Prognostic Value of BisoprololInduced Hemodynamic Effects in Heart Failure During the Cardiac Insufficiency BIsoprolol Study (CIBIS)
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Abstract
Background To further evaluate the mechanism of βblocker–induced benefits in heart failure, the relationships between bisoprololinduced hemodynamic effects and survival were studied during the Cardiac Insufficiency BIsoprolol Study (CIBIS).
Methods and Results In 557 patients studied, bisoprolol significantly reduced heart rate (−16.3±15.3 versus −1.6±13.4 bpm, respectively; P<.001) compared with placebo at 2 months after inclusion in the study. Heart rate change over time had the highest predictive value for survival (P<.01). Left ventricular fractional shortening (LVFS) significantly increased in the bisoprolol group compared with the placebo group 5 months after inclusion (+0.04±0.06 versus −0.001±0.05, respectively; P<.001; n=160). LVFS change over time was also significantly correlated with further survival (P<.02 by Cox analysis). Using a nonparametric approach, we demonstrated a significant interaction between study treatment group and LVFS over time. Patients who demonstrated improvement of LVFS over time (82% and 51% of patients in the bisoprolol and the placebo groups, respectively; P<.02) were at lower risk, but the hazard did not further decrease with a further increase of fractional shortening, and there was no significant difference between study treatment groups. Finally, it could be demonstrated that each of the three factors (heart rate change over time, LVFS change over time, and bisoprolol treatment) made a specific contribution to mortality rate.
Conclusions Preservation of left ventricular function appears to play a key role in the bisoprololinduced beneficial effects on prognosis in heart failure. Shortterm βblocker–induced cardiac effects could provide a means to identify those patients who will experience improved survival over the long term.
The potential therapeutic effect of βblocker treatment on survival in heart failure is still being actively investigated through largescale, randomized trials such as CIBIS II^{1} with bisoprolol, BEST^{2} with bucindolol, and MERIT with metoprolol. The underlying cardioprotective mechanism relies on protection against the deleterious consequences of cardiac sympathetic stimulation.^{3} Most randomized clinical trials have already shown that βblocker therapy, when administrated with a progressively increasing dose, can improve symptoms, reduce heart failure–related events, and increase left ventricular ejection fraction.^{4} ^{5} ^{6} ^{7} ^{8} ^{9} ^{10} ^{11}
Encouraging results concerning improved survival have been recorded with bisoprolol during CIBIS^{7} and more recently with carvedilol,^{10} ^{11} but convincing data from largescale trials with survival as a primary end point are still needed to confirm a benefit.
Survival improvement might be related to the amplitude of hemodynamic effects initially induced by βblockers, especially heart rate reduction and left ventricular ejection fraction increase. To better understand the mechanism of this benefit and to characterize patients who could benefit most from such a treatment, we evaluated the relationships between these treatmentinduced hemodynamic effects and survival in CIBIS using a standard Cox multivariate model^{12} and a new, nonparametric approach.^{13}
Methods
Study Design, Hemodynamics, and Left Ventricular Function Measurements
The CIBIS study design and inclusion and exclusion criteria have been described previously.^{7} Briefly, CIBIS was a doubleblind, placebocontrolled, randomized, multicenter European trial that compared two parallel groups of patients with chronic heart failure. Symptomatic (NYHA class III or IV) ambulatory patients with a left ventricular ejection fraction <40% were included.
Baseline Hemodynamics
The following parameters were recorded at baseline and at rest before randomization for all patients included in CIBIS: blood pressure (systolic and diastolic), heart rate, left ventricular ejection fraction, and isotopic or contrast angiography results (within 4 weeks before randomization). Left ventricular echocardiographic evaluation was also performed and the following measurements obtained: EDD, ESD, and fractional shortening [(EDD−ESD)/EDD]. Measurements of left ventricular dimensions were obtained by Mmode echocardiography recording guided by twodimensional imaging in the shortaxis, parasternal view at the level of the mitral chordae at the edge of the mitral valve leaflets. Measurements were performed according to the leading edge–to–leading edge technique.
Evaluations of Change Over Time
Blood pressure and heart rate were recorded at rest at each followup visit for all patients. Differences between baseline measurements and measurements made after a 2month period of treatment after randomization were studied in all included patients still alive and receiving study treatment at that time (per protocol patients), ie, 1 month after the last dose increment. Echocardiographic left ventricular function evaluation was repeated 5 months after randomization in a subgroup of 160 patients still alive and receiving treatment at that time.
