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Blood pressure, fatigue, and the pathogenesis of aneurysmal subarachnoid hemorrhage
Surgical Neurology 66 (2006) 574 – 580 www.surgicalneurology-online.com
Aneurysm
Blood pressure, fatigue, and the pathogenesis of aneurysmal subarachnoid hemorrhage Patrick Mitchell, FRCS4, Danny Birchall, FRCR, A. David Mendelow, MD Neurosurgery (School of Surgical and Reproductive Sciences), Newcastle University, Newcastle General Hospital, NE4 6BE Newcastle upon Tyne, UK Received 9 February 2006; accepted 13 June 2006
Abstract
Background: Endovascular embolization and BP reduction appear to protect cerebral aneurysms from rupture, although they do not totally eliminate aneurysm wall stress as does surgical clipping. We investigate the possible mechanisms of rupture to offer an explanation for this. Methods: Fatigue modeling of cerebral aneurysms using BP as the source of wall stress loading and calculating the increase in cycles to failure produced by a reduction in stress loading. Results: A modest reduction in the stress loading of aneurysms leads to a disproportionately large increase in the time taken for them to rupture. This result is based on the following assumptions and is thus restricted to aneurysm for which they may reasonably be true. 1. 2. 3.
Fatigue is the dominant mode of failure. The final decline in strength is rapid. The aneurysm lasts at least 6 weeks before rupturing.
Conclusions: A slight reduction in the stress on aneurysm walls can dramatically reduce their risk of rupture. D 2006 Elsevier Inc. All rights reserved. Keywords:
Cerebral aneurysm; Subarachnoid hemorrhage; Fatigue modeling
1. Introduction Endovascular coil embolization and BP reduction appear to protect cerebral aneurysms from rupture [10,11,21]. The effect of embolization on aneurysm walls is complex, but reduction in wall stress is likely to be significant. Unlike surgical clipping, coiling can only reduce rather than eliminate aneurysm wall stress because the coil ball does not totally isolate the aneurysm from the circulation. A reduction in wall stress is also likely to be significant in the protection afforded by BP reduction [28]. Nimodipine given immediately after SAH is associated with reduced BP and rehemorrhage rates [23]. We advance the conjecture that fatigue is the dominant mechanism of failure in aneurysms Abbreviations: BP, blood pressure; FSDF, failure stress distribution function; SAH, subarachnoid hemorrhage. 4 Corresponding author. Tel.: +44 191 2562506; fax: +44 191 2563139. E-mail address: [email protected] (P. Mitchell). 0090-3019/$ – see front matter D 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.surneu.2006.06.063
with thin, avascular, acellular walls and describe a model relating wall stress reduction to prolongation of time to aneurysm failure. The processes that lead to the rupture of cerebral aneurysms can be classified into 3 groups: catastrophic failure, fatigue failure, and active remodeling. The stress that would provoke either catastrophic or fatigue failure is the pressure difference between the inside and the outside of the cerebral arteries. This is dominated by BP. Catastrophic failure is sudden failure provoked by a transmural pressure gradient that exceeds the bursting strength of the vessel. Although it must reflect the ultimate stage in the development of SAH, it is probably not the only mechanism at work. Fatigue failure involves the progressive weakening of a structure by sustained or repeated application of a stress below what is necessary to cause catastrophic failure. The catastrophic failure stress becomes progressively lower as fatigue develops. This process is physical in its nature.
P. Mitchell et al. / Surgical Neurology 66 (2006) 574–580
Active remodeling is a process of living systems and involves the breakdown and formation of macromolecules. Healing, fibrosis, and lytic processes are included. Evidence of these processes can be found within the walls of a subset of aneurysms [24,25,29], but there remains a large group with thin avascular and acellular walls where fatigue may dominate behavior [1]. Fatigue failure is an important issue in many engineering fields, and computer-based models are widely used to predict and control for it. Standard models start with the material characteristics and form of the stressed component under study and the characteristics of the sustained or varying stress to which it is subject. From this, they calculate the time-to-failure. In the context of aneurysms, such an application is not feasible because the material properties are not sufficiently well defined. The fatigue behavior of aneurysm walls has not been determined experimentally, and the size, shape, and wall thickness are highly variable and not well characterized. The limits that can be placed on these values are so wide that no useful information about the absolute time-to-failure can be derived. Although the absolute time-to-rupture cannot be addressed, the relative effect of BP reduction can be because of a convenient property of the fatigue model: a specific reduction in the magnitude of cyclic stress (ie, BP) results in the time-to-failure being multiplied by a factor that is independent of time-to-failure itself over a wide range. This bfailure delay factorQ is the number by which the expected time-to-failure of the aneurysm is multiplied when a specific reduction systolic BP is applied. An effect of using failure delay factor rather than time-tofailure as the model output is that constraints can be placed on the time-to-failure without resulting in a circular argument.
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2. Methods A model was developed using the systolic/diastolic BP cycle as the dynamic stress load. Published data on diurnal, age- and stress-related excursions of BP were used to construct a time profile of systolic BP variation (Fig. 1). A 10-mm spherical aneurysm whose structural integrity is derived largely from collagen that behaves as a bundle of independent microfibers was assumed. Aneurysm walls are composite structures, and this model closely approximates composite behavior [17]. For each BP cycle, the stress on the whole aneurysm wall is divided by the number of fibers that remain intact to find the stress on individual fibers. The probability of failure of individual fibers is a function of this stress. We use a mathematical function that we call the bfailure stress distribution functionQ (FSDF) to calculate this probability. The FSDF of the aneurysm wall takes the stress on individual collagen fibers as its argument and gives the probability of the fiber failing in that cycle. Such function is known for many materials, and in general, they are sigmoid, giving a probability of failure close to 0 for low stress, 1 for stress larger than the bursting strength (the stress needed to provoke catastrophic failure), and changing from 0 to 1 fairly abruptly close to the bursting stress. Aneurysm walls have fibers of varying thickness and geometric loading. These can be accommodated by adjusting the FSDF to allow for more gradual transition from 0 to 1. Experiments with collagen [13,14,30,31] show that time-to-failure declines exponentially with increasing levels of applied stress, and this places a constraint on the type of FSDF that is appropriate. The logistic function fits these requirements and was used as the FSDF [Eq. (1)]. Two parameters determine the average and range of stresses over which the probability of failure changes from near 0 to 1. These cannot be accurately defined from available measurements, and wide ranges were
Fig. 1. This figure illustrates the logic behind one of the most important assumptions in the model: that the final decline in bursting strength of aneurysm walls is rapid. A 30-day profile of systolic BP is simulated using published data on diurnal BP fluctuations. Hypothetical plots of declining bursting strength are also included (lines 1-3). Were these declining plots real, then the aneurysms they represent would burst at the point when they cross over the BP plot. The figure shows that aneurysmal hemorrhages occurring in association with occasional high excursions of BP (the minority observed) could be weakening very slowly as for line 1. More rapid weakening allows hemorrhages to occur during the day between hypertensive excursions, but this would imply very short aneurysm duration (lines 2) unless accelerating weakening was occurring as is characteristic of the fatigue process. Line 3 is of such a profile of weakening and shows how accelerating weakening allows not only for hemorrhages to occur during the day but also during the relatively low-BP period of the night, as has been observed. The slow linear decline would result in almost all SAHs being associated with stress responses. The fatigue and fast linear declines allow SAHs to occur at other times. The fact that some hemorrhages occur during the night and aneurysms appear to have a duration of some weeks at least before rupturing requires that some aneurysms at least weaken with a profile similar to line 3. Note that if all aneurysms behaved like line 3, it would be consistent with observations of the timing of SAHs that have been made.
