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Wall Stress Studies of Abdominal Aortic Aneurysm in a Clinical Model
Wall Stress Studies of Abdominal Aortic Aneurysm in a Clinical Model Mano J. Thubrikar, PhD, Jihad Al-Soudi, MS, and Francis Robicsek, MD, PhD, Charlotte, North Carolina
To estimate when an abdominal aortic aneurysm (AAA) may rupture, it is necessary to understand the forces responsible for this event. We investigated the wall stresses in an AAA in a clinical model. Using CT scans of the AAA, the diameter and wall thickness were measured and the model of the aneurysm was created. The wall stresses were determined using a finite element analysis in which the aorta was considered isotropic with linear material properties and was loaded with a pressure of 120 mmHg. The AAA was eccentric with a length of 10.5 cm, a diameter of 2.5 to 5.9 cm, and a wall thickness of 1.0 to 2.0 mm. The aneurysm had specific areas of high stress. On the inner surface the highest stress was 0.4 N/mm2 and occurred along two circumferentially oriented belts—one at the bulb and the other just below. The stress was longitudinal at the anterior region of the bulb and circumferential elsewhere, suggesting that a rupture caused by this stress will result in a circumferential tear at the anterior portion of the bulb and a longitudinal tear elsewhere. In the mid-surface the highest stress was 0.37 N/mm2 and occurred at two locations: the posterior region of the bulb and anteriorly just below. The stress was circumferential, suggesting that the rupture caused by this stress will produce a longitudinal tear. The location and orientation of the maximum stress were influenced more by the tethering force than by the wall thickness, luminal pressure, or wall stiffness. In conclusion, the rupture of an AAA is most likely to occur on the inner surface at the bulb. Such analytical approaches could lead to a better understanding of the aneurysm rupture and may be instrumental in planning surgical interventions.
INTRODUCTION The timing of surgical intervention for an abdominal aortic aneurysm (AAA) continues to pose a challenge. Although presence of an AAA of a 7-cm or greater diameter is generally regarded as indication for surgery, aneurysms of smaller diameter also
The Heineman Medical Research Laboratory, Carolinas Heart Institute, Carolinas Medical Center, Charlotte, NC. Correspondence to: M.J. Thubrikar, PhD, Heineman Medical Research Laboratory, Carolinas Medical Center, 1000 Blythe Boulevard, Charlotte, NC 28203, USA, E-mail: [email protected]. Ann Vasc Surg 2001; 15: 355-366 DOI: 10.1007/s100160010080 © Annals of Vascular Surgery Inc. Published online: April 26, 2001
rupture, though less frequently. While the rupture of an AAA carries a very high mortality, surgical intervention is also associated with a significant risk. The search thus continues for the best assessment of which aneurysm is likely to rupture and when. Since the rupture occurs when the stress in the wall exceeds its strength, the more we know about this stress the closer we come to predicting the risk of rupture. In reviews on pathophysiology and surgical treatment of an AAA,1-4 it has been established that the risk of rupture increases with the size of the aneurysm.5-8 Darling et al. reported that for an AAA of diameter <4, 4-5, 5-7, and 7-10 cm, the frequency of rupture was 8%, 25%, 50%, and 64%, respectively.5 Scott et al.8 reported that for an AAA of diameter 3-4.4 cm and 4.5-5.9 cm, the ac355
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tual rupture plus the elective surgery rate was, respectively, 2.1%/year and 10.2%/year. Such studies have led to the use of the diameter as the main criterion for judging the necessity of surgical intervention in asymptomatic AAAs. Although an AAA of 7 cm or greater diameter is considered a definite candidate for operation by most surgeons, Cronenwett et al.7 suggested that elective repair should be carried out when the diameter reaches 5-6 cm, whereas Scott et al.8 suggested that for patients with aneurysms between 4.5 and 6 cm, elective surgery should be considered only if symptoms develop. Following the above guidelines leaves a large number of aneurysms of 4-6 cm in diameter untreated, and unfortunately a significant number of these do rupture. The challenge is to identify those 4- to 6-cm aneurysms that carry a high risk of rupture, so as to avoid immediate death or the need of emergency repair. To better understand the chances of rupture, several authors have investigated parameters other than aneurysm diameter, such as the rate of aneurysm expansion.9,10 and the presence of thrombus.3,6,9,11-15 A faster increase in diameter is considered to indicate a higher risk of rupture. Wolf et al.9 describe a higher rate of expansion, >0.5 cm/ year, in 19% of AAA cases and also suggest that the “arc of thrombus” is one of the predictors of rapid expansion. Scott et al.8 regarded a “growth” rate of ⱖ 1 cm/year for AAAs as one of the principal indicators for operation. Compared to enlargement, the role of thrombus in the rupture of an AAA has remained controversial. Dobrin3 suggested that mural thrombus in the aneurysm neither reduces the luminal pressure exerted on the wall nor offers a retractive force and thus it has no effect on the wall stress. Schurink et. al.14 reported that thrombus does not reduce the pressure near the aneurysmal wall and thus will not reduce the risk of aneurysm rupture. Contrary to this, Mower et al.12 stated that thrombus indeed reduces wall stress by decreasing the diameter of the effective lumen. On the basis of numerical analysis, Di Martino et al.15 predicted that thrombus reduces the effect of the pressure load on the aneurysmal wall. According to Satta et al.11 thrombus thickness is greater in ruptured (3.5 cm) than in expanding (2.0 cm) AAAs. Pillari et al.6 found that for aneurysms >7 cm, the increase in sac diameter was associated with an increase in lumen diameter without any change in the thrombus volume; in contrast, for an AAA of <5 to 7 cm, the increase in sac diameter was associated with the increase in
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thrombus volume without any change in the lumen diameter. Stress analysis of the aneurysm through the use of models has been carried out by us as well as by others and has provided insight into this problem.16-22 In the present study, we have created an aneurysm model based on computed tomography (CT) scans of one clinical case of an AAA. This model represents the geometry in vivo. Wall stress was determined using finite element analysis (FEA), a method previously applied by us 20,23,24 and others16,17,19,22 to investigate similar problems.
MATERIALS AND METHODS Stresses in the aneurysm wall produced by internal pressure were determined using FEA. This required the following information: geometry of the aneurysm, material properties of the aneurysm wall, internal pressure, and boundary conditions. Geometry Several CT scans of an AAA of one clinical case under investigation were obtained. The patient was an 80-year-old male, 5 ft 6 in. in height and weighing 150 lbs. His aneurysm occupied most of the infrarenal region and extended close to the aortic bifurcation. From these scans 11 sections, each 1 cm apart, were selected (Fig. 1). These sections were enlarged and used for the measurement of the diameter, wall thickness, and distance of the center of each cross section from the examining table. Whenever a section was not perfectly circular, an average diameter was used for that section. This approximation is justified because the cross sections were almost circular. To define the geometry more accurately, 11 additional cross sections were created and placed alternately with the measured sections so that, in the composite model, each section was only 5 mm apart. Of the 11 additionally created cross sections, 10 had dimensions that were average of the two adjacent measured cross sections. The bottom one had dimensions assigned to produce a smooth geometry. These 22 cross sections were connected to create the surface of the aneurysm. Thus, the surface of the AAA was made up of 21 circumferential belts or areas, each bordered by two cross sections (Fig. 2). The values of the wall thickness and radius of various cross sections are shown in Figure 2 and Table I. Figure 2 shows the midsurface (half the wall thickness is added on each side of this surface for analysis) of the aneurysm. The presence of the clot in the aneurysm (Fig. 1) is ignored, primarily because our own (unreported)
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Fig. 1. Selected CT scans (9 of 11) from a patient showing an AAA. Top row, left to right, then the middle and bottom rows represent the aorta from proximal to distal end. The presence of the aneurysm is easily established as there is a substantial increase in the cross-sectional area of the
aorta. The aorta appears fairly circular. The thickness of the aortic wall can be seen clearly in some cross sections. Thrombus within the aneurysm can also be seen. All scans are 1 cm apart except scans 1, 2, and 3 (top row); they are, respectively, 0.5 and 2.5 cm apart.
data suggest that the clot does not reduce wall stress significantly.
