Eur J VascSurg 6, 551-557 (1992)
The Management of Small Abdominal Aortic Aneurysms: a Computer Simulation Using Monte Carlo Methods* Jonathan A. Michaels
Nuffield Department of Surgery, John Radcliffe Hospital, Headington, Oxford, OX3 9DU, U.K. Many small abdominal aortic aneurysms can now be identified by ultrasound screening and it is necessary to decide whether the risks of enlargement and rupture justify elective surgery. A computer simulation of the behaviour of small aneurysms has been constructed using Monte Carlo methods to model patterns of enlargement and rupture. The effect of policy with regard to the observation and timing of intervention has been evaluatedfor different patient groups. The results demonstrate the value of early intervention in otherwisefit patients with expected operative mortality rates of 5% or below, whereas, for older patients (over 70 years old) 6-monthly screening and operation if the aneurysm exceeds 5cm is suggested. Higher risk patients with expected operative mortality of over 10 % may be better treated conservatively up to an aneurysm diameter of 7 or 8 cm. The method used for simulation is flexible, being easily adjusted to take account of new itiformation, and may have applications in other areas of clinical decision making. Key Words: Abdominal aortic aneurysms; Rupture rate; Mathematical modelling.
Introduction
One of the most critical decisions that a surgeon must make is to recommend a major operation to an asymptomatic patient. Abdominal aortic aneurysm is a serious condition which, if untreated, will frequently lead to death through rupture. 1 Most palpable aneurysms are large with a relatively high risk of rupture, so that surgical treatment is usually indicated whenever one is discovered. 2 In recent years the improvements in imaging techniques have made it possible to screen the "at risk" population and this has led to the identification of a large number of small aneurysms before they become clinically evident. 3 Decisions must be made concerning the size at which operative treatment is warranted and the policy regarding follow-up and intervention in the unoperated cases. As more small aneurysms are identified, and followed up by imaging, data is emerging about their expansion and rupture rates. Most previous policies for the management of small aneurysms have been of a general nature, although attempts have been made to use more quan* Winner of the 1991 Bard Prize. 0950-821X/92/050551+ 07 $08.00/0© 1992Grune & Stratton Ltd.
titative methods such as decision analysis. 4 However, such analytic methods require many simplifications and approximations as the probability distribution functions for events such as expansion and rupture are not easily represented by simple mathematical formulae. An alternative approach, which has previously been used in some biomedical applications, is the use of "Monte Carlo modelling". In this technique a single member of a population has a series of events simulated by the generation of random numbers with an appropriate distribution. The process is repeated many times to build up a large population and .the results of different policies and probabilities can be compared. Such methods have previously been used to study the distribution of laser light within tissue 5 and the efficacy of protection against X-rays. 6 The aims of this study are to apply these techniques to modelling the expansion and rupture of small abdominal aortic aneurysms and to predict the effect of different policies with respect to follow-up and intervention. Method
The computer simulation was carried out using a
552
J.A. Michaels
standard spreadsheet program (Excel 2.2, Microsoft, U.K.) running on an Apple Macintosh computer. Subroutines and functions were developed, using the program's macro facility, to generate the possible events with the required probability distribution. The probabilities were obtained from published data as described below. In carrying out the simulation the computer program generates a patient whose age and aneurysm size are within a preset range. Consecutive periods of 1 month are considered and aneurysm enlargement, death or rupture, are determined based upon random number generation with the required probability distribution. This process is repeated until the time of death for the patient. The events and aneurysm sizes are recorded on the spreadsheet, as is the timing of operation for each of a series of policies relating to intervention. The process is carried out repeatedly to build up a large number of patient histories, with the number of operations required and distribution of survival times being calculated for each of the policies. In carrying out the simulation it is possible to set operative and general mortality to any desired level. In the present simulation two populations have been considered. The first is a low-risk group with a general mortality from non-aneurysm causes which is based upon the OPCS data for standardised agerelated mortality for males in England and Wales. 7 Figures of 2, 5 and 8% were considered for the operative mortality of elective aneurysm repair, in keeping with the range quoted in the larger published series. 8'9 A second, higher risk, population was considered using similar data which was scaled to give survival similar to that previously described for higher risk patients with peripheral vascular disease 1° and a correspondingly higher operative mortality was set at 10, 15 and 20% in keeping with data relating to risk assessment for major surgery. 11 Rupture rate was based upon a number of published reports 12-14 and was related to aneurysm size as shown in Table 1. Expansion rate was also based upon a number of literature reports 14-22 and a skewed distribution was produced which was related to aneurysm size. Negative expansion rates and those of less than 2 m m per year were taken to be zero expansion. The program was tested by generating simulated data and comparing this with the intended distribution or published data to assess the accuracy of the simulation. The spreadsheet was used to generate 500 patients in the 60-65 year and 70-75 year age ranges, with aneurysms between 3 and 4cm in diameter. Simulations were carried out for the general and opEur J VascSurg Vol 6, September1992
Table 1. Estimates of rupture rates for different sizes of aneurysm as used for the computer simulation
Diameter (cm)
Rupture rate (% per year)
3-3.9
0.5
4-4.9
1.0
5-5.9
5.0
6-6.9
9.0
7-7.9
12.5
8-8.9
25.0
9-9.9
50.0
>10
90.0
erative mortalities given above. The effect of different policies of intervention and follow-up were assessed with measurements of aneurysm diameter at intervals of 6 or 12 months and using outcome measures of mean survival, median survival, death rate from rupture and number of operations. The policies tested were: immediate operation, no operation, operation when the measured increase since diagnosis is more than 1 cm, and operation when the measured diameter reaches 4, 5, 6, 7 or 8 cm.
Results
Cumulative survival, generated by simulation of patients without aneurysm mortality is shown in Figure 1 for groups of 500 patients aged 60-65. Median survival for the normal and high risk groups IOO
s y_ "6 g
0
I
I
I
5
I0
15
20
Time (years)
Fig. 1. Cumulative survival of 500 simulated patients aged 60 to 65 years, having low (11) or high ([:3) risk as defined in the text.
