense of increasing axial stretches (i.e. parameter values). It is clear from upper diagram in Fig. 1 that the abdominal cava is extremely compliant at low pressure but becomes increasingly inextensible as the transmural pressure approaches some 2–3 mmHg (0.25–0.4 kPa). The dependence of axial force on pressure shown in Fig. 1B is typical for blood vessels: decrease with pressure at small but increase with pressure at large levels of axial stretch (Weizsa¨cker and Pascale, 1977). For lz ¼ 1:84 the force is nearly independent of pressure. The average value of this ‘‘crossover stretch’’ for the six experiments is 1:82  0:07 (SD), nearly the same value as the measured in vivo axial pre-stretch. Around this prestretch and for a ‘‘physiological’’ transmural pressure of 2 mmHg (0.27 kPa), assuming wall incompressibility, the mid-wall diameter to wall thickness ratio resulted to be approximately 100. Hence the vena cava abdominalis is thin-walled indeed.

3

3. The model

2.8 2.6

3.1. Formulation of the model

2.4 2.2

In the analytical treatment we will use the following notation:

2 1.8

Symbol

1.6 1.4 0

500

1000

1500

2000

2500

Transmural Pressure [Pa] 0.05

Reference state: D m H m

Description Reference (outer) diameter Reference wall thickness

State during experiment: d m (outer) Diameter f N Axial force p Pa Transmural pressure lz n.d. Axial extension ratio ly n.d. Circumferential extension ratio tz Pa First Piola–Kirchhoff stress in axial direction ty Pa First Piola–Kirchhoff stress, circumferential

0.045 0.04

Axial Force [N]

Unit

0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 0

500

1000

1500

2000

2500

Transmural Pressure [Pa] Fig. 1. Deformation of a cylindrical segment of vena cava (Sample 4) when it is pressurised while held at 9 successive levels of axial stretch, lz . Here lz ¼ 1:28; 1:42; 1:56; 1:63; 1:70; 1:77; 1:84; 1:91, and 1.98. The inset arrows indicate the sense of increasing axial stretches.

Considering the large value of the diameter-to-wall thickness ratio and above all the manifold uncertainties regarding the zero-stress state of the vena cava, the membrane approximation is used in the present theoretical approach. Hence, possible residuals present in the wall are neglected and the load-free geometry is taken as the reference state. Moreover, we assume that the vessel is cylindrical. This is justified by the observation that the

ARTICLE IN PRESS G.W. Desch, H.W. Weizsa¨cker / Journal of Biomechanics 40 (2007) 3130–3145

vessel’s cross-section is almost circular even in the cases of low, positive transmural pressures. The deformation is assumed to be orthotropic, no shear strains and stresses are considered. The quantities D; H; d; f ; p; lz are observed in the experiments. For the circumferential stretch we have the obvious relation ly ¼

d . D

(1)

The first Piola–Kirchhoff stresses in the wall are in equilibrium with the axial force and the transmural pressure. This leads to the following relations, which are independent of the material’s particular constitutive law and hold in any thin-walled cylindrical vessel: p d lz , 2H   1 d 2 pp tz ¼ f þ . DpH 4 ty ¼

ð2Þ ð3Þ

Stress ratio circumferential / axial (n.d.)

Using these relations, we obtain immediately ly , tz and ty , so that these quantities will be considered like experimental data in the following treatment. The key observation, which is the starting point of our model, is a surprising observation made on closer inspection of the data. It is shown in Fig. 2, where we plot the ratio of stresses ty =tz , obtained from (2) and (3), versus the circumferential extension ratio ly . These data correspond to nine different axial extension ratios (see Fig. 1). Nevertheless, with exception of the data taken at very low stretches, all data points are located remarkably close to a single curve. For comparison the figure shows a curve fitted to the data using the model (8). We conclude that, at least for large strains, the stress ratio depends almost exclusively on the circumferential extension ratio. The data

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in Fig. 2 are taken from Sample 4, which show this phenomenon most strikingly. However, most of the other samples exhibit a quite similar picture. To motivate the further procedure and to get an intuitive feeling for the implications of this observation, we consider first a simple structure depicted in Fig. 3. We imagine a network consisting of elastic fibers in axial direction hooked up to inextensible fibers in circumferential and oblique directions. The elastic fibers try to align the network fibers in axial direction, thus causing a circumferential contraction. Since in the hypothetical structure considered in this example all oblique and circumferential fibers are inextensible, the circumferential extension of the vessel depends exclusively on the angle of the oblique fibers. The axial extension, however, is the sum of two parts, one depending on the angle of the oblique fibers, and the second part depending on the extension of the elastic axial fibers. Thus the first part is dependent on the circumferential extension, while the second part depends on the axial stress and the stored elastic energy. It is this decomposition of the axial strain, on which our model relies. Of course, in biological tissue we expect a much more irregular and complex arrangement of fibers, and our model will not adhere to the geometry depicted in Fig. 3, nor will we try to construct more complex geometries of fiber arrangements. However, we will keep in mind the one feature that the axial strain can be decomposed into the sum of an ‘‘elastic’’ and an ‘‘inelastic’’ component as above. In fact, the following theorem states that this decomposition follows from the independence of the stress ratio ty =tz of lz , which we have observed in Fig. 2.