Internal Pressure
Material Properties The aneurysmal aorta was assumed to be homogeneous and isotropic with linear elastic material properties. The modulus of elasticity used was 4.66 N/mm2 with a Poisson’s ratio of 0.49. Although human arterial tissue acts like nonlinear material, above a pressure load of 80 mmHg, the aorta behaves more like a linearly elastic material.25 The value of the elastic modulus used in the present study, suggested by our preliminary studies on the human aneurysmal aorta, is the larger number in the range reported for the normal aorta,23,25-28 and it is closer to the values reported for the aneurysmal aorta.27
An internal pressure of 120 mmHg (0.016 N/mm2, systolic) was applied to the aneurysm. To account for the tethering force that allows the aneurysm to increase in length, the aneurysm was considered to have a closed-end geometry. A force equivalent to the pressure acting on the closed end was applied to the lower end. This force mimics the tethering force on the aneurysm in an approximate sense since the exact value is unknown. To further understand the influence of this force, another model was created in which an additional tethering force was applied to the aneurysm. Boundary Conditions The model was constrained at the upper end while the bottom end was allowed to move in the longi-
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Table I. Dimensions of the aneurysm
Fig. 2. Model of the aortic aneurysm showing values of wall thickness and radius of cross section at various levels. The wall thickness is kept constant between two adjacent cross sections. Eccentricity of the aneurysm is obvious. The bottom represents the distal end near the aortic bifurcation.
Ring no.
Radius (mm)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
12.4 13.3 15 18.8 20.5 21.9 23.5 25 26.8 28.8 29.3 29.3 28.9 27.8 26.8 25.5 23.8 22 20 17 15 14.6
Measured thickness (mm)
1.1 1.2 1.4 1.7 1.9 2.0 1.4 1.2 1.0 1.0 1.0 1.1 1.1 1.1 1.2 1.1 1.4 1.6 1.6 1.5 1.3
t/R (%)
Thickness 1.31 mm t/R (%)
Thickness 1.58 mm t/R (%)
9 9 9 9 9 9 6 5 3.7 3.5 3.4 3.8 3.8 4 4.5 4.3 6 7.3 8 8.8 8.7
10.6 9.9 8.7 6.9 6.4 6 5.6 5.2 4.9 4.6 4.5 4.5 4.5 4.7 4.9 5.1 5.5 5.9 6.6 7.7 8.7
12.8 11.9 10.5 8.4 7.7 7.2 6.7 6.3 5.9 5.5 5.4 5.4 5.5 5.7 5.9 6.2 6.7 7.2 7.9 9.3 10.5
R, radius; t, thickness. Rings are 5 mm apart.
tudinal (Z) direction—i.e., the aneurysm could increase in length. Due to symmetry, only half of the model was analyzed, and therefore both the edges had to be restricted to prevent out-of-plane rotation. One edge was fixed in two directions (X and Y) but allowed to move longitudinally (Z), whereas another edge was fixed only in one direction (X) and allowed to move in the other two directions (Y and Z)—i.e., the aneurysm was allowed to increase also in diameter.
to simulate pressure at the closed end of the aneurysm. Nonlinear analysis, which allows for geometric nonlinearity and step-wise loading, was carried out. Results were obtained in terms of parameters such as wall stress on the inner and middle surface of the aneurysm, equivalent stress, direction of stress, deformations, and displacements. Parametric Studies
Mesh Generation and Analysis For the FEA we used the ANSYS 5.3 program (ANSYS, Inc., Houston, PA). The geometric parameters shown in Figure 2 were used to create a model of half of the aneurysm (Fig. 3). Mesh size was varied and a finer mesh was used in the bulb region (Fig. 4). Bulb is defined as the region of the greatest diameter. 2-D Shell element type 63, of both quadrilateral and triangular shapes, was used with a total of 3950 elements. Each belt was assigned a thickness according to the measured value. An internal pressure of 120 mmHg was applied. Also, a force at each of the nodes on the bottom edge was applied
Several parameters, including modulus of elasticity, local variation in the wall thickness, and tethering force, still remain unknown in vivo. Therefore, it became necessary to determine how they might influence the results. We thus carried out the following parametric studies. (1) We determined how the wall thickness might affect the results by creating two other models in which wall thicknesses of 1.31 and 1.58 mm were used for the entire aneurysm. The choice of these two thicknesses is explained in the Results section. (2) Since, in a patient, blood pressure does vary, we applied a higher internal pressure of 160 mmHg and determined the stresses. (3) Since the aneurysmal aorta is reported to be
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Fig. 3. Various views of model of half of aneurysm considering the symmetry: 1, right lateral view; 2, posteroanterior view (the aneurysm is considered symmetric along the posteroanterior plane); 3, an oblique view.
“stiffer,”26,27 we determined the effect of a still higher modulus by creating another model in which the modulus was five times that used in the original model. (4) The effect of the tethering force was determined by increasing the length of the aneurysm by almost twice the amount seen in the analysis of the original model.
Fig. 4. Distribution of nodes and the mesh used in original undeformed geometry. Displacement of aneurysm geometry due to a pressure load of 120 mmHg is also shown. The pressure load causes predominantly circumferential distension.
RESULTS The aneurysm was fairly typical; it was eccentric, 10.5 cm long, about 6 cm in diameter, and had a wall thickness of 1.0-2.0 mm. The model of half of the aneurysm was analyzed since the aneurysm was fairly symmetric (Figs. 2 and 3). Model I (The Original Model): Measured Wall Thickness Figure 5 shows the contours of the first principal stress (maximum stress) on the inner surface and on the middle surface of the aortic wall. The maximum stress on the inner surface is located along two circumferential belts—one at and the other below the bulb (Fig. 5A). Maximum stress on the posterior side (location A) was 0.4 N/mm2, on the anterior side just below the bulb (location B) it was 0.3 N/mm2, and on the anterior side at the bulb (loca-
tion C) it was 0.4 N/mm2 (Figs. 5 and 6, Table II). The maximum stress on the middle surface of the wall occurred at two locations—one just above the bulb on the posterior wall and the other just below the bulb on the anterior wall (Fig. 5B). The maximum stress on the posterior wall (location A) was 0.37 N/mm2, on the anterior wall just below the bulb (location B) it was 0.37 N/mm2,and on the anterior wall at the bulb (location C), 0.24 N/mm2 (Figs. 5 and 6 Table II). The directions of these stresses are shown in Figure 6. On the inner surface, the direction of maximum stress was longitudinal at the anterior region of the bulb (location C) and circumferential at other locations (A and B). On the middle surface the direction of the maximum stress was similar, i.e., longitudinal at the bulb anteriorly (location C) and circumferential at other locations (A and B) (Fig. 6). It is notable that even
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Fig. 5. First principal stress (maximum stress) on inner surface A and middle surface B of aneurysmal aortic wall. Stress values are color coded and shown in insert in N/mm2. A On the inner surface, maximum stress occurs along two circumferentially oriented belts, one at the
though the stress anteriorly at the bulb (location C) was lower than that at the other two locations, it should be considered significant because of its longitudinal orientation (The undilated aorta has only half the stress in the longitudinal direction compared to that in the circumferential direction). Hence, in the aneurysm the actual increase in stress is likely to be the most at the anterior region of the bulb in the longitudinal direction. These findings suggest that if the wall of the aneurysm tears by the above-mentioned stresses, then the tear at the anterior region of the bulb (location C) will be oriented in the circumferential direction, whereas the tear elsewhere will be oriented in the longitudinal direction (Table III). In other words, if we knew how the tear was oriented we could determine which stress caused it. Equivalent stress on the inner and middle surface of the wall was also examined. This stress had a distribution pattern similar to that of the first principal stress (Fig. 5), but was slightly lower in magnitude. We also examined the second principal stress and judged it to be less important because of
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level of the bulb and the other just below the bulb. B Maximum stress occurs posteriorly at the bulb and anteriorly just below the bulb. Stress distribution is for luminal pressure of 120 mmHg.
Fig. 6. Orientation of maximum stress (first principal stress) on inner surface of aorta in three regions of interest. At locations A and B the maximum stress is oriented in the circumferential direction, while at location C it is oriented in the longitudinal direction.