Management of Abdominal Aortic Aneurysms
a r e 15 a n d 7 y e a r s r e s p e c t i v e l y . S i m u l a t i o n o f d i c h o t o m o u s v a r i a b l e s s u c h as o p e r a t i v e m o r t a l i t y b e h a v e d as p r e d i c t e d w i t h a m a x i m u m o f o n e t o t w o p e r c e n t variation between the expected and observed event r a t e o v e r 500 s i m u l a t i o n s . T h e y e a r l y e x p a n s i o n r a t e o f 100 s i m u l a t e d a n e u r y s m s of d i a m e t e r s f r o m 3 - 6 c m is s h o w n g r a p h i c a l l y i n F i g u r e 2. C h a r a c t e r i s t i c s of t h e e x p a n s i o n
553
rates for different size aneurysms are shown in Table 2 with data from some published series for comparison. T h e d e t a i l e d r e s u l t s of t h e s i m u l a t i o n f o r a g r o u p of p a t i e n t s a g e d 7 0 - 7 5 w i t h 3 - 4 c m a n e u r y s m s , f o r each of the possible policies that were tested, are s h o w n i n T a b l e 3. B a s e d o n t h e s e f i g u r e s a n d a n o p e r a t i v e m o r t a l i t y of 5 % t h e l o n g e s t m e a n s u r v i v a l is
Table 2. Aneurysm expansion rates derived from the computer simulation and similar figures from published data
Expansion rate (mm per year) Source
n
Size (cm)
Maximum
No increase (%)
Computer simulation
100
3-3.9
2.8
0
0
15.8
53
100
4-4.9
6.0
5.7
0
16.7
22
100
5-5.9
6.8
6.1
0
18.5
19
100
6-6.9
9.6
9.1
0
20.2
5
100
7-7.9
12.6
12.1
2.1
24.5
0
Collin15
50
2.5-5
2.8
2.2
- 0.4
10.0
22
Walsh 16
459
Mean 4.2
3.3
Nevitt TM
103
1 cm increase in diameter
Diameter >5 cm
Diameter >6 cm
Deaths*
Operations
Ruptures
Mean
Median
10th centile
90th centile
0
500
0
115.1
99
26
223
6
67
428
5
116.0
98
31
221
12
76
417
7
115.4
98
31
217
6
103
387
10
115.5
98
33
216
12
109
381
10
115.3
98
32
216
6
130
358
12
115.6
98
35
216
12
136
351
13
114.9
97
33
215
6
175
292
33
112.8
94
35
209
12
180
279
41
111.2
92
33
209
All aneurysms Diameter >4 cm
Outcome
* Deaths are those occurring due to causes other than aneurysm rupture, prior to elective aneurysm repair. Eur J Vasc Surg Vol 6, September 1992
55 4
J.A. Michaels
30
E 2O
several levels of operative mortality. Six-monthly m e a s u r e m e n t s give a better result than 12-monthly m e a s u r e m e n t s in all cases but with marginal differences w h e r e the a n e u r y s m is less than 5 cm. Median survival is lower than m e a n survival t h r o u g h o u t and, for the higher risk groups, the policy giving the longest m e d i a n survival is more conservative than that giving the greatest mean.
x x
o
x
•
I0-
x
~
~ x
0
~
x
x x
x
x
x
x
x
XxYX
~,l(X X
-IO 3
x
x x xx
~x
x
x ~
x x x
I
I
4
5
x
x
x
x x x
•
Discussion
6
Initial size (cm)
Fig. 2. Yearly expansion rate vs size for 100 aneurysms derived from the computer simulation. given b y the policy of scanning every 6 m o n t h s a n d operating w h e n the m e a s u r e d diameter is over 4 cm. The best m e d i a n survival is given b y immediate operation a l t h o u g h for all the policies w h i c h involve operation at 5 cm or below the m e a n or m e d i a n survival m e a s u r e s vary by less than one m o n t h . The m o s t conservative of these policies saves 30% of the elective operations at a cost of a r u p t u r e rate of about 2.5%. S u m m a r y data in Tables 4 a n d 5 s h o w s the predicted results for low a n d high risk patient g r o u p s at
In m a k i n g surgical decisions there is a great t e n d e n c y to think in "black a n d w h i t e " terms as, for example, in declaring patients "fit" or " u n f i t " for general anaesthesia. In practice, there is a c o n t i n u o u s range of probabilities of operative mortality a n d with the w i d e s p r e a d use of c o m p u t e r i s e d audits a n d the characterisation of risk factors 23 it is b e c o m i n g possible to p u t patients into fairly accurate risk categories. 24 To design a r a n d o m i s e d trial to evaluate the difference b e t w e e n operating at 4 or 5 cm diameter on patients in a particular age range a n d risk g r o u p w o u l d be impractical d u e to the limited recruitment a n d small differences that w o u l d be expected. The alternatives are to make general rules, w h i c h m a y not give the
Table 4. Median survival and required number of operations for each policy for intervention and various risks of operative mortality, for 500 simulated patients aged 60-65 years in the low-risk category having an initial aneurysm diameter of 3.0-4.0 cm
Indications for elective operation
Scanning interval (months)
2%*
5%*
8%*
500
187
183
178
6
465
187
183
176
12
460
187
183
176
6
451
187
183
176
12
445
186
181
174
6
435
183
180
173
12
430
183
180
173
6
403
178
177
170
12
394
177
174
169
6
365
169
168
162
12
365
166
164
160
6
325
151
147
140
12
311
148
138
133
All aneurysms Diameter :>4cm
:>1 cm increase in diameter
Diameter :> 5 cm
Diameter :> 6 cm
Diameter :> 7 cm
Diameter :> 8 cm
* Operative mortality. Eur J Vasc Surg Vol 6, September 1992
Median survival (months) Operations required
Management of Abdominal Aortic Aneurysms
555
Table 5. Median survival and required n u m b e r of operations for each policy for intervention and various risks of operative mortality, for 500 simulated patients aged 60-65 years in the h i g h risk category h a v i n g an initial a n e u r y s m diameter of 3.0-4.0 cm
Indications for elective operation
Scanning interval (months)
10%*
15%*
20%*
500
87
73
69
6
397
87
73
72
12
386
86
73
72
6
356
88
74
74
12
350
87
74
73
6
326
88
78
78
12
318
87
77
73
6
265
87
82
82
12
249
84
79
74
6
214
87
84
84
12
209
81
79
77
6
179
86
84
84
12
167
79
79
77
6
142
86
82
82
12
130
79
78
77
6
108
85
82
82
12
93
78
78
76
All a n e u r y s m s Diameter > 4 cm
>1 cm increase in diameter
Diameter > 5 cm
Diameter > 6 cm
Diameter > 7 cm
Diameter > 8 cm
Diameter > 9 cm
Diameter > 10 cm
Median survival (months) Operations required
* Operative mortality.
