An Improved Statistical Method for Assessing the Results of Operation Gary L. Grunkemeier, Ph.D., Louis E. Lambert, M.S., Lawrence I. Bonchek, M.D., a n d Albert Starr, M.D.
ABSTRACT The advantages of the actuarial method of analyzing postoperative survival are now widely accepted. In evaluating the late results of heart valve replacement operations, however, certain nonfatal complications must also be considered. The use of individual event-free rates does not portray individual risk, since some valves have multiple complications. By using cumulative complication-free rates, however, it is possible to estimate the percentage of valves active and free from certain major complications for each postoperative interval. A series of actuarial curves is used, each of which represents a different major complication. These curves are subtracted progressively from the total of valves at risk, and an actuarial representation of complication-free survival results.
T
his report extends the actuarial method for analysis of postoperative events to include the use of cumulative complicationfree rates (or curves). The new method is compared with standard methods of analyzing survival and complication-free rates, and the appropriate applications of each are illustrated with data from the long-term follow-up of prosthetic heart valve replacements.
Statistical Methods SURVIVAL RATES
The use of actuarial survival rates (life-table techniques) for analyzing postoperative follow-up data is widely accepted. Briefly, a point of entry into the study is established, and the amount of experience in the study contributed by each patient and the most recent status of each (i.e., dead or alive) are used to estimate the percentage of patients expected to survive to each year after operation. These estimates can then be plotted against time as a survival curve for the population of patients. Standard errors can be attached to these estimates to provide approximate confidence intervals. INDIVIDUAL EVENT-FREE RATES
In evaluating the results of certain surgical procedures, not only must mortality be taken into consideration, but also nonfatal postoperative complications From the Division of Cardiopulmonary Surgery, University of Oregon Medical School, Portland, Ore. Supported by U. S. Public Health Service Grant no. HL 16461. Presented at the Eleventh Annual Meeting of The Society of Thoracic Surgeons, Montreal, Que., Canada, Jan. 20-22, 1975. Address reprint requests to Dr. Grunkemeier, 3181 S.W. Sam Jackson Park Rd., Portland, Ore. 97201.
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GRUNKEMEIER E T AL. introduced by the treatment itself. For example, the time-related incidence of thromboembolism, endocarditis, reoperation, and similar complications must be accounted for in any complete assessment of the results of prosthetic valve replacement. In a previous report [ 11 we described a method of analyzing nonfatal postoperative complications using “event-free” rates. This is a generalization of the survival rate method in that the first Occurrence of a given event is substituted for the event “death.” For each postoperative complication, the event-free percentage can be plotted against time to provide a series of event-free curves. Such individual curves for nonfatal events show the time-related occurrences of the events, but they do not show the actual risk as it affects the population since they estimate the percentage of the original population that would be event free if no patients died and no prostheses were replaced. Thus, these individual rates measure the risk of each complication but do not together give the overall risk to each patient, since some patients experience more than one type of nonfatal complication. For example, let us assume that in a group of patients the five-year survival rate is 50%and the five-year embolus-free rate is also 50%.From this information, all that can be said about the percentage of patients who are alive and free from embolus at five years is that it may be zero, SO%, or somewhere in between, depending on whether none, all, or some of the deaths occurred in individuals who had previously suffered an embolus. This in turn depends on whether the events are independent or related. To cite just two examples of related multiple complications in a single valve, almost all patients with infected prostheses progress to subsequent reoperation or death, and those with hemolytic anemia are often reoperated upon for cloth tear. Isolated event-free curves will not convey data of this type, yet this information is vital for making accurate comparisons of two alternative prostheses or of various possible treatments of any malady with which major nonfatal complications are associated. CUMULATIVE COMPLICATION-FREE RATES
The solution to this problem is the use of cumulative complication-free rates. A simple example will illustrate the principle involved. Suppose there are two complications: death (D) and thromboembolism (E). The yearly survival (D-free) rates are determined in the usual manner. Then these two complications are combined into a singlecompound event, death and/or thromboembolism (D/E),and the yearly event-free rates are determined for this compound event, that is, the D/E-free rates, in the same manner as if D/E were a single event. This means counting as the event whichever occurs first to a given valve, D or E, since this is the first occurrence of the event D/E and provides the percentage of patients free from both death and thromboembolism. These rates may be depicted graphically in the usual way and the resulting two curves superimposed (Fig. 1). Subtracting the D/E-free rate from the D-free survival rate calculated for the same time postoperatively gives the percentage of the original group still alive but having suffered a postoperative embolus by that time. In this way, the population at any time after operation is divided into three groups: the percentage dead, the
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. s