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Table II. First principal stress in aneurysm wall (N/mm2) Posterior wall at bulba (location A) Model no.
Model parameter
1
Measured thickness (t) Constant thickness (t = 1.31 mm) Constant thickness (t = 1.58 mm) Luminal pressure (P = 160 mmHg) Elastic modulus (E = 5 Einitial) Displacement (Uz = 5 mm)
2 3 4 5 6 a
Anterior wall just below bulba (location B)a
Anterior wall at bulba (location C)a
Inner surface
Middle surface
Inner surface
Middle surface
Inner surface
Middle surface
0.4 ↔ 0.34
0.37 ↔ 0.34
0.3 ↔ 0.26
0.37 ↔ 0.32
0.4 ↕ 0.34
0.24 ↕ 0.18
0.27
0.28
0.23
0.26
0.27
0.15
0.49
0.48
0.39
0.48
0.49
0.31
0.44
0.41
0.36
0.41
0.44
0.24
0.65 ↕b
0.43 ↕b
0.4
0.36
0.5
0.33
Arrow indicates the direction of stress. Indicates the only place where the direction has changed.
b
Table III. Relationship between cause, location, and orientation of the tear Orientation of tear Model no.
1 2 3 4 5 6
Model parameter
Stress causing the tear
Posterior wall at bulb ruptures
Anterior wall below bulb ruptures
Anterior wall at bulb ruptures
Measured thickness (t) Constant thickness (t = 1.31 mm) Constant thickness (t = 1.58 mm) Luminal pressure (P = 160 mmHg) Elastic modulus (E = 5 Einitial) Displacement (Uz = 5 mm)
Inner surface Middle surface Inner surface Middle surface Inner surface Middle surface Inner surface Middle surface Inner surface Middle surface Inner surface Middle surface
Longitudinal Longitudinal Longitudinal Longitudinal Longitudinal Longitudinal Longitudinal Longitudinal Longitudinal Longitudinal Circumferential Circumferential
Longitudinal Longitudinal Longitudinal Longitudinal Longitudinal Longitudinal Longitudinal Longitudinal Longitudinal Longitudinal Longitudinal Longitudinal
Circumferential Circumferential Circumferential Circumferential Circumferential Circumferential Circumferential Circumferential Circumferential Circumferential Circumferential Circumferential
its lower magnitude. The second principal stress was oriented 90° to the first principal stress (Fig. 6). We determined the displacement of the aneurysm wall and found that when the pressure increased from 0 to 120 mmHg, the aneurysm increased in diameter by a maximum of 3 mm (maximum displacement) (Fig. 4). Models II and III: Wall Thickness 1.31 mm and 1.58 mm We also determined how the stresses might change if the wall thickness was greater than that mea-
sured. Two other thickness values, 1.31 mm (model II) and 1.58 mm (model III), were used for constant wall thickness for the entire aneurysm. The thickness 1.31 mm was chosen because it is close to the thickness of the undilated aorta at the ends of the aneurysm (Table I). The thickness 1.58 mm was selected because it gives a ratio t/R of 1/8 for the undilated segment, a ratio often used for the normal aorta (Table I).25 It may be noted that a constant thickness implies that as the aneurysm “grows” there is a buildup in the wall thickness to counterbalance the thinning that must accompany the increase in diameter.
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Fig. 7. Maximum stress (first principal stress) on inner surface A and middle surface B of aorta for model in which wall thickness of 1.31 mm is constant for entire
aneurysm. The stress distribution appears only slightly altered compared to that seen in the original model I (Fig. 5).
With the constant wall thickness of 1.31 mm (model II) the first principal stress on the inner surface at locations A, B, and C was 0.34, 0.26, and 0.34 N/mm2, respectively (Fig. 7, Table II). On the middle surface, the respective stresses were 0.34, 0.32, and 0.18 N/mm2 (Fig. 7, Table II). The pattern of stress distribution and the orientation of the stress were reasonably similar to those in the original model I (Figs. 5 and 6). The equivalent stress and the second principal stress were also determined but then considered less important because of their lower magnitude. The maximum displacement was 2.82 mm. With the wall thickness of 1.58 mm (model III), the first principal stress on the inner surface at locations A, B, and C was 0.27, 0.23, and 0.27 N/mm2, respectively, and on the middle surface was 0.28, 0.26, and 0.15 N/mm2, respectively (Table II). The pattern of stress distribution and the orientation of the stress were similar to those in the original model I (Figs. 5-7). The maximum displacement was 2.38 mm.
Model IV: Intraluminal Pressure of 160 mmHg Because AAA patients are often hypertensive, stresses were also determined at an internal pressure load of 160 mmHg (conservative estimate) in the initial model I. In this case, the bottom edge of the model was loaded with a force proportional to an internal pressure of 160 mmHg. The first principal stress (maximum stress) on the inner surface at locations A, B, and C was 0.49, 0.39, and 0.49 N/mm2, respectively (Table II). The stress on the middle surface was 0.48, 0.48, and 0.31 N/mm2, respectively. As expected, the rise in pressure caused all of the stresses to increase. The pattern of stress distribution and the orientation of the stress were similar to those in the original model I (Figs. 5 and 6). The maximum displacement was 3.83 mm. Model V: Increased Wall Stiffness The wall of the aneurysmal aorta is stiffer26,27,28,29 and when it is expressed in terms of pressure-
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Fig. 8. Maximum stress (first principal stress) on inner surface A and middle surface B of aorta for model in which luminal pressure and tethering force, which causes
5 mm increase in length, are applied. The stress distribution and orientation are altered substantially compared to that in original model I (Figs. 5 and 6).
diameter relationships, the stiffness may be greater by a factor of 2 to 4.28,29 We studied the effect of stiffness by increasing the modulus five times (E = 5Eo). The first principal stress on the inner surface at locations A, B, and C was 0.44, 0.36, and 0.44 N/mm2, respectively, and on the middle surface it was 0.41, 0.41, and 0.24 N/mm2, respectively (Table II)—i.e., the stresses were slightly increased. Once again, the pattern of stress distribution and the orientation of the stress were similar to those in the original model I (Figs. 5 and 6). The maximum displacement was 0.89 mm. The increased stiffness had more influence on the displacement than on the stresses.
amount observed in the aneurysm, which lengthens freely under pressure (model I). Working with the original model, we now applied both the internal pressure of 120 mmHg and an axial displacement of 5 mm to the bottom edge. The results were as follows. The areas of high stresses shifted to the posterior wall of the aneurysm (Fig. 8). The first principal stress on the inner surface at locations A, B, and C was 0.65, 0.4, and 0.5 N/mm2, respectively, and on the middle surface was 0.43, 0.36, and 0.33 N/mm2, respectively (Table II). All of the stresses were increased considerably and both the pattern of stress distribution (Fig. 8) and the orientation of stress (Table II) were altered. The orientation of the first principal stress on both the inner and the middle surface on the posterior wall, at location A, changed and became longitudinal. For instance, in Figure 6 at location A, the three bottom arrows appear vertically oriented. The orientation of the stresses just below the bulb (location B) and at the anterior region of the bulb (location C), however, remain the same as that in the original model (model I). The maximum displacement increased to 5.27 mm and
Model VI: Effect of Pull (Tethering Force) The aorta may be under a greater pull along the length due to tethering than what may occur by free expansion under internal pressure. Therefore, we determined stresses under conditions in which the aorta is pulled down by 5 mm along its length. This displacement is approximately twice the
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was along the long axis. These results show that if the tear on the posterior wall was oriented in the circumferential direction, then it would have been caused by the forces where the tethering force was dominant (Table III).
