Signs of Aneurysm Formation Learned from Rubber Balloon Inflation

Fumio Nogata^1*, Yasunari Yokota², Yoko Kawamura², Hiroyuki Morita³, Yoshihiro Uno⁴, William R. Walsh⁵, Takahiko Kawamura⁶, Nigishi Hotta^7,8, Kenji Kagechika⁹

¹Gifu University, Gifu, Japan.
²Faculty of Engineering, Gifu University, Gifu, Japan.
³Department of General Medicine and General Internal Medicine, Gifu University School of Medicine, Gifu, Japan.
⁴Department of General Medicine, Gifu Prefectural General Medical Center, Gifu, Japan.
⁵Prince of Wales Hospital, University of New South Wales, Kensington, Australia.
⁶Diabetes and Endocrine Internal Medicine, Chubu Rosai Hospital, Nagoya, Japan.
⁷Nagoya University, Nagoya, Japan.
⁸Chubu Rosai Hospital, Nagoya, Japan.
⁹Anamizu General Hospital, Ishikawa, Japan.

DOI: 10.4236/jbm.2024.1211042                

Abstract

Aneurysms can be classified into two main types based on their shape: saccular (spherical) and fusiform (cylindrical). In order to clarify the formation of aneurysms, we analyzed and examined the relationship between external force (internal pressure) and deformation (diameter change) of a spherical model using the Neo-Hookean model, which can be used for hyperelastic materials and is similar to Hooke’s law to predict the nonlinear stress-strain behavior of materials with large deformation. For a cylindrical model, we conducted an experiment using a rubber balloon. In the spherical model, the magnitude of the internal pressure Δp value is proportional to G (modulus of rigidity) and t (thickness), and inversely proportional to R (radius of the sphere). In addition, the maximum pressure Δp (max) is reached when λ (=expanded diameter/original diameter) is approximately 1.2, and the change in diameter becomes unstable (nonlinear change) thereafter. In the cylindrical model, localized expansion occurred at λ = 1.32 (λ = 1.98 when compared to the diameter at internal pressure Δp = 0) compared to the nearby uniform diameter, followed by a sudden rapid expansion (unstable expansion jump), forming a distinct bulge, and the radial and longitudinal deformations increased with increasing Δp, leading to the rupture of the balloon. Both models have a starting point where nonlinear deformation changes (rapid expansion) occur, so quantitative observation of the artery’s shape and size is important to prevent aneurysm formation.

Keywords

Aneurysm Formation, Pressure-Inflation Characteristics, Elastic Instability, Expansion Signs, Saccular and Fusiform

1. Introduction

Aneurysms are balloon-like bulges in the walls of arteries and other blood vessels that are asymptomatic until they become significantly larger or rupture [1]. They occur in locally weakened part of the blood vessel wall, and the part that expands to more than 1.5 times the size of normal arterial diameter is called aneurysms [2]-[4]. Aneurysms can occur anywhere in the circulatory system, but are most common in the brain, thoracic aneurysms, abdominal aortic aneurysms (AAA) and legs [5]-[8]. Furthermore, because aneurysms can be fatal if they rupture, it is believed that it is very important to recognize the signs and prevent aneurysms before they occur.

On the other hand, Michael Faraday invented a rubber balloon in 1824 to conduct experiments on gas laws [9]. Since then, many studies have been reported on the mechanism of expansion and contraction of rubber balloons [10]-[14], their structure and deformation behavior, etc. These results (external forces and deformation characteristics) are very useful because rubber balloons resemble organs such as the heart, lungs, bladder, and arteries.

However, because the deformation behavior is complex, various theoretical models have been proposed [15]-[18]. Meanwhile, aneurysms can be classified into two main types based on their shape, saccular (spherical) and fusiform (cylindrical).

In this report, to clarify the formation of aneurysms, we analyzed and considered the relationship between the external force (internal pressure) and deformation (diameter change) for a spherical model using the Neo-Hookean model [17] [18], which can be used for hyperelastic materials and is similar to Hooke’s law for predicting the nonlinear stress-strain behavior of materials with large deformation. For the cylindrical model, experiments were conducted using a rubber balloon. Both models have a starting point where nonlinear expansion deformation occurs with an increase in internal pressure, so we have shown the importance of detailed detection of gradual diameter changes that progress at both the macro- and micro-levels, especially changes just before a sudden increase in diameter.

2. Inflation of a Rubber Balloon: Pressure-Diameter Relation

2.1. Spherical Model

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.

Figure 1. A spherical model of radius λR and thickness t subjected to internal pressure Δp.

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].

Figure 2. Pressure-expansion ratio (λ) characteristics of a spherical rubber balloon from the kinetic theory [10]. The rigidity modulus G value (0.24 GPa) was calculated from the authors’ experimental data in the initial small deformation range (Appendix Figure A1, [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.

Graph 1. (a) Aneurysm size and rupture risk classification according to the UCAS Japan report [21] [22]; (b) Changes in wall thickness and wall stress for the sizes in (a), detailed calculations are given in the authors’ report [23].

Figure 4 shows the increase in wall stress (nominal and true stress) from an initial aneurysm diameter of 3 mm to 25 mm, t = 0.8 mm (average thickness for adults, [24]) at internal pressures of 120 (16 kPa) and 150 mmHg (20 kPa).

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.