f
optimum management in all subgroups of patients, or to attempt to calculate or simulate the results based upon the available data. In using the latter approach two major problems arise. The first is to determine the most appropriate of the available data upon which to base the calculations. The second is to decide upon the best measures of outcome. The use of a computer simulation makes it possible to experiment with different possible values for each parameter and calculations can be altered to take account of differing circumstances. This also makes it easier to identify the data that are most critical to the outcome and thus to direct future research. In the present study the figures were derived from a review of published data. The survival following repair of aneurysm has been reported to be normal 2s or slightly reduced 26 compared with agematched controls. In higher risk groups, most of the excess mortality is related to coronary events and
patients can be identified who may benefit from further investigation and treatment of coronary artery disease. 1° The higher rate of mortality used for the simulation corresponds fairly closely to the survival of patients with peripheral vascular disease or angina. 27 Operative mortality is reported at less than 5% 28 in many of the large series, although larger scale studies give figures of around 10% or higher. 29 A range of values have been used in this simulation and different values may be appropriate for particular circumstances. No account has been taken of possible relationship of operative mortality to age or aneurysm size as there are no reliable data on which to base such calculations. The data for expansion rates of small aneurysms are fairly consistent (see Table 2) but reliable evidence of rupture rate is sparse. One of the most critical aspects of the simulation is the rupture rate for very small aneurysms. With a rate of only a few per cent it will require the accumulated experience of several Eur J Vasc Surg Vol 6, September 1992
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J.A. Michaels
centres to put accurate confidence limits on such data. It is difficult to draw valid conclusions from post-mortem data 3° as there is a sampling bias and the time over which the patient was "at risk" is not known. A policy of operating upon aneurysms at 6cm diameter in the higher risk groups has been previously proposed. 31 This would appear to be largely based upon the finding that the majority of aneurysms discovered at post-mortem are unruptured. 3° However, it seems likely that many such aneurysms have developed soon before death whilst younger, otherwise healthy patients with aneurysms discovered on screening would be expected to be at greater risk of dying from aneurysm rupture. There are few aneurysms treated conservatively over a prolonged period and the small number of patients who refuse surgery, or are considered unfit, will rarely have repeated investigations. The estimates used in the study are only approximate and are lower than some previously reported figures 9 to take account of the probable underestimate in the size of the aneurysm at the time of rupture. The choice of a suitable measure of outcome is difficult. Most previous policies have been based upon survival. However, as can be seen from the results, there is a difference between using median and mean survival figures as early operative deaths skew the survival curve. Some account should probably be taken of the view that the reduction in life expectancy due to risk of early death (e.g. operative mortality) is more important than the same reduction due to a higher risk of later death (i.e. early years are more valuable than later ones). 32 The use of median survival takes some account of this as the mean is affected to a greater extent by a few long-term survivors. An alternative would be to discount the value of subsequent years if an appropriate discount rate could be determined by some form of utility analysis. 33 The current analysis takes no account of quality of life, and there may be an argument for using QALYs as a suitable measure of outcome. 32 There is little evidence regarding the quality of life after aneurysm repair, and it has been suggested that, even after repair of a ruptured aneurysm, no reduction is evident. 25 It is also possible that knowledge of the presence of aneurysm and the anxiety of repeated scans would reduce the quality of life in unoperated patients. Without more evidence it would be impossible to produce a useful comparison based on this measure. Cost effectiveness is another measure upon which decisions might be based. However, the costEur J VascSurg Vol 6, September1992
ing of different policies is not a simple matter as it is not easy to define exactly which costs are to be included (e.g. treatment of ruptures or other terminal disease, geriatric care, etc.). Since major vascular surgery causes a temporary reduction in the quality of life, and has a definite financial cost, it would seem reasonable to select a policy which results in the need for fewest operations when the differences in survival are marginal. Using the results of Table 3 the "best" policy would be immediate operation, based upon median survival, or operation at 4 cm diameter based upon mean survival. The differences in survival are marginal and a policy of 6-monthly screening with operation when the size exceeds 5 cm would result in a change in expected survival of less than 1 month with nearly 30% fewer operations being required. A still more conservative policy would produce a far greater reduction in expected survival. Survival following rupture has not been taken into account in this study. Estimates of the overall mortality following rupture of an aneurysm in the community vary from 789 to 94% 34 and are probably an underestimate due to those dying without the diagnosis being made. At this level survivors would make little difference to the overall results. Based upon the results of the simulation and the considerations above, a policy for relating management to age and operative risk can be suggested by reference to Tables 4 and 5. A suggested policy would be based upon providing the best median survival, but with a reduction of up to I month being accepted if it reduces the expected number of required operations, for the reasons suggested above. In general terms, operation is to be recommended at an early stage in younger patients if a low operative mortality can be expected (5% or lower), whereas operation when the aneurysm reaches 5 cm is reasonable for an older population. Very high risk patients (with a higher mortality likely from other causes and a probable operative mortality over 10%), may be better treated conservatively up to a size of 7 or 8 cm. In such cases investigations should probably be repeated at intervals of less than a year when the aneurysm is 5 cm or larger. It is important to recognise that there is no specific threshold size above which surgery should be recommended. The optim u m policy is a function of several variables, the most important of which is the expected operative mortality which will be specific for the individual patient and surgeon. Figure 3 shows a graphical representation of such a policy based upon the results of this model. The use of computer simulation to investigate such problems provides a possible means of making a
Management of Abdominal Aortic Aneurysms
2O
A
o< I0 °
0
4
5
6
7
8
Size (crn)
Fig. 3. Graphical representation of a possible management policy relating estimated operative mortality to aneurysm diameter.