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used. The ranges were narrowed by constraining the timeto-failure as explained in the caption of Fig. 3. After the number of fiber failures in one BP cycle is calculated from the FSDF, this number is subtracted from the number of intact fibers. The process is then repeated for the next BP cycle and so on until the number of remaining fibers falls to 0. The process is illustrated in the flow diagram of Fig. 2. 2.1. The assumptions on which the model depends 2.1.1. The number of cycles to failure The average life span of aneurysms before rupture has been calculated to be less than 40 weeks (around 30 million cycles) [19], although much longer delays have been recorded [12]. Failure was constrained to occur after not less than 5 million cycles (approximately 6 weeks) because the issue of treatment is unlikely to arise for aneurysms bleeding sooner that this. Calculations were truncated at 40 years because they became impractical beyond that. These constraints restrict the parameters used in the logistic FSDF (Fig. 3). 2.1.2. Aneurysm bursting strength Aneurysm bursting strength was derived via 2 independent means, and the 2 results were compared as a model check. The breaking or failure stress of human tendon collagen ranges from 20 to 100 MPa [14]. In tendons, the collagen density is high, and all fibers are parallel. In pressure vessels such as aneurysms, fibers are oriented in different directions to resist the stress that occurs 3608 around any point. Theoretical models of this situation perform poorly against experiment [9], but a reasonable approximation is that spherical failure stress is around half of linear failure stress [5] assuming optimal fiber orientation, which is found in aneurysms [3]. This and the lower density of collagen fibers found in aneurysm walls [4,16] suggest lower failure stress than tendon collagen. No direct testing of aneurysm walls is available, but estimates based on the microscopic examination of thickness, collagen birefringence and fiber orientation in cadaver, and operative aneurysm give failure stresses ranging from 0.7 to 2 MPa. From this range, the catastrophic bursting pressure of an aneurysm can be calculated from size and wall thickness: p ¼ 2st=r where p is bursting pressure, t is wall thickness, r is aneurysm radius, and s is collagen failure stress. The study where aneurysm wall strength estimates were made also reported diameters and wall thicknesses enabling bursting pressures to be calculated [4,16]. For 4 aneurysms, they were 61, 105, 46, and 360 kPa (460, 790, 346, and 2700 mm Hg). The bursting pressure of cerebral vessels has been measured directly to be around 200 kPa (1500 mm Hg) [20], and we assume that aneurysms are weaker than this,
Fig. 2. Flow diagram of the computer program used to calculate the number of cycles to failure.
which accords well with the results above, adding confidence in the model. Unruptured aneurysms when found must be strong enough to withstand systolic BP; hence, a minimum busting pressure of 16 kPa (120 mm Hg) is imposed. The FSDF is thus constrained to be higher than 0.5 for all pressures greater than 200 kPa and lower than 0.5 for all pressures less than 16 kPa. This translates as bthe probability of sudden rupture of an aneurysms before the fatigue process starts must be more than 50/50 for BP over 1500 mm Hg and less than 50/50 for BP under 120 mm Hg.Q 2.1.3. Blood pressure variation Continuous ambulant intra-arterial BP has been measured over 24- to 48-hour periods [2,8,18]. This showed
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the circadian BP cycle. The Framingham study shows an increase of systolic BP of 0.35% per year with advancing age [7]. Another source of BP variation is physiological stress. The frequency and extent of stress-related hypertensive episodes are poorly documented, but an extraordinary case report gives some insight [8]. A 24 -year-old man with hypertension was undergoing ambulatory intraarterial BP monitoring when he was attacked by an assailant with a knife. He survived, and a record of BP during the event showed a transient rise in systolic BP from 190 to 312 mm Hg, a 64% increase. These data allow a long-term BP profile to be constructed consisting of a steady background increase in systolic BP with age with added circadian rhythm and occasional stress-related hypertensive episodes.
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3. Results For a specific value of a, the failure delay factor is not sensitive to changes in b. It is this feature that allows us to conclude that a 10% decrease in stress loading leads to at least a 3-fold increase in time-to-failure (Fig. 4). The minimum failure delay factor is 3. Extreme degrees of sensitivity to stress loading are consistent with our
2.1.4. Diurnal BP variation It is assumed that the rate of decline in bursting pressure in the last few hours before aneurysm rupture is greater than 30 mm Hg per 6 hours. This is an extrapolation from the observation that some SAHs occur during the night [6,15] (illustrated in Fig. 3). The FSDF used was f ðsÞ ¼
1 1 þ e abs
ð1Þ
where s is the magnitude of cyclic stress; a and b are parameters of the model that determine the position and steepness of the transition from probabilities near 0 to near 1. f(s) takes the value of 0.5 at the mean bursting pressure when a ¼ bs:
ð2Þ
s is the stress magnitude, and we have constrained this to be between 15 and 200 kPa. It is the geometric range that is important in this application, and this represents a geometric range of 40 3 fold. Applying this range to s in Eq. (2) allows a to be constrained by b, specifically: a is in the range 3b to 40b. A range of values was selected for b. The model was run for this range of b with values of a ranging from 3b to 40b. Combinations of a and b that resulted in times-to-failure under 6 weeks or over 40 years were rejected. Combinations that resulted in less than 4 kPa (30 mm Hg) decline in bursting pressure in the last 6 hours of the aneurysm life were likewise rejected. The resulting more constrained range of values of a and b were used to calculate the failure delay factors. The total number of cycles necessary to reduce the residual fiber number to 0 is noted as x h (for high BP) and the process repeated with a 10% reduction in BP loading (Fig. 1). Ten percent was chosen for illustrative purposes because it is an achievable treatment target. This results in more cycles to failure x l (for low BP). The failure delay factor is obtained by dividing x l by x h.
