Since the 1960s, the linear elasticity theory has been widely used to analyse inflation experiments performed with blood vessels or vascular biomaterials (a non-specialist reader can consult, for example, the resource book of Y. C. Fung [ 1 ]). Vessel walls must have enough strength in the circumferential direction to withstand arterial pressures and enough compliance to allow pulsatile flow. Classical mechanical tests consist of longitudinal or circumferential tensile tests and inflation experiments. In some studies, ultimate tensile strength, elongation at break and suture retention force are also measured. The most straightforward information deduced from inflation experiments is the compliance of the organ’s wall (relative increase in diameter consecutive to a given increase in internal pressure) and the burst pressure. In order to infer more detailed mechanical parameters, a constitutive law for the wall is necessary. Linear elasticity approach is acceptable as long as the deformations remain small (corresponding to the linear part of the stress-strain curve). If the material is assumed isotropic, it is possible to infer its Young’s modulus (E) and Poisson ratio (ν) from tube equilibrium equations [ 2 ]. However, the structural organization (layered microstructure composed of collagen, elastin, and SMCs (smooth muscle cells)) of vessel walls confers to them some anisotropic properties [ 3 , 4 ]. Mechanical responses to some solicitations are not the same in any direction. The studies of Dobrin and Doyle [ 5 ], Dobrin [ 6 ], Lillie et al. [ 7 ] provide calculations for circumferential and longitudinal elastic modulus, based on the assumption that the radial stress (the stress which tends to compress the wall) amounts to only a few percents of the longitudinal and azimuthal stress. For that reason, they neglect a priori the σ_rr contribution in their analysis.

Concomitantly, mechanical tests were also performed on vascular grafts in order to check the graft patency. Some compliance mismatch between the graft and the host artery may lead to graft thrombosis [ 8 , 9 ]. Dynamic compliance can be evaluated as ((D_s − D_d)/D_d)/(P_s − P_d) where subscripts s and d mean “systolic” and “diastolic”, D is the graft diameter, and P is the internal pressure. Circumferential compliance allows the prosthesis to expand in response to the pulsatile pressure. The trend is to optimize materials and tubular bio-structures before surgery [ 10 , 11 ]. Many other motivations may be cited: modeling arterial clamping or balloon dilatation, helping understand pathologies and improving treatments, fabrication of manikins for medical training or adaptation to robotic surgery.

More recently, other tubular organs have also been studied, such as esophagus or trachea. A detailed description of these organs and of their physiological functions in relation with their mechanical requirements may be found in the reviews of Pien et al. [ 4 ] and of Saksena et al. [ 12 ]. These organs may be affected by a variety of diseases or injuries: congenital malformation, autoimmune disease, inflammation, infection, and cancer. Resection and replacement are sometimes unavoidable, and may be associated with other problems (leakage, rejection, stricture). The esophagus has to be compliant enough in order to convey food to the stomach by peristalsis and contractions of the muscle layer. The trachea is submitted to air pressure variations during the respiratory cycle (inspiration/expiration). Two recent review papers are focused more precisely on the gastrointestinal tissues [ 13 , 14 ]. They point out the anisotropy of these tissues due to their multi-layered structure and indicate that inflation-extension tests may be appropriate to characterize both the distension of the tissue in the circumferential direction and the stretch in the longitudinal direction.

Many groups now use tissue engineering technologies to fabricate synthetic constructs as tissue replacements for hollow tubular organs. Several review papers give an extensive description of the fabrication processes (scaffolds, cells, mechanical and biological tests required for the regenerated organ) [ 3 , 4 , 12 , 15 ]. Mechanical and biological performances are closely linked since the scaffold stiffness influences cell-material interactions and cell differentiation. In view of implantation, anatomical accuracy, suturability, autoimmune acceptance, and long-term patency are also required for the tubular constructs. Bio-engineered tubular organs may also be used for fundamental research: basic science, a better understanding of some diseases, and drug testing. Examples of organ targeted studies may be found in De Mel et al. [ 16 ] (3D tubular scaffolds for paediatric organ production), Farhat et al. [ 17 ] (3D bioprinting for esophageal tissue repair and reconstruction), Lee et al. [ 18 ] (3D printing of vascularized tissues). In each case, the mechanical properties of the tubular structures are checked to match those of the human tissues.