rational management policy, which can be easily modified to take new information into account. Clinical trials are now taking place which will provide interesting data to compare with the predictions made here. References 1 BOWERSD, CAVE WS. Aneurysms of the abdominal aorta: a 20year study. J R Soc Med 1985; 78: 812-820. 2 THOMPSON JE, GARRETT WV, PATMAN RD, TALKINGTON CM, WILLIAMS SJ. Elective surgery for abdominal aortic aneurysms. In: BERGAN JJ, YAO JST (eds). Aneurysms, Diagnosis and Treatment. New York: Grune & Stratton, 1982; 287-301. 3 BERGQVISTD, JENDTEG S, LINDGREN B. Standards for the costbenefit approach to vascular surgery. Acta Chir Scand 1990; Suppl 555: 105-110. 4 KITSLAARPJEHM, KIEVITJ. The treatment policy in asymptomatic abdominal aneurysms: a Markov decision analysis. Br J Surg; in press. 5 KEIJZER M, JACQUES SL, PRAHL SA, WELCH AJ. Light distributions in artery tissue: Monte Carlo simulations for finitediameter laser beams. Lasers Surg Med 1989; 9: 148-154. 6 SIMPKIN DJ. Shielding requirements for constant-potential d i a g nostic X-ray beams determined by a Monte Carlo calculation. Health Phys 1989; 56: 151-164. 7 Standardised mortality tables for England and Wales, i988. Series DH2 No. 15. Office of Population Censuses and Surveys, DHSS: London, 1988. 8 CRAWFORD ES, PALAMARAAE, SALEFI SA, ROEHM JOF. Aortic aneurysms: current status of surgical treatment. Surg Clin North Am 1979; 59: 597-636. 9 TAYLORLM, PORTER JM. Basic data related to clinical decisionmaking in abdominal aortic aneurysms. Ann Vasc Surg 1986; 1: 502-504. 10 HERTZERNR, YOUNGJR, BEVENEG, et al. Late results of coronary bypass in patients presenting with lower extremity ischemia: the Cleveland Clinic study. Ann Vasc Surg 1986; 1: 411-420. 11 LUNDELL L, NORB.KCKB. Abdominal aortic aneurysms--results of treatment in nonspecialized units. Acta Chit Scand 1983; 149: 695-702.
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12 CRONENWETT JL, MURPHY TF, ZELENOCK GB, et al. Actuarial analysis of variables associated with rupture of small abdominal aortic aneurysms. Surgery 1985; 98: 472-483. 13 BERNSTEIN EF, CHAN EL. Abdominal aortic aneurysm in highrisk patients. Ann Surg 1984; 200: 255-263. 14 NEVITT MP, BALLARDDJ, HALLETTJW. Prognosis of abdominal aortic aneurysms: a population based study. N Engl J Med 1989; 321: 1009-1014. 15 COLLIN J, ARAUIO L, WALTON J. How fast do very small aortic aneurysms grow? Eur ] Vasc Surg 1989; 3: 15-17. 16 WALSH AKM, BRIEEA N, NASH JR, CALLUM KG. The natural history of small abdominal aortic aneurysms: an ultrasound study. Eur J Vasc Surg 1990; 4: 459-461. 17 GLIMAKER H, HOLMBERG L, ELVIN A, et al. Natural history of patients with abdominal aortic aneurysm. Eur J Vasc Surg 1991; 5: 125-130. 18 BERGQVISTD, BENGTSSONH. Risk factors for rupture of abdominal aortic aneurysm. Acta Chir Scand 1990; 156: 63-68. 19 CRONENWETT JL, SARGENT SK, WALL MH, et al. Variables that affect the expansion rate and outcome of small abdominal aortic aneurysms. J Vasc Surg 1990; 11: 260-269. 20 HEATHERB, COLLIN J, WALTONJ. Optimum rescreening interval for abdominal aortic aneurysm. Br J Surg 1991; 78:3652 21 DELIN A, OHLSI~N H, SWEDENBORGJ. Growth rate of abdominal aortic aneurysms as measured by computed tomography. Br J Surg 1985; 72: 530-532. 22 BERNSTEIN EF, DILLEY RB, GOLDENBERGER LE, GOSINK BB, LEOPOLD GR. Growth rates of small abdominal aortic aneurysms. Surgery 1976; 80: 765-773. 23 GOLDMAN L, CALDERA DL, NUSSBAUM SR, et aI. Multifactorial index of cardiac risk in noncardiac surgical procedures. N Engl J Med 1977; 297: 845-850. 24 EAGLEKA, BOUCHER CA. Cardiac risk of noncardiac surgery. N Engl J Med 1989; 321: 1330-1332. 25 ROHRER MJ, CUTLER BS, WHEELER B. Long-term survival and quality of life following ruptured abdominal aortic aneurysm. Arch Surg 1988; 123: 1213-1217. 26 JOHNSON G, GURRI JA, BURNHAM SJ. Life expectancy after abdominal aneurysm repair. In: BERGANJJ, YAO JST, eds. Aneurysms: Diagnosis and Treatment. New York: Grune & Stratton, 1982; 279-285. 27 EUROPEANCORONARYSURGERYSTUDYGROUP. Long-term results of prospective randomised study of coronary artery bypass surgery in stable angina pectoris. Lancet 1982; ii: 1173-1180. 28 CRAWFORD ES, SALEH SA, BABBJW, GLAESERDH, VACCARO PS, SILVERS A. Infrarenal abdominal aortic aneurysm: factors influencing survival after operation performed over a 25-year period. Ann Surg 1981; 193: 699-709. 29 PILCHERDB, DAVISJH, ASHIKAGART, etal. Treatment ofabdominal aortic aneurysm in a single state over 7½ years. Am J Surg 1980; 139: 487-494. 30 DARLING RC, MESSINA CR, BREWSTER DC, OTTINGER L W . Autopsy study of unoperated aortic aneurysms. Circulation 1977; 56: 161-164. 31 SCOTTRAP, ASHTON HA, KAY DN. Screening and patient selection in aortic aneurysms. Current Prac Surg 1990; 2: 68-71. 32 MIYAMOTO JM, ERAKER SA. Parameter estimates for a QALY utility model. Med Decis Making 1985; 5: 191-213, 33 LLEWELLYN-THOMASH, SUTHERLANDHJ, TIBSHIRANIR, CIAMPIA, TILL JE, BOYDNF. The measurement of patients' values in medicine. Med Decis Making 1982; 2: 449-462. 34 JOHANSSONG, SWEDENBORGJ. Ruptured abdominal aortic aneurysms: a study of incidence and mortality. Br J Surg 1986; 73: 101-103.