Fig. 3. The process by which the 2 parameters (a and b) required for the logistic FSDF were inferred. The values of the 2 parameters are on the vertical and horizontal axes. Points representing allowed combinations are printed in black. To begin with, panel (A) shows the entire graph to be covered as constraints have not yet been imposed. The first constraints are that the initial bursting strength is less than 200 kPa, and the final decline in bursting strength is greater than 30 mm Hg in 6 hours. The effect is shown in panel (B). The final constraint is that aneurysms should last at least 6 weeks. The effect of all constraints is shown in panel (C).
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Fig. 4. Sensitivity to load reduction. The gray line shows the factor by which time-to-rupture increases with a 10% reduction in loading (for example, by 10% reduction in BP) on a log scale versus the value given to parameter 1. The black area is copied from Fig. 2 for reference and shows the allowed range of parameters. It turns out that the factor plotted is insensitive to parameter b within this range but is sensitive to parameter a, so only one graph is needed. The minimum factor is 3. Although not plotted, the line extends to infinity.
assumptions and would in practice imply complete protection from rupture. The minimum time-to-rupture was arbitrarily specified as 6 weeks. If the minimum time-to-rupture is raised, the minimum failure delay factor for a 10% stress reduction is also raised. For example, if the minimum time-to-rupture is 5 years, the minimum failure delay factor is 7.4 extending survival to 37 years. 4. Discussion The principal limitation of this study is that the contribution of fatigue to aneurysm failure remains conjectural, there being no direct evidence for it, but there is reason to believe that it is important. A model of aneurysm behavior involving catastrophic failure only predicts that bursting pressure is constant over time. Failure would occur on the first occasion that the systolic BP reaches the bursting pressure and not before, or later at a lower BP. This could apply to arterial wall failure and aneurysm formation but not to aneurysm rupture and SAH (Fig. 1). With constant failure pressure, most SAHs would be associated with stressful circumstances, the remainder occurring during the morning. This is not observed. There is a peak incidence in the morning, but a substantial number of SAHs occur at night, and few are associated with stressful episodes [6,15]. Fatigue processes are characterized by an early phase of slow weakening that accelerates as failure approaches. Collagen, the principle structural component of aneurysm walls [29,32], has been shown to behave this way [31]. Fig. 1 includes a plot derived from the model showing that a fatigue-based mechanism is consistent with the observed circadian distribution of SAH. Although it is reasonable that fatigue plays a role in aneurysm rupture, it cannot be the only factor. Active remodeling is known to occur in some cases [25,29].
Many more sophisticated fatigue models than this have been developed, aiming to improve the accuracy of time-tofailure predictions [5]. With aneurysms, no matter how accurate the model, there is little hope of predicting time-tofailure reliably because the initial conditions are poorly defined and variable from one aneurysm to another. The present results depend on a basic property of fatigue systems—gradual weakening that accelerates as failure approaches—and not on the specific model or FSDF used. The minimum failure delay factor of 3 is subject to revision, but the conclusion that the failure delay factor ranges from a value greater than 1 upward with no upper limit and that a small change in stress produce a disproportionately large failure delay factor is quite robust. If reducing stress loading disproportionately delays rupture, the converse is also true. Raising stress disproportionately hastens it. The modest rise in BP with age could greatly magnify the likelihood of failure. This effect could be augmented by, or could even explain, the observed decline in cerebral artery wall strength with age of 2% per year [20]. Even a modest reduction in cyclical stress loading could dramatically influence the prognosis in persons prone to aneurysmal SAH. Such a reduction could be effected by medical lowering of BP [26-28] or by coiling. Aneurysm formation and aneurysm failure causing SAH are clinically distinct processes. They must also represent distinct failure processes as opposed to different periods in one failure process. Fig. 5 shows the profile of strain versus number of stress cycles from our model. Exactly comparable strain/cycle profiles have been found for collagen [14,30]. Little strain is apparent until failure approaches, when strain increases rapidly with no stabilization before failure. This implies that if aneurysm formation and SAH were derived from one failure process, aneurysms would be extremely short-lived, and we would be most unlikely to find unruptured ones at all. Because this is not observed, there must be at least 2 failure processes involved. It would appear that aneurysm formation involves complete failure of one part of the artery wall. After this, the remaining wall components are subject to fatigue stresses and may fail causing an SAH. These failure processes could involve anatomically distinct parts of the vessel wall such as the media, internal elastic lamina, or adventitia. Alternatively,
Fig. 5. Profile of aneurysm strain (expansion) during the fatigue process.
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the separately failing components could exist within the same anatomical layers. Histological studies suggest that the aneurysm wall is largely composed of fibrous tissue, mainly collagen [32]. These fibers could have existed at any location within the artery wall before aneurysm formation [29]. Remodeling could strengthen aneurysms and cancel or reverse the weakening effect of fatigue in a way that cannot be predicted, but certain hypotheses can be advanced. Many small aneurysms have thin walls that are acellular and avascular [25] with seemingly little potential for remodeling, and if any aneurysms can be accurately modeled by fatigue processes, this group would be included. By contrast, all large aneurysms must be remodeled to some degree. It is likely that there is insufficient material in the wall of the parent artery to account for them, so some must be added during their formation. Despite diversity of the causes of SAH, aneurysm formation is a ubiquitous precursor. Medial and intimal defects have been observed [32], as has derangement of cerebral artery fiber architecture [22] in persons prone to SAH. Mycotic aneurysms form another group with a mechanism of formation that is different again. It would seem that the diverse mechanisms of artery wall failure leave similar aneurysms that subsequently fail in similar ways. 5. Conclusions We conjecture that fatigue failure is a significant mechanism in the rupture of cerebral aneurysms. In cases where fatigue is important, modest BP reduction or alteration of aneurysm hemodynamics, for example, by coiling, may be highly effective in delaying or preventing rupture. References [1] Abruzzo T, Shengelaia GG, Dawson III RC, Owens DS, Cawley CM, Gravanis MB. Histologic and morphologic comparison of experimental aneurysms with human intracranial aneurysms. AJNR Am J Neuroradiol 1998;19(7):1309 - 14. [2] Broadhurst P, Brigden G, Dasgupta P, Lahiri A, Raftery EB. Ambulatory intra-arterial blood pressure in normal subjects. Am Heart J 1990;120(1):160 - 6. [3] Canham PB, Finlay HM, Dixon JG, Ferguson SE. Layered collagen fabric of cerebral aneurysms quantitatively assessed by the universal stage and polarized light microscopy. Anat Rec 1991; 231(4):579 - 92. [4] Canham PB, Finlay HM, Kiernan JA, Ferguson GG. Layered structure of saccular aneurysms assessed by collagen birefringence. Neurol Res 1999;21(7):618 - 26. [5] Davila CC, Camanho PP. Failure criteria for FRP laminates in plane stress. National Aeronautics and Space Administration [Langley Research Center, Hampton, VA 2003 NASA/TM-2003-212663]. [6] Feigin VL, Anderson CS, Rodgers A, Bennett DA. Subarachnoid hemorrhage occurrence exhibits a temporal pattern—evidence from meta-analysis. Eur J Neurol 2002;9(5):511 - 6. [7] Franklin SS, Gustin Wt, Wong ND, Larson MG, Weber MA, Kannel D, Levy D. Hemodynamic patterns of age-related changes in blood pressure. The Framingham Heart Study. Circulation 1997; 96(1):308 - 15.