2.5

2

1.5

1

0.5

0 0.5

0.75

1

1.25

1.5

Circumferential stretch (n.d.) Fig. 2. Stress ratio ty =tz for a rat vena cava, plotted against the circumferential extension ratio ly (Sample 4). The diagram shows the data for 9 different axial extension ratios. The solid line is a curve fitted by model equation (8).

Fig. 3. The ‘‘bedspring model’’: a simple fiber network. Elastic elements are arranged in axial direction and connected by a network of inextensible fibers.

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G.W. Desch, H.W. Weizsa¨cker / Journal of Biomechanics 40 (2007) 3130–3145

(In order to keep the exposition fluent, we postpone a proof to the appendix.) Theorem 1. Consider an elastic body subject to twodimensional orthotropic deformation in directions z and y. (1) Suppose that the constitutive law can be stated by a strain-energy density function (no viscoelastic or viscoplastic effects), and that the ratio of the first Piola– Kirchhoff stresses ty =tz depends exclusively on the extension ratio ly in direction y. Then the extension ratio in direction z can be decomposed into a sum of two parts lz ¼ lz;inel þ lz;el , so that lz;inel depends exclusively on ly , while both the energy density and the stress tz depend only on lz;el . In fact, up to an integration constant C, one has Z   ty lz;inel ¼  (4) ðly Þ dly þ C. tz (2) Conversely, assume that there is a decomposition lz ¼ ~ ðlz;el Þ such that lz;inel ðly Þ þ lz;el ðlz ; ly Þ and a function W d ~ tz ¼ W ðlz;el Þ; dlz;el

ty d ¼ lz;inel ðly Þ. dly tz ~ ðlz;el ðlz ; ly ÞÞ is an Then the function W ðlz ; ly Þ ¼ W energy density function for the material. This observation leads to a two-step approach for modeling the elastic constitutive law of the venous wall: (1) How does the ratio of the first Piola–Kirchhoff stresses ty =tz depend on the circumferential extension ratio ly ? By integration, this step also gives the relation between ly and the inelastic component lz;inel of the axial extension ratio. (2) How does the axial stress tz depend on the elastic component lz;el ¼ lz  lz;inel of the axial extension ratio? Utilizing the model from Step 1, this yields also the circumferential stress via the stress ratio. The two-step approach simplifies the modeling procedure considerably: it is much easier to find the laws to describe two separate scalar functions than to find a single law to describe a function from the two-dimensional space into the two-dimensional space. For shorthand we will refer to the relations between ly , ty =tz and lz;inel as the ‘‘inelastic system’’, while the relations between lz;el and tz will be referred to as the ‘‘elastic system’’. Much room is left to speculation what these systems are physically. One might possibly think of the inelastic system as a network of (almost) inextensible fibers, while the elastic system might consist of elastic elements arranged in axial direction. However, no such assumptions are made in the model. Theorem 1 only implies that formally there is some mechanical system which reacts on axial extension by

circumferential contraction and vice-versa without storage or loss of energy, and this system is serially connected to a hyperelastic system reacting by axial stress on axial strain. To set up the relation between the circumferential stretch and the stress ratio, we were initially tempted to mimic the geometry of Fig. 3 and use the lengths of the various fibers as parameters. It turns out, that this approach results in poor reproduction of data. Instead, we have resorted to a purely phenomenological approach, inspecting the data and trying various plots and transformations. A good fit can be obtained by a power law ty ¼ cinel ðly;max  ly Þ2 . tz The power 2 is close to optimal, and the model is not very sensitive to the power. An attempt to leave the power as a free parameter has not improved the reproduction of the data significantly. From (4) we obtain, up to an integration constant, lz;inel ¼ cinel ðly;max  ly Þ1 þ C.

(5)

At first glance it strikes strangely that lz;inel is negative. However, there is room for an integration constant C, which can be added to lz;inel and subtracted from lz;el , so that it finally cancels from all computations. This model has two dimensionless parameters ly;max and cinel . The circumferential stretch will never exceed ly;max . In fact, as ly approaches ly;max , the inelastic part lz;inel of the axial stretch goes to 1. We admit that we would have expected a model with a finite limit. Possibly quite different model equations would turn out from data with vessels pressurized close to the limit of rupture. The fraction cinel =ly;max quantifies the gain, how strongly the axial contraction reacts on circumferential extension, if axial stress is kept constant. In fact, at the point ly ¼ 1=2ly;max , where the vessel is stretched circumferentially to half of its theoretical maximal extensibility, we have d=dly lz;inel ¼ cinel =4ly;max . We emphasize that this is a blackbox model of the global relation between circumferential and axial stretch of the inelastic system. No assumptions are made on any arrangement of fibers. To model the relation between the elastic part of the axial stretch and the axial stress, we have again taken a phenomenological attitude. The quantity lz;el can be computed from lz and lz;inel , the latter being obtained from the model of the inelastic system. (For comparison, we have also estimated lz;inel by a numerical integration procedure using (4), independent of the model for the inelastic system. We have obtained quite similar results.) Again, trial and error have led to a power law tz ¼ cel ðlz;max  lz;el Þ2 . No significant improvement has been obtained by leaving the power a free parameter. The equation above does in fact not admit a zero-stress state. A third parameter would be needed to compensate for this unrealistic feature. We have not pursued this approach, since our measurements