Accepted 14 April 1992
Eur J Vasc Surg Vol 6, September 1992
The Management of Small Abdominal Aortic Aneurysms: a Computer Simulation Using Monte Carlo Methods* Jonathan A. Michaels
Nuffield Department of Surgery, John Radcliffe Hospital, Headington, Oxford, OX3 9DU, U.K. Many small abdominal aortic aneurysms can now be identified by ultrasound screening and it is necessary to decide whether the risks of enlargement and rupture justify elective surgery. A computer simulation of the behaviour of small aneurysms has been constructed using Monte Carlo methods to model patterns of enlargement and rupture. The effect of policy with regard to the observation and timing of intervention has been evaluatedfor different patient groups. The results demonstrate the value of early intervention in otherwisefit patients with expected operative mortality rates of 5% or below, whereas, for older patients (over 70 years old) 6-monthly screening and operation if the aneurysm exceeds 5cm is suggested. Higher risk patients with expected operative mortality of over 10 % may be better treated conservatively up to an aneurysm diameter of 7 or 8 cm. The method used for simulation is flexible, being easily adjusted to take account of new itiformation, and may have applications in other areas of clinical decision making. Key Words: Abdominal aortic aneurysms; Rupture rate; Mathematical modelling.
Introduction
One of the most critical decisions that a surgeon must make is to recommend a major operation to an asymptomatic patient. Abdominal aortic aneurysm is a serious condition which, if untreated, will frequently lead to death through rupture. 1 Most palpable aneurysms are large with a relatively high risk of rupture, so that surgical treatment is usually indicated whenever one is discovered. 2 In recent years the improvements in imaging techniques have made it possible to screen the "at risk" population and this has led to the identification of a large number of small aneurysms before they become clinically evident. 3 Decisions must be made concerning the size at which operative treatment is warranted and the policy regarding follow-up and intervention in the unoperated cases. As more small aneurysms are identified, and followed up by imaging, data is emerging about their expansion and rupture rates. Most previous policies for the management of small aneurysms have been of a general nature, although attempts have been made to use more quan* Winner of the 1991 Bard Prize. 0950-821X/92/050551+ 07 $08.00/0© 1992Grune & Stratton Ltd.
titative methods such as decision analysis. 4 However, such analytic methods require many simplifications and approximations as the probability distribution functions for events such as expansion and rupture are not easily represented by simple mathematical formulae. An alternative approach, which has previously been used in some biomedical applications, is the use of "Monte Carlo modelling". In this technique a single member of a population has a series of events simulated by the generation of random numbers with an appropriate distribution. The process is repeated many times to build up a large population and .the results of different policies and probabilities can be compared. Such methods have previously been used to study the distribution of laser light within tissue 5 and the efficacy of protection against X-rays. 6 The aims of this study are to apply these techniques to modelling the expansion and rupture of small abdominal aortic aneurysms and to predict the effect of different policies with respect to follow-up and intervention. Method
The computer simulation was carried out using a
552
J.A. Michaels
standard spreadsheet program (Excel 2.2, Microsoft, U.K.) running on an Apple Macintosh computer. Subroutines and functions were developed, using the program's macro facility, to generate the possible events with the required probability distribution. The probabilities were obtained from published data as described below. In carrying out the simulation the computer program generates a patient whose age and aneurysm size are within a preset range. Consecutive periods of 1 month are considered and aneurysm enlargement, death or rupture, are determined based upon random number generation with the required probability distribution. This process is repeated until the time of death for the patient. The events and aneurysm sizes are recorded on the spreadsheet, as is the timing of operation for each of a series of policies relating to intervention. The process is carried out repeatedly to build up a large number of patient histories, with the number of operations required and distribution of survival times being calculated for each of the policies. In carrying out the simulation it is possible to set operative and general mortality to any desired level. In the present simulation two populations have been considered. The first is a low-risk group with a general mortality from non-aneurysm causes which is based upon the OPCS data for standardised agerelated mortality for males in England and Wales. 7 Figures of 2, 5 and 8% were considered for the operative mortality of elective aneurysm repair, in keeping with the range quoted in the larger published series. 8'9 A second, higher risk, population was considered using similar data which was scaled to give survival similar to that previously described for higher risk patients with peripheral vascular disease 1° and a correspondingly higher operative mortality was set at 10, 15 and 20% in keeping with data relating to risk assessment for major surgery. 11 Rupture rate was based upon a number of published reports 12-14 and was related to aneurysm size as shown in Table 1. Expansion rate was also based upon a number of literature reports 14-22 and a skewed distribution was produced which was related to aneurysm size. Negative expansion rates and those of less than 2 m m per year were taken to be zero expansion. The program was tested by generating simulated data and comparing this with the intended distribution or published data to assess the accuracy of the simulation. The spreadsheet was used to generate 500 patients in the 60-65 year and 70-75 year age ranges, with aneurysms between 3 and 4cm in diameter. Simulations were carried out for the general and opEur J VascSurg Vol 6, September1992
Table 1. Estimates of rupture rates for different sizes of aneurysm as used for the computer simulation
Diameter (cm)
Rupture rate (% per year)
3-3.9
0.5
4-4.9
1.0
5-5.9
5.0
6-6.9
9.0
7-7.9
12.5
8-8.9
25.0
9-9.9
50.0
>10
90.0
erative mortalities given above. The effect of different policies of intervention and follow-up were assessed with measurements of aneurysm diameter at intervals of 6 or 12 months and using outcome measures of mean survival, median survival, death rate from rupture and number of operations. The policies tested were: immediate operation, no operation, operation when the measured increase since diagnosis is more than 1 cm, and operation when the measured diameter reaches 4, 5, 6, 7 or 8 cm.
Results
Cumulative survival, generated by simulation of patients without aneurysm mortality is shown in Figure 1 for groups of 500 patients aged 60-65. Median survival for the normal and high risk groups IOO
s y_ "6 g
0
I
I
I
5
I0
15
20
Time (years)
Fig. 1. Cumulative survival of 500 simulated patients aged 60 to 65 years, having low (11) or high ([:3) risk as defined in the text.