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[8] Heber ME, Raftery EB, Thompson D. Intra-arterial ambulatory blood pressure monitoring can save your life—or, acute haemodynamic response to a murderous assault. Int J Cardiol 1988;20(1):138 - 41. [9] Hinton M, Kaddour A, Soden P. A comparison of the predictive capabilities of current failure theories for composite laminates judged against experimental evidence. Compos Sci Technol 2002; 62(12-13):1725 - 97. [10] Anonymous. Unruptured intracranial aneurysms—risk of rupture and risks of surgical intervention. International Study of Unruptured Intracranial Aneurysms Investigators. N Engl J Med 1998;339(24): 1725 - 33. [11] Wiebers DO, Whisnant JP, Huston 3rd J, et al. Unruptured intracranial aneurysms: natural history, clinical outcome, and risks of surgery and endovascular treatment. Lancet 2003;362:103 - 10. [12] Juvela S, Porras M, Poussa K. Natural history of unruptured intracranial aneurysms: probability of and risk factors for aneurysm rupture. J Neurosurg 2000;93(3):379 - 87. [13] Ker RF. The design of soft collagenous load-bearing tissues. J Exp Biol 1999;202 Pt 23:3315 - 24. [14] Ker RF, Wang XT, Pike AV. Fatigue quality of mammalian tendons. J Exp Biol 2000;203 Pt 8:1317 - 27. [15] Kleinpeter G, Schatzer R, Bock F. Is blood pressure really a trigger for the circadian rhythm of subarachnoid hemorrhage? Stroke 1995; 26(10):1805 - 10. [16] MacDonald DJ, Finlay HM, Canham PB. Directional wall strength in saccular brain aneurysms from polarized light microscopy. Ann Biomed Eng 2000;28(5):533 - 42. [17] McArtney L. Model of composite degradation due to fatigue damage. UK7 National Physics Laboratory; 1999. [18] Millar-Craig MW, Bishop CN, Raftery EB. Circadian variation of blood-pressure. Lancet 1978;1(8068):795 - 7. [19] Mitchell P, Jakubowski J. Estimate of the maximum time interval between formation of cerebral aneurysm and rupture. J Neurol Neurosurg Psychiatry 2000;69(6):760 - 7. [20] Mitchell P, Jakubowski J. Failure testing cerebral arteries: are branch points weaker than unbranched vessels? Br J Neurosurg 2002;16(6): 578 - 82. [21] Molyneux A. International Subarachnoid Aneurysm Trial (ISAT) of neurosurgical clipping versus endovascular coiling in 2143 patients with ruptured intracranial aneurysms: a randomised trial. Lancet 2002;360(9342):1267 - 74. [22] Ostergaard JR, Reske-Nielsen E, Oxlund H. Histological and morphometric observations on the reticular fibers in the arterial beds of patients with ruptured intracranial saccular aneurysms. Neurosurgery 1987;20(4):554 - 8. [23] Pickard JD, Murray GD, Illingworth R, Shaw MD, Teasdale GM, Foy PM, Humphrey PM, Lang DA, Nelson R, Richards P, et al. Effect of oral nimodipine on cerebral infarction and outcome after subarachnoid hemorrhage: British aneurysm nimodipine trial. BMJ 1989;298(6674): 636 - 42. [24] Scanarini M, Mingrino S, Giordano R, Baroni A. Histological and ultrastructural study of intracranial saccular aneurysmal wall. Acta Neurochir (Wien) 1978;43(3-4):171 - 82. [25] Schlote W, Gaus C. Histologic aspects from ruptured and nonruptured aneurysms. Neurol Res 1994;16(1):59 - 62. [26] Slosberg PS. Nonoperative management of ruptured intracranial aneurysms. Clin Neurosurg 1974;21:90 - 9. [27] Slosberg PS. Zero percent mortality due to recurrent hemorrhage in follow-up of medically treated ruptured single intracranial aneurysms: a 23-year study. Trans Am Neurol Assoc 1979;104: 180 - 3. [28] Slosberg PS. Unexpected results in long-term medically treated ruptured intracranial aneurysm including data on 14 patients followed more than 30 years each. Acta Neurochir (Wien) 1997; 139(8):697 - 705. [29] Stehbens WE. Histopathology of cerebral aneurysms. Arch Neurol 1963;8:272 - 85.
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[30] Wang XT, Ker RF. Creep rupture of wallaby tail tendons. J Exp Biol 1995;198(3):831 - 45. [31] Wang XT, Ker RF, Alexander RM. Fatigue rupture of wallaby tail tendons. J Exp Biol 1995;198(3):847 - 52. [32] Yong-Zhong G, van Alphen HA. Pathogenesis and histopathology of saccular aneurysms: review of the literature. Neurol Res 1990; 12(4):249 - 55.