ARTICLE IN PRESS G.W. Desch, H.W. Weizsa¨cker / Journal of Biomechanics 40 (2007) 3130–3145

for low stretches are rather uncertain. It should therefore be recorded, that our model is only applicable to sufficiently large stretches and stresses. This model has two parameters, namely the dimensionless quantity lel;max , and cel with the dimension of a pressure. The model implies that the elastic system cannot be stretched beyond lz;el ¼ lel;max . The quantities lz;el and lz;inel , however, are only determined up to a common integration constant which has been set to zero arbitrarily, and they are hidden quantities, not accessible to measurement. Therefore an interpretation of the numerical value of lel;max in terms of the measured data cannot be given: it changes depending on the choice of the integration constant. The elastic system is highly nonlinear. The quantity 4cel =l3el;max is the Young modulus of the elastic system, if the constitutive law is linearized at a state when lz;el ¼ 12lel;max . At this state, d=dlz;el tz ¼ 4cel =l3el;max . ~ 0 ðlz;el Þ Notice also that tz ¼ W with ~ ðlz;el Þ ¼ cel ðlel;max  lz;el Þ1 . Therefore, according to W Theorem 1, Part 2, the combination of the inelastic and the elastic models above yields a constitutive law which admits an energy density W ðlz ; ly Þ ¼

cel . lel;max  lz  cinel ðly;max  ly Þ1

Below we give a summary of the whole model: Model 1 (Four parameter model): We introduce the following model for the elastic behavior of the wall of a vena cava under orthotropic deformation: Model parameters: ly;max ðn:d:Þ; cinel ðn:d:Þ; lel;max ðn:d:Þ; cel ðPaÞ. Model equations: lz ¼ lz;el þ lz;inel , cinel , lz;inel ¼  ly;max  ly ty cinel ¼ , tz ðly;max  ly Þ2 cel tz ¼ . ðlel;max  lz;el Þ2

ð6Þ ð7Þ ð8Þ ð9Þ

Here the state variables lz , ly , tz , ty are the stretch ratios and first Piola–Kirchhoff stresses in axial ðzÞ and circumferential ðyÞ direction. The quantities lz;el , lz;inel are hidden (not measurable), dimensionless variables depending on the deformation state of the vessel. The reader may notice that the integration constant C in (5) has not been included in the model. In fact, including C in (7) could be compensated by changing (9) to tz ¼

cel . ðlel;max þ C  lz;el Þ2

(10)

Although C now seems to appear as a fifth model parameter, it cancels out from all computations, and the model with (7) and (9) replaced by (5) and (10) yields exactly the same reproduction of data as Model 1.

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3.2. Fitting the data For each of the 6 Samples, the model parameters were determined using the following fitting procedure: (1) From the measured data (lz , p, f , d, H, D), the circumferential stretch ly and the first Piola–Kirchhoff stresses ty , tz are computed by Eqs. (1)–(3). (2) The parameters for the network, i.e., cinel and ly;max are fitted on behalf of Eq. (8). We use a nonlinear least squares procedure with ly as the independent variable and ty =tz as the dependent variable. (By transformation of the dependent variable, linear regression could be used. This was utilized to get an initial guess for the parameters. However, the nonlinear optimizer improved the reproduction of the data considerably.) (3) Given cinel and ly;max , the decomposition of the axial stretch (lz;inel , lz;el ) can be computed using Eqs. (7) and (6). (4) Another nonlinear least squares procedure based on Eq. (9) with lz;el as independent variable and tz as dependent variable yields the parameters for the elastic elements (lel;max and cel ). (5) For a comparison of the fitted model to the experimental data, the stretches lz and ly were considered as the independent variables, from which the predicted stresses tz;model ; ty;model were computed by the model. Using the fitted parameters, successive computations yield  the decomposition of the axial stretch lz;el , lz;inel by Eqs. (7) and (6)  the predicted first Piola–Kirchhoff stress in axial direction tz;model by Eq. (9),  the predicted first Piola–Kirchhoff stress in circumferential direction ty;model by Eq. (8). (6) The stresses obtained by the model are graphically compared to the stresses ty ; tz computed from the experimental data using Eqs. (2) and (3). (7) To evaluate the goodness of the fit numerically, the relative error was computed by the formula ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi sP P ðtz  tz;model Þ2 ðty  ty;model Þ2 P P error ¼ . þ 2 t2z 2 t2y

(11)

In order to cope with the uncertainty of the data at low stretches, we removed the 20% of data with the lowest stretches before fitting the parameters (steps 2–4). However, for the final comparison of the model with the data (steps 5–7 above), all data have been used. We emphasize that for each of the 6 samples one set of four parameters was fitted to reproduce both, axial and circumferential stresses simultaneously for the whole range of axial and circumferential stretches covered by the experiment. Table 1 shows the fitted model parameters and the modeling error according to Eq. (11) for all six samples.