Management of Abdominal Aortic Aneurysms
a r e 15 a n d 7 y e a r s r e s p e c t i v e l y . S i m u l a t i o n o f d i c h o t o m o u s v a r i a b l e s s u c h as o p e r a t i v e m o r t a l i t y b e h a v e d as p r e d i c t e d w i t h a m a x i m u m o f o n e t o t w o p e r c e n t variation between the expected and observed event r a t e o v e r 500 s i m u l a t i o n s . T h e y e a r l y e x p a n s i o n r a t e o f 100 s i m u l a t e d a n e u r y s m s of d i a m e t e r s f r o m 3 - 6 c m is s h o w n g r a p h i c a l l y i n F i g u r e 2. C h a r a c t e r i s t i c s of t h e e x p a n s i o n
553
rates for different size aneurysms are shown in Table 2 with data from some published series for comparison. T h e d e t a i l e d r e s u l t s of t h e s i m u l a t i o n f o r a g r o u p of p a t i e n t s a g e d 7 0 - 7 5 w i t h 3 - 4 c m a n e u r y s m s , f o r each of the possible policies that were tested, are s h o w n i n T a b l e 3. B a s e d o n t h e s e f i g u r e s a n d a n o p e r a t i v e m o r t a l i t y of 5 % t h e l o n g e s t m e a n s u r v i v a l is
Table 2. Aneurysm expansion rates derived from the computer simulation and similar figures from published data
Expansion rate (mm per year) Source
n
Size (cm)
Maximum
No increase (%)
Computer simulation
100
3-3.9
2.8
0
0
15.8
53
100
4-4.9
6.0
5.7
0
16.7
22
100
5-5.9
6.8
6.1
0
18.5
19
100
6-6.9
9.6
9.1
0
20.2
5
100
7-7.9
12.6
12.1
2.1
24.5
0
Collin15
50
2.5-5
2.8
2.2
- 0.4
10.0
22
Walsh 16
459
Mean 4.2
3.3
Nevitt TM
103
1 cm increase in diameter
Diameter >5 cm
Diameter >6 cm
Deaths*
Operations
Ruptures
Mean
Median
10th centile
90th centile
0
500
0
115.1
99
26
223
6
67
428
5
116.0
98
31
221
12
76
417
7
115.4
98
31
217
6
103
387
10
115.5
98
33
216
12
109
381
10
115.3
98
32
216
6
130
358
12
115.6
98
35
216
12
136
351
13
114.9
97
33
215
6
175
292
33
112.8
94
35
209
12
180
279
41
111.2
92
33
209
All aneurysms Diameter >4 cm
Outcome
* Deaths are those occurring due to causes other than aneurysm rupture, prior to elective aneurysm repair. Eur J Vasc Surg Vol 6, September 1992
55 4
J.A. Michaels
30
E 2O
several levels of operative mortality. Six-monthly m e a s u r e m e n t s give a better result than 12-monthly m e a s u r e m e n t s in all cases but with marginal differences w h e r e the a n e u r y s m is less than 5 cm. Median survival is lower than m e a n survival t h r o u g h o u t and, for the higher risk groups, the policy giving the longest m e d i a n survival is more conservative than that giving the greatest mean.
x x
o
x
•
I0-
x
~
~ x
0
~
x
x x
x
x
x
x
x
XxYX
~,l(X X
-IO 3
x
x x xx
~x
x
x ~
x x x
I
I
4
5
x
x
x
x x x
•
Discussion
6
Initial size (cm)
Fig. 2. Yearly expansion rate vs size for 100 aneurysms derived from the computer simulation. given b y the policy of scanning every 6 m o n t h s a n d operating w h e n the m e a s u r e d diameter is over 4 cm. The best m e d i a n survival is given b y immediate operation a l t h o u g h for all the policies w h i c h involve operation at 5 cm or below the m e a n or m e d i a n survival m e a s u r e s vary by less than one m o n t h . The m o s t conservative of these policies saves 30% of the elective operations at a cost of a r u p t u r e rate of about 2.5%. S u m m a r y data in Tables 4 a n d 5 s h o w s the predicted results for low a n d high risk patient g r o u p s at
In m a k i n g surgical decisions there is a great t e n d e n c y to think in "black a n d w h i t e " terms as, for example, in declaring patients "fit" or " u n f i t " for general anaesthesia. In practice, there is a c o n t i n u o u s range of probabilities of operative mortality a n d with the w i d e s p r e a d use of c o m p u t e r i s e d audits a n d the characterisation of risk factors 23 it is b e c o m i n g possible to p u t patients into fairly accurate risk categories. 24 To design a r a n d o m i s e d trial to evaluate the difference b e t w e e n operating at 4 or 5 cm diameter on patients in a particular age range a n d risk g r o u p w o u l d be impractical d u e to the limited recruitment a n d small differences that w o u l d be expected. The alternatives are to make general rules, w h i c h m a y not give the
Table 4. Median survival and required number of operations for each policy for intervention and various risks of operative mortality, for 500 simulated patients aged 60-65 years in the low-risk category having an initial aneurysm diameter of 3.0-4.0 cm
Indications for elective operation
Scanning interval (months)
2%*
5%*
8%*
500
187
183
178
6
465
187
183
176
12
460
187
183
176
6
451
187
183
176
12
445
186
181
174
6
435
183
180
173
12
430
183
180
173
6
403
178
177
170
12
394
177
174
169
6
365
169
168
162
12
365
166
164
160
6
325
151
147
140
12
311
148
138
133
All aneurysms Diameter :>4cm
:>1 cm increase in diameter
Diameter :> 5 cm
Diameter :> 6 cm
Diameter :> 7 cm
Diameter :> 8 cm
* Operative mortality. Eur J Vasc Surg Vol 6, September 1992
Median survival (months) Operations required
Management of Abdominal Aortic Aneurysms
555
Table 5. Median survival and required n u m b e r of operations for each policy for intervention and various risks of operative mortality, for 500 simulated patients aged 60-65 years in the h i g h risk category h a v i n g an initial a n e u r y s m diameter of 3.0-4.0 cm
Indications for elective operation
Scanning interval (months)
10%*
15%*
20%*
500
87
73
69
6
397
87
73
72
12
386
86
73
72
6
356
88
74
74
12
350
87
74
73
6
326
88
78
78
12
318
87
77
73
6
265
87
82
82
12
249
84
79
74
6
214
87
84
84
12
209
81
79
77
6
179
86
84
84
12
167
79
79
77
6
142
86
82
82
12
130
79
78
77
6
108
85
82
82
12
93
78
78
76
All a n e u r y s m s Diameter > 4 cm
>1 cm increase in diameter
Diameter > 5 cm
Diameter > 6 cm
Diameter > 7 cm
Diameter > 8 cm
Diameter > 9 cm
Diameter > 10 cm
Median survival (months) Operations required
* Operative mortality.