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percentage alive with a previous embolus, and the percentage alive and embolus free. It is the latter two percentages whichcannot be estimated from the individual survival and thromboernbolism-free curves and which we consider important in analyzing valve performance from the point of view of assessing the overall risk. This method can be extended to cover several nonfatal complications by first assigning to them a hierarchy of severity or importance, forming nested subsets of these complications beginning with the worst and adding one at a time in order of their severity, and finally superimposing the family of complication-free curves resulting from these nested sets. We call these cumulative complication-free curves since one complication is added at a time to form the compound sets until the remainders are free from all complications. Since the breakdown of the population at any given time postoperatively adds up to loo%, it is possible to present the experience graphically by a “pie” diagram showing those free from each successive complication. REREPLACEMENT VERSUS REMOVAL
At this point it is worthwhile to consider a concept fundamental to these analyses: the distinction between an analysis of results based on patients and one based on valves. This difference becomes evident when attention is focused on valve rereplacement. If the patient’s death were the only end-point for a valve, the analysis would be unambiguous: the patient and valve would both be at risk from the time of implantation, and death would constitute the (unique) end-point for both. But in fact, valves can fail without death occurring because the prosthesis is replaced, i.e., the patient’s own valve is rereplaced. For emphasis, we now choose words which indicate that the analysis is in terms of prostheses rather than patients, such as removal instead of rereplacement, and active instead of alive. However, for convenience, we continue to use death to refer to valves, even though it is the patient who dies. The survival rate as it is calculated in this paper and in that by Anderson and associates [ 11 is not the crude or actually observed rate but the net or underlying rate that would obtain if removals were eliminated as a cause of valve failure. Thus
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Embolus - f r e e Removal - f r e e
Survival
1
0 op
2
I
I
I
I
3
4
5
FIG. 2 . Individual survival, thromboembolism-free, and removal-free curves for 2 2 0 Model 6 3 1 0 - 6 3 2 0 StarrEdwards mitral valves.
Years Postop
the difference between the survival rate and the deathhemoval-free rate represents the percentage of valves that were removed but would still be active if “removal” were eliminated as a cause of valve failure.
Example An illustration of this method follows, using the valve series from the University of Oregon Medical School. This is the same series used in our previous report [ 11so that the current life tables and resulting curves may be compared with the earlier work. That paper presented a life table for the totally cloth covered Model 6310-6320 Starr-Edwards mitral valve based on 206 implants since 1968 and a modified life table for thromboembolism, then used them to construct individual survival and embolus-free curves. Using the same techniques we have constructed updated curves for this same series, now totaling 220 valves. The individual event-free curves for death, removal, and thromboembolism are superimposed in Figure 2. Figure 3 contains the cumulative complication-free curves for the same three events based on the same series. (Life tables and details of construction are given in the Appendix.) As mentioned, curves for additional complicating events could be added in an analogous way. For comparing series by this method it is essential that the hierarchy of complications be taken in the same order, as this affects the final decomposition of the groups. It should begin with death and FIG. 3 . Cumulative survival, death1 removal-free, and deathlremovall embolus-free curves for 220 Model 631 0 -6320 mitral valves.
100
“Dead”
70
-
960
-
Q
ki
$
8
o w
+SE
Active, Embolus-free
N.220 I
I
I
2
I
3
4
5
Years Postoo
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I
Statistical Assessment
loo
of Results of Operation
I
, 0 op
I
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3
4
I 4
Active, 1 Embolus-free
' O - -
-0--
I _
,
5
_ --0 I
6
7
8
9
Years Postop
FIG. 4 . Cumulative survival and deathlembolus-free curves for 86 Model 6 1 2 0 mitral valves. (Dashed lines indicate fewer than 25 valves at risk.)
removal (both terminal events) and proceed, in order, to the other, nonterminal complications. We performed similar analyses on the non-cloth-covered Model 6 120 mitral prosthesis, in use since 1965, based on our total series of 86 valves. Figure 4 contains the cumulative complication-free curves for death and thromboembolism (there were no removals). These two groups of patients differ in several ways, including year of operation and percentage of patients receiving various types of anticoagulant management, so that conclusions regarding differences in the performance of these prostheses should be drawn cautiously. Nevertheless, for comparison, Figures 5 and 6 present pie diagrams representing breakdown by postoperative status for each series at three and five years, respectively, with the deaths further separated into operative and late and the emboli separated into those with residual neurological deficits and those without.
model 6120 N = 86 (N$= 84)
model 6310-20 N = 220 (N; = 166)
FIG. 5 . Comparison of cumulative breakdown of mitral valve experience three years postoperatively, Model 6120 (left) versus Model 6 3 1 0 - 6 3 2 0 (right). (N* = effective sample size; see text.)
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model 6120 N = 86 (Nj= 80)
-
model 6310 20 N = 220 ( N g = 116)
FIG. 6. Comparison of cumulative breakdown of mitral valve experience f i v e years postoperatively, Model 6 1 2 0 (left) versus Model 6 3 1 0 - 6 3 2 0 (right). (N* = effective sample size; see text.)
