Modeling Nontraumatic Aneurysm Evolution Growth and Rupture.
Abstract: We have presented a mathematical model to study the evolution growth and risk rupture of nontraumatic aneurysms contained within a cylindrical region of blood vessels. Analytical and numerical solutions are studied. Results affirmed that the intra-aneurysmal pressure and bloodstream flow account for the evolution and growth of aneurysms and we find that an aneurysm may rupture when the ratio of the lateral membrane contraction to longitudinal membrane extension approaches one. Numerical properties of intra-aneurysmal pressure impact fluid velocity membrane displacement and the deformed radius with respect to the Poisson ratio membrane thickness and extensional rigidity are studied. The importance of the findings is rested on the fact that they can be used to improve noninvasive means for predicting aneurysm rupture and treatment and management decisions after rupture.
Keywords: Elastodynamics Filtration Navier-Stokes Numerical Solutions Permeability Poisson Ratio Aneurysm.
In this paper we describe a mathematical model which may lead to understanding the evolution growth and rupture potential of nontraumatic aneurysms. The model describes a quasi-static non-convectional acceleration axi-symmetric Navier-Stokes equations in cylindrical coordinates coupled with the Camenschi- Fung [1 2] type linear elastodynamic system of equations with filtration. The profile of solutions of the system of equations described may help to provide insights in developing noninvasive means for detecting when a nontraumatic aneurysm may rupture and deciding the best treatment and management strategies of ruptured aneurysms.
The genesis of a nontraumatic aneurysm is contingent on any condition that causes the walls of the blood vessel to weaken [3-8]. The most commonly investigated forms of aneurysms are the aortic cerebral artery and intracranial aneurysm[3 5-7 9-11]. The development of the cerebral aneurysm for example is contingent on various physical factors associated with blood flow [3 5 10]. Studies suggest that the inertial forces of the bloodstream result in the local elevation of intravascular pressure and the flow impact . This means that the impacting forces and the local pressure elevation at the aneurysm have a large contribution to the development of cerebral aneurysms. The other contingent factor is the wall shear stress which is the viscous friction of the bloodstream that acts parallel to the vessel wall [3 9
12]. The overall impact of these forces on the thinning of the aneurysm wall has not been suggested in the literature. In this study we have shown that when the lateral contraction of the membrane wall is in balance with the longitudinal extension an aneurysm may rupture and that at the rupture point about 80% of the membrane wall thins out. The analytical and numerical results presented here affirms the results in the literature [3 9 10] that the intra-aneurysmal pressure and the bloodstream flow contribute to the evolution and development of aneurysms.
Modern neuroimaging techniques often detect unruptured cerebral artery aneurysm which is estimated to be present in 3%-6% of the population [9
10]. Aggressive rupture preventing treatment is often an option but may lead to morbidity. The specific risk for rupture of a nontraumatic aneurysm is unknown and risk assessments are based on general knowledge of factors leading to subarachnoid hemorrhage deduced from epidemiological studies [6 9 10]. Additionally aneurysms of large size proximal location and small neck or fundus ratio are associated with increased risk for rupture [5 9 10]. Thus more reliable parameters to predict the risk of aneurysmal rupture are needed. Intra-aneurysmal pressure gradients bloodstream flow profiles membrane displacement profiles membrane thickness and Poisson ratio could provide additional information regarding the risk of rupture. Moftakher R. et al. hypothesized in  that Phase Contrast with vastly undersampled isotropic projection reconstruction could accurately assess intra-aneurysmal pressure gradients in a canine aneurysmal model when compared with invasive measurements.
Isaken J. G. et al. developed in  a computational model for
simulation of fluid-structured interaction in cerebral aneurysms based on patient specific lesion geometry with emphasis on wall tension.