Besides, noticeable progresses have been done with experimental characterizing tools that allow three-dimensional measurements of local strains and stresses. Bernal et al. [ 19 ] proposed a technique that uses sonometry data from piezoelectric elements to measure the strain in the longitudinal and circumferential directions of pressurized arteries. A review by Macrae et al. [ 20 ] is especially dedicated to “methods in mechanical testing of arterial tissues”. In addition to improved optical measuring systems, marker tracking, CT scan or electronic speckle pattern interferometry are quoted as techniques that allow to evaluate stresses and strains. Sanders et al. [ 21 ] used ultrasound strain imaging in whole-vessel-inflation experiments. Strain maps were obtained through intravascular ultrasound elastography. They aimed at characterizing mechanical properties of the different atherosclerotic plaque constituents in order to evaluate the rupture risk of the plaque. An optical clearing method is adopted by Maeda et al. [ 22 ]. It is coupled with imaging and microscopy techniques and allows some three-dimensional characterization of the mechanical behavior of rat thoracic aorta under intraluminal pressurization. Microstructural analysis of the inflated tissue is provided. Urban et al. [ 23 ] demonstrate that the ultra-sound technique can be precise enough to capture the diameter variation of an artery due to the pressure pulse transmitted by blood flow. This allows to evaluate the compliance of the artery wall. In future works, they plan to use different US propagating modes in the axial and circumferential directions in order to capture anisotropic mechanical properties of the vessel wall.

In this context, elasticity laws are revisited and completed in this paper. If coupled with microstructural measurements provided by modern experimental tools, the proposed three-dimensional analysis of the deformation of anisotropic tubular organs under moderate inflation can allow a more detailed comprehension of the mechanical behavior of native and fabricated tissues.

Let us consider a hollow tubular organ with the following known initial dimensions: wall thickness h₀, external diameter d_e0, (d_e0 = 2r_e0), length l₀. Classical cylindrical coordinates (r, θ, z) are used, and z is the longitudinal axis ( Figure 1 ). If an internal pressure P is applied inside the tube (the external pressure is taken as zero), its diameter will increase due to the compliance of the wall.

This increase in diameter is associated with a decrease in wall thickness and a length variation. The dimensions in the deformed state are denoted h for the wall thickness, d_e (= 2r_e) for the diameter, and l for the length. ( Figure 2 )

The internal radii at time t = 0 are easily obtained as:

$r_{i0}=r_{e0}-h_0$ (1)

The initial cross section of the esophagus wall is:

$S_0=\pi \left(r_{e0}^2-r_{i0}^2\right)$ (2)

Wall incompressibility [ 24 , 25 ] yields:

$\pi \left(r_e^2-r_i^2\right)l=\pi \left(r_{e0}^2-r_{i0}^2\right)l_0$ (3)

The deformed internal radius r_i is deduced from this wall volume conservation. Then, the deformed wall thickness, h, may be evaluated as:

$h=r_e-r_i$ (4)

The initial volume of the esophagus is:

$V_0=\pi r_{e0}^2{l}_0$ (5)

and its volume in the inflated state is:

$V=\pi r_e^{2}l$ (6)

This allows to calculate a compliance, based on the relative volume variation in response to the imposed internal pressure P:

$C_v=\frac{\left(\frac{V-V_0}{V_0}\right)}{P}$ (7)

The inverse of C_v (E_p = 1/C_v) may be interpreted as a “pressure elastic modulus”. A compliance based on diameter variation, C_d, may also be defined:

$C_d=\frac{\left(\frac{d_e-d_{e0}}{d_{e0}}\right)}{P}$ (8)

However, these compliances C_d and C_v do not account for the changes in wall thickness [ 26 ]. A possible length variation during inflation is taken into account in C_v, but not in C_d.