Commentary The authors present a very interesting mathematical model that describes a biophysical mechanism that may explain the pathophysiology of a special group of aneurysms—those that grow and rupture quickly. The presence of this group of aneurysms best explains the apparent paradox that small aneurysms have very low rupture rates in
prospective studies yet make up the majority of ruptured aneurysms in large series. Although their property of rupturing soon after growth makes proof of the existence of these aneurysms difficult, perhaps widespread screening for intracranial aneurysms, which is increasingly being considered as a public health measure, will yield information regarding their mechanics. Some readers may criticize the highly theoretical nature of this study. However, mathematical models are useful in circumstances such as these where measurements of physical qualities are difficult. The authors should be commended for the original approach and imaginative analysis. Phillip S. Dickey, MD New Haven, CT 06510 USA
Aneurysm
Blood pressure, fatigue, and the pathogenesis of aneurysmal subarachnoid hemorrhage Patrick Mitchell, FRCS4, Danny Birchall, FRCR, A. David Mendelow, MD Neurosurgery (School of Surgical and Reproductive Sciences), Newcastle University, Newcastle General Hospital, NE4 6BE Newcastle upon Tyne, UK Received 9 February 2006; accepted 13 June 2006
Abstract
Background: Endovascular embolization and BP reduction appear to protect cerebral aneurysms from rupture, although they do not totally eliminate aneurysm wall stress as does surgical clipping. We investigate the possible mechanisms of rupture to offer an explanation for this. Methods: Fatigue modeling of cerebral aneurysms using BP as the source of wall stress loading and calculating the increase in cycles to failure produced by a reduction in stress loading. Results: A modest reduction in the stress loading of aneurysms leads to a disproportionately large increase in the time taken for them to rupture. This result is based on the following assumptions and is thus restricted to aneurysm for which they may reasonably be true. 1. 2. 3.
Fatigue is the dominant mode of failure. The final decline in strength is rapid. The aneurysm lasts at least 6 weeks before rupturing.
Conclusions: A slight reduction in the stress on aneurysm walls can dramatically reduce their risk of rupture. D 2006 Elsevier Inc. All rights reserved. Keywords:
Cerebral aneurysm; Subarachnoid hemorrhage; Fatigue modeling
1. Introduction Endovascular coil embolization and BP reduction appear to protect cerebral aneurysms from rupture [10,11,21]. The effect of embolization on aneurysm walls is complex, but reduction in wall stress is likely to be significant. Unlike surgical clipping, coiling can only reduce rather than eliminate aneurysm wall stress because the coil ball does not totally isolate the aneurysm from the circulation. A reduction in wall stress is also likely to be significant in the protection afforded by BP reduction [28]. Nimodipine given immediately after SAH is associated with reduced BP and rehemorrhage rates [23]. We advance the conjecture that fatigue is the dominant mechanism of failure in aneurysms Abbreviations: BP, blood pressure; FSDF, failure stress distribution function; SAH, subarachnoid hemorrhage. 4 Corresponding author. Tel.: +44 191 2562506; fax: +44 191 2563139. E-mail address: [email protected] (P. Mitchell). 0090-3019/$ – see front matter D 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.surneu.2006.06.063
with thin, avascular, acellular walls and describe a model relating wall stress reduction to prolongation of time to aneurysm failure. The processes that lead to the rupture of cerebral aneurysms can be classified into 3 groups: catastrophic failure, fatigue failure, and active remodeling. The stress that would provoke either catastrophic or fatigue failure is the pressure difference between the inside and the outside of the cerebral arteries. This is dominated by BP. Catastrophic failure is sudden failure provoked by a transmural pressure gradient that exceeds the bursting strength of the vessel. Although it must reflect the ultimate stage in the development of SAH, it is probably not the only mechanism at work. Fatigue failure involves the progressive weakening of a structure by sustained or repeated application of a stress below what is necessary to cause catastrophic failure. The catastrophic failure stress becomes progressively lower as fatigue develops. This process is physical in its nature.
P. Mitchell et al. / Surgical Neurology 66 (2006) 574–580
Active remodeling is a process of living systems and involves the breakdown and formation of macromolecules. Healing, fibrosis, and lytic processes are included. Evidence of these processes can be found within the walls of a subset of aneurysms [24,25,29], but there remains a large group with thin avascular and acellular walls where fatigue may dominate behavior [1]. Fatigue failure is an important issue in many engineering fields, and computer-based models are widely used to predict and control for it. Standard models start with the material characteristics and form of the stressed component under study and the characteristics of the sustained or varying stress to which it is subject. From this, they calculate the time-to-failure. In the context of aneurysms, such an application is not feasible because the material properties are not sufficiently well defined. The fatigue behavior of aneurysm walls has not been determined experimentally, and the size, shape, and wall thickness are highly variable and not well characterized. The limits that can be placed on these values are so wide that no useful information about the absolute time-to-failure can be derived. Although the absolute time-to-rupture cannot be addressed, the relative effect of BP reduction can be because of a convenient property of the fatigue model: a specific reduction in the magnitude of cyclic stress (ie, BP) results in the time-to-failure being multiplied by a factor that is independent of time-to-failure itself over a wide range. This bfailure delay factorQ is the number by which the expected time-to-failure of the aneurysm is multiplied when a specific reduction systolic BP is applied. An effect of using failure delay factor rather than time-tofailure as the model output is that constraints can be placed on the time-to-failure without resulting in a circular argument.
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2. Methods A model was developed using the systolic/diastolic BP cycle as the dynamic stress load. Published data on diurnal, age- and stress-related excursions of BP were used to construct a time profile of systolic BP variation (Fig. 1). A 10-mm spherical aneurysm whose structural integrity is derived largely from collagen that behaves as a bundle of independent microfibers was assumed. Aneurysm walls are composite structures, and this model closely approximates composite behavior [17]. For each BP cycle, the stress on the whole aneurysm wall is divided by the number of fibers that remain intact to find the stress on individual fibers. The probability of failure of individual fibers is a function of this stress. We use a mathematical function that we call the bfailure stress distribution functionQ (FSDF) to calculate this probability. The FSDF of the aneurysm wall takes the stress on individual collagen fibers as its argument and gives the probability of the fiber failing in that cycle. Such function is known for many materials, and in general, they are sigmoid, giving a probability of failure close to 0 for low stress, 1 for stress larger than the bursting strength (the stress needed to provoke catastrophic failure), and changing from 0 to 1 fairly abruptly close to the bursting stress. Aneurysm walls have fibers of varying thickness and geometric loading. These can be accommodated by adjusting the FSDF to allow for more gradual transition from 0 to 1. Experiments with collagen [13,14,30,31] show that time-to-failure declines exponentially with increasing levels of applied stress, and this places a constraint on the type of FSDF that is appropriate. The logistic function fits these requirements and was used as the FSDF [Eq. (1)]. Two parameters determine the average and range of stresses over which the probability of failure changes from near 0 to 1. These cannot be accurately defined from available measurements, and wide ranges were
Fig. 1. This figure illustrates the logic behind one of the most important assumptions in the model: that the final decline in bursting strength of aneurysm walls is rapid. A 30-day profile of systolic BP is simulated using published data on diurnal BP fluctuations. Hypothetical plots of declining bursting strength are also included (lines 1-3). Were these declining plots real, then the aneurysms they represent would burst at the point when they cross over the BP plot. The figure shows that aneurysmal hemorrhages occurring in association with occasional high excursions of BP (the minority observed) could be weakening very slowly as for line 1. More rapid weakening allows hemorrhages to occur during the day between hypertensive excursions, but this would imply very short aneurysm duration (lines 2) unless accelerating weakening was occurring as is characteristic of the fatigue process. Line 3 is of such a profile of weakening and shows how accelerating weakening allows not only for hemorrhages to occur during the day but also during the relatively low-BP period of the night, as has been observed. The slow linear decline would result in almost all SAHs being associated with stress responses. The fatigue and fast linear declines allow SAHs to occur at other times. The fact that some hemorrhages occur during the night and aneurysms appear to have a duration of some weeks at least before rupturing requires that some aneurysms at least weaken with a profile similar to line 3. Note that if all aneurysms behaved like line 3, it would be consistent with observations of the timing of SAHs that have been made.