In the proposed model we have introduced additional parameters the Poisson ratio and membrane wall thickness as determining measures predicting the potential for an aneurysm to rupture. The mathematical model is constituted by equations (1) - (14) of section 2. We used the Camenschi dimensionless variable transforms and quasi-static conditions  to reduce the problem to parameters that can be measured by noninvasive means. The analytical solutions provide conditions for membrane enlargement to minimum and maximum stretches that depends on the intra-aneurysmal pressure gradients and rates. It also reveals that a nontraumatic aneurysm may rupture when the Poisson ratio v0 approaches anisotropic material values. In section 3 we provide numerical analysis of the solutions based on experimental data of parameters derived from the cited literature. The analysis confirms that the profile of the deformed radius and the displacement components of the membrane becomes discontinuous as the Poisson
ratio approaches anisotropic material values. Numerical analysis of intra-aneurysmal pressure membrane displacement and thickness affirm that their profile before and after the rupture of an aneurysm are consistent with in vitro and in vivo observations. In particular it shows that when 80% of the membrane wall thins out then the aneurysm may rupture.
There are many hypotheses regarding aneurysm enlargement and rupture [3-10]. However the roles of the Poisson ratio and the thinning of the membrane wall are not well understood. Shah and Humphrey suggested in  that studies on the mechanics of saccular aneurysms should be focused on quasi-static analyses that investigate the roles of lesion geometry and material properties including growth and remodeling. The model presented here has taken the material properties growth and remodeling into account. Remodeling of the membrane wall in responses to tears is incorporated in the filtration coefficient and retain the normal balance in the micro- vascular mechanism in supplying tissues or the surrounding fluids with nutrients and clearing waste products. However the model does not take into consideration the healing process of the membrane layers. Another interesting finding is that when the Poisson ratio is in the range 0 less than v0 less than 1 and v0 greater than 2 the aneurysm may stretch or shrink but not rupture. In fact most experimental studies choose values of Poisson ratio in the isotropic material range 0 less than v0 less than 0.5 and do not factor the filtration process and anisotropic material composition of the membrane's inner and outer layers which remain in place after the media has deflated (aneurysm has ruptured) into consideration. We have considered the Poisson ratio in the anisotropic material range 1 less than v0 less than 2 in this study to indicate that the membrane extensional rigidity is weakened by generating a substantial anisotropy in stiffness. This is discussed in section 2 in the activities leading to the development of an aneurysm sac in the media layer of the membrane which is surrounded by the inner and outer layers made of dominantly polyurethane material which is anisotropic . In section 4 we provided the post rupture analysis and demonstrated how the membrane extensional rigidity deduced from the membrane wall thickness and Poisson ratio can be used to design a grading scale to measure the severity of aneurysm rupture.
2. MATHEMATICAL MODEL
We will derive a model and then illustrate how it can be used to model the evolution growth and predict the rupture potential of nontraumatic aneurysms. It is known that inertial forces of the bloodstream results in local elevation of intra-vascular pressure and the flow impacting force together with local pressure elevation at the aneurysm contributes to the development of aneurysms [10 12]. It is also hypothesized that aneurysms rupture when the wall tension exceeds the strength of the wall tissue. We shall consider an axi- symmetric motion of equation in two media; the Newtonian fluid coupled with the linear elastic membrane. Due to the nature of the origin of the blood flow into the blood vessels which is imposed by the heart pumping blood into the circulatory system we shall consider non-convection acceleration axi- symmetric viscous incompressible pressure driven Navier-Stokes equations in cylindrical coordinates coupled with the Camenschi-Fung type elastic membrane equations [1 2 16]. We shall assume that the aneurysm occurs within a cylindrical portion of the vessel of length L . We let a be the undeformed radius of the cylinder and a( z t ) be the deformed radius describing the curvature of the aneurysm. We let P( z t ) be the intra-aneurysmal pressure with P (t ) and
P2 (t ) the end pressures at z = 0 and z = L respectively (See Figure 1).