A three-dimensional analysis may be proposed as follows:

Classical notations in a cylindrical frame are used for the extension ratios: λ_z = l/l₀ is the longitudinal extension ratio; λ_θ = d_e/d_e0 is the circumferential extension ratio (perimeter of the inflated tube/perimeter of the initial tube) and λ_r = h/h₀ is the radial stretch ratio (change of dimension in the radial direction). As explained by Patel et al. [ 27 ] and Dobrin and Doyle [ 5 ], only orthogonal elongating strains in the three principal geometric directions need to be considered when the tubular sample is loaded by inflation and longitudinal traction.

The corresponding deformations are denoted e_rr, e_θθ, and e_zz, with

$e_{rr}={\lambda }_r-1,e_{\theta \theta }={\lambda }_{\theta }-1,e_{zz}={\lambda }_z-1$ (9)

We adopt the following approached formulas in order to compute the stresses in the esophagus wall:

- The radial normal stress, σ_rr, is estimated as:

${\sigma }_{rr}=-P/2$ (10)

(This is the mean value between the internal pressure P, and the external pressure (0)).

- The circumferential stress, σ_θθ, is:

${\sigma }_{\theta \theta }=P\frac{r_i}{h}$ (11)

The longitudinal stress, σ_zz, is calculated as the sum of two components:

${\sigma }_{zz}=\frac{P\pi r_i^2}{\pi \left(r_e^2-r_i^2\right)}+{\sigma }_{zext}$ (12)

where the first term corresponds to the longitudinal stress due to pressure and the second term (σ_zext) is the stress due to an eventual external longitudinally applied traction.

These formulas have been widely used for inflated elastic tissues (for example, blood vessels) [ 5 - 7 , 27 ]. They rely on the assumption that the wall thickness h is small when compared to the vessel radius r, and that σ_rr varies linearly across the wall. They were initially deduced from the equations describing the equilibrium of an isotropic inflated elastic tube, that are recalled in [ 10 ]. As explained above, in such inflation experiments, shear stresses and shear deformations are classically neglected [ 19 ].

Constitutive equations for static anisotropic elasticity may be derived from the paper of Patel et al. [ 27 ]:

$\left(\begin{array}{c}{e}_{rr}\\ e_{\theta \theta }\\ e_{zz}\end{array}\right)=\left(\begin{array}{ccc}\frac{1}{E_r}& -\frac{{\nu }_{r\theta }}{E_{\theta }}& -\frac{{\nu }_{rz}}{E_z}\\ -\frac{{\nu }_{\theta r}}{E_r}& \frac{1}{E_{\theta }}& -\frac{{\nu }_{\theta z}}{E_z}\\ -\frac{{\nu }_{zr}}{E_r}& -\frac{{\nu }_{z\theta }}{E_{\theta }}& \frac{1}{E_z}\end{array}\right)\left(\begin{array}{c}{\sigma }_{rr}\\ {\sigma }_{\theta \theta }\\ {\sigma }_{zz}\end{array}\right)$ (13)

where E_r (Pa), E_θ (Pa) and E_z (Pa) denote the elastic moduli in the radial, circumferential and longitudinal directions.

Due to symmetry conditions, some additional relations exist:

$\frac{{\nu }_{z\theta }}{E_{\theta }}=\frac{{\nu }_{\theta z}}{E_z}$; $\frac{{\nu }_{zr}}{E_r}=\frac{{\nu }_{rz}}{E_z}$; $\frac{{\nu }_{r\theta }}{E_{\theta }}=\frac{{\nu }_{\theta r}}{E_r}$ (14)

The quantities ν_rθ, ν_rz, ν_θz, ν_θr, ν_zr, ν_zθ are Poisson ratios. For example, ν_rθ represents the ratio of the contractile strain in the radial direction due to an elongating strain in the circumferential direction, ν_rz represents the ratio of the contractile strain in the radial direction due to an elongating strain in the longitudinal direction, etc. More generally, ν_ij characterize the strain in the i-direction produced by a loading in the j-direction.