DISCUSSION Justification of the Model Our model is the first in which the geometry of an actual AAA of one clinical case is used, to account for true eccentricity, evaluate the effect of various parameters, and map the details of stress distribution throughout the aneurysm. The present model, therefore, is an advancement over those in which axisymmetric geometry has been used.16-20 The aneurysmal aorta was considered isotropic, homogeneous, and incompressible; these assumptions have also been made by others.16-20 The studies of Raghavan et al.27 support the postulate that the aneurysmal aortic tissue is isotropic. Although the material properties of the AAA are nonlinear,26,27 at a pressure load of 60 mmHg or greater, they are fairly linear. Since the geometry of the aorta used in the present study is that which existed in vivo, i.e., at a systemic pressure of the patient, at this geometry the AAA wall behaves almost linearly. Others have used similar assumptions in their analysis.16-19
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As models II and III show, the increase in wall thickness reduces the stress but changes neither the distribution pattern nor the orientation of the stress (Table II). An increase in intraluminal pressure raises the wall stress, while an increase in the wall stiffness raises the wall stress only minimally. Hence, the stress depends more on wall thickness and luminal pressure than on the material properties. This is advantageous to patients because the material properties of the AAA are not known. The change in the aneurysm diameter predicted by the model is also of interest, since pulsatile wall motion has been measured in patients.31 The model predicts that the aneurysm diameter increases by 0.83 mm when the pressure in the aneurysm increases from 120 mmHg (model I, maximum displacement 3.0 mm) to 160 mmHg (model IV, maximum displacement 3.83 mm). Malina et al.33 reported that in 47 patients pulsatile wall motion of the aneurysm was on average 1.0 mm. Considering that the pressure change associated with this motion is not known, the reported wall motion is close to that predicted by the model. This further shows the validity of the model. The tethering force does affect the stress distribution and the orientation. If a substantial tethering force is acting on the AAA, the tear in the posterior wall should be oriented circumferentially, otherwise it will be oriented longitudinally (Table II).
Wall Stress
Significance of Wall Stress
In our model of in vivo AAA geometry, we found that the maximum wall stress should be considered important in three regions: on the posterior and anterior wall of the bulb and on the anterior wall just below the bulb (Figs. 5 and 6, Table II). Mower et al.19 reported maximum stress at the bulb to be 0.057 N/mm2 circumferentially and 0.016 N/mm2 longitudinally. Inzoli et al.17 found the maximum stress to be 0.2 N/mm2 and Stringfellow et al.16 reported the maximum circumferential stress to be 0.11 N/mm2 and the longitudinal stress to be 0.08 N/mm2. Both16,17 reported that maximum stress occurred near the neck of the aneurysm where the dilated aorta meets the undilated segment. Mower et al.19 pointed out, however, that ruptures often do not occur at this location according to the report by Darling et al.5 Stress values in the bulb reported by us are higher than those reported by Mower et al.,19 mainly because of differences in the geometry and wall thickness used in the two studies. Overall, the magnitude and distribution of wall stress observed in the present study were comparable to those reported by Vorp et al.30 and Raghavan et al.22 for eccentric aneurysm models.
The present study explains which stresses may be responsible for the AAA rupture. First, the stress on the inner wall is greater than that on the middle wall and therefore the tear has a predilection to the inner surface. Second, the highest stresses occur at the bulb and thus the rupture is likely to begin here. Third, the rupture in the posterior wall region (location A) will produce a longitudinal tear whereas that in the anterior wall will produce a longitudinal tear below (location B) and a circumferential tear at (location C) the bulb (Fig. 6, Table III). This location and orientation of the tear is not likely to be altered by a change in wall thickness, luminal pressure, or wall stiffness (Tables II and III). Also, the orientation of the tear is the same, whether it is caused by the stress on the inner surface or by the stress on the middle surface (Table III). If the tear on the posterior wall is circumferential, then it must have occurred because of a significant tethering force (Tables II and III). Yield Stress and Location of Rupture Sites The results of the present study predict the likely location and orientation of the tear in the aneu-
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rysm. The aortic wall starts to yield when the stress reaches the yield stress. Raghavan et al.27 reported the yield stress to be 0.65 ± 0.09 N/mm2 in the longitudinal direction and 0.707 ± 0.12 N/mm2 in the circumferential direction for the anterior wall of the AAA. Our own unreported data suggest slightly lower values. The stress obtained in the present study is lower than the yield stress and that is why the aneurysm has not ruptured in the patient. However, if we consider additional parameters such as localized thinning, calcification, buckling, or weakening in the aneurysm, then the stress could equal or even exceed the yield stress. Thus, the present results can be used to determine the probability of rupture of an AAA on the basis of wall stress. Darling et al.5 reported that 62% of ruptures occur near the maximum diameter (bulb) region, 34% at the midpoint, and only 4% at the junction of the dilated and undilated aorta. Our results predict that, in the absence of any localized abnormality, almost all of the ruptures should occur in the bulb region. Hence, our results agree with the rupture locations reported by Darling et al. in a vast majority of cases.5 Our results also predict the orientation of the tear, however, there are no published reports on the tear orientation in the AAA for comparison. In the aortic dissection, our previous studies did suggest that the intimal tear is oriented orthogonal to the most rapidly increasing wall stress.20, 32,33
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CONCLUSIONS This study describes the wall stress in the AAA of one clinical case and identifies the locations and orientation of maximum stress. The stress is highest at the bulb. The stress is elevated in the posterior and anterior region of the bulb, and in the anterior region below the bulb. The orientation of the stress suggests that the tear is most likely to be circumferential at the anterior region of the bulb and longitudinal elsewhere. The parametric studies suggest that these results could be applied to a broader population of patients. These types of studies could lead to the determination of probability of rupture of an AAA and could assist in deciding the timing of surgical intervention.
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eurysms. Current concept. Surg Clin North Am 1989;69: 687-703. Wills A, Thompson MM, Crowther M, Sayers RD, Bell PRF. Pathogenesis of abdominal aortic aneurysms—cellular and biochemical mechanisms. Eur J Vasc Endovasc Surg 1996; 12:391-400. Darling RC, Messina CR, Brewster DC, Ottinger LW. Autopsy study of unoperated abdominal aortic aneurysms. Circulation 1977;56 (Suppl 2):161-164. Pillari G, Chang JB, Zito J. Computed tomography of abdominal aortic aneurysm. Arch Surg 1988;123:727-732. Cronenwett JL, Sargent SK, Wall MH. Variables that affect the expansion rate and outcome of small abdominal aortic aneurysms. J Vasc Surg 1990;11:260-269. Scott RAP, Tisi PV, Ashton HA, Allen DR. Abdominal aortic aneurysm rupture rates: a 7-year follow-up of the entire abdominal aortic aneurysm population detected by screening. J Vasc Surg 1998;28:124-128. Wolf YG, Thomas WS, Brennan FJ, Goff WG, Sise MJ, Bernstein EF. Computed tomography scanning findings associated with rapid expansion of abdominal aortic aneurysms. J Vasc Surg 1994;20:529-538. Schneiderman J, Bordin GM, Engelberg I. Expression of fibrinolytic genes in atherosclerotic abdominal aortic aneurysms wall. A possible mechanism for aneurysm expansion. J Clin Invest 1995;96:639-645. Satta J, Laara E, Juvonen T. Intraluminal thrombus predicts rupture of an abdominal aortic aneurysm. Letter to the editor. J Vasc Surg 1996;23:737-739. Mower WR, Quin˜ones WJ, Gambhir SS. Effect of intraluminal thrombus on abdominal aortic aneurysm wall stress. J Vasc Surg 1997;26:602-608. Vorp DA, Mandarino WA, Webster MW, Gorcsan III J. Potential influence of intraluminal thrombus on abdominal aortic aneurysm as assessed by a new non-invasive method. Cardiovasc Surg 1996;4:732-739. Schurink GWH, van Baalen JM, Visser MJT, van Bockel JH. Thrombus within an aortic aneurysm does not reduce pressure on the aneurysmal wall. J Vasc Surg 2000;31:501-506. Di Martino E, Mantero S, Inzoli F. Biomechanics of abdominal aortic aneurysm in the presence of endoluminal thrombus: experimental characterisation and structural static computational analysis. Eur J Vasc Endovasc Surg 1998;15:290299. Stringfellow MM, Lawrence PF, Stringfellow RG. The influence of aorta-aneurysm geometry upon stress in the aneurysm wall. J Surg Res 1987;42:425-433. Inzoli F, Boschetti F, Zappa M, Longo T, Fumero R. Biomechanical factors in abdominal aortic aneurysm rupture. Eur J Vasc Surg 1993;7:667-674. Elger DF, Blackketter DM, Budwig RS, Johansen KH. The influence of shape on the stresses in model abdominal aortic aneurysms. J Biomech Eng 1996;118:326-332. Mower WR, Baraff LJ, Sneyd J. Stress distributions in vascular aneurysms: factors affecting risk of aneurysm rupture. J Surg Res 1993;55:155-161. Thubrikar MJ, Agali P, Robicsek F. Wall stress as a possible mechanism for the development of transverse intimal tears in aortic dissections. J Med Eng Technol 1999;23:17-134. Hall AJ, Busse EFG, McCarville DJ, Burgess JJ. Aortic wall tension as a predictive factor for abdominal aortic aneurysm rupture: improving the selection of patients for abdominal aortic aneurysm repair. Ann Vasc Surg 2000;14:152-157. Raghavan ML, Vorp DA, Federle MP, Makaroun MS, Webster MW. Wall stress distribution on three-dimentionally re-
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constructed models of human abdominal aortic aneurysm. J Vasc Surg 2000;31:760-769. Salzar RS, Thubrikar MJ, Eppink RT. Pressure-induced mechanical stress in the carotid artery bifurcation: a possible correlation to atherosclerosis. J Biomech 1995;28:13331340. Thubrikar MJ, Roskelley SK, Eppink RT. Study of stress concentration in the walls of the bovine coronary arterial branch. J Biomech 1996;23:15-26. McDonald, DA. The elastic properties of the arterial wall. In: Blood Flow in Arteries, 2nd ed. Baltimore: Williams & Wilkins, 1974, pp 238-282. He CM, Roach MR. The composition and mechanical properties of abdominal aortic aneurysms. J Vasc Surg 1994;20: 6-13. Raghavan ML, Webster MW, Vorp DA. Ex vivo biomechanical behavior of abdominal aortic aneurysm: assessment using a new mathematical model. Ann Biomed Eng 1996; 24:573-582. MacSweeney STR, Young G, Greenhalgh RM, Powell JT.