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Table 1 Model parameters and error fitting the 4-parameter Model 1 Number of data

Network model

Elastic model

Relative error

4 Par. model

51 96 113 108 78 73

ly;max

cinel

lel;max

cel

lel;max

cel

m

1.4280 1.6359 1.5182 1.5657 1.4362 1.4479

0.00796 0.02638 0.01887 0.01885 0.01426 0.01185

2.4364 2.2450 2.3321 2.2479 2.1841 2.2015

3821.64 2530.64 3906.24 2099.50 1252.77 1582.79

2.3468 2.3902 2.5199 2.2993 2.1852 2.2211

726.37 2780.46 3749.85 1921.76 721.46 1057.79

2.2440 1.7126 2.1785 1.4036 1.3875 1.4889

For this model, the relative error is between 18% (for Sample 4) and 34% (for Sample 1). Figs. 4 and 5 give a graphic comparison between the simulated stresses and the experimental stresses for Sample 4 and the somewhat atypical Sample 1. While the data from Sample 4 were the ones best reproduced by the model, most of the other samples have created a similar picture. With exception to Sample 1, the model seems to reflect the qualitative behavior quite reasonably, in particular with respect to the fact that only 4 parameters were used to adapt the model to the data. Remember that we have fitted the parameter using only 80% of the data, removing the data with the least stretches. A fifth parameter would be in order to adapt the model to low stretches. Numerically, there are still substantial deviations. The best agreement is observed with Sample 4. The properties of Sample 1 deviate from those of the other samples remarkably. We will return to this topic in Section 3.3 below. The performance of the potential on all six samples is summarized in Figs. 10 and 11 at the end of the paper.

4 Par.

5 Par.

0.3343 0.2262 0.3093 0.1769 0.2875 0.2274

0.1018 0.1772 0.2104 0.1747 0.2010 0.1238

Sample 4

4

12

x 10

Data Fit

10

Axial Stress (Pa)

1 2 3 4 5 6

5 Par. model

8

6

4

2

0 0.5

1

1.5

Circumferential Stretch (n.d.) 4

6

Sample 1

x 10

Data 5

Fit

Model 1 is based on the assumption, that the stress ratio depends only on the circumferential stretch, which holds, of course, only approximately. As a consequence (Theorem 1), the axial stretch can be decomposed in an elastic and an inelastic part, with the axial stress depending exclusively on the elastic stretch. An inspection of the relation between the elastic stretch (computed by the model) and the axial stress can therefore be used to validate or disvalidate the model. In Fig. 6 we plot the axial stress against the elastic axial stretch for two of our samples. Sample 4 exhibits what is also typical for most others. While the data are not located exactly on a single curve, at least a reasonable fit by a single curve is possible. Sample 1, as we have already mentioned, behaves quite differently than all other specimens in our series. Here, the data are far from being located on a single curve. With respect to this graph, the poor reproduction of the data for this sample by Model 1 is no longer surprising.

Axial Stress (Pa)

3.3. An additional parameter? 4

3

2

1

0 0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

Circumferential Stretch (n.d.)

Fig. 4. Dependence of the axial stress on circumferential stretch. The diagrams show the data (circles) and the values predicted by Model 1 (asterisk) for Samples 4 (upper plot) and 1 (lower plot). Each curve corresponds to a constant axial stretch.

It turns out that a single additional parameter can change this picture. Remember that lz;inel is related to the stress ratio by Eq. (4). From this we get lz;el ¼ lz  lz;inel . Introducing a

ARTICLE IN PRESS G.W. Desch, H.W. Weizsa¨cker / Journal of Biomechanics 40 (2007) 3130–3145 4

Sample 4

4

14

x 10

12

Data

Fit

Fit

10

10

8

Axial Stress (Pa)

Circumferential Stress (Pa)

Sample 4

x 10

Data 12

8

6

6

4

4 2

2

0

0 0.5

1

1.3

1.5

Sample 1

x 10

6

1.6

1.7

1.8

1.9

2

2.1

2.2

Sample 1

x 10

Data

Data 9

1.5

4

4

10

1.4

Elastic Part of Axial Stretch (n.d.)

Circumferential Stretch (n.d.)