The incompressibility of the tissue imposes the following condition:

$e_{rr}+e_{\theta \theta }+e_{zz}=0$ (15)

The condition (15) may be expressed using Equation (13).

This writes:

$\frac{{\sigma }_{rr}}{E_r}-\frac{{\nu }_{r\theta }}{E_{\theta }}{\sigma }_{\theta \theta }-\frac{{\nu }_{rz}}{E_z}{\sigma }_{zz}-\frac{{\nu }_{\theta r}}{E_r}{\sigma }_{rr}+\frac{{\sigma }_{\theta \theta }}{E_{\theta }}-\frac{{\nu }_{\theta z}}{E_z}{\sigma }_{zz}-\frac{{\nu }_{zr}}{E_r}{\sigma }_{rr}-\frac{{\nu }_z{}_{\theta }}{E_{\theta }}{\sigma }_{\theta \theta }+\frac{{\sigma }_{zz}}{E_z}=0$ (16)

that is:

$\frac{{\sigma }_{rr}}{E_r}\left(1-{\nu }_{\theta r}-{\nu }_{zr}\right)+\frac{{\sigma }_{\theta \theta }}{E_{\theta }}\left(1-{\nu }_{r\theta }-{\nu }_{z\theta }\right)+\frac{{\sigma }_{zz}}{E_z}\left(1-{\nu }_{rz}-{\nu }_{\theta z}\right)=0$ (17)

In order that Equation (17) can be satisfied in any case, the factor of σ_rr/E_r in this sum has to be null, idem for the factor of σ_θθ/E_θ, and the factor of σ_zz/E_z. Consequently,

$\{\begin{array}{l}{\nu }_{\theta r}+{\nu }_{zr}=1\\ {\nu }_{r\theta }+{\nu }_{z\theta }=1\\ {\nu }_{rz}+{\nu }_{\theta z}=1\end{array}$ (18)

or equivalently (using (14)):

$\{\begin{array}{l}{\nu }_{r\theta }\frac{E_r}{E_{\theta }}+{\nu }_{rz}\frac{E_r}{E_z}=1\\ {\nu }_{r\theta }+{\nu }_{\theta z}\frac{E_{\theta }}{E_z}=1\\ {\nu }_{rz}+{\nu }_{\theta z}=1\end{array}$ (19)

The Poisson coefficients ν_rθ, ν_rz, and ν_θz thus depend on the orthogonal elastic moduli and may be deduced from the system of Equation (19):

${\nu }_{rz}=\frac{1}{2}-\frac{1}{2}{E}_z\left(\frac{1}{E_{\theta }}-\frac{1}{E_r}\right)$ (20)

${\nu }_{r\theta }=\frac{1}{2}-\frac{1}{2}{E}_{\theta }\left(\frac{1}{E_z}-\frac{1}{E_r}\right)$ (21)

${\nu }_{\theta z}=\frac{1}{2}-\frac{1}{2}{E}_z\left(\frac{1}{E_r}-\frac{1}{E_{\theta }}\right)$ (22)

and then

${\nu }_{\theta r}=\frac{1}{2}-\frac{1}{2}{E}_r\left(\frac{1}{E_z}-\frac{1}{E_{\theta }}\right)$ (23)

${\nu }_{zr}=\frac{1}{2}-\frac{1}{2}{E}_r\left(\frac{1}{E_{\theta }}-\frac{1}{E_z}\right)$ (24)

${\nu }_{z\theta }=\frac{1}{2}-\frac{1}{2}{E}_{\theta }\left(\frac{1}{E_r}-\frac{1}{E_z}\right)$ (25)

One can notice that:

$\frac{1}{6}\left({\nu }_{r\theta }+{\nu }_{rz}+{\nu }_{\theta z}+{\nu }_{\theta r}+{\nu }_{zr}+{\nu }_{z\theta }\right)=\frac{1}{2}$ (26)

and that Equation (22) is also given in Lillie et al. [ 7 ], with the physical meaning of “a decrease in radius obtained on the increase in length under uniaxial load”. Equations (20) to (25) thus develop and confirm the approach initiated by Patel et al. [ 27 ].