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Mechanical properties of the aneurysmal aorta. Br J Surg 1992;79:1281-1284. Sonesson B, Hansen F, La¨nne T. Abdominal aortic aneurysm: a general defect in the vasculature with focal manifestations in the abdominal aorta? J Vasc Surg 1997;26:247254. Vorp DA, Raghavan ML, Webster MW. Mechanical wall stress in abdominal aortic aneurysm: influence of diameter and asymmetry. J Vasc Surg 1998;27:632-639. Malina M, La¨nne T, Ivancev K, Lindblad B, Brunkwall J. Reduced pulsatile wall motion of abdominal aortic aneurysms after endovascular repair. J Vasc Surg 1998;27:624631. Robicsek F, Thubrikar MJ. Hemodynamic considerations regarding the mechanism and prevention of aortic dissection. Ann Thorac Surg 1994;58:1247-1253. Robicsek, F, Thubrikar, M. The mechanism and prevention of aortic dissection in Marfan syndrome. In: Hetzer, R, Gehle, P, Ennker, J, eds. Cardiovascular Aspects of Marfan Syndrome. New York: Springer-Verlag, 1995, pp 61-70.
To estimate when an abdominal aortic aneurysm (AAA) may rupture, it is necessary to understand the forces responsible for this event. We investigated the wall stresses in an AAA in a clinical model. Using CT scans of the AAA, the diameter and wall thickness were measured and the model of the aneurysm was created. The wall stresses were determined using a finite element analysis in which the aorta was considered isotropic with linear material properties and was loaded with a pressure of 120 mmHg. The AAA was eccentric with a length of 10.5 cm, a diameter of 2.5 to 5.9 cm, and a wall thickness of 1.0 to 2.0 mm. The aneurysm had specific areas of high stress. On the inner surface the highest stress was 0.4 N/mm2 and occurred along two circumferentially oriented belts—one at the bulb and the other just below. The stress was longitudinal at the anterior region of the bulb and circumferential elsewhere, suggesting that a rupture caused by this stress will result in a circumferential tear at the anterior portion of the bulb and a longitudinal tear elsewhere. In the mid-surface the highest stress was 0.37 N/mm2 and occurred at two locations: the posterior region of the bulb and anteriorly just below. The stress was circumferential, suggesting that the rupture caused by this stress will produce a longitudinal tear. The location and orientation of the maximum stress were influenced more by the tethering force than by the wall thickness, luminal pressure, or wall stiffness. In conclusion, the rupture of an AAA is most likely to occur on the inner surface at the bulb. Such analytical approaches could lead to a better understanding of the aneurysm rupture and may be instrumental in planning surgical interventions.
INTRODUCTION The timing of surgical intervention for an abdominal aortic aneurysm (AAA) continues to pose a challenge. Although presence of an AAA of a 7-cm or greater diameter is generally regarded as indication for surgery, aneurysms of smaller diameter also
The Heineman Medical Research Laboratory, Carolinas Heart Institute, Carolinas Medical Center, Charlotte, NC. Correspondence to: M.J. Thubrikar, PhD, Heineman Medical Research Laboratory, Carolinas Medical Center, 1000 Blythe Boulevard, Charlotte, NC 28203, USA, E-mail: [email protected]. Ann Vasc Surg 2001; 15: 355-366 DOI: 10.1007/s100160010080 © Annals of Vascular Surgery Inc. Published online: April 26, 2001
rupture, though less frequently. While the rupture of an AAA carries a very high mortality, surgical intervention is also associated with a significant risk. The search thus continues for the best assessment of which aneurysm is likely to rupture and when. Since the rupture occurs when the stress in the wall exceeds its strength, the more we know about this stress the closer we come to predicting the risk of rupture. In reviews on pathophysiology and surgical treatment of an AAA,1-4 it has been established that the risk of rupture increases with the size of the aneurysm.5-8 Darling et al. reported that for an AAA of diameter <4, 4-5, 5-7, and 7-10 cm, the frequency of rupture was 8%, 25%, 50%, and 64%, respectively.5 Scott et al.8 reported that for an AAA of diameter 3-4.4 cm and 4.5-5.9 cm, the ac355
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tual rupture plus the elective surgery rate was, respectively, 2.1%/year and 10.2%/year. Such studies have led to the use of the diameter as the main criterion for judging the necessity of surgical intervention in asymptomatic AAAs. Although an AAA of 7 cm or greater diameter is considered a definite candidate for operation by most surgeons, Cronenwett et al.7 suggested that elective repair should be carried out when the diameter reaches 5-6 cm, whereas Scott et al.8 suggested that for patients with aneurysms between 4.5 and 6 cm, elective surgery should be considered only if symptoms develop. Following the above guidelines leaves a large number of aneurysms of 4-6 cm in diameter untreated, and unfortunately a significant number of these do rupture. The challenge is to identify those 4- to 6-cm aneurysms that carry a high risk of rupture, so as to avoid immediate death or the need of emergency repair. To better understand the chances of rupture, several authors have investigated parameters other than aneurysm diameter, such as the rate of aneurysm expansion.9,10 and the presence of thrombus.3,6,9,11-15 A faster increase in diameter is considered to indicate a higher risk of rupture. Wolf et al.9 describe a higher rate of expansion, >0.5 cm/ year, in 19% of AAA cases and also suggest that the “arc of thrombus” is one of the predictors of rapid expansion. Scott et al.8 regarded a “growth” rate of ⱖ 1 cm/year for AAAs as one of the principal indicators for operation. Compared to enlargement, the role of thrombus in the rupture of an AAA has remained controversial. Dobrin3 suggested that mural thrombus in the aneurysm neither reduces the luminal pressure exerted on the wall nor offers a retractive force and thus it has no effect on the wall stress. Schurink et. al.14 reported that thrombus does not reduce the pressure near the aneurysmal wall and thus will not reduce the risk of aneurysm rupture. Contrary to this, Mower et al.12 stated that thrombus indeed reduces wall stress by decreasing the diameter of the effective lumen. On the basis of numerical analysis, Di Martino et al.15 predicted that thrombus reduces the effect of the pressure load on the aneurysmal wall. According to Satta et al.11 thrombus thickness is greater in ruptured (3.5 cm) than in expanding (2.0 cm) AAAs. Pillari et al.6 found that for aneurysms >7 cm, the increase in sac diameter was associated with an increase in lumen diameter without any change in the thrombus volume; in contrast, for an AAA of <5 to 7 cm, the increase in sac diameter was associated with the increase in
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thrombus volume without any change in the lumen diameter. Stress analysis of the aneurysm through the use of models has been carried out by us as well as by others and has provided insight into this problem.16-22 In the present study, we have created an aneurysm model based on computed tomography (CT) scans of one clinical case of an AAA. This model represents the geometry in vivo. Wall stress was determined using finite element analysis (FEA), a method previously applied by us 20,23,24 and others16,17,19,22 to investigate similar problems.