The effective sample sizes are indicated by N* in Figures 5 and 6. The effective sample size is the number of valves that would have to be at risk and under continuous observation with no withdrawals in order to give an estimate having the same precision as that actually obtained in the face of withdrawals [6].
Comment The actuarial method of survival analysis has been described by several authors [ 1 , 2 , 6 ]and is becoming widely used in reporting long-term follow-up of prosthetic valve replacements [4,5,8]. Cumulative complication-free curves constitute a valuable addition to the standard individual curves by providing estimates of the percentage free from all complications (and various subsets of them) at given times postoperatively. The present report was inspired by early work from the Mayo Clinic [71 which used this approach but did not explain the method. We have given the procedures involved and examples of their usefulness in describing and comparing two series of mitral valves (see Figs. 5 , 6 ) . Some implications from the diagrams will be discussed, but complete clinical information on these valve series is outside the scope of this report and is being published elsewhere [3]. Our figures contain operative mortality. For the analysis of valve performance it is perhaps more meaningful to exclude this variable, which is a function of nonprosthetic factors, and to include only operative survivors. In fact, when the group under consideration is defined by a specific medical regimen, e.g., chronic anticoagulant therapy, only late survival is, in general, meaningful. On the other hand, exclusion of operative mortality may also bias comparisons between valves, since operative deaths probably correspond to the sickest patients, who, if they survive operation, will be more prone to die early postoperatively. This could explain the differences in the breakdown of deaths between the two models of mitral valves. The other notable difference between these valves, as shown in Figures 5 and 6, is that there were some removals but also a compensating reduction in the number of embolic episodes with the cloth-covered prosthesis. The majority of the
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Statistical Assessment of Results .f Operation emboli produced no residual neurological deficit, however, and if these are ignored, the percentage active and free from complicationsat five years increases to 68% for Model 6120 and 69% for Model 6310-6320 valves. In any case, cumulative complication reporting provides valuable comparative descriptions and assists one in making a reasonable choice among various prostheses having comparable overall results.
Appendix: Technical Details As in the previous report for individual event-free rates [l], we divide the worksheet into two tables, one for death and removal (Table 1) and the other for thromboembolism (Table 2).We begin by introducing the notation used in Table 1. The subscript k denotes the postoperative year, though it could refer to an interval of any other length (and actually does in the case of the operative period).
Nk = number of valves entering the kth year. dk = number of valves “dying” in the kth year. rk = number of valves removed in the kth year. wk = number of valves withdrawn in the kth year, i.e., only completing part of thekth year but not falling into either dkor rk(see below for further discussion). = proportion of valves not dying in the kth year. The formula for computing the proportion from the above variables is given below. The superscripts on these proportions (and the rates below) refer to the event(s) in question, in this case death only. = the kth year survival rate. Throughout, we refer to this as a rate (and use capital P) to distinguish it from the intervalpofmrtions (lowercase p) above, from which it is derived using Equation 4 below. $= proportion not dying or removed in the kth year. l‘i?= the kth year deathhemoval-free (d/r-free) rate. Note that N, is the total number of valves entering the study and
The formulas for computing the interval event-free proportions are:
and for the corresponding kth year event-free rates:
which holds for all superscripts. Equation 4 shows that thekth year event-free rate VOL. 20, NO. 3, SEPTEMBER, 1975
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GRUNKEMEIER ET AL. TABLE 1 . VARIABLES NEEDED FOR DEATH AND REMOVAL ANALYSIS OF 220 MITRAL VALVES*
op. 1 2 3 4 5
220 21 1 172 123 86 52
7 12 9 7
1 5 3 0 0 1
1
1
1 22 37 30 33 35
0.97 0.94 0.94 0.94 0.99 0.97
0.97 0.91 0.86 0.80 0.79 0.77
0.96 0.92 0.92 0.94 0.99 0.94
0.96 0.88 0.81 0.76 0.75 0.71
*See Appendix for definitions.
is computed by multiplying together the event-free proportions for all years up to and including the kth. This is the basis of the actuarial method and allows the survival curve to be calculated without the assumption of an underlying curve or decrement function. The values for these variables are listed in Table 1. The chief departure from the corresponding table in our previous report is that the “removed for cause” group (referring to patients) has been separated into (i) “removed” (referring to prostheses), which we consider a complication, and (ii)“stagewise”multiple cases, which are included, along with the “lost to follow-up” and “surviving an incomplete interval” groups, into the composite group we call “withdrawn.”This combination conforms with standard practice, since these groups are all treated similarly in the formulas, and is done for notational simplicity.We recognize, however, the practical importance of considering the (clinically)different groups that compose the withdrawn valves. The variables needed to compute the death/removaV thromboembolism-free rates are given below, a prime sign being used-to denote embolus-free.