In the case of isotropy E_r = E_θ = E_z and, as expected for an incompressible material,

${\nu }_{r\theta }={\nu }_{rz}={\nu }_{\theta z}={\nu }_{\theta r}={\nu }_{zr}={\nu }_{z\theta }=\frac{1}{2}$.

Equation (13) coupled with Equations (20) to (25) can be re-written as follows:

$\{\begin{array}{l}{e}_{rr}=\frac{1}{E_r}{\sigma }_{rr}-{\sigma }_{\theta \theta }\left(\frac{1}{2E_{\theta }}-\frac{1}{2E_z}+\frac{1}{2E_r}\right)-{\sigma }_{zz}\left(\frac{1}{2E_z}-\frac{1}{2E_{\theta }}+\frac{1}{2E_r}\right)\\ e_{\theta \theta }=-{\sigma }_{rr}\left(\frac{1}{2E_{\theta }}-\frac{1}{2E_z}+\frac{1}{2E_r}\right)+\frac{1}{E_{\theta }}{\sigma }_{\theta \theta }-{\sigma }_{zz}\left(\frac{1}{2E_z}-\frac{1}{2E_r}+\frac{1}{2E_{\theta }}\right)\\ e_{zz}=-{\sigma }_{rr}\left(\frac{1}{2E_z}-\frac{1}{2E_{\theta }}+\frac{1}{2E_r}\right)-{\sigma }_{\theta \theta }\left(\frac{1}{2E_z}-\frac{1}{2E_r}+\frac{1}{2E_{\theta }}\right)+\frac{1}{E_z}{\sigma }_{zz}\end{array}$ (27)

The relative deformations e_rr, e_θθ, e_zz can be deduced from the experiments (Equation (9)); the normal stresses σ_rr, σ_θθ, and σ_zz can be estimated using Equations (10) to (12).

One may introduce the quantities: X_r = 1/E_r, X_θ = 1/E_θ, and X_z = 1/E_z. The system to solve thus becomes:

$\{\begin{array}{l}2e_{rr}=X_r(2{\sigma }_{rr}-{\sigma }_{\theta \theta }-{\sigma }_{zz})+\left({\sigma }_{zz}-{\sigma }_{\theta \theta }\right)X_{\theta }-\left({\sigma }_{zz}-{\sigma }_{\theta \theta }\right)X_z\\ 2e_{\theta \theta }=\left({\sigma }_{zz}-{\sigma }_{rr}\right)X_r+X_{\theta }\left(-{\sigma }_{rr}+2{\sigma }_{\theta \theta }-{\sigma }_{zz}\right)-\left({\sigma }_{zz}-{\sigma }_{rr}\right)X_z\\ 2e_{zz}=-\left({\sigma }_{rr}-{\sigma }_{\theta \theta }\right)X_r+\left({\sigma }_{rr}-{\sigma }_{\theta \theta }\right)X_{\theta }+X_z\left(-{\sigma }_{rr}-{\sigma }_{\theta \theta }+2{\sigma }_{zz}\right)\end{array}$ (28)

Since the 3 equations of this system are linked by the relation: e_rr + e_θθ + e_zz = 0, one of the unknown has to be taken as a parameter or determined by another method (for example, traction experiments performed with the studied tissue samples). If we suppose that the longitudinal modulus, E_z, is known, X_r and X_θ are deduced from Equation (28) and expressed as functions of e_rr, e_θθ, and X_z.