If we let . = (v u) be the velocity vector of the fluid flowing through the portion of the vessel containing the aneurysms where v ( z r t ) and u( z r t ) are the
transversal and longitudinal components and . = (CC ) is the displacement vector of the membrane. C( z t ) and C ( z t ) are the transversal and longitudinal components of the membrane displacement respectively as illustrated in Figure 1 and furthermore A normal artery wall consist of three layers. The innermost endothelial layer is called the intima the middle layer consisting of smooth muscle is called the media and the other layer consisting of connecting tissues is called the adventitia . The deformed composed of only the intima and adventitia. The material composition of the intima and adventitia is made up of dominantly anisotropic polyurethane. The intima based on in vitro and in vivo observations [17
18] remains normal but subintimal cellular proliferation also occurs when aneurysm developed. The internal elastic membrane responsible for the thinning of the artery wall is either reduced in size or absent causing the media to retract to the junction of the aneurysm neck with the parent blood vessel. This development transforms the membrane within the aneurysm region from pseudoelastic isotropic to anisotropic media. Thus the deformation transforms the curvature of the weakened portion of the blood vessel into an aneurysmal sac and hence tolerating the extension of the membrane and transverse shear curving and pressure difference. We shall consider the blood vessels as permeating deformable shell filled with a congenital . Nevertheless we shall consider the membrane as a pseudoelastic isotropic material in the biological sense  that the material properties of the membrane remain the same throughout the deformation process becoming anisotropic only during the development of aneurysm sac. This feature holds only for nontraumatic aneurysms. The theory of isotropic elasticity allows the Poisson ratio in the range
1 S v0 S 0.5 for an object with surface with no constraint. Physically this means that for the material to be stable the stiffness must be positive. That is it is required that both the bulk and shear modulus be greater than or equal to zero [20 21]. On the other hand isotropic objects that are constrained at the surface can have Poisson ratios outside the above range and be stable [20 21]. Since the blood vessels are constrained by the surrounding tissue and fluid we shall consider values of Poisson ratio to include values outside the isotropic range 1 S v0 S 0.5 . This consideration also includes the regime of anisotropic deformation when the constituted material of the membrane within the aneurysm region is made up of the intima and adventitia since the concept of Poisson ratio can be extended to anisotropic materials with values outside the isotropic range 1 S v0 S 0.5 [15 20- 23].
We shall also assume that the natural vascular process of exchange of nutrients and waste from the evolution and development of aneurysms are by filtration. Thus the boundary conditions expressing the adherence of the fluid to the membrane wall and the fluid filtration through the membrane are:Equation
We will now justify (10). In a normal vascular system of functions there is a free exchange of nutrients water electrolytes and microphage between the intra-vascular and extra-vascular components of the blood vessels. Several mechanisms are responsible for this critical function of the vascular system. Physiologists including Michel [12 25 26] investigated the mechanism by which plasma and its solutes cross the vascular barrier. They discovered that capillaries are the vascular segment responsible for molecular exchange in normal tissues and that gases water and microphages cross the capillary endothelial cell barrier freely but the passage of larger molecules such as plasma proteins are tightly restricted. They also discovered that several mechanisms are involved in this exchange. The most important though are the bulk flow and diffusion. The rate of change in either direction is determined by physical factors such as hydrostatic pressure osmotic pressure and the physical nature of the
arrier separating the blood and the interstitium of the tissue. That is the permeability of the membrane wall. While the diffusion process is deemed the most important mechanism in this exchange the diffusion coefficient in the Fick equation  depends on molecular size . It is important for the exchange of small molecules which is driven by molecular concentration gradient across vascular endothelium defined by the Fick equation
J = k (C C ) . C C is the concentration difference.
Sample and Golovin employed this condition in [29 30] to study the dynamics of a double-lipid bilayer membrane by coupling intermembrane separation and the lipid chemical composition of a two-component membrane and dependence on the membrane curvatures. They focused on the thermodynamical
equilibrium in  and non-equilibrium in  of fluxes across the membrane. In this derivation we are considering the impacting forces on the membrane rather than the concentration of the fluid content of the blood vessels and hence admit the exchange of large molecular fluxes such as plasma proteins.
Consequently filtration is much more important than diffusion for flux of large molecules such as plasma proteins and is governed by the Starling equation [12 13 25 26]:
Vascular permeability is essential for the health of normal tissues and it is also an important characteristic of many disease state in which it is greatly increased [12 25 26 28]. Since aneurysm is caused by the
weakening of the membrane which results in high levels of plasma-protein escape activity we shall take the osmotic reflection coefficient to be zero and therefore arrive at the Darcy-Starling filtration velocity given in equation (10). That is when an aneurysm evolved the reabsorption process is stopped.