NL = number of valves entering kth year embolus-free. d,: = number of embolus-free deaths in kth year. rl: = number of embolus-free removals in kth year. el, = number of (first) emboli in kth year. wl: = number withdrawn embolus-free in kth year. # = proportion of valves deathhemovaVembolus-freeduring kth year. P? = kth year deathlremovaVembolus-free rate. TABLE 2. VARIABLES NEEDED FOR THROMBOEMBOLISM ANALYSIS OF 220 MITRAL VALVES*
~
op. 1 2 3 4 5
220 204 162 113 81 49
7 12 9 6 1 1
1 5 3 0 0 1
*See Appendix for definitions. 296
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6
5 1 0 0 0
1 21 36 26 31 32
0.94 0.89 0.9 1 0.94 0.98 0.94
0.94 0.83 0.75 0.71 0.70 0.66
Statistical Assessment of Results of Operation As before, N; is the total number of valves entering the study, and N,'+l = NI: - (ek + d{ p"k"= 1 - (ek + d,'
+
rl:
+ wi)
+ rL)/(Nk)- %w;)
with Ppobtained, as before, by multiplication (Equation 4).The values of these variables for our example are given in Table 2. The curves resulting from these three sets of cumulative complication-free rates are graphed in Figure 3.
Note: In our previous publication [l], the word cumulative was sometimes used to distinguish rates (P) from proportions (p) and did not have the technical implications it has throughout this paper.
References 1 . Anderson, R. P., Bonchek, L. I., Grunkemeier, G. E., Lambert, L. E., and Starr, A. T h e analysis and presentation of surgical results by actuarial methods.J Surg Res 16:224, 1974. 2. Berkson, J., and Gage, R. P. Calculation of survival rates for cancer. Proc Staff Meet Mayo Clin 25:270, 1950. 3. Bonchek, L. I., and Starr, A. Ball valve prostheses: Current appraisal of late results. A m J Cardiol 35:843, 1975. 4. Braun, L. O., Kincaid, 0. W., and McGoon, D. C. Prognosis of aortic valve replacement in relation to the preoperative heart size.J Thorac Cardiovasc Surg 65:381, 1973. 5. Carpentier, A., Deloche, A., Fabiani, J. N., Forman, J., Camilleri, J. P., Soyer, R., and Dubost, C. Six-year follow-up of glutaraldehyde-preserved heterografts. J Thorac Cardiovasc Surg 68:771, 1974. 6. Cutler, S.J., and Elderer, F. Maximum utilization of the life table method in analyzing survival. J Chronic DZr 8:699, 1958. 7. Duvoisin, G. E., Wallace, R. B., Ellis, F. H., Jr., Anderson, M. W., and McGoon, D. C. Late results of cardiac-valve replacement. Circulation 37,38 (Suppl II):75, 1968. 8. Isom, 0.W., Dembrow, J. M., Glassman, E., Pasternak, B. S., Sackler,J. P., and Spencer, F. C. Factors influencing long-term survival after isolated aortic valve replacement. Circulation 49, 50 (Suppl 11):154, 1974. 9. Starr, A., Bonchek, L. I., Anderson, R. P., Wood, J. A., and Chapman, R. D. Late complications of aortic valve replacement with cloth-covered composite-seat prostheses: A six-year appraisal. A n n ThorRc Surg 19:289, 1975.
Discussion DR. DONALD A. BARNHORST (Rochester, Minn.): T h e points made by the authors concern a series from the University of Oregon which is a fine example of excellent, meticulous follow-up. Their experience with prosthetic valve replacement is a model for assessing the complications and benefits accompanying these devices. As Dr. Grunkemeier mentioned, we at the Mayo Clinic became interested in publishing our data in actuarial fashion in 1968. In brief, the paper he alluded to, authored by George Duvoisin and others, reported not only death but also the various complications attendant upon aortic valve replacement at that point in time. It was principally through the urging of Dr. Lila Elveback, of our section of statistics, that we began reporting data in this fashion. In their two most recent papers, Dr. Grunkemeier and his colleagues have extended these concepts and, perhaps even more importantly, have put into readily understood detail the mechanics of how these computations should be done, so that those of us less expert in statistics can apply them to o u r own data. T h e concept of cumulative complication-free rate introduced by Dr. Grunkemeier is another advance in our ability to VOL. 20, NO. 3, SEPTEMBER, 1975
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GRUNKEMEIER E T AL. assess the benefits and complications that have been discussed in such detail at today’s session. I thank Dr. Barnhorst for his favorable comments and again DR. GRUNKEMEIER: acknowledge our indebtedness to the Mayo Clinic for initially influencing us to adopt the use of survival rate analysis. Our message is that we must obtain the maximum information from our data, by applying the most exact and efficient method of analysis, in order to make the ultimate decision ofwhat course to follow with the next patient. This goal pertains to all areas of cardiac surgery but especially to valve replacement, in which a controlled clinical trial with randomized patients is not feasible due to the evolutionary nature of prosthesis development.