$X_r=\frac{1}{E_r}=e_{rr}\frac{{\sigma }_{rr}-2{\sigma }_{\theta \theta }+{\sigma }_{zz}}{{\left({\sigma }_{rr}-{\sigma }_{\theta \theta }\right)}^2}+e_{\theta \theta }\frac{{\sigma }_{zz}-{\sigma }_{\theta \theta }}{{\left({\sigma }_{rr}-{\sigma }_{\theta \theta }\right)}^2}+X_z\frac{{\left({\sigma }_{zz}-{\sigma }_{\theta \theta }\right)}^2}{{\left({\sigma }_{rr}-{\sigma }_{\theta \theta }\right)}^2}$ (29)

$X_{\theta }=\frac{1}{E_{\theta }}=e_{rr}\frac{{\sigma }_{zz}-{\sigma }_{rr}}{{\left({\sigma }_{rr}-{\sigma }_{\theta \theta }\right)}^2}-e_{\theta \theta }\frac{2{\sigma }_{rr}-{\sigma }_{\theta \theta }-{\sigma }_{zz}}{{\left({\sigma }_{rr}-{\sigma }_{\theta \theta }\right)}^2}+X_z\frac{{\left({\sigma }_{rr}-{\sigma }_{zz}\right)}^2}{{\left({\sigma }_{rr}-{\sigma }_{\theta \theta }\right)}^2}$ (30)

Coming back to Equation (20) to (25), it is then possible to calculate all the Poisson coefficients.

Intraluminal pressurization of rat thoracic aorta is considered. The data are taken from Maeda et al. [ 22 ]. The specimen initial geometry is h₀ = 150 μm, r_e0 = 1 mm, l₀ = 20 mm (so that the initial internal radius r_i0 = 0.85 mm). For an internal pressure P = 40 mmHg, the deformations e_θθ and e_zz are reported as: e_θθ = 0.26 and e_zz = 0.008. It is thus possible to calculate the deformed geometry: r_e = 1.26 mm, r_i = 1.146 mm, h = 114.3 μm. These geometric data are summarized in Table 1 .

The volume compliance (Equation (7)) is 1.12 × 10⁻⁴ Pa⁻¹ and the diameter compliance (Equation (8)) is 4.87 × 10⁻⁵ Pa⁻¹. The radial stretch ratio λ_r is 0.762 so that e_rr = −0.238. The stresses are obtained through Equations (10)-(12) and are summarized in Table 2 . It is observed that, as expected, σ_θθ ≈ 2σ_zz. The radial stress (the stress which tends to compress the wall) amounts to only 10 % of the longitudinal stress and 5% of the azimuthal stress. For that reason, some authors [ 5 , 7 , 28 ] neglect a priori the σ_rr contribution in their analysis. Using the value of Assoul et al. [ 25 ] for the longitudinal elastic modulus E_z of the rat thoracic aorta (E_z = 1 × 10⁵ Pa), the radial and circumferential elastic moduli are deduced from Equation (29) and (30): E_r = 1.53 × 10⁵ Pa, and E_θ = 1.36 × 10⁵ Pa. These results are summarized in Table 3 . The corresponding Poisson coefficients (Equations (20)-(25)) are presented in Table 4 .

No sophisticated interpretation of these results can be made because they are based on rough estimations of the input data. However, the orders of magnitude are in good agreement with similar results that can be found in the literature. Cox [ 29 ] studied the anisotropic properties of the canine carotid artery with significant axial pre-stress (λ_z > 1.5) and pressures up to 200 mmHg. He mentions that the elastic moduli (in the range 10⁵ - 10⁶ Pa) and the various Poisson ratios (between 0 and 1) are complex functions of the extension ratios. For the elastic moduli of aortas (estimated in living dogs), Patel et al. [ 27 ] report values in the range 3.9 × 10⁵ Pa to 8.8 × 10⁵ Pa, with E_r < E_θ and E_z, depending on the extension ratio values (λ_θ and λ_z varying between 1.4 and 1.6). Nahon et al. [ 28 ] examined the mechanical properties of canine iliac arteries under operating pressures of 80 to 160 mmHg. Their approach is two-dimensional. For an internal pressure of 80 mmHg, they obtain E_θ = 2 × 10⁵ Pa, E_z = 4.6 × 10⁵ Pa and ν_θz = 0.56. They also indicate a compliance value of 2.25 × 10⁻⁵ Pa⁻¹ for some canine femoral artery. More recently, an attempt was made by Sugita et al. [ 30 ] to evaluate 3D local strains (at the cell scale) in mice thoracic aortas during intraluminal pressurization. In their work, one can also find an estimation of the volume compliance of the studied vessels around 0.37 × 10⁻⁴ Pa⁻¹. Skacel and Bursa [ 31 ] tried to establish some relations between the Poisson ratios and the internal layered structure of the arteries walls. Their demonstration is based on uniaxial traction experiments of porcine aortic wall. They show that “in-plane” Poisson ratios are between 0 and 0.5 and “out-of-plane” Poisson ratios can get values between 0.5 and 1. They also point out the importance of the transversal contraction (in the thickness direction).