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used. The ranges were narrowed by constraining the timeto-failure as explained in the caption of Fig. 3. After the number of fiber failures in one BP cycle is calculated from the FSDF, this number is subtracted from the number of intact fibers. The process is then repeated for the next BP cycle and so on until the number of remaining fibers falls to 0. The process is illustrated in the flow diagram of Fig. 2. 2.1. The assumptions on which the model depends 2.1.1. The number of cycles to failure The average life span of aneurysms before rupture has been calculated to be less than 40 weeks (around 30 million cycles) [19], although much longer delays have been recorded [12]. Failure was constrained to occur after not less than 5 million cycles (approximately 6 weeks) because the issue of treatment is unlikely to arise for aneurysms bleeding sooner that this. Calculations were truncated at 40 years because they became impractical beyond that. These constraints restrict the parameters used in the logistic FSDF (Fig. 3). 2.1.2. Aneurysm bursting strength Aneurysm bursting strength was derived via 2 independent means, and the 2 results were compared as a model check. The breaking or failure stress of human tendon collagen ranges from 20 to 100 MPa [14]. In tendons, the collagen density is high, and all fibers are parallel. In pressure vessels such as aneurysms, fibers are oriented in different directions to resist the stress that occurs 3608 around any point. Theoretical models of this situation perform poorly against experiment [9], but a reasonable approximation is that spherical failure stress is around half of linear failure stress [5] assuming optimal fiber orientation, which is found in aneurysms [3]. This and the lower density of collagen fibers found in aneurysm walls [4,16] suggest lower failure stress than tendon collagen. No direct testing of aneurysm walls is available, but estimates based on the microscopic examination of thickness, collagen birefringence and fiber orientation in cadaver, and operative aneurysm give failure stresses ranging from 0.7 to 2 MPa. From this range, the catastrophic bursting pressure of an aneurysm can be calculated from size and wall thickness: p ¼ 2st=r where p is bursting pressure, t is wall thickness, r is aneurysm radius, and s is collagen failure stress. The study where aneurysm wall strength estimates were made also reported diameters and wall thicknesses enabling bursting pressures to be calculated [4,16]. For 4 aneurysms, they were 61, 105, 46, and 360 kPa (460, 790, 346, and 2700 mm Hg). The bursting pressure of cerebral vessels has been measured directly to be around 200 kPa (1500 mm Hg) [20], and we assume that aneurysms are weaker than this,
Fig. 2. Flow diagram of the computer program used to calculate the number of cycles to failure.
which accords well with the results above, adding confidence in the model. Unruptured aneurysms when found must be strong enough to withstand systolic BP; hence, a minimum busting pressure of 16 kPa (120 mm Hg) is imposed. The FSDF is thus constrained to be higher than 0.5 for all pressures greater than 200 kPa and lower than 0.5 for all pressures less than 16 kPa. This translates as bthe probability of sudden rupture of an aneurysms before the fatigue process starts must be more than 50/50 for BP over 1500 mm Hg and less than 50/50 for BP under 120 mm Hg.Q 2.1.3. Blood pressure variation Continuous ambulant intra-arterial BP has been measured over 24- to 48-hour periods [2,8,18]. This showed
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the circadian BP cycle. The Framingham study shows an increase of systolic BP of 0.35% per year with advancing age [7]. Another source of BP variation is physiological stress. The frequency and extent of stress-related hypertensive episodes are poorly documented, but an extraordinary case report gives some insight [8]. A 24 -year-old man with hypertension was undergoing ambulatory intraarterial BP monitoring when he was attacked by an assailant with a knife. He survived, and a record of BP during the event showed a transient rise in systolic BP from 190 to 312 mm Hg, a 64% increase. These data allow a long-term BP profile to be constructed consisting of a steady background increase in systolic BP with age with added circadian rhythm and occasional stress-related hypertensive episodes.
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3. Results For a specific value of a, the failure delay factor is not sensitive to changes in b. It is this feature that allows us to conclude that a 10% decrease in stress loading leads to at least a 3-fold increase in time-to-failure (Fig. 4). The minimum failure delay factor is 3. Extreme degrees of sensitivity to stress loading are consistent with our
2.1.4. Diurnal BP variation It is assumed that the rate of decline in bursting pressure in the last few hours before aneurysm rupture is greater than 30 mm Hg per 6 hours. This is an extrapolation from the observation that some SAHs occur during the night [6,15] (illustrated in Fig. 3). The FSDF used was f ðsÞ ¼
1 1 þ e abs
ð1Þ
where s is the magnitude of cyclic stress; a and b are parameters of the model that determine the position and steepness of the transition from probabilities near 0 to near 1. f(s) takes the value of 0.5 at the mean bursting pressure when a ¼ bs:
ð2Þ
s is the stress magnitude, and we have constrained this to be between 15 and 200 kPa. It is the geometric range that is important in this application, and this represents a geometric range of 40 3 fold. Applying this range to s in Eq. (2) allows a to be constrained by b, specifically: a is in the range 3b to 40b. A range of values was selected for b. The model was run for this range of b with values of a ranging from 3b to 40b. Combinations of a and b that resulted in times-to-failure under 6 weeks or over 40 years were rejected. Combinations that resulted in less than 4 kPa (30 mm Hg) decline in bursting pressure in the last 6 hours of the aneurysm life were likewise rejected. The resulting more constrained range of values of a and b were used to calculate the failure delay factors. The total number of cycles necessary to reduce the residual fiber number to 0 is noted as x h (for high BP) and the process repeated with a 10% reduction in BP loading (Fig. 1). Ten percent was chosen for illustrative purposes because it is an achievable treatment target. This results in more cycles to failure x l (for low BP). The failure delay factor is obtained by dividing x l by x h.