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ABSTRACT The advantages of the actuarial method of analyzing postoperative survival are now widely accepted. In evaluating the late results of heart valve replacement operations, however, certain nonfatal complications must also be considered. The use of individual event-free rates does not portray individual risk, since some valves have multiple complications. By using cumulative complication-free rates, however, it is possible to estimate the percentage of valves active and free from certain major complications for each postoperative interval. A series of actuarial curves is used, each of which represents a different major complication. These curves are subtracted progressively from the total of valves at risk, and an actuarial representation of complication-free survival results.
T
his report extends the actuarial method for analysis of postoperative events to include the use of cumulative complicationfree rates (or curves). The new method is compared with standard methods of analyzing survival and complication-free rates, and the appropriate applications of each are illustrated with data from the long-term follow-up of prosthetic heart valve replacements.
Statistical Methods SURVIVAL RATES
The use of actuarial survival rates (life-table techniques) for analyzing postoperative follow-up data is widely accepted. Briefly, a point of entry into the study is established, and the amount of experience in the study contributed by each patient and the most recent status of each (i.e., dead or alive) are used to estimate the percentage of patients expected to survive to each year after operation. These estimates can then be plotted against time as a survival curve for the population of patients. Standard errors can be attached to these estimates to provide approximate confidence intervals. INDIVIDUAL EVENT-FREE RATES
In evaluating the results of certain surgical procedures, not only must mortality be taken into consideration, but also nonfatal postoperative complications From the Division of Cardiopulmonary Surgery, University of Oregon Medical School, Portland, Ore. Supported by U. S. Public Health Service Grant no. HL 16461. Presented at the Eleventh Annual Meeting of The Society of Thoracic Surgeons, Montreal, Que., Canada, Jan. 20-22, 1975. Address reprint requests to Dr. Grunkemeier, 3181 S.W. Sam Jackson Park Rd., Portland, Ore. 97201.
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GRUNKEMEIER E T AL. introduced by the treatment itself. For example, the time-related incidence of thromboembolism, endocarditis, reoperation, and similar complications must be accounted for in any complete assessment of the results of prosthetic valve replacement. In a previous report [ 11 we described a method of analyzing nonfatal postoperative complications using “event-free” rates. This is a generalization of the survival rate method in that the first Occurrence of a given event is substituted for the event “death.” For each postoperative complication, the event-free percentage can be plotted against time to provide a series of event-free curves. Such individual curves for nonfatal events show the time-related occurrences of the events, but they do not show the actual risk as it affects the population since they estimate the percentage of the original population that would be event free if no patients died and no prostheses were replaced. Thus, these individual rates measure the risk of each complication but do not together give the overall risk to each patient, since some patients experience more than one type of nonfatal complication. For example, let us assume that in a group of patients the five-year survival rate is 50%and the five-year embolus-free rate is also 50%.From this information, all that can be said about the percentage of patients who are alive and free from embolus at five years is that it may be zero, SO%, or somewhere in between, depending on whether none, all, or some of the deaths occurred in individuals who had previously suffered an embolus. This in turn depends on whether the events are independent or related. To cite just two examples of related multiple complications in a single valve, almost all patients with infected prostheses progress to subsequent reoperation or death, and those with hemolytic anemia are often reoperated upon for cloth tear. Isolated event-free curves will not convey data of this type, yet this information is vital for making accurate comparisons of two alternative prostheses or of various possible treatments of any malady with which major nonfatal complications are associated. CUMULATIVE COMPLICATION-FREE RATES
The solution to this problem is the use of cumulative complication-free rates. A simple example will illustrate the principle involved. Suppose there are two complications: death (D) and thromboembolism (E). The yearly survival (D-free) rates are determined in the usual manner. Then these two complications are combined into a singlecompound event, death and/or thromboembolism (D/E),and the yearly event-free rates are determined for this compound event, that is, the D/E-free rates, in the same manner as if D/E were a single event. This means counting as the event whichever occurs first to a given valve, D or E, since this is the first occurrence of the event D/E and provides the percentage of patients free from both death and thromboembolism. These rates may be depicted graphically in the usual way and the resulting two curves superimposed (Fig. 1). Subtracting the D/E-free rate from the D-free survival rate calculated for the same time postoperatively gives the percentage of the original group still alive but having suffered a postoperative embolus by that time. In this way, the population at any time after operation is divided into three groups: the percentage dead, the
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Statistical Assessment of Results of Operation 100
9, 9,
I
80
. s
c
Deod
I
60
Gi
20
0 ‘o
,
Alive, Post-embolus
-7-
rn
: Survivol
0
= Deoth/embolur-tree