f
optimum management in all subgroups of patients, or to attempt to calculate or simulate the results based upon the available data. In using the latter approach two major problems arise. The first is to determine the most appropriate of the available data upon which to base the calculations. The second is to decide upon the best measures of outcome. The use of a computer simulation makes it possible to experiment with different possible values for each parameter and calculations can be altered to take account of differing circumstances. This also makes it easier to identify the data that are most critical to the outcome and thus to direct future research. In the present study the figures were derived from a review of published data. The survival following repair of aneurysm has been reported to be normal 2s or slightly reduced 26 compared with agematched controls. In higher risk groups, most of the excess mortality is related to coronary events and
patients can be identified who may benefit from further investigation and treatment of coronary artery disease. 1° The higher rate of mortality used for the simulation corresponds fairly closely to the survival of patients with peripheral vascular disease or angina. 27 Operative mortality is reported at less than 5% 28 in many of the large series, although larger scale studies give figures of around 10% or higher. 29 A range of values have been used in this simulation and different values may be appropriate for particular circumstances. No account has been taken of possible relationship of operative mortality to age or aneurysm size as there are no reliable data on which to base such calculations. The data for expansion rates of small aneurysms are fairly consistent (see Table 2) but reliable evidence of rupture rate is sparse. One of the most critical aspects of the simulation is the rupture rate for very small aneurysms. With a rate of only a few per cent it will require the accumulated experience of several Eur J Vasc Surg Vol 6, September 1992
556
J.A. Michaels
centres to put accurate confidence limits on such data. It is difficult to draw valid conclusions from post-mortem data 3° as there is a sampling bias and the time over which the patient was "at risk" is not known. A policy of operating upon aneurysms at 6cm diameter in the higher risk groups has been previously proposed. 31 This would appear to be largely based upon the finding that the majority of aneurysms discovered at post-mortem are unruptured. 3° However, it seems likely that many such aneurysms have developed soon before death whilst younger, otherwise healthy patients with aneurysms discovered on screening would be expected to be at greater risk of dying from aneurysm rupture. There are few aneurysms treated conservatively over a prolonged period and the small number of patients who refuse surgery, or are considered unfit, will rarely have repeated investigations. The estimates used in the study are only approximate and are lower than some previously reported figures 9 to take account of the probable underestimate in the size of the aneurysm at the time of rupture. The choice of a suitable measure of outcome is difficult. Most previous policies have been based upon survival. However, as can be seen from the results, there is a difference between using median and mean survival figures as early operative deaths skew the survival curve. Some account should probably be taken of the view that the reduction in life expectancy due to risk of early death (e.g. operative mortality) is more important than the same reduction due to a higher risk of later death (i.e. early years are more valuable than later ones). 32 The use of median survival takes some account of this as the mean is affected to a greater extent by a few long-term survivors. An alternative would be to discount the value of subsequent years if an appropriate discount rate could be determined by some form of utility analysis. 33 The current analysis takes no account of quality of life, and there may be an argument for using QALYs as a suitable measure of outcome. 32 There is little evidence regarding the quality of life after aneurysm repair, and it has been suggested that, even after repair of a ruptured aneurysm, no reduction is evident. 25 It is also possible that knowledge of the presence of aneurysm and the anxiety of repeated scans would reduce the quality of life in unoperated patients. Without more evidence it would be impossible to produce a useful comparison based on this measure. Cost effectiveness is another measure upon which decisions might be based. However, the costEur J VascSurg Vol 6, September1992
ing of different policies is not a simple matter as it is not easy to define exactly which costs are to be included (e.g. treatment of ruptures or other terminal disease, geriatric care, etc.). Since major vascular surgery causes a temporary reduction in the quality of life, and has a definite financial cost, it would seem reasonable to select a policy which results in the need for fewest operations when the differences in survival are marginal. Using the results of Table 3 the "best" policy would be immediate operation, based upon median survival, or operation at 4 cm diameter based upon mean survival. The differences in survival are marginal and a policy of 6-monthly screening with operation when the size exceeds 5 cm would result in a change in expected survival of less than 1 month with nearly 30% fewer operations being required. A still more conservative policy would produce a far greater reduction in expected survival. Survival following rupture has not been taken into account in this study. Estimates of the overall mortality following rupture of an aneurysm in the community vary from 789 to 94% 34 and are probably an underestimate due to those dying without the diagnosis being made. At this level survivors would make little difference to the overall results. Based upon the results of the simulation and the considerations above, a policy for relating management to age and operative risk can be suggested by reference to Tables 4 and 5. A suggested policy would be based upon providing the best median survival, but with a reduction of up to I month being accepted if it reduces the expected number of required operations, for the reasons suggested above. In general terms, operation is to be recommended at an early stage in younger patients if a low operative mortality can be expected (5% or lower), whereas operation when the aneurysm reaches 5 cm is reasonable for an older population. Very high risk patients (with a higher mortality likely from other causes and a probable operative mortality over 10%), may be better treated conservatively up to a size of 7 or 8 cm. In such cases investigations should probably be repeated at intervals of less than a year when the aneurysm is 5 cm or larger. It is important to recognise that there is no specific threshold size above which surgery should be recommended. The optim u m policy is a function of several variables, the most important of which is the expected operative mortality which will be specific for the individual patient and surgeon. Figure 3 shows a graphical representation of such a policy based upon the results of this model. The use of computer simulation to investigate such problems provides a possible means of making a
Management of Abdominal Aortic Aneurysms
2O
A
o< I0 °
0
4
5
6
7
8
Size (crn)
Fig. 3. Graphical representation of a possible management policy relating estimated operative mortality to aneurysm diameter.