Fig. 3. The process by which the 2 parameters (a and b) required for the logistic FSDF were inferred. The values of the 2 parameters are on the vertical and horizontal axes. Points representing allowed combinations are printed in black. To begin with, panel (A) shows the entire graph to be covered as constraints have not yet been imposed. The first constraints are that the initial bursting strength is less than 200 kPa, and the final decline in bursting strength is greater than 30 mm Hg in 6 hours. The effect is shown in panel (B). The final constraint is that aneurysms should last at least 6 weeks. The effect of all constraints is shown in panel (C).
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Fig. 4. Sensitivity to load reduction. The gray line shows the factor by which time-to-rupture increases with a 10% reduction in loading (for example, by 10% reduction in BP) on a log scale versus the value given to parameter 1. The black area is copied from Fig. 2 for reference and shows the allowed range of parameters. It turns out that the factor plotted is insensitive to parameter b within this range but is sensitive to parameter a, so only one graph is needed. The minimum factor is 3. Although not plotted, the line extends to infinity.
assumptions and would in practice imply complete protection from rupture. The minimum time-to-rupture was arbitrarily specified as 6 weeks. If the minimum time-to-rupture is raised, the minimum failure delay factor for a 10% stress reduction is also raised. For example, if the minimum time-to-rupture is 5 years, the minimum failure delay factor is 7.4 extending survival to 37 years. 4. Discussion The principal limitation of this study is that the contribution of fatigue to aneurysm failure remains conjectural, there being no direct evidence for it, but there is reason to believe that it is important. A model of aneurysm behavior involving catastrophic failure only predicts that bursting pressure is constant over time. Failure would occur on the first occasion that the systolic BP reaches the bursting pressure and not before, or later at a lower BP. This could apply to arterial wall failure and aneurysm formation but not to aneurysm rupture and SAH (Fig. 1). With constant failure pressure, most SAHs would be associated with stressful circumstances, the remainder occurring during the morning. This is not observed. There is a peak incidence in the morning, but a substantial number of SAHs occur at night, and few are associated with stressful episodes [6,15]. Fatigue processes are characterized by an early phase of slow weakening that accelerates as failure approaches. Collagen, the principle structural component of aneurysm walls [29,32], has been shown to behave this way [31]. Fig. 1 includes a plot derived from the model showing that a fatigue-based mechanism is consistent with the observed circadian distribution of SAH. Although it is reasonable that fatigue plays a role in aneurysm rupture, it cannot be the only factor. Active remodeling is known to occur in some cases [25,29].
Many more sophisticated fatigue models than this have been developed, aiming to improve the accuracy of time-tofailure predictions [5]. With aneurysms, no matter how accurate the model, there is little hope of predicting time-tofailure reliably because the initial conditions are poorly defined and variable from one aneurysm to another. The present results depend on a basic property of fatigue systems—gradual weakening that accelerates as failure approaches—and not on the specific model or FSDF used. The minimum failure delay factor of 3 is subject to revision, but the conclusion that the failure delay factor ranges from a value greater than 1 upward with no upper limit and that a small change in stress produce a disproportionately large failure delay factor is quite robust. If reducing stress loading disproportionately delays rupture, the converse is also true. Raising stress disproportionately hastens it. The modest rise in BP with age could greatly magnify the likelihood of failure. This effect could be augmented by, or could even explain, the observed decline in cerebral artery wall strength with age of 2% per year [20]. Even a modest reduction in cyclical stress loading could dramatically influence the prognosis in persons prone to aneurysmal SAH. Such a reduction could be effected by medical lowering of BP [26-28] or by coiling. Aneurysm formation and aneurysm failure causing SAH are clinically distinct processes. They must also represent distinct failure processes as opposed to different periods in one failure process. Fig. 5 shows the profile of strain versus number of stress cycles from our model. Exactly comparable strain/cycle profiles have been found for collagen [14,30]. Little strain is apparent until failure approaches, when strain increases rapidly with no stabilization before failure. This implies that if aneurysm formation and SAH were derived from one failure process, aneurysms would be extremely short-lived, and we would be most unlikely to find unruptured ones at all. Because this is not observed, there must be at least 2 failure processes involved. It would appear that aneurysm formation involves complete failure of one part of the artery wall. After this, the remaining wall components are subject to fatigue stresses and may fail causing an SAH. These failure processes could involve anatomically distinct parts of the vessel wall such as the media, internal elastic lamina, or adventitia. Alternatively,
Fig. 5. Profile of aneurysm strain (expansion) during the fatigue process.
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the separately failing components could exist within the same anatomical layers. Histological studies suggest that the aneurysm wall is largely composed of fibrous tissue, mainly collagen [32]. These fibers could have existed at any location within the artery wall before aneurysm formation [29]. Remodeling could strengthen aneurysms and cancel or reverse the weakening effect of fatigue in a way that cannot be predicted, but certain hypotheses can be advanced. Many small aneurysms have thin walls that are acellular and avascular [25] with seemingly little potential for remodeling, and if any aneurysms can be accurately modeled by fatigue processes, this group would be included. By contrast, all large aneurysms must be remodeled to some degree. It is likely that there is insufficient material in the wall of the parent artery to account for them, so some must be added during their formation. Despite diversity of the causes of SAH, aneurysm formation is a ubiquitous precursor. Medial and intimal defects have been observed [32], as has derangement of cerebral artery fiber architecture [22] in persons prone to SAH. Mycotic aneurysms form another group with a mechanism of formation that is different again. It would seem that the diverse mechanisms of artery wall failure leave similar aneurysms that subsequently fail in similar ways. 5. Conclusions We conjecture that fatigue failure is a significant mechanism in the rupture of cerebral aneurysms. In cases where fatigue is important, modest BP reduction or alteration of aneurysm hemodynamics, for example, by coiling, may be highly effective in delaying or preventing rupture. References [1] Abruzzo T, Shengelaia GG, Dawson III RC, Owens DS, Cawley CM, Gravanis MB. Histologic and morphologic comparison of experimental aneurysms with human intracranial aneurysms. AJNR Am J Neuroradiol 1998;19(7):1309 - 14. [2] Broadhurst P, Brigden G, Dasgupta P, Lahiri A, Raftery EB. Ambulatory intra-arterial blood pressure in normal subjects. Am Heart J 1990;120(1):160 - 6. [3] Canham PB, Finlay HM, Dixon JG, Ferguson SE. Layered collagen fabric of cerebral aneurysms quantitatively assessed by the universal stage and polarized light microscopy. Anat Rec 1991; 231(4):579 - 92. [4] Canham PB, Finlay HM, Kiernan JA, Ferguson GG. Layered structure of saccular aneurysms assessed by collagen birefringence. Neurol Res 1999;21(7):618 - 26. [5] Davila CC, Camanho PP. Failure criteria for FRP laminates in plane stress. National Aeronautics and Space Administration [Langley Research Center, Hampton, VA 2003 NASA/TM-2003-212663]. [6] Feigin VL, Anderson CS, Rodgers A, Bennett DA. Subarachnoid hemorrhage occurrence exhibits a temporal pattern—evidence from meta-analysis. Eur J Neurol 2002;9(5):511 - 6. [7] Franklin SS, Gustin Wt, Wong ND, Larson MG, Weber MA, Kannel D, Levy D. Hemodynamic patterns of age-related changes in blood pressure. The Framingham Heart Study. Circulation 1997; 96(1):308 - 15.