Statistical Analysis and Relationships With Survival
All descriptive statistics are presented as mean±SD.
Baseline values and changes over time of studied variables were compared between groups with the use of a χ^{2} test or Fisher’s exact test for categorical variables and Student’s t test for continuous variables. Relationships between change over time and baseline values were studied in each treatment group, taking into account the phenomenon of regression to the mean.^{14}
Relationships With Survival
Standard approach. Relationships between baseline variables, recorded change over time, and survival during followup were studied on an intentiontotreat basis. Survival curves were established by use of the KaplanMeier estimation method.^{15} Comparison of survival rates between the two treatment groups were performed by use of the logrank test. The Cox proportional hazards model was used to assess the risk ratio between the two treatment groups and the associations of variables with survival.^{12} A specific Cox model based on the findings obtained from the nonparametric approach was also specified (see below).
Nonparametric estimation of covariate effects. Data were analyzed further with an original, nonparametric approach to the hazard function estimation.^{13} Its technical aspects are presented in Appendix A. For the Cox model, the hazard function at time t for an individual with a covariate z is λ (t; z)=λ_{0}(t) exp(βz). This formulation assumes a linear relationship between the covariate and the logarithm of the hazard (instantaneous mortality rate), summarized by the parameter β. The logrisk ratio expresses the prognostic value of the covariate. Because the effect of z may not be a linear one, we propose a hazard model in which the linear predictor βz is replaced by a predictor function f(z). The function f expresses a general effect of the covariate on the logarithm of the hazard; it is unspecified and not restricted to any parametric specification.
If the problem is multidimensional, that is, if z is a pdimensional covariate vector, then f(z) is f(z^{1},…, z^{p}). In the Cox model, βz becomes β_{1}z^{1} +…+ β_{p}z^{p}, which assumes additive effects of covariate components on log hazard. Conversely, in the proposed model, function f is some f(z^{1},…, z^{p}) without any constraints; in particular, no additive assumption is done, nor are any interaction terms specified in advance.
On the basis of the nonparametric estimation, different kinds of graphs may be drawn. First, one can plot values of the log hazard for different values of a given quantitative covariate and obtain a function displaying the effect of this covariate. Second, one can plot the same function but make it conditional on either value of a binary covariate (such as placebo versus active drug); this may reveal interaction between effects of the two covariates. Third, one can plot a threedimensional representation displaying the relationships between log hazard and two quantitative covariates.
Because the accuracy of the nonparametric estimation relies on the number of events, its use was questionable for extreme values of covariates. Therefore, the graphic representation excluded the lowest and highest 2.5% values of the covariates.
Moreover, the estimation method also provides an estimator for the variance of the hazard, both marginal and conditional on one or several covariates. On the basis of these calculations, one can then compute how the variance is reduced, that is, how hazard fluctuations are explained, by the inclusion of successive covariates; if the reduction is important, it means that the covariate plays a great explanatory role given the other covariates already included.
Because the general nonparametric method does not allow computation of confidence intervals, a Cox regression model was specified to analyze the relationships between fractional shortening changes and hazard rates, as suggested by the nonparametric approach. This model specifies that the fractional shortening changes could be a prognostic factor, the strength of which could depend on whether or not fractional shortening increased within the first 5 months after inclusion and whether the patient belonged to the bisoprolol or placebo group.
Study Organization
All CIBIS investigators and participants have been listed in a previous publication.^{7} All investigators who participated in the CIBIS echocardiographic substudy are listed in the “Acknowledgments.”
Results
Population Characteristics
Population characteristics of the study treatment groups have been reported previously. Of the 641 patients included in the study, 95% (n=609) were in NYHA functional class III and 5% (n=32) were in class IV at inclusion. Mean age was 59.6±10.4 years. The causes of heart failure were idiopathic dilated cardiomyopathy (36%), ischemia (55%), hypertension (5%), and valvular disease (4%); these causes were well balanced between the placebo and bisoprolol groups.
Of the 641 patients initially included in CIBIS, 36 died before 60 days of followup, 26 withdrew from the study prematurely, and 22 were not seen on schedule 2 months after randomization. Therefore, differences versus baseline with regard to heart rate and blood pressure were available for only 557 patients.