Alive, Embolus-free
I
percentage alive with a previous embolus, and the percentage alive and embolus free. It is the latter two percentages whichcannot be estimated from the individual survival and thromboernbolism-free curves and which we consider important in analyzing valve performance from the point of view of assessing the overall risk. This method can be extended to cover several nonfatal complications by first assigning to them a hierarchy of severity or importance, forming nested subsets of these complications beginning with the worst and adding one at a time in order of their severity, and finally superimposing the family of complication-free curves resulting from these nested sets. We call these cumulative complication-free curves since one complication is added at a time to form the compound sets until the remainders are free from all complications. Since the breakdown of the population at any given time postoperatively adds up to loo%, it is possible to present the experience graphically by a “pie” diagram showing those free from each successive complication. REREPLACEMENT VERSUS REMOVAL
At this point it is worthwhile to consider a concept fundamental to these analyses: the distinction between an analysis of results based on patients and one based on valves. This difference becomes evident when attention is focused on valve rereplacement. If the patient’s death were the only end-point for a valve, the analysis would be unambiguous: the patient and valve would both be at risk from the time of implantation, and death would constitute the (unique) end-point for both. But in fact, valves can fail without death occurring because the prosthesis is replaced, i.e., the patient’s own valve is rereplaced. For emphasis, we now choose words which indicate that the analysis is in terms of prostheses rather than patients, such as removal instead of rereplacement, and active instead of alive. However, for convenience, we continue to use death to refer to valves, even though it is the patient who dies. The survival rate as it is calculated in this paper and in that by Anderson and associates [ 11 is not the crude or actually observed rate but the net or underlying rate that would obtain if removals were eliminated as a cause of valve failure. Thus
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Embolus - f r e e Removal - f r e e
Survival
1
0 op
2
I
I
I
I
3
4
5
FIG. 2 . Individual survival, thromboembolism-free, and removal-free curves for 2 2 0 Model 6 3 1 0 - 6 3 2 0 StarrEdwards mitral valves.
Years Postop
the difference between the survival rate and the deathhemoval-free rate represents the percentage of valves that were removed but would still be active if “removal” were eliminated as a cause of valve failure.
Example An illustration of this method follows, using the valve series from the University of Oregon Medical School. This is the same series used in our previous report [ 11so that the current life tables and resulting curves may be compared with the earlier work. That paper presented a life table for the totally cloth covered Model 6310-6320 Starr-Edwards mitral valve based on 206 implants since 1968 and a modified life table for thromboembolism, then used them to construct individual survival and embolus-free curves. Using the same techniques we have constructed updated curves for this same series, now totaling 220 valves. The individual event-free curves for death, removal, and thromboembolism are superimposed in Figure 2. Figure 3 contains the cumulative complication-free curves for the same three events based on the same series. (Life tables and details of construction are given in the Appendix.) As mentioned, curves for additional complicating events could be added in an analogous way. For comparing series by this method it is essential that the hierarchy of complications be taken in the same order, as this affects the final decomposition of the groups. It should begin with death and FIG. 3 . Cumulative survival, death1 removal-free, and deathlremovall embolus-free curves for 220 Model 631 0 -6320 mitral valves.
100
“Dead”
70
-
960
-
Q
ki
$
8
o w
+SE
Active, Embolus-free
N.220 I
I
I
2
I
3
4
5
Years Postoo
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I
Statistical Assessment
loo
of Results of Operation
I
, 0 op
I
2
3
4
I 4
Active, 1 Embolus-free
' O - -
-0--
I _
,
5
_ --0 I
6
7
8
9
Years Postop
FIG. 4 . Cumulative survival and deathlembolus-free curves for 86 Model 6 1 2 0 mitral valves. (Dashed lines indicate fewer than 25 valves at risk.)
removal (both terminal events) and proceed, in order, to the other, nonterminal complications. We performed similar analyses on the non-cloth-covered Model 6 120 mitral prosthesis, in use since 1965, based on our total series of 86 valves. Figure 4 contains the cumulative complication-free curves for death and thromboembolism (there were no removals). These two groups of patients differ in several ways, including year of operation and percentage of patients receiving various types of anticoagulant management, so that conclusions regarding differences in the performance of these prostheses should be drawn cautiously. Nevertheless, for comparison, Figures 5 and 6 present pie diagrams representing breakdown by postoperative status for each series at three and five years, respectively, with the deaths further separated into operative and late and the emboli separated into those with residual neurological deficits and those without.
model 6120 N = 86 (N$= 84)
model 6310-20 N = 220 (N; = 166)
FIG. 5 . Comparison of cumulative breakdown of mitral valve experience three years postoperatively, Model 6120 (left) versus Model 6 3 1 0 - 6 3 2 0 (right). (N* = effective sample size; see text.)
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model 6120 N = 86 (Nj= 80)
-
model 6310 20 N = 220 ( N g = 116)
FIG. 6. Comparison of cumulative breakdown of mitral valve experience f i v e years postoperatively, Model 6 1 2 0 (left) versus Model 6 3 1 0 - 6 3 2 0 (right). (N* = effective sample size; see text.)