Fit

Fit

5

8 7

Axial Stress (Pa)

Circumferential Stress (Pa)

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6 5 4

4

3

2

3 2

1

1 0

0 0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

Circumferential Stretch (n.d.)

Fig. 5. Dependence of the circumferential stress on circumferential stretch. The diagrams show the data (circles) and the values predicted by Model 1 (asterisk) for Samples 4 (left plot) and 1 (lower plot). Each curve corresponds to a constant axial stretch.

new parameter m we change the latter equation to lz ¼ lz;el þ mlz;inel .

1.8

1.85

1.9

1.95

2

2.05

2.1

2.15

Elastic Part of Axial Stretch (n.d.)

Fig. 6. Dependence of axial stress on elastic stretch, computed by Model 1. The diagrams show the data (circles) and a least-squares fit using model (9) for Samples 4 (upper plot) and 1 (lower plot).

Model equations (7)–(9) are taken from Model 1 unchanged. The decomposition equation (6) is changed to (12).

(12)

For most of our samples, m can be fitted such that the data points of tz plotted against lz;el lie close to a single curve. Fig. 7 shows for two samples, that now the axial stress at least approximately depends as a function only on lz;el . Even Sample 1 yields a fit which is not good, but acceptable. We modify Model 1 just by introducing the correction parameter m. With m ¼ 1 fixed, the modified model is again reduced to Model 1. Model 2 (Five parameter model): We introduce the following modifications in Model 1: A parameter m is added to the other model parameters ly;max , cy , lel;max , cel .

The fitting and evaluation of Model 2 is done quite similarly as for Model 1. Table 1 shows the fitted parameters and errors. The correction parameter m takes mostly values between 1.3 and 1.8, with two highs above 2 for Samples 1 and 3. Compared to Model 1, the relative errors have declined by factor between 0.5 and 0.8, with an extreme 0.30 for Sample 1 (where Model 1 showed very poor performance) and only little improvement (0.99) for Sample 4. Figs. 8 and 9 display, how the stresses are reproduced by Model 2. At least the qualitative behavior of the data is reproduced quite satisfactorily by the model. Regarding the fitted data, we would be tempted to draw a very optimistic conclusion about the correcting factor m,

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4

Sample 4

4

12

x 10

12

Data

Data Fit

Fit

10

Axial Stress (Pa)

Axial Stress (Pa)

10

Sample 4

x 10

8

6

4

8

6

4

2

2

0

0 1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

2.1

2.2

0.5

1

4

7

Sample 1

x 10

Fit

6

Fit

5

Axial Stress (Pa)

Axial Stress (Pa)

x 10

Data

5

4

3

4

3

2

2

1

1

0

Sample 1

4

7

Data 6

1.5

Circumferential Stretch (n.d.)

Elastic Part of Axial Stretch (n.d.)

1.8

1.85

1.9

1.95

2

2.05

2.1

2.15

2.2

2.25

Elastic Part of Axial Stretch (n.d.)

Fig. 7. Dependence of axial stress on elastic stretch, computed by Model 2. The diagrams show the data (circles) and a least-squares fit using model (9) for Samples 4 (upper plot) and 1 (lower plot).

but as theorists we rise objections. Of course, the stress ratio’s independence of the axial stretch holds only approximately. Talking in terms of a fiber model, the physical reasons may be manifold. For instance, circumferential fibers will not be completely inextensible, elastic elements will not be arranged exactly in axial direction, reallocation of fibers may be a very complex process, which may not even be globally modelled by an elastic system without including plastic effects. Even a network with all elastic fibers arranged exactly in axial direction and completely inextensible fibers in oblique and circumferential directions does not necessarily exhibit the decomposition behavior described in Theorem 1, since in a more complicated network, various axial fibers may be extended to different stretch ratios. Thus there are many reasons to stick not too rigidly to the assumption of a decomposition of axial stretch in the sense of Theorem 1 and to introduce at least some correction in the model. However, there is no

0 0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

Circumferential Stretch (n.d.)

Fig. 8. Dependence of the axial stress on circumferential stretch. The diagrams show the data (circles) and the values predicted by Model 2 (asterisk) for Samples 4 (upper plot) and 1 (lower plot). Each curve corresponds to a constant axial stretch.

theoretical justification, beware explanation, for the way the correcting parameter m is introduced in the formula. In fact, Model 2 as it stands cannot be associated with any potential, since it violates the assumption of energy conservation. On the other hand, the reproduction of the data is improved considerably by the introduction of m. We will return to this issue in the discussion. 4. Discussion In the present paper a systematic analysis of experimental stress–strain data and a fiber-net model based on this analysis were used to establish, for first time in the literature, a two-dimensional constitutive model for the pseudoelastic behavior of the vena cava abdominalis from rats. A membrane approximation using the load-free

ARTICLE IN PRESS G.W. Desch, H.W. Weizsa¨cker / Journal of Biomechanics 40 (2007) 3130–3145

Sample 4

4

15

x 10

Data

Circumferential Stress (Pa)

Fit

10

5

0 0.5

1

1.5

Circumferential Stretch (n.d.)

