We consider from a mechanical perspective the expansion of a rubber balloon of original outer radius R and thickness t, as shown in Figure 1 , using the Neo-Hookean which is a strain energy density function used to analyze hyperelastic materials such as rubbers.

Suppose that the internal pressure (Δp) increases and the radius of the rubber wall of the spherical balloon increases λ time (the expansion ratio (d’o/do) of circumferences of the balloon in the inflated and uninflated states). Where do (=2R) is the outer diameter of the uninflated balloon, t’ is the wall thickness after expansion, d’o is the outer diameter after expansion, σ_θ is circumferential stress, σ_z is stress in z direction, and σ_r is radial stress.

Under incompressible and the volume is constant conditions, the wall thickness becomes t/λ², because the sphere deforms equally in three directions:

${\lambda }_i={\lambda }_r{\lambda }_{\theta }{\lambda }_z=1.$ (1)

Therefore, from the elastic energy due to deformation of the rubber wall and the energy due to pressure change (Δp), the total free energy (E_total) can be calculated as follows [10] .

$E_{total}=4\pi R^{2}t\left(G/2\right)\left[2{\lambda }^2+\left(1/{\lambda }^4\right)-3\right]-\Delta P\langle 4\pi R^3\left({\lambda }^3-1\right)\rangle /3.$ (2)

where G is modulus of rigidity, λ (=d’o/do) is the ratio of the initial diameter (do) to expanded diameter (d’o), and the radius is R = do/2. The conditions for both to be in equilibrium are

$\partial E_{total}/\partial \lambda =0.$ (3)

Therefore,

$\Delta p=\frac{2tG}{R}\left[\left(1/\lambda \right)-\left(1/{\lambda }^7\right)\right]$ (4)

This equation shows the relationship between Δp and balloon diameter, and also shows that beyond a certain threshold (maximum pressure Δp*) there is no stable solution.

That is, depending on the mechanical properties of the material, it reaches the maximum value Δp*, after which the diameter increases nonlinearly: a mechanical instability that increases rapidly with diameter.

The magnitude of the value of internal pressure Δp is proportional to G and t, and inversely proportional to R. As an example, we considered the relationship between pressure and expansion.

Figure 2 shows the relationship between the internal pressure Δp and the expansion ratio λ (=di/do) for two different diameters (do = 2R) and t, where G = 0.24 kPa was used [19] .

The two curves were almost the same λ values at the Δp* (λ = 1.16 and 1.18), even though their diameters and thicknesses differed by several times. For real rubber materials, simple forms of the kinetic theory of rubber, e.g. uniaxial force-extension curves, are limited in validity to approximately λ < 3.5, because at larger extensions crystallization occurs and the elasticity is no longer purely entropic ( [20] : footnote, p. 23). For this reason, as an example, the red dashed line shows the actual behavior of rubber diagrammatically.

Based on the above mechanical knowledge, we reviewed the published risk data regarding aneurysm rupture by the report of the UCAS Japan [21] .

The Japanese Society of Neurosurgery conducted an eight-year investigation into the rupture of unruptured cerebral aneurysms of 3 mm or larger discovered during examinations, with and without special treatment (natural observation) and with treatment. They investigated 6697 aneurysms (5720 patients) and presented the risk of rupture (%) in five size-based categories [22] . This report suggests that the size, location, and shape of the aneurysm are important risk factors, with larger sizes associated with higher rupture rates. Data compiled by the UCAS Japan group regarding the size and rupture risk of cerebral aneurysms are shown in Graph 1(a) , and changes in wall thickness and stress due to aneurysm expansion are shown in Graph 1(b) : These wall stresses were calculated for the sizes in Graph 1(a) assuming a sphere under internal pressure. Details are given in the authors’ report [23] .

Figure 3 shows the relationship between pressure Δp and λ using Equation (2), assuming that the single sphere model with initial diameter do = 3 is extended to 25 mm, from the data in Graph 1(a) and Graph 1(b) . This graph means that expansion deformation becomes mechanically unstable when the maximum pressure value Δp* = 167 mmHg (22.2 kPa) or λ = 1.33 is exceeded as shown in the arrow whether or not the aneurysm ruptures will depend on the mechanical properties of the subject’s arteries.