The effective sample sizes are indicated by N* in Figures 5 and 6. The effective sample size is the number of valves that would have to be at risk and under continuous observation with no withdrawals in order to give an estimate having the same precision as that actually obtained in the face of withdrawals [6].
Comment The actuarial method of survival analysis has been described by several authors [ 1 , 2 , 6 ]and is becoming widely used in reporting long-term follow-up of prosthetic valve replacements [4,5,8]. Cumulative complication-free curves constitute a valuable addition to the standard individual curves by providing estimates of the percentage free from all complications (and various subsets of them) at given times postoperatively. The present report was inspired by early work from the Mayo Clinic [71 which used this approach but did not explain the method. We have given the procedures involved and examples of their usefulness in describing and comparing two series of mitral valves (see Figs. 5 , 6 ) . Some implications from the diagrams will be discussed, but complete clinical information on these valve series is outside the scope of this report and is being published elsewhere [3]. Our figures contain operative mortality. For the analysis of valve performance it is perhaps more meaningful to exclude this variable, which is a function of nonprosthetic factors, and to include only operative survivors. In fact, when the group under consideration is defined by a specific medical regimen, e.g., chronic anticoagulant therapy, only late survival is, in general, meaningful. On the other hand, exclusion of operative mortality may also bias comparisons between valves, since operative deaths probably correspond to the sickest patients, who, if they survive operation, will be more prone to die early postoperatively. This could explain the differences in the breakdown of deaths between the two models of mitral valves. The other notable difference between these valves, as shown in Figures 5 and 6, is that there were some removals but also a compensating reduction in the number of embolic episodes with the cloth-covered prosthesis. The majority of the
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Statistical Assessment of Results .f Operation emboli produced no residual neurological deficit, however, and if these are ignored, the percentage active and free from complicationsat five years increases to 68% for Model 6120 and 69% for Model 6310-6320 valves. In any case, cumulative complication reporting provides valuable comparative descriptions and assists one in making a reasonable choice among various prostheses having comparable overall results.
Appendix: Technical Details As in the previous report for individual event-free rates [l], we divide the worksheet into two tables, one for death and removal (Table 1) and the other for thromboembolism (Table 2).We begin by introducing the notation used in Table 1. The subscript k denotes the postoperative year, though it could refer to an interval of any other length (and actually does in the case of the operative period).
Nk = number of valves entering the kth year. dk = number of valves “dying” in the kth year. rk = number of valves removed in the kth year. wk = number of valves withdrawn in the kth year, i.e., only completing part of thekth year but not falling into either dkor rk(see below for further discussion). = proportion of valves not dying in the kth year. The formula for computing the proportion from the above variables is given below. The superscripts on these proportions (and the rates below) refer to the event(s) in question, in this case death only. = the kth year survival rate. Throughout, we refer to this as a rate (and use capital P) to distinguish it from the intervalpofmrtions (lowercase p) above, from which it is derived using Equation 4 below. $= proportion not dying or removed in the kth year. l‘i?= the kth year deathhemoval-free (d/r-free) rate. Note that N, is the total number of valves entering the study and
The formulas for computing the interval event-free proportions are:
and for the corresponding kth year event-free rates:
which holds for all superscripts. Equation 4 shows that thekth year event-free rate VOL. 20, NO. 3, SEPTEMBER, 1975
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GRUNKEMEIER ET AL. TABLE 1 . VARIABLES NEEDED FOR DEATH AND REMOVAL ANALYSIS OF 220 MITRAL VALVES*
op. 1 2 3 4 5
220 21 1 172 123 86 52
7 12 9 7
1 5 3 0 0 1
1
1
1 22 37 30 33 35
0.97 0.94 0.94 0.94 0.99 0.97
0.97 0.91 0.86 0.80 0.79 0.77
0.96 0.92 0.92 0.94 0.99 0.94
0.96 0.88 0.81 0.76 0.75 0.71
*See Appendix for definitions.
is computed by multiplying together the event-free proportions for all years up to and including the kth. This is the basis of the actuarial method and allows the survival curve to be calculated without the assumption of an underlying curve or decrement function. The values for these variables are listed in Table 1. The chief departure from the corresponding table in our previous report is that the “removed for cause” group (referring to patients) has been separated into (i) “removed” (referring to prostheses), which we consider a complication, and (ii)“stagewise”multiple cases, which are included, along with the “lost to follow-up” and “surviving an incomplete interval” groups, into the composite group we call “withdrawn.”This combination conforms with standard practice, since these groups are all treated similarly in the formulas, and is done for notational simplicity.We recognize, however, the practical importance of considering the (clinically)different groups that compose the withdrawn valves. The variables needed to compute the death/removaV thromboembolism-free rates are given below, a prime sign being used-to denote embolus-free.