Sample 1

4

10

x 10

Data

Circumferential Stress (Pa)

9

Fit

8 7 6 5 4 3 2 1 0 0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

Circumferential Stretch (n.d.)

Fig. 9. Dependence of the circumferential stress on circumferential stretch. The diagrams show the data (circles) and the values predicted by Model 2 (asterisk) for Samples 4 (left plot) and 1 (lower plot). Each curve corresponds to a constant axial stretch.

configuration as the reference state was used to compute stresses and strains. This approach was motivated by the huge diameter to wall thickness ratio, by the tremendous flexibility of the caval wall but above all by the realization that the stress-free configuration of the cava abdominalis remains essentially unknown. In fact, as mentioned earlier, it is questionable whether the radially cut-open configuration was stress-free and it remains unclear whether the observed irregular geometry of rings and sectors were due to remanent residuals, material heterogeneity, difficulty in dissection or else. Future histological analyses accompanying mechanical tests may shed some light on this complex issue. Elastic properties of arteries have been fitted successfully by several elastic potentials (see, e.g., Carboni et al., 2007 and the references therein). The vena cava, however,

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behaves quite differently than arteries. In particular, at low stretches it is extremely soft and sloggy, and it stiffens very rapidly when filled beyond a certain threshold. Our attempts to fit the elastic properties of vena cava by artery potentials have shown that these constitutive laws cannot reproduce such rapid increase of stiffness. A fiber network as shown in Fig. 3 is much better suited to describe such behavior. However, a literal transformation of Fig. 3 in a mathematical model has shown poor agreement of the numerical values. All we have maintained from this approach is, intuitively spoken, that the elastic parts of the structure are oriented in axial direction. This is reflected in the decomposition of the axial stretch ratio in an ‘‘elastic part’’ and an ‘‘inelastic part’’, with a one-to-one correspondence of the latter to the circumferential strain. According to Fig. 2, it is at least reasonable to claim that stress ratios are (almost) independent of the axial stretch. This, in turn, is equivalent to the fact that the axial stretch admits a decomposition as above. The precise formulae to model the behavior of the elastic elements and the relation of the inelastic axial stretch to the circumferential stretch have been obtained phenomenologically on behalf of the data, and collected in Model 1. Four parameters are required, but if the model should be tuned to fit also low stretches, a fifth parameter would be needed. In the range of very low stretches, our data seem too uncertain to support any mathematical description. In particular, the load-free reference state could not be determined very precisely. We have therefore withstood from adapting the model to small stretches. Each of the four parameters can be given an intuitive meaning: the parameter cel reflects the strength of the axial elastic elements, lel;max is related to the maximal stretch of these elements, cinel measures the strength of interaction between axial and circumferential stretch, and ly;max is an upper bound for the circumferential stretch. The qualitative behavior of the experimental data is well reflected by the model (c.f. Figs. 4, 5). The numerical values for the stresses predicted by the model deviate from the measured data by some 18% in the best case, and 33% in the worst case (c.f. Table 1). It turns out that in some cases the numerical performance of the model can be increased significantly by introducing a fifth parameter m. The five-parameter Model 2, however, does no longer describe a material with an elastic potential. On the other hand, the presence of m improves the reproduction of the data so much, that one is tempted to conjecture some physical reality hidden behind this purely phenomenological correction. Although presently it is still difficult to unequivocally identify the reason for the improvement, one likely cause may be relaxation or even fatigue of the vessel wall in the axial direction. As mentioned in Weizsa¨cker (1988), because of difficulties in handling vessels so thin-walled and soft as the rats’ cavae, more than half a dozen of pressurization experiments had to be performed (and discarded) before obtaining the first viable result, namely Sample 1. During this still laborious test and due to various experimental maneuvers, the vessel

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Sample 1 4

6

x 10

4

10

Data

Circumferential Stress (Pa)

Axial Stress (Pa)

Data 9

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5

x 10

4

3

2

Fit

8 7 6 5 4 3 2

1

1

0

0

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1

1.1

1.2

1.3

1.4

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0.8

Sample 2

4

12

x 10

Circumferential Stress (Pa)

Axial Stress (Pa)

4

2

1.3

1.4

Fit

8

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0

0 0.7

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0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

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0.8

Circumferential Stretch (n.d.)

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

Circumferential Stretch (n.d.)

Sample 3

4

x 10

4

14 Data

x 10

Data

Fit

12

Circumferential Stress (Pa)

Axial Stress (Pa)

1.2

x 10

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6

10

1.1

Data

Fit

8

12

1

4

12 Data

10

0.9

Circumferential Stretch (n.d.)

Circumferential Stretch (n.d.)

8

6

4

2

Fit

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8

6

4

2

0

0 0.7

0.8

0.9

1

1.1

1.2

Circumferential Stretch (n.d.)