Thus equations (1) - (14) constitute the mathematical model describing the evolution and development of aneurysms contained within a cylindrical portion of the blood vessel. If any form of aneurysm occurs within the cylindrical portion of the arteries or blood vessels in general then the deformed radius a( z t ) defines the geometry of the aneurysm. Since as Figure 1 shows the extension or stretch of a( z t ) is measured through the neck of the aneurysm or the opening into the aneurysm (fundus aspect ratio)
from the bloodstream vessel. Consequently the model given here can be used to study the characteristics of all forms of aneurysms within a cylindrical portion of the vessel.
Assumption 2 in (14) leads to a quasi-static problem. That is initial conditions are not needed to solve and analyze the problems of aneurysm evolution and development. We now non-dimensionalize equations (1) - (13) using (14) and the following Camenschi  dimensionless variable transformationsEquation
The dimensional equivalent of the aneurysm size is given in (13). The velocity and membrane components and the aneurysm size are completely determined if the pressure functions P( z t ) P (t ) and P (t ) are found.
for the pressures P (t ) at the beginning of the aneurysm region z = 0 and P2 (t ) at the end of the aneurysm region z = L we will use the Milnor [3 31 32] pressure waveform
We shall employ the perturbation method of [1 33] by developing the pressure function P( z t ) in a power series with respect to the small parameter s and neglecting the O(s 2 ) terms. That is we assume thatEquation
Results and Analysis
Computations revealed discontinuity in the profiles of the impact fluid velocity and its components the membrane displacement and its components the intra- aneurysmal pressure the deformed radius the rate of filtration and the flux on any cross-section of the membrane within the aneurysm region when the Poisson ratio approaches 1. We shall interpret the points of discontinuity in the solution profiles as indicators for aneurysm rupture. The dynamic profiles of the bloodstream impacting forces intra-aneurysmal pressure elevations and the deformed radius exhibit similar characteristics for f = 0.1 0.2 0.511.2 and 5 .
However we shall present the results for the cases f = 1.2 corresponding to the regular heart beat of 72 beats per minute and f = 0.2 the experimental value at irregular heart beat of 12 beats per minute. In each case we considered two scenarios the case where a fixed point ( z t ) is considered within the aneurysm and one where all points ( z t ) are considered within
the aneurysm region. In both cases we see that for f = 0.2 and f = 1.2 time changes of the simulations of the intra-aneurysmal pressure P( z t ) are in synchrony with the membrane displacement components its magnitude and the deformed radius (See Figures 2 3 12 13 and 13). For the figures presented here we
maintained P (t ) and P (t ) at the same oscillating values. Even when one end is maintained at higher pulsating pressure values there were no significant differences in regards to rupture potential indicators in the characteristics of the profiles of the intra- aneurysmal pressure membrane displacement deformed radius filtration velocity filtration rate and fluid flux.
In Figures 2 and 3 the profile of the deformed radius a( z t ) is flat before and after the discontinuity. However in Figure 10 which is associated with the case of a regular heart beat it shows local elevations before the discontinuity at the time v = 1 and then it oscillates with time afterwards to vanish. In Figure 11 corresponding to the case of irregular heart beat the profile of the deformed radius increases before the discontinuity and then drops gradually after the discontinuity and then oscillates and die out. In Figures 10 and 11 the intra-aneurysmal pressure P( z t )
shows local pressure elevations before the discontinuity. Following the point of discontinuity the pressure profile drops and then elevates and then oscillates and die out. The transversal component of the membrane displacement shows the same characteristics with the deformed radius a( z t ) . But this is not surprising since a( z t ) = 1 + sC( z t ) . The longitudinal component of the membrane displacement elevates slightly followed by a drop leading into the point of discontinuity then delays afterwards slightly and elevates then oscillates to vanish. The profiles of the impact fluid velocity in Figure 10 and 11 shows elevated values leading into the point of discontinuity
and oscillates to zero afterward. Similarly the profile of the magnitude of the membrane displacement elevates rapidly leading to the point of discontinuity and then oscillates to zero afterward. There are more oscillations following the point of discontinuity for the case of a regular heart beat than the case of irregular heart beat. Impact of the Poisson Ratio on the Relationship of a( z t ) with P( z t )
The impact of the Poisson ratio on the relationship between the deformed radius and intra-aneurysmal pressure can be seen from the linear relationship between a( z t ) and P( z t ) given in equation (34) for values of v
This research is partially supported by the National Science Foundation (NSF) Grant Award DMS- 1214359.EQUATION
Camenschi G. A Mathematical Model of the Permeable Transporting Systems. Rev. Roum. Math. PUres et Appli Tome XXVIII Bucarest 1983. 1981; 4: 275-82.