Figure 3. Pressure (Δp)—expansion ratio (λ) characteristic of human artery (the UCAS Japan data) according to the kinetic theory: when the λ value exceeds 1.33, the diameter increases unstably, as shown by the red arrow. The human stiffness modulus, G = 0.08 MPa (common carotid artery, CCA), was obtained from the authors’ clinical data by echo-video analysis (Appendix Figure A2).

Figure 4. Wall stress and human arterial strength with increasing aneurysm size in the UCAS Japan model: AB is the range based on arterial strength, and CD is the range of elongation (in the 80s) at rupture. Note that even if strength is sufficient, reduced elongation (embrittlement) can significantly contribute to rupture.

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.

2.2. Cylindrical Model

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

Figure 5. The change in contour and dimensions of a balloon until it ruptures. The true stress and the nominal stress at the rupture: Δp = 306 mmHg = 40.7 kPa, t' = 1/(λ_θ) = (0.4/8.55) = 0.046 mm, σ_z = (σ_θ/2), σ_r = −Δp at the inner wall, σ_r = −Δp = 0 at the outer wall, nominal stress σ_θ = σ_θ = ΔpR/to = (40.7 × (53/2)/0.4)/1000 = 2.7 MPa, true stress σ_θ = pr/t' = (40.7 × (53/2)/0.046)/1000 = 23.4 MPa.

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 6. A photograph of the moment the bulge formed (from d’o = 9.3, and 12.3 to 22.6 mm in Figure 5). The largest bulge was generated at λ’ = b/a = 22.6/9.3 = 2.43.

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.

Figure 7. A Photograph shown in the research by Kugo et al. on the development of a drug to shrink aneurysms [31].

Graph 2. Signs of aneurysm formation and its changes in shape.

In this paper, we discussed the mechanical behavior of rubber balloons and human arteries from the same perspective. Historically, this has been done by analogy with biological organs such as the heart, lungs, and bladder [10]-[14]. Here too, the emphasis has been on finding signs of aneurysm formation. One of the reasons for this is the similarity of the behavior of the stress-elongation λ graphs of human arteries, young sheep arteries, and rubber balloons in uniaxial tensile tests, as shown in Figure A5 in the Appendix.

Graph 2 summarizes shape changes in the signs of aneurysm formation. The diameter ratio (λ) at the onset of aneurysm formation due to kinetic instability, partial slight expansion, and instability jump is shown. Although the data is limited, looking at these figures, it can be said that a slight bulge of about 1.3 times or more indicates the possibility of aneurysm formation. Therefore, to prevent the development of aneurysms, it is important to detect and monitor partial bulges of uniform diameter, and attention to these diameter ratios λ may be an indicator. Monitoring of blood pressure is also important.

3. Concluding Remarks

In order to clarify the process of aneurysm formation, the external force (internal pressure, Δp) versus deformation (diameter expansion) characteristics of a spherical model were analyzed and examined using the Neo-Hookean model. For a cylindrical model, experiments were conducted using a rubber balloon.

1) For the spherical model, the magnitude of the Δp value is proportional to G (modulus of rigidity) and t (thickness), and inversely proportional to R (radius of the sphere).

The internal pressure-deformation relationship according to the kinetic theory reaches the maximum pressure Δp (max) when λ (=expanded diameter/original diameter) is approximately 1.2 or more, and then the change in diameter becomes unstable (nonlinear change).

In other words, the diameter may expand significantly depending on the mechanical properties of the material.

2) For the cylindrical model, when a small local bulge exceeds λ (the ratio to the surrounding uniform diameter) of 1.32, it expands rapidly to form a distinct bulge, and as Δp increases, the diameter and longitudinal deformation increase, leading to the balloon bursting.

Both models have a starting point where nonlinear deformation changes (rapid expansion) occur, so if we can detect in detail the gradual partial diameter changes that progress globally and locally, especially the changes just before the diameter ratio λ increases suddenly, it may be possible to prevent the expansion of aneurysms.

Acknowledgements

The authors would like to express their gratitude to the undergraduate and graduate students who participated in related research in our laboratory from 2005 to 2012. The clinical studies were conducted in accordance with the regulations of the ethical committees of Gifu University Hospital and the hospital to which the co-authors are affiliated.

Appendix

Figure A1. Nominal strain-stress curves of a rubber balloon: Young’s modulus E = 0.72 GPa and rigidity modulus G = 0.24 GPa were obtained in the small deformation range.

Figure A2. In vivo strength estimation data using B-mode ultrasound images from our laboratory. Data of E = 0.23 MPa, G = 0.24 GPa (20 - 30 years) were used for the discussion.

Figure A3. Tensile strength (true stress) range of the common carotid artery (CCA) in the 80s: it is clear that there is a large degree of individual variation.

Figure A4. Tensile elongation range at break of common carotid artery (CCA) in the 80s. The red line shows true stress and black is nominal stress.

Figure A5. Comparison of the circumferential tensile elongation (λf) at ruptured for humans, sheep, and balloons: the curves showing the elastic to elastoplastic region show similar changes. The arteries of 80’s are about 50% of the arteries of young sheep and about 30% of the arteries of balloons, showing significant stretch stiffening (arteriosclerosis). This is one of the reasons why balloons are used in mechanical property analysis studies of biological organs such as the heart, lungs, and bladder [10]-[14].

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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