1.3

1.4

1.5

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

Circumferential Stretch (n.d.)

Fig. 10. Comparison of the 4-parameter Model 1 to the data (I). The diagrams display the axial (left plot) and circumferential (right plot) first Piola–Kirchhoff stresses, measured (circles) and fitted by the model (asterisk) for the first three samples. Each curve belongs to a constant axial stretch. With exception to the first sample, the qualitative behavior of the data is simulated reasonably well by the model.

ARTICLE IN PRESS G.W. Desch, H.W. Weizsa¨cker / Journal of Biomechanics 40 (2007) 3130–3145

Sample 4

4

12

4

x 10

14

x 10

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Fit

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Circumferential Stress (Pa)

Axial Stress (Pa)

Data

Fit

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3141

8

6

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Circumferential Stretch (n.d.)

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Circumferential Stretch (n.d.)

Sample 5 4

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4

x 10

12

x 10

Data 9

Data

Fit

Circumferential Stress (Pa)

7

Axial Stress (Pa)

Fit

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8

6 5 4 3 2

8

6

4

2

1 0

0 0.4

0.6

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1.2

1.4

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Circumferential Stretch (n.d.)

Sample 6

4

10

12

1.2

1.4

x 10

Data

Fit

10

Circumferential Stress (Pa)

8 7

Axial Stress (Pa)

1

4

x 10

Data 9

0.8

Circumferential Stretch (n.d.)

6 5 4 3 2

Fit

8

6

4

2

1 0

0 0.7

0.8

0.9

1

1.1

1.2

Circumferential Stretch (n.d.)

1.3

1.4

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

Circumferential Stretch (n.d.)

Fig. 11. Comparison of the 4-parameter Model 1 to the data (II). The diagrams display the axial (left plot) and circumferential (right plot) first Piola–Kirchhoff stresses, measured (circles) and fitted by the model (asterisk) for the samples 4–6. Each curve belongs to a constant axial stretch. Sample 4 gives a particularly good agreement between data and model.

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Sample 1

4

7

x 10

4

10

Data

Data 9

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Circumferential Stress (Pa)

6

5

Axial Stress (Pa)

x 10

4

3

2

Fit

8 7 6 5 4 3 2

1 1

0

0

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0.8

0.9

1

1.1

1.2

1.3

1.4

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0.8

Circumferential Stretch (n.d.)

Sample 2

4

12

x 10

Fit

Circumferential Stress (Pa)

Axial Stress (Pa)

4

2

1.3

1.4

Fit

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6

4

2

0

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1.1

1.2

1.3

1.4

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Circumferential Stretch (n.d.)

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1.1

1.2

1.3

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Circumferential Stretch (n.d.)

Sample 3

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x 10

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14 Data

x 10

Data

Fit

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x 10

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10

1.1

Data

8

12

1

4

12 Data

10

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Circumferential Stretch (n.d.)

8

6

4

2

Fit

10

8

6

4

2

0

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0.8

0.9

1

1.1

1.2

Circumferential Stretch (n.d.)

1.3

1.4

1.5

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

Circumferential Stretch (n.d.)

Fig. 12. Comparison of the 5-parameter Model 2 to the data (I). The diagrams display the axial (left plot) and circumferential (right plot) first Piola–Kirchhoff stresses, measured (circles) and fitted by the model (asterisk) for the first three samples. Each curve belongs to a constant axial stretch.

ARTICLE IN PRESS G.W. Desch, H.W. Weizsa¨cker / Journal of Biomechanics 40 (2007) 3130–3145

Sample 4

4

12

x 10

4

15

x 10

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Data

Fit

Fit

Circumferential Stress (Pa)

Axial Stress (Pa)

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8

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4

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10

5

2

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1

Circumferential Stretch (n.d.)

Sample 5

4

10

x 10

4

12

x 10

Data 9

Data

Fit

Fit

10

Circumferential Stress (Pa)

8 7

Axial Stress (Pa)

1.5

Circumferential Stretch (n.d.)

6 5 4 3

8

6

4

2

2

1 0

0 0.4

0.6

0.8

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1.2

1.4

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0.6

Circumferential Stretch (n.d.)

0.8

1

1.2

1.4

Circumferential Stretch (n.d.)

Sample 6 4

10

4

x 10

12 Data

9

Data

Fit

10

Circumferential Stress (Pa)

8 7

Axial Stress (Pa)

x 10

6 5 4 3 2

Fit

8

6

4

2

1 0

0 0.7

0.8

0.9

1

1.1

1.2

Circumferential Stretch (n.d.)

1.3

1.4

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

Circumferential Stretch (n.d.)

Fig. 13. Comparison of the 5-parameter Model 2 to the data (II). The diagrams display the axial (left plot) and circumferential (right plot) first Piola–Kirchhoff stresses, measured (circles) and fitted by the model (asterisk) for the samples 4–6. Each curve belongs to a constant axial stretch.