i) It may be of interest to evaluate if the assumption of isotropy in the analysis of inflation experiments would yield very different results. If the mechanical properties are supposed to be identical in all directions, the incompressibility condition leads to a value of 0.5 for ν, and the Young’s modulus may be found in Bergel [ 2 ]:

$E_B=\frac{\Delta P}{\left(\frac{\Delta r_e}{r_{e0}}\right)}\frac{2\left(1-{\nu }^2\right)r_i^2}{r_e^2-r_i^2}$ (31)

This equation is established with the hypothesis that no change in length occurs during the inflation experiment. Since ν = 0.5, Equation (31) becomes:

$E_B=\frac{\Delta P}{\left(\frac{\Delta r_e}{r_{e0}}\right)}\frac{3r_i^2}{2\left(r_e^2-r_i^2\right)}$ (32)

A change in length Δl may be easily taken into account as a boundary condition in the longitudinal displacement. This leads to the following formula for the elastic modulus:

$E_{Bmodified}=\frac{\Delta P}{\frac{\Delta r_e}{r_{e0}}+\nu \frac{\text{Δ}l}{l_0}}\frac{2\left(1-{\nu }^2\right)r_i^2}{r_e^2-r_i^2}$ (33)

One can notice that if Δl = 0, Equation (33) reduces to Equation (31). Using the notations of this paper, ΔP = P-0, Δr_e/r_e0 = e_θθ, and Δl/l₀ = e_zz.

The E_B modulus as defined in Equation (31) may be closely related to a circumferential modulus, since it is inversely proportional to the wall compliance C_d (Equation (8)) and proportional to a geometric factor of order (r_i/h):

$E_B=\frac{1}{C_d}\frac{2\left(1-{\nu }^2\right)r_i^2}{\left(r_e-r_i\right)\left(r_e+r_i\right)}\approx \frac{1}{C_d}\frac{\left(1-{\nu }^2\right)r_i}{h}$ (34)

With the numerical data adopted in Section 3, the value of E_B from Equation (32) is 1.47 × 10⁵ Pa and the value from Equation (34) is 1.54 × 10⁵ Pa. Both values are close to the E_θ value obtained in Section 3 (E_θ = 1.36 × 10⁵ Pa).

ii) If internal pressures and tube radii are increased further, the linear elasticity approach is no longer valid. The stiffness of the tissue increases with the load, and this produces an exponential stress-strain relationship. Different models have been proposed in the literature to describe this hyperelastic behavior. One of the most popular is the “HGO-model” [ 32 ].

In spite of the simplifying hypotheses that are made, the theoretical approaches presented in this paper may be useful for the interpretation of inflation experiments of tubular elastic biological tissues. Inflation induces deformations of the organ’s wall in three directions (radial, circumferential, and longitudinal). Each deformation influences the two others. The Poisson ratios are the physical quantities that put in evidence of this coupling. This paper stresses the importance of analyzing the evolution of the wall thickness during inflation and suggests the possibility of combining mechanical results with very detailed microscopic or imaging results.