MATERIALS AND METHODS Stresses in the aneurysm wall produced by internal pressure were determined using FEA. This required the following information: geometry of the aneurysm, material properties of the aneurysm wall, internal pressure, and boundary conditions. Geometry Several CT scans of an AAA of one clinical case under investigation were obtained. The patient was an 80-year-old male, 5 ft 6 in. in height and weighing 150 lbs. His aneurysm occupied most of the infrarenal region and extended close to the aortic bifurcation. From these scans 11 sections, each 1 cm apart, were selected (Fig. 1). These sections were enlarged and used for the measurement of the diameter, wall thickness, and distance of the center of each cross section from the examining table. Whenever a section was not perfectly circular, an average diameter was used for that section. This approximation is justified because the cross sections were almost circular. To define the geometry more accurately, 11 additional cross sections were created and placed alternately with the measured sections so that, in the composite model, each section was only 5 mm apart. Of the 11 additionally created cross sections, 10 had dimensions that were average of the two adjacent measured cross sections. The bottom one had dimensions assigned to produce a smooth geometry. These 22 cross sections were connected to create the surface of the aneurysm. Thus, the surface of the AAA was made up of 21 circumferential belts or areas, each bordered by two cross sections (Fig. 2). The values of the wall thickness and radius of various cross sections are shown in Figure 2 and Table I. Figure 2 shows the midsurface (half the wall thickness is added on each side of this surface for analysis) of the aneurysm. The presence of the clot in the aneurysm (Fig. 1) is ignored, primarily because our own (unreported)
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Fig. 1. Selected CT scans (9 of 11) from a patient showing an AAA. Top row, left to right, then the middle and bottom rows represent the aorta from proximal to distal end. The presence of the aneurysm is easily established as there is a substantial increase in the cross-sectional area of the
aorta. The aorta appears fairly circular. The thickness of the aortic wall can be seen clearly in some cross sections. Thrombus within the aneurysm can also be seen. All scans are 1 cm apart except scans 1, 2, and 3 (top row); they are, respectively, 0.5 and 2.5 cm apart.
data suggest that the clot does not reduce wall stress significantly.
Internal Pressure
Material Properties The aneurysmal aorta was assumed to be homogeneous and isotropic with linear elastic material properties. The modulus of elasticity used was 4.66 N/mm2 with a Poisson’s ratio of 0.49. Although human arterial tissue acts like nonlinear material, above a pressure load of 80 mmHg, the aorta behaves more like a linearly elastic material.25 The value of the elastic modulus used in the present study, suggested by our preliminary studies on the human aneurysmal aorta, is the larger number in the range reported for the normal aorta,23,25-28 and it is closer to the values reported for the aneurysmal aorta.27
An internal pressure of 120 mmHg (0.016 N/mm2, systolic) was applied to the aneurysm. To account for the tethering force that allows the aneurysm to increase in length, the aneurysm was considered to have a closed-end geometry. A force equivalent to the pressure acting on the closed end was applied to the lower end. This force mimics the tethering force on the aneurysm in an approximate sense since the exact value is unknown. To further understand the influence of this force, another model was created in which an additional tethering force was applied to the aneurysm. Boundary Conditions The model was constrained at the upper end while the bottom end was allowed to move in the longi-
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Table I. Dimensions of the aneurysm
Fig. 2. Model of the aortic aneurysm showing values of wall thickness and radius of cross section at various levels. The wall thickness is kept constant between two adjacent cross sections. Eccentricity of the aneurysm is obvious. The bottom represents the distal end near the aortic bifurcation.
Ring no.
Radius (mm)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
12.4 13.3 15 18.8 20.5 21.9 23.5 25 26.8 28.8 29.3 29.3 28.9 27.8 26.8 25.5 23.8 22 20 17 15 14.6
Measured thickness (mm)
1.1 1.2 1.4 1.7 1.9 2.0 1.4 1.2 1.0 1.0 1.0 1.1 1.1 1.1 1.2 1.1 1.4 1.6 1.6 1.5 1.3
t/R (%)
Thickness 1.31 mm t/R (%)
Thickness 1.58 mm t/R (%)
9 9 9 9 9 9 6 5 3.7 3.5 3.4 3.8 3.8 4 4.5 4.3 6 7.3 8 8.8 8.7
10.6 9.9 8.7 6.9 6.4 6 5.6 5.2 4.9 4.6 4.5 4.5 4.5 4.7 4.9 5.1 5.5 5.9 6.6 7.7 8.7
12.8 11.9 10.5 8.4 7.7 7.2 6.7 6.3 5.9 5.5 5.4 5.4 5.5 5.7 5.9 6.2 6.7 7.2 7.9 9.3 10.5
R, radius; t, thickness. Rings are 5 mm apart.
tudinal (Z) direction—i.e., the aneurysm could increase in length. Due to symmetry, only half of the model was analyzed, and therefore both the edges had to be restricted to prevent out-of-plane rotation. One edge was fixed in two directions (X and Y) but allowed to move longitudinally (Z), whereas another edge was fixed only in one direction (X) and allowed to move in the other two directions (Y and Z)—i.e., the aneurysm was allowed to increase also in diameter.
to simulate pressure at the closed end of the aneurysm. Nonlinear analysis, which allows for geometric nonlinearity and step-wise loading, was carried out. Results were obtained in terms of parameters such as wall stress on the inner and middle surface of the aneurysm, equivalent stress, direction of stress, deformations, and displacements. Parametric Studies
Mesh Generation and Analysis For the FEA we used the ANSYS 5.3 program (ANSYS, Inc., Houston, PA). The geometric parameters shown in Figure 2 were used to create a model of half of the aneurysm (Fig. 3). Mesh size was varied and a finer mesh was used in the bulb region (Fig. 4). Bulb is defined as the region of the greatest diameter. 2-D Shell element type 63, of both quadrilateral and triangular shapes, was used with a total of 3950 elements. Each belt was assigned a thickness according to the measured value. An internal pressure of 120 mmHg was applied. Also, a force at each of the nodes on the bottom edge was applied
Several parameters, including modulus of elasticity, local variation in the wall thickness, and tethering force, still remain unknown in vivo. Therefore, it became necessary to determine how they might influence the results. We thus carried out the following parametric studies. (1) We determined how the wall thickness might affect the results by creating two other models in which wall thicknesses of 1.31 and 1.58 mm were used for the entire aneurysm. The choice of these two thicknesses is explained in the Results section. (2) Since, in a patient, blood pressure does vary, we applied a higher internal pressure of 160 mmHg and determined the stresses. (3) Since the aneurysmal aorta is reported to be
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Fig. 3. Various views of model of half of aneurysm considering the symmetry: 1, right lateral view; 2, posteroanterior view (the aneurysm is considered symmetric along the posteroanterior plane); 3, an oblique view.
“stiffer,”26,27 we determined the effect of a still higher modulus by creating another model in which the modulus was five times that used in the original model. (4) The effect of the tethering force was determined by increasing the length of the aneurysm by almost twice the amount seen in the analysis of the original model.
Fig. 4. Distribution of nodes and the mesh used in original undeformed geometry. Displacement of aneurysm geometry due to a pressure load of 120 mmHg is also shown. The pressure load causes predominantly circumferential distension.