 Fung YC. Biomechanics mechanical properties of of living tissues. Springer Verlag New York Inc. 1993.
 Ferguson GG. Physical factors in the initiation growth and rupture of human intracranial saccular aneurysms. J Neurosurg 1972; 37: 666-77. http://dx.doi.org/10.3171/jns.1972.37.6.0666
 Kwembe TA Jones SN. A mathematical model of cylindrical shaped aneurysms. Biomat 2006 Proceedings of the International Symposium on Mathematical and Computational Biology. Rio de Janeiro Brazil. world Scientific Publishers 2006; pp. 35-48.
 Sekhar LN Heros RC. Origgin growth and rupture of saccular aneurysms: a review. Neurosurgery 1981; 8: 248- 60. http://dx.doi.org/10.1227/00006123-198102000-00020
 Shah AD Humphrey JD. Finite strain elastodynamics of intracranial saccular aneurysms. J Biomech 1999; 32: 593- 500. http://dx.doi.org/10.1016/S0021-9290(99)00030-5
 Stehbans WE. Etiology of intracranial berry aneurysms. J Neurysurg 1989; 70: 823-31.
 Steiger HJ. Pathophysiology of development and rupture of cerebral aneurysms. Acta Neurochir Suppl 1990; Wein: 48: 1-57.
 Isaken JG Bazileus Y Kuamsdal T Zhang Y Kaspersen JH Waterloo K Romner B Ingebrigtsen T. Determination of wall tension in cerebral artery aneurysm by numerical simulation. Stroke 2008; 39: 3172-78. http://dx.doi.org/10.1161/STROKEAHA.107.503698
 Shojima M Oshima M Takagi K Torii R Nagata K Ichiro S Morita A Kirino T. Role of the bloodstream impacting force and the local pressure elevation in the rupture of cerebral aneurysms. Stroke 2005; 36: 1933-38. http://dx.doi.org/10.1161/01.STR.0000177877.88925.06  Brisman JL Song JK Newell DW. Cerebral aneurysms. N Engl J Med 2006; 355(9): 928-39.
 Greaves GN Greer AL Lakes RS Rouxel T. Poisson's Ratio and Modern Materials. Nat Mater 2011; 10: 823-36. http://dx.doi.org/10.1038/nmat3134
 Lakes RS Wineman A. On Poisson's Ration in Linearly Viscoelastic Solids. J Elasticity 2006; 85: 45-63. http://dx.doi.org/10.1007/s10659-006-9070-4
 Norris AN. Extreme values of Poisson's ratio and other engineering moduli in anisotropic materials. J Mechan Mater Struct 2006; 1(4): 793-12.
 Olafsson E Hauser WA Gudmundsson G. A population- based study of prognosis of ruptured cerebral aneurysm: mortality and recurrence of subarachnoid hemorrhage. Neurology 1997; 48(5): 1191-95.
 West J. Respiratory Physiology: the essentials 9th edition. Baltimore: Lippincott Williams and Wilkins 2012; p. 177.
 Michel CC. Fluid exchange in the microcirculation. Physiol 2004; 557(3): 701-702.
 Michel CC. Microvsular permeability ultrafiltration and restricted diffusion. Am J Physiol Heart Circ Physiol 2004; 287: H1887-H1888. http://dx.doi.org/10.1152/classicessays.00012.2004
 Fick A. Annalen der Physik 1855; 170(1): 59-86. Nagy JA Benjamin L Zeng H Dvorak AM Dvorak HF. Vascular permeability vascular hyperpermeability and angiogenesis. Angiogenesis 2008; 11: 109-19. http://dx.doi.org/10.1007/s10456-008-9099-z
 Sample C Golovin AA. Morphological and chemical oscillations in a couple-membrane system. SIAM J Appl Math 2011; 71: 622-34. http://dx.doi.org/10.1137/100800154
 Sample C Golovin AA. Nonlinear dynamics of a double bilipid membrane. Phys Rev E 2007; 76: article 031925.
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|Date:||Dec 31, 2014|
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