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was kept stretched at various levels of axial strain for rather long time spans which caused appreciable relaxation in axial direction. In other words, Sample 1 likely is the least reliable data while Sample 4, stemming from a ‘‘fluent experiment’’ is the most reliable of the set presented. Comparing results for these two samples in Table 1, one realizes that the introduction of the parameter greatly improves the fit of Sample 1, less so that of Sample 4. Hence one is tempted to regard m as a correcting factor compensating for some inelastic aspects of the vessel wall rheology (see Figs. 10–13). We have also used our model to fit data from tests on 6 segments of caval veins from 6 rabbits (Weizsa¨cker, 1992; Weizsa¨cker and Zierler, 1993). These experiments were performed with a protocol similar to that of Carboni et al. (2007): pressurization was done with weights attached to the lower end of vertically suspended tubular segments of vena cava. Unfortunately, only 2 or 3 levels of axial force (including force zero i.e. no weight attached) were used in those experiments, too few as it turned out, to prove or disprove our model. The only observation worthwhile to be mentioned from plots analog to Fig. 2 is that for force levels larger than zero the data points were indeed located close to one and the same curve, while data for vanishing axial force departed significantly from this curve. It must be emphasized that the present approach is essentially phenomenological and that the ‘‘bedspring model’’ shown in Fig. 3 is first and foremost meant to mimic the peculiar stress–strain behavior shown in Fig. 2, rather than to mirror actual histological structures of the caval wall. It is nevertheless interesting to note that our structural model roughly resembles the architectural arrangement suggested by Azuma and Hasegawa (1973) and by Hasegawa (1983) in which elastin fibers are preferentially oriented in the axial direction. Bundles of elastin fibers running in the axial direction and embedded in a felty network of collagen fibers are also reported in Rhodin (1980). In the inferior vena cava these longitudinal elastin fibers are thought to provide strong longitudinal traction, which counteracts the transverse pressure of the viscera. Yet, despite the clinical relevance, presently much of these histomechanical interpretations still remain highly speculative. Very few if any data exist in the literature that could directly be compared to the present ones. Most results on the cava stem from simple elongation tests with too little information on the boundary conditions as to allow some extrapolation to the two-dimensional behavior (Pianosi et al., 1999a, b). Azuma and Hasegawa (1973) observe that in the low stress range circumferential strips of veins were in general more distensible than axial ones, while the reverse was true in the range of high stress. Such change in the direction of anisotropy was also found in the present data when considering states of equibiaxial strain (or, alternatively, states of equibiaxial stress). But the transition occurred at quite small deformations and was not observed in the experiments on the cava from rabbits

mentioned above. In these latter experiments the caval wall was consistently more distensible in the axial direction. Obviously these aspects of anisotropy need further investigation, also with regard to the influence of the time-history of loading applied to the vessel and thus to the limits of the pseudoelastic approach. Under normal conditions, the pressure in the vena cava is thought to be a few mmHg. Unfortunately, little actual measurements can be found in the literature on the range over which the transmural pressure may change under various pathophysiological and interventional conditions. The pressure range used here, namely 0–20 mmHg (0–2.6 kPa), should cover most of the part of the physiologically relevant conditions where the transmural pressure is positive. On the other hand, it avoids harm to the vessel. In this context it is interesting to mention that Wesly et al. (1975) could pressurize canine jugular and human saphenous veins up to 220 mmHg (30 kPa) without apparent irreversible effects or rupture. Monos et al. (1995) quote saphenous vein rupture pressures as incredibly high as 2873 mmHg (383 kPa). The histological structure of veins differs from that of arteries and so does the mechanical behavior. This is particularly true for proximal vessels such as the abdominal cava and the abdominal aorta (Rhodin, 1980; Weizsa¨cker, 1988). Hence it does not surprise that pseudoelastic descriptors developed for arterial mechanics are inadequate for the venous wall. The mechanical properties of arteries depend on their topographical location and potentials have been developed to properly mirror this dependency (Weizsa¨cker et al., 1995). There is much evidence that both structure and rheology of the veins also vary with topography (Rhodin, 1980; Azuma and Hasegawa, 1973; Hasegawa, 1983; Ohhashi et al., 1992). Yet a clear and coherent picture of these mechanical variations, based on multiaxial tests, has still to emerge. Although comparable sets of data for more distal veins qualitatively resemble the present ones (Brossollet and Vito, 1995) it remains to be seen whether the constitutive formulation used in this paper can be extended to the whole venous system. Similarly, even though it is reasonable to assume that the low pressure system operates in a similar manner in different mammals interspecies extrapolations above all to humans must be done with caution. Conflict of interest The authors confirm that the publication of this paper involves no conflict of interest of any kind. Acknowledgment Thanks are due to Stephen E. Greenwald for suggesting the term ‘‘bedspring model’’ at the 5th World Congress of Biomechanics.

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Appendix A. Supplementary data Supplementary data associated with this article can be found in the online version at doi:10.1016/j.jbiomech.2007.03.028

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