RESULTS The aneurysm was fairly typical; it was eccentric, 10.5 cm long, about 6 cm in diameter, and had a wall thickness of 1.0-2.0 mm. The model of half of the aneurysm was analyzed since the aneurysm was fairly symmetric (Figs. 2 and 3). Model I (The Original Model): Measured Wall Thickness Figure 5 shows the contours of the first principal stress (maximum stress) on the inner surface and on the middle surface of the aortic wall. The maximum stress on the inner surface is located along two circumferential belts—one at and the other below the bulb (Fig. 5A). Maximum stress on the posterior side (location A) was 0.4 N/mm2, on the anterior side just below the bulb (location B) it was 0.3 N/mm2, and on the anterior side at the bulb (loca-
tion C) it was 0.4 N/mm2 (Figs. 5 and 6, Table II). The maximum stress on the middle surface of the wall occurred at two locations—one just above the bulb on the posterior wall and the other just below the bulb on the anterior wall (Fig. 5B). The maximum stress on the posterior wall (location A) was 0.37 N/mm2, on the anterior wall just below the bulb (location B) it was 0.37 N/mm2,and on the anterior wall at the bulb (location C), 0.24 N/mm2 (Figs. 5 and 6 Table II). The directions of these stresses are shown in Figure 6. On the inner surface, the direction of maximum stress was longitudinal at the anterior region of the bulb (location C) and circumferential at other locations (A and B). On the middle surface the direction of the maximum stress was similar, i.e., longitudinal at the bulb anteriorly (location C) and circumferential at other locations (A and B) (Fig. 6). It is notable that even
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Fig. 5. First principal stress (maximum stress) on inner surface A and middle surface B of aneurysmal aortic wall. Stress values are color coded and shown in insert in N/mm2. A On the inner surface, maximum stress occurs along two circumferentially oriented belts, one at the
though the stress anteriorly at the bulb (location C) was lower than that at the other two locations, it should be considered significant because of its longitudinal orientation (The undilated aorta has only half the stress in the longitudinal direction compared to that in the circumferential direction). Hence, in the aneurysm the actual increase in stress is likely to be the most at the anterior region of the bulb in the longitudinal direction. These findings suggest that if the wall of the aneurysm tears by the above-mentioned stresses, then the tear at the anterior region of the bulb (location C) will be oriented in the circumferential direction, whereas the tear elsewhere will be oriented in the longitudinal direction (Table III). In other words, if we knew how the tear was oriented we could determine which stress caused it. Equivalent stress on the inner and middle surface of the wall was also examined. This stress had a distribution pattern similar to that of the first principal stress (Fig. 5), but was slightly lower in magnitude. We also examined the second principal stress and judged it to be less important because of
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level of the bulb and the other just below the bulb. B Maximum stress occurs posteriorly at the bulb and anteriorly just below the bulb. Stress distribution is for luminal pressure of 120 mmHg.
Fig. 6. Orientation of maximum stress (first principal stress) on inner surface of aorta in three regions of interest. At locations A and B the maximum stress is oriented in the circumferential direction, while at location C it is oriented in the longitudinal direction.
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Table II. First principal stress in aneurysm wall (N/mm2) Posterior wall at bulba (location A) Model no.
Model parameter
1
Measured thickness (t) Constant thickness (t = 1.31 mm) Constant thickness (t = 1.58 mm) Luminal pressure (P = 160 mmHg) Elastic modulus (E = 5 Einitial) Displacement (Uz = 5 mm)
2 3 4 5 6 a
Anterior wall just below bulba (location B)a
Anterior wall at bulba (location C)a
Inner surface
Middle surface
Inner surface
Middle surface
Inner surface
Middle surface
0.4 ↔ 0.34
0.37 ↔ 0.34
0.3 ↔ 0.26
0.37 ↔ 0.32
0.4 ↕ 0.34
0.24 ↕ 0.18
0.27
0.28
0.23
0.26
0.27
0.15
0.49
0.48
0.39
0.48
0.49
0.31
0.44
0.41
0.36
0.41
0.44
0.24
0.65 ↕b
0.43 ↕b
0.4
0.36
0.5
0.33
Arrow indicates the direction of stress. Indicates the only place where the direction has changed.
b
Table III. Relationship between cause, location, and orientation of the tear Orientation of tear Model no.
1 2 3 4 5 6
Model parameter
Stress causing the tear
Posterior wall at bulb ruptures
Anterior wall below bulb ruptures
Anterior wall at bulb ruptures
Measured thickness (t) Constant thickness (t = 1.31 mm) Constant thickness (t = 1.58 mm) Luminal pressure (P = 160 mmHg) Elastic modulus (E = 5 Einitial) Displacement (Uz = 5 mm)
Inner surface Middle surface Inner surface Middle surface Inner surface Middle surface Inner surface Middle surface Inner surface Middle surface Inner surface Middle surface
Longitudinal Longitudinal Longitudinal Longitudinal Longitudinal Longitudinal Longitudinal Longitudinal Longitudinal Longitudinal Circumferential Circumferential
Longitudinal Longitudinal Longitudinal Longitudinal Longitudinal Longitudinal Longitudinal Longitudinal Longitudinal Longitudinal Longitudinal Longitudinal
Circumferential Circumferential Circumferential Circumferential Circumferential Circumferential Circumferential Circumferential Circumferential Circumferential Circumferential Circumferential
its lower magnitude. The second principal stress was oriented 90° to the first principal stress (Fig. 6). We determined the displacement of the aneurysm wall and found that when the pressure increased from 0 to 120 mmHg, the aneurysm increased in diameter by a maximum of 3 mm (maximum displacement) (Fig. 4). Models II and III: Wall Thickness 1.31 mm and 1.58 mm We also determined how the stresses might change if the wall thickness was greater than that mea-
sured. Two other thickness values, 1.31 mm (model II) and 1.58 mm (model III), were used for constant wall thickness for the entire aneurysm. The thickness 1.31 mm was chosen because it is close to the thickness of the undilated aorta at the ends of the aneurysm (Table I). The thickness 1.58 mm was selected because it gives a ratio t/R of 1/8 for the undilated segment, a ratio often used for the normal aorta (Table I).25 It may be noted that a constant thickness implies that as the aneurysm “grows” there is a buildup in the wall thickness to counterbalance the thinning that must accompany the increase in diameter.
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Fig. 7. Maximum stress (first principal stress) on inner surface A and middle surface B of aorta for model in which wall thickness of 1.31 mm is constant for entire
aneurysm. The stress distribution appears only slightly altered compared to that seen in the original model I (Fig. 5).
With the constant wall thickness of 1.31 mm (model II) the first principal stress on the inner surface at locations A, B, and C was 0.34, 0.26, and 0.34 N/mm2, respectively (Fig. 7, Table II). On the middle surface, the respective stresses were 0.34, 0.32, and 0.18 N/mm2 (Fig. 7, Table II). The pattern of stress distribution and the orientation of the stress were reasonably similar to those in the original model I (Figs. 5 and 6). The equivalent stress and the second principal stress were also determined but then considered less important because of their lower magnitude. The maximum displacement was 2.82 mm. With the wall thickness of 1.58 mm (model III), the first principal stress on the inner surface at locations A, B, and C was 0.27, 0.23, and 0.27 N/mm2, respectively, and on the middle surface was 0.28, 0.26, and 0.15 N/mm2, respectively (Table II). The pattern of stress distribution and the orientation of the stress were similar to those in the original model I (Figs. 5-7). The maximum displacement was 2.38 mm.