NL = number of valves entering kth year embolus-free. d,: = number of embolus-free deaths in kth year. rl: = number of embolus-free removals in kth year. el, = number of (first) emboli in kth year. wl: = number withdrawn embolus-free in kth year. # = proportion of valves deathhemovaVembolus-freeduring kth year. P? = kth year deathlremovaVembolus-free rate. TABLE 2. VARIABLES NEEDED FOR THROMBOEMBOLISM ANALYSIS OF 220 MITRAL VALVES*
~
op. 1 2 3 4 5
220 204 162 113 81 49
7 12 9 6 1 1
1 5 3 0 0 1
*See Appendix for definitions. 296
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6
5 1 0 0 0
1 21 36 26 31 32
0.94 0.89 0.9 1 0.94 0.98 0.94
0.94 0.83 0.75 0.71 0.70 0.66
Statistical Assessment of Results of Operation As before, N; is the total number of valves entering the study, and N,'+l = NI: - (ek + d{ p"k"= 1 - (ek + d,'
+
rl:
+ wi)
+ rL)/(Nk)- %w;)
with Ppobtained, as before, by multiplication (Equation 4).The values of these variables for our example are given in Table 2. The curves resulting from these three sets of cumulative complication-free rates are graphed in Figure 3.
Note: In our previous publication [l], the word cumulative was sometimes used to distinguish rates (P) from proportions (p) and did not have the technical implications it has throughout this paper.
References 1 . Anderson, R. P., Bonchek, L. I., Grunkemeier, G. E., Lambert, L. E., and Starr, A. T h e analysis and presentation of surgical results by actuarial methods.J Surg Res 16:224, 1974. 2. Berkson, J., and Gage, R. P. Calculation of survival rates for cancer. Proc Staff Meet Mayo Clin 25:270, 1950. 3. Bonchek, L. I., and Starr, A. Ball valve prostheses: Current appraisal of late results. A m J Cardiol 35:843, 1975. 4. Braun, L. O., Kincaid, 0. W., and McGoon, D. C. Prognosis of aortic valve replacement in relation to the preoperative heart size.J Thorac Cardiovasc Surg 65:381, 1973. 5. Carpentier, A., Deloche, A., Fabiani, J. N., Forman, J., Camilleri, J. P., Soyer, R., and Dubost, C. Six-year follow-up of glutaraldehyde-preserved heterografts. J Thorac Cardiovasc Surg 68:771, 1974. 6. Cutler, S.J., and Elderer, F. Maximum utilization of the life table method in analyzing survival. J Chronic DZr 8:699, 1958. 7. Duvoisin, G. E., Wallace, R. B., Ellis, F. H., Jr., Anderson, M. W., and McGoon, D. C. Late results of cardiac-valve replacement. Circulation 37,38 (Suppl II):75, 1968. 8. Isom, 0.W., Dembrow, J. M., Glassman, E., Pasternak, B. S., Sackler,J. P., and Spencer, F. C. Factors influencing long-term survival after isolated aortic valve replacement. Circulation 49, 50 (Suppl 11):154, 1974. 9. Starr, A., Bonchek, L. I., Anderson, R. P., Wood, J. A., and Chapman, R. D. Late complications of aortic valve replacement with cloth-covered composite-seat prostheses: A six-year appraisal. A n n ThorRc Surg 19:289, 1975.
Discussion DR. DONALD A. BARNHORST (Rochester, Minn.): T h e points made by the authors concern a series from the University of Oregon which is a fine example of excellent, meticulous follow-up. Their experience with prosthetic valve replacement is a model for assessing the complications and benefits accompanying these devices. As Dr. Grunkemeier mentioned, we at the Mayo Clinic became interested in publishing our data in actuarial fashion in 1968. In brief, the paper he alluded to, authored by George Duvoisin and others, reported not only death but also the various complications attendant upon aortic valve replacement at that point in time. It was principally through the urging of Dr. Lila Elveback, of our section of statistics, that we began reporting data in this fashion. In their two most recent papers, Dr. Grunkemeier and his colleagues have extended these concepts and, perhaps even more importantly, have put into readily understood detail the mechanics of how these computations should be done, so that those of us less expert in statistics can apply them to o u r own data. T h e concept of cumulative complication-free rate introduced by Dr. Grunkemeier is another advance in our ability to VOL. 20, NO. 3, SEPTEMBER, 1975
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GRUNKEMEIER E T AL. assess the benefits and complications that have been discussed in such detail at today’s session. I thank Dr. Barnhorst for his favorable comments and again DR. GRUNKEMEIER: acknowledge our indebtedness to the Mayo Clinic for initially influencing us to adopt the use of survival rate analysis. Our message is that we must obtain the maximum information from our data, by applying the most exact and efficient method of analysis, in order to make the ultimate decision ofwhat course to follow with the next patient. This goal pertains to all areas of cardiac surgery but especially to valve replacement, in which a controlled clinical trial with randomized patients is not feasible due to the evolutionary nature of prosthesis development.
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