Now let us compare these wall stresses to the strength of blood vessels. The authors previously performed uniaxial tensile and internal pressure tests on human common carotid artery specimens (CCA, 80-year-old) to obtain data on the strength ( [25] [26] , Appendix Figure A3 ). The results showed that the true stress was 0.37 to 2.38 MPa in tensile test, the elongation ratio at fractured λf = 1.40 to 2.18, and 0.86 to 1.70 MPa in internal pressure test, and the elongation ratio at ruptured λf = 1.20 to 1.87 (Appendix, Figure A4 ).

When these measurement data are matched with the wall stress curve in Figure 4 , the range is λ = 3.5 to 5.6 (do = 10 to 17 mm), and AB within this range has a hazard ratio of 9.1 as specified by UCAS Japan, which is considered to have sufficient strength to accommodate aneurysm diameters up to approximately 17 mm.

On the other hand, the stress value λf (1.40 - 2.18: Appendix Figure A4 ) in the CD range was very small, with a hazard ratio of 1.1 (d’o = 4 - 7 mm).

This means that the possibility of blood vessel rupture increases as the tensile deformation (brittleness) decreases. For reference, Appendix Figure A5 compares a young sheep’s artery and a rubber balloon with a human artery. It is clear that the burst elongation of human arteries decreases with age. This may be one of the reasons why blood vessels burst without warning. Therefore, it is important to measure stiffness, strength, and tensile deformation of arteries from young adulthood.

Assuming the shape of a fusiform aneurysm, an internal pressure load experiment was conducted using a cylindrical rubber balloon whose axial length was sufficiently longer than the wall thickness so that the mechanical effects due to the shape of both ends could be ignored.

Figure 5 shows the changes in the contour shape and dimensions of a rubber

balloon (diameter do = 6.2 mm, thickness t = 0.2 mm) until it ruptures. From the equation for a thin-walled cylinder subjected to internal pressure Δp, the circumferential stress σ_θ = ΔpR/t, the axial stress σ_z = ΔpR/2t =σ_θ/2, the radial stress σ_r = −ΔpR/t on the inner wall, and σ_r = 0 on the outer wall. Therefore, σ_z is half the circumferential stress σ_θ. The photographs above show (a) the initial shape, (b) the expansion of diameter with increasing internal pressure and the appearance of small local bulges, (c) the sudden expansion of diameter with increasing internal pressure, followed by (d) nearly sphere, (e) expansion in diameter and length in the Z direction, and (f) just before it ruptures.

The characteristic change was that the pressure increased from Δp = 70 mmHg (9.3 kPa, 23% of the rupture pressure) and suddenly a large bulge formed at 128.5 mmHg (17 kPa), changing in diameter from do’ = 12.3 mm (λ’ = 1.98) to 22.6 mm (λ’ = 22.6/6.2 = 3.6). Then, at about 155 mmHg (20.6 kPa), it became a sphere with a diameter of about 32 mm. Then, the elongation in the z direction increased and it rupture at a diameter of $\varphi$ 53 mm, z = about 82 mm. These data show that, like the spherical model mentioned above, the cylindrical model also has a region of mechanical instability that increases rapidly with diameter.

Figure 6 shows a photograph of the moment when the bulge formed due to instability expansion jump. It was taken from a recorded video frame (1/30 s). From the left, we see the size of the original shape ( $\varphi$ 6.2 mm), uniform expansion at 70 mmHg ( $\varphi$ 9.3 mm), slight partial expansion of about d’o = $\varphi$ 12.3 mm, and the instability jump at λ’ = 2.43 at that point. Analysis by Muller et al. ( [27] , p: 38) reports that from kinetic theory [10] , the onset of mechanical instability ΔP* occurs at λ’ = 1.7 or more.

Similar to the pressure-inflation ratio (λ = 1.18 in Figure 2 ) characteristic of the spherical rubber balloon mentioned above, even in the cylindrical shape, an increase in diameter can occur due to a sudden change in the instability expansion jump [28] - [30] . It is important to note that aneurysm formation does not begin as a slow swelling, but rather as slight partial expansion until mechanical instability occurs and then grows rapidly.

Figure 7 is a photograph shown in a study by Kugo et al. to the development of a drug to shrink aneurysms of human abdominal aortic aneurysms (AAA) [30] . From the figure, the ratio of the diameter of the uniform part (a) to the aneurysm diameter (b) was λ = b/a = 2.1.