rational management policy, which can be easily modified to take new information into account. Clinical trials are now taking place which will provide interesting data to compare with the predictions made here. References 1 BOWERSD, CAVE WS. Aneurysms of the abdominal aorta: a 20year study. J R Soc Med 1985; 78: 812-820. 2 THOMPSON JE, GARRETT WV, PATMAN RD, TALKINGTON CM, WILLIAMS SJ. Elective surgery for abdominal aortic aneurysms. In: BERGAN JJ, YAO JST (eds). Aneurysms, Diagnosis and Treatment. New York: Grune & Stratton, 1982; 287-301. 3 BERGQVISTD, JENDTEG S, LINDGREN B. Standards for the costbenefit approach to vascular surgery. Acta Chir Scand 1990; Suppl 555: 105-110. 4 KITSLAARPJEHM, KIEVITJ. The treatment policy in asymptomatic abdominal aneurysms: a Markov decision analysis. Br J Surg; in press. 5 KEIJZER M, JACQUES SL, PRAHL SA, WELCH AJ. Light distributions in artery tissue: Monte Carlo simulations for finitediameter laser beams. Lasers Surg Med 1989; 9: 148-154. 6 SIMPKIN DJ. Shielding requirements for constant-potential d i a g nostic X-ray beams determined by a Monte Carlo calculation. Health Phys 1989; 56: 151-164. 7 Standardised mortality tables for England and Wales, i988. Series DH2 No. 15. Office of Population Censuses and Surveys, DHSS: London, 1988. 8 CRAWFORD ES, PALAMARAAE, SALEFI SA, ROEHM JOF. Aortic aneurysms: current status of surgical treatment. Surg Clin North Am 1979; 59: 597-636. 9 TAYLORLM, PORTER JM. Basic data related to clinical decisionmaking in abdominal aortic aneurysms. Ann Vasc Surg 1986; 1: 502-504. 10 HERTZERNR, YOUNGJR, BEVENEG, et al. Late results of coronary bypass in patients presenting with lower extremity ischemia: the Cleveland Clinic study. Ann Vasc Surg 1986; 1: 411-420. 11 LUNDELL L, NORB.KCKB. Abdominal aortic aneurysms--results of treatment in nonspecialized units. Acta Chit Scand 1983; 149: 695-702.
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12 CRONENWETT JL, MURPHY TF, ZELENOCK GB, et al. Actuarial analysis of variables associated with rupture of small abdominal aortic aneurysms. Surgery 1985; 98: 472-483. 13 BERNSTEIN EF, CHAN EL. Abdominal aortic aneurysm in highrisk patients. Ann Surg 1984; 200: 255-263. 14 NEVITT MP, BALLARDDJ, HALLETTJW. Prognosis of abdominal aortic aneurysms: a population based study. N Engl J Med 1989; 321: 1009-1014. 15 COLLIN J, ARAUIO L, WALTON J. How fast do very small aortic aneurysms grow? Eur ] Vasc Surg 1989; 3: 15-17. 16 WALSH AKM, BRIEEA N, NASH JR, CALLUM KG. The natural history of small abdominal aortic aneurysms: an ultrasound study. Eur J Vasc Surg 1990; 4: 459-461. 17 GLIMAKER H, HOLMBERG L, ELVIN A, et al. Natural history of patients with abdominal aortic aneurysm. Eur J Vasc Surg 1991; 5: 125-130. 18 BERGQVISTD, BENGTSSONH. Risk factors for rupture of abdominal aortic aneurysm. Acta Chir Scand 1990; 156: 63-68. 19 CRONENWETT JL, SARGENT SK, WALL MH, et al. Variables that affect the expansion rate and outcome of small abdominal aortic aneurysms. J Vasc Surg 1990; 11: 260-269. 20 HEATHERB, COLLIN J, WALTONJ. Optimum rescreening interval for abdominal aortic aneurysm. Br J Surg 1991; 78:3652 21 DELIN A, OHLSI~N H, SWEDENBORGJ. Growth rate of abdominal aortic aneurysms as measured by computed tomography. Br J Surg 1985; 72: 530-532. 22 BERNSTEIN EF, DILLEY RB, GOLDENBERGER LE, GOSINK BB, LEOPOLD GR. Growth rates of small abdominal aortic aneurysms. Surgery 1976; 80: 765-773. 23 GOLDMAN L, CALDERA DL, NUSSBAUM SR, et aI. Multifactorial index of cardiac risk in noncardiac surgical procedures. N Engl J Med 1977; 297: 845-850. 24 EAGLEKA, BOUCHER CA. Cardiac risk of noncardiac surgery. N Engl J Med 1989; 321: 1330-1332. 25 ROHRER MJ, CUTLER BS, WHEELER B. Long-term survival and quality of life following ruptured abdominal aortic aneurysm. Arch Surg 1988; 123: 1213-1217. 26 JOHNSON G, GURRI JA, BURNHAM SJ. Life expectancy after abdominal aneurysm repair. In: BERGANJJ, YAO JST, eds. Aneurysms: Diagnosis and Treatment. New York: Grune & Stratton, 1982; 279-285. 27 EUROPEANCORONARYSURGERYSTUDYGROUP. Long-term results of prospective randomised study of coronary artery bypass surgery in stable angina pectoris. Lancet 1982; ii: 1173-1180. 28 CRAWFORD ES, SALEH SA, BABBJW, GLAESERDH, VACCARO PS, SILVERS A. Infrarenal abdominal aortic aneurysm: factors influencing survival after operation performed over a 25-year period. Ann Surg 1981; 193: 699-709. 29 PILCHERDB, DAVISJH, ASHIKAGART, etal. Treatment ofabdominal aortic aneurysm in a single state over 7½ years. Am J Surg 1980; 139: 487-494. 30 DARLING RC, MESSINA CR, BREWSTER DC, OTTINGER L W . Autopsy study of unoperated aortic aneurysms. Circulation 1977; 56: 161-164. 31 SCOTTRAP, ASHTON HA, KAY DN. Screening and patient selection in aortic aneurysms. Current Prac Surg 1990; 2: 68-71. 32 MIYAMOTO JM, ERAKER SA. Parameter estimates for a QALY utility model. Med Decis Making 1985; 5: 191-213, 33 LLEWELLYN-THOMASH, SUTHERLANDHJ, TIBSHIRANIR, CIAMPIA, TILL JE, BOYDNF. The measurement of patients' values in medicine. Med Decis Making 1982; 2: 449-462. 34 JOHANSSONG, SWEDENBORGJ. Ruptured abdominal aortic aneurysms: a study of incidence and mortality. Br J Surg 1986; 73: 101-103.
Accepted 14 April 1992
Eur J Vasc Surg Vol 6, September 1992