Model IV: Intraluminal Pressure of 160 mmHg Because AAA patients are often hypertensive, stresses were also determined at an internal pressure load of 160 mmHg (conservative estimate) in the initial model I. In this case, the bottom edge of the model was loaded with a force proportional to an internal pressure of 160 mmHg. The first principal stress (maximum stress) on the inner surface at locations A, B, and C was 0.49, 0.39, and 0.49 N/mm2, respectively (Table II). The stress on the middle surface was 0.48, 0.48, and 0.31 N/mm2, respectively. As expected, the rise in pressure caused all of the stresses to increase. The pattern of stress distribution and the orientation of the stress were similar to those in the original model I (Figs. 5 and 6). The maximum displacement was 3.83 mm. Model V: Increased Wall Stiffness The wall of the aneurysmal aorta is stiffer26,27,28,29 and when it is expressed in terms of pressure-
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Fig. 8. Maximum stress (first principal stress) on inner surface A and middle surface B of aorta for model in which luminal pressure and tethering force, which causes
5 mm increase in length, are applied. The stress distribution and orientation are altered substantially compared to that in original model I (Figs. 5 and 6).
diameter relationships, the stiffness may be greater by a factor of 2 to 4.28,29 We studied the effect of stiffness by increasing the modulus five times (E = 5Eo). The first principal stress on the inner surface at locations A, B, and C was 0.44, 0.36, and 0.44 N/mm2, respectively, and on the middle surface it was 0.41, 0.41, and 0.24 N/mm2, respectively (Table II)—i.e., the stresses were slightly increased. Once again, the pattern of stress distribution and the orientation of the stress were similar to those in the original model I (Figs. 5 and 6). The maximum displacement was 0.89 mm. The increased stiffness had more influence on the displacement than on the stresses.
amount observed in the aneurysm, which lengthens freely under pressure (model I). Working with the original model, we now applied both the internal pressure of 120 mmHg and an axial displacement of 5 mm to the bottom edge. The results were as follows. The areas of high stresses shifted to the posterior wall of the aneurysm (Fig. 8). The first principal stress on the inner surface at locations A, B, and C was 0.65, 0.4, and 0.5 N/mm2, respectively, and on the middle surface was 0.43, 0.36, and 0.33 N/mm2, respectively (Table II). All of the stresses were increased considerably and both the pattern of stress distribution (Fig. 8) and the orientation of stress (Table II) were altered. The orientation of the first principal stress on both the inner and the middle surface on the posterior wall, at location A, changed and became longitudinal. For instance, in Figure 6 at location A, the three bottom arrows appear vertically oriented. The orientation of the stresses just below the bulb (location B) and at the anterior region of the bulb (location C), however, remain the same as that in the original model (model I). The maximum displacement increased to 5.27 mm and
Model VI: Effect of Pull (Tethering Force) The aorta may be under a greater pull along the length due to tethering than what may occur by free expansion under internal pressure. Therefore, we determined stresses under conditions in which the aorta is pulled down by 5 mm along its length. This displacement is approximately twice the
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was along the long axis. These results show that if the tear on the posterior wall was oriented in the circumferential direction, then it would have been caused by the forces where the tethering force was dominant (Table III).
DISCUSSION Justification of the Model Our model is the first in which the geometry of an actual AAA of one clinical case is used, to account for true eccentricity, evaluate the effect of various parameters, and map the details of stress distribution throughout the aneurysm. The present model, therefore, is an advancement over those in which axisymmetric geometry has been used.16-20 The aneurysmal aorta was considered isotropic, homogeneous, and incompressible; these assumptions have also been made by others.16-20 The studies of Raghavan et al.27 support the postulate that the aneurysmal aortic tissue is isotropic. Although the material properties of the AAA are nonlinear,26,27 at a pressure load of 60 mmHg or greater, they are fairly linear. Since the geometry of the aorta used in the present study is that which existed in vivo, i.e., at a systemic pressure of the patient, at this geometry the AAA wall behaves almost linearly. Others have used similar assumptions in their analysis.16-19
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As models II and III show, the increase in wall thickness reduces the stress but changes neither the distribution pattern nor the orientation of the stress (Table II). An increase in intraluminal pressure raises the wall stress, while an increase in the wall stiffness raises the wall stress only minimally. Hence, the stress depends more on wall thickness and luminal pressure than on the material properties. This is advantageous to patients because the material properties of the AAA are not known. The change in the aneurysm diameter predicted by the model is also of interest, since pulsatile wall motion has been measured in patients.31 The model predicts that the aneurysm diameter increases by 0.83 mm when the pressure in the aneurysm increases from 120 mmHg (model I, maximum displacement 3.0 mm) to 160 mmHg (model IV, maximum displacement 3.83 mm). Malina et al.33 reported that in 47 patients pulsatile wall motion of the aneurysm was on average 1.0 mm. Considering that the pressure change associated with this motion is not known, the reported wall motion is close to that predicted by the model. This further shows the validity of the model. The tethering force does affect the stress distribution and the orientation. If a substantial tethering force is acting on the AAA, the tear in the posterior wall should be oriented circumferentially, otherwise it will be oriented longitudinally (Table II).
Wall Stress
Significance of Wall Stress
In our model of in vivo AAA geometry, we found that the maximum wall stress should be considered important in three regions: on the posterior and anterior wall of the bulb and on the anterior wall just below the bulb (Figs. 5 and 6, Table II). Mower et al.19 reported maximum stress at the bulb to be 0.057 N/mm2 circumferentially and 0.016 N/mm2 longitudinally. Inzoli et al.17 found the maximum stress to be 0.2 N/mm2 and Stringfellow et al.16 reported the maximum circumferential stress to be 0.11 N/mm2 and the longitudinal stress to be 0.08 N/mm2. Both16,17 reported that maximum stress occurred near the neck of the aneurysm where the dilated aorta meets the undilated segment. Mower et al.19 pointed out, however, that ruptures often do not occur at this location according to the report by Darling et al.5 Stress values in the bulb reported by us are higher than those reported by Mower et al.,19 mainly because of differences in the geometry and wall thickness used in the two studies. Overall, the magnitude and distribution of wall stress observed in the present study were comparable to those reported by Vorp et al.30 and Raghavan et al.22 for eccentric aneurysm models.
The present study explains which stresses may be responsible for the AAA rupture. First, the stress on the inner wall is greater than that on the middle wall and therefore the tear has a predilection to the inner surface. Second, the highest stresses occur at the bulb and thus the rupture is likely to begin here. Third, the rupture in the posterior wall region (location A) will produce a longitudinal tear whereas that in the anterior wall will produce a longitudinal tear below (location B) and a circumferential tear at (location C) the bulb (Fig. 6, Table III). This location and orientation of the tear is not likely to be altered by a change in wall thickness, luminal pressure, or wall stiffness (Tables II and III). Also, the orientation of the tear is the same, whether it is caused by the stress on the inner surface or by the stress on the middle surface (Table III). If the tear on the posterior wall is circumferential, then it must have occurred because of a significant tethering force (Tables II and III). Yield Stress and Location of Rupture Sites The results of the present study predict the likely location and orientation of the tear in the aneu-
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rysm. The aortic wall starts to yield when the stress reaches the yield stress. Raghavan et al.27 reported the yield stress to be 0.65 ± 0.09 N/mm2 in the longitudinal direction and 0.707 ± 0.12 N/mm2 in the circumferential direction for the anterior wall of the AAA. Our own unreported data suggest slightly lower values. The stress obtained in the present study is lower than the yield stress and that is why the aneurysm has not ruptured in the patient. However, if we consider additional parameters such as localized thinning, calcification, buckling, or weakening in the aneurysm, then the stress could equal or even exceed the yield stress. Thus, the present results can be used to determine the probability of rupture of an AAA on the basis of wall stress. Darling et al.5 reported that 62% of ruptures occur near the maximum diameter (bulb) region, 34% at the midpoint, and only 4% at the junction of the dilated and undilated aorta. Our results predict that, in the absence of any localized abnormality, almost all of the ruptures should occur in the bulb region. Hence, our results agree with the rupture locations reported by Darling et al. in a vast majority of cases.5 Our results also predict the orientation of the tear, however, there are no published reports on the tear orientation in the AAA for comparison. In the aortic dissection, our previous studies did suggest that the intimal tear is oriented orthogonal to the most rapidly increasing wall stress.20, 32,33
Wall stress in aneurysm 365
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CONCLUSIONS This study describes the wall stress in the AAA of one clinical case and identifies the locations and orientation of maximum stress. The stress is highest at the bulb. The stress is elevated in the posterior and anterior region of the bulb, and in the anterior region below the bulb. The orientation of the stress suggests that the tear is most likely to be circumferential at the anterior region of the bulb and longitudinal elsewhere. The parametric studies suggest that these results could be applied to a broader population of patients. These types of studies could lead to the determination of probability of rupture of an AAA and could assist in deciding the timing of surgical intervention.
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