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[8] Heber ME, Raftery EB, Thompson D. Intra-arterial ambulatory blood pressure monitoring can save your life—or, acute haemodynamic response to a murderous assault. Int J Cardiol 1988;20(1):138 - 41. [9] Hinton M, Kaddour A, Soden P. A comparison of the predictive capabilities of current failure theories for composite laminates judged against experimental evidence. Compos Sci Technol 2002; 62(12-13):1725 - 97. [10] Anonymous. Unruptured intracranial aneurysms—risk of rupture and risks of surgical intervention. International Study of Unruptured Intracranial Aneurysms Investigators. N Engl J Med 1998;339(24): 1725 - 33. [11] Wiebers DO, Whisnant JP, Huston 3rd J, et al. Unruptured intracranial aneurysms: natural history, clinical outcome, and risks of surgery and endovascular treatment. Lancet 2003;362:103 - 10. [12] Juvela S, Porras M, Poussa K. Natural history of unruptured intracranial aneurysms: probability of and risk factors for aneurysm rupture. J Neurosurg 2000;93(3):379 - 87. [13] Ker RF. The design of soft collagenous load-bearing tissues. J Exp Biol 1999;202 Pt 23:3315 - 24. [14] Ker RF, Wang XT, Pike AV. Fatigue quality of mammalian tendons. J Exp Biol 2000;203 Pt 8:1317 - 27. [15] Kleinpeter G, Schatzer R, Bock F. Is blood pressure really a trigger for the circadian rhythm of subarachnoid hemorrhage? Stroke 1995; 26(10):1805 - 10. [16] MacDonald DJ, Finlay HM, Canham PB. Directional wall strength in saccular brain aneurysms from polarized light microscopy. Ann Biomed Eng 2000;28(5):533 - 42. [17] McArtney L. Model of composite degradation due to fatigue damage. UK7 National Physics Laboratory; 1999. [18] Millar-Craig MW, Bishop CN, Raftery EB. Circadian variation of blood-pressure. Lancet 1978;1(8068):795 - 7. [19] Mitchell P, Jakubowski J. Estimate of the maximum time interval between formation of cerebral aneurysm and rupture. J Neurol Neurosurg Psychiatry 2000;69(6):760 - 7. [20] Mitchell P, Jakubowski J. Failure testing cerebral arteries: are branch points weaker than unbranched vessels? Br J Neurosurg 2002;16(6): 578 - 82. [21] Molyneux A. International Subarachnoid Aneurysm Trial (ISAT) of neurosurgical clipping versus endovascular coiling in 2143 patients with ruptured intracranial aneurysms: a randomised trial. Lancet 2002;360(9342):1267 - 74. [22] Ostergaard JR, Reske-Nielsen E, Oxlund H. Histological and morphometric observations on the reticular fibers in the arterial beds of patients with ruptured intracranial saccular aneurysms. Neurosurgery 1987;20(4):554 - 8. [23] Pickard JD, Murray GD, Illingworth R, Shaw MD, Teasdale GM, Foy PM, Humphrey PM, Lang DA, Nelson R, Richards P, et al. Effect of oral nimodipine on cerebral infarction and outcome after subarachnoid hemorrhage: British aneurysm nimodipine trial. BMJ 1989;298(6674): 636 - 42. [24] Scanarini M, Mingrino S, Giordano R, Baroni A. Histological and ultrastructural study of intracranial saccular aneurysmal wall. Acta Neurochir (Wien) 1978;43(3-4):171 - 82. [25] Schlote W, Gaus C. Histologic aspects from ruptured and nonruptured aneurysms. Neurol Res 1994;16(1):59 - 62. [26] Slosberg PS. Nonoperative management of ruptured intracranial aneurysms. Clin Neurosurg 1974;21:90 - 9. [27] Slosberg PS. Zero percent mortality due to recurrent hemorrhage in follow-up of medically treated ruptured single intracranial aneurysms: a 23-year study. Trans Am Neurol Assoc 1979;104: 180 - 3. [28] Slosberg PS. Unexpected results in long-term medically treated ruptured intracranial aneurysm including data on 14 patients followed more than 30 years each. Acta Neurochir (Wien) 1997; 139(8):697 - 705. [29] Stehbens WE. Histopathology of cerebral aneurysms. Arch Neurol 1963;8:272 - 85.
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[30] Wang XT, Ker RF. Creep rupture of wallaby tail tendons. J Exp Biol 1995;198(3):831 - 45. [31] Wang XT, Ker RF, Alexander RM. Fatigue rupture of wallaby tail tendons. J Exp Biol 1995;198(3):847 - 52. [32] Yong-Zhong G, van Alphen HA. Pathogenesis and histopathology of saccular aneurysms: review of the literature. Neurol Res 1990; 12(4):249 - 55.
Commentary The authors present a very interesting mathematical model that describes a biophysical mechanism that may explain the pathophysiology of a special group of aneurysms—those that grow and rupture quickly. The presence of this group of aneurysms best explains the apparent paradox that small aneurysms have very low rupture rates in
prospective studies yet make up the majority of ruptured aneurysms in large series. Although their property of rupturing soon after growth makes proof of the existence of these aneurysms difficult, perhaps widespread screening for intracranial aneurysms, which is increasingly being considered as a public health measure, will yield information regarding their mechanics. Some readers may criticize the highly theoretical nature of this study. However, mathematical models are useful in circumstances such as these where measurements of physical qualities are difficult. The authors should be commended for the original approach and imaginative analysis. Phillip S. Dickey, MD New Haven, CT 06510 USA