Baseline characteristics of these per protocol patients for whom hemodynamic effects were studied 2 months after randomization (n=557) and baseline characteristics of patients for whom echocardiographic examination was repeated after a 5month period of treatment (n=160) were both similar to the entire CIBIS population, as shown in Table 1⇓. In this echocardiographic study population, however, the proportion of ischemic patients was slightly lower than in the remaining CIBIS patients who were not included in the echocardiography substudy (χ^{2} test, P<.02).
Study Treatment Effects on Heart Rate and Blood Pressure
After the first 2 months of treatment, bisoprolol slightly but nonsignificantly reduced systolic blood pressure (−2.7±17.9 versus −0.0±17.6 mm Hg for placebo; P=.08). It significantly (P<.0001) reduced diastolic blood pressure (−2.7±10.7 versus +1.3±11.4 mm Hg for placebo) and heart rate (−16.3±15.3 versus −1.6±13.4 bpm for placebo). These effects were obtained with just 2.5 mg of bisoprolol. The last dose increment from 2.5 to 5 mg 4 weeks after inclusion had no relevant effects on heart rate or diastolic blood pressure (Fig 1⇓). Bisoprololinduced hemodynamic effects on diastolic blood pressure and heart rate were maintained during followup. The heart rate change over time recorded 2 months after randomization was significantly linked with baseline heart rate in both study treatment groups. The unadjusted regression parameters were −0.651±0.053 in the bisoprolol group and −0.302±0.056 in the placebo group. After adjustment according to Blomqvist,^{14} the parameters became −0.600±0.061 and −0.196±0.065, respectively (we have supposed for this calculation that the intraindividual variance of the initial heart rate measurement was equal to 25). Therefore, the level of regression was attenuated to a greater extent by this adjustment in the placebo group than in the bisoprolol group.
Left Ventricular Function
In the 160 per protocol patients in whom echocardiographic measurements could be obtained after a 5month period of treatment, left ventricular EDD did not significantly change with bisoprolol or placebo administration, but left ventricular ESD significantly decreased and left ventricular fractional shortening significantly increased in the bisoprolol group compared with the placebo group (Table 2⇓). Left ventricular fractional shortening increased (positive change) in a significantly higher proportion of patients in the bisoprolol group (68 [82%] of 74 patients) than in the placebo group (39 [51%] of 77 patients) (P<.02).
Among patients in the bisoprolol group, fractional shortening change over time was not significantly correlated to bisoprololinduced heart rate change over time at 5 months (r=−.11, P=NS).
For both groups considered together (n=160), the fractional shortening change over time was not correlated to baseline heart rate but appeared significantly correlated to heart rate change over time (r=−0.3, P=.02).
Relationships With Survival
Results of analysis of relationships between baseline variables and survival based on the Cox model have been reported previously for the entire CIBIS population.^{7} Bisoprolol was not found to significantly reduce mortality either with univariate or multivariate analysis.
With univariate analysis (Cox model), the heart rate change over time but not blood pressure changes (between baseline and 2 months of treatment) was significantly related to longer survival (P<.002; n=557 patients; RR=1.022 per bpm).
With a multivariate Cox regression including the study treatment group, baseline characteristics, and hemodynamic changes over time, the heart rate reduction appeared to be the most powerful predictor of survival with the following increasing order of probability values for the different covariates: heart rate change over time (P<.001), NYHA class (P<.02), age (P<.05), presence of ventricular arrhythmias on standard 12lead ECG (P<.05), and baseline systolic blood pressure (P=.055). The probability value associated with the study treatment group was 80.
The graphic representation of the nonparametric analysis (displayed in Fig 2⇓ for the 160 patients of the echocardiographic study) shows that the relationship between heart rate change over time and log hazard is almost linear between −40 and 10 bpm, the range of values that included 90% of patients. The overall slope is in agreement with the estimated RR obtained with the Cox model (RR=1.022 bpm) (Fig 2⇓). Patients receiving bisoprolol were at lower risk than those receiving placebo, whatever their heart rate change over time. In the placebo group, the risk of death decreased only when heart rate decreased (Fig 3⇓).
In the 160 patients for whom echocardiographic parameters could be obtained after a 5month period of treatment, fractional shortening change over time, whatever the study treatment group, was significantly related to survival when we used a Cox regression model with separation of population into two groups around the median value of fractional shortening change over time (P=.013; Fig 4⇓). Patients with the best improvement in fractional shortening had a better survival rate. The median value was 0.014 with a minimal value of −0.17 and a maximal value of +0.22. Within this subgroup of 160 patients (echo study), 8% (n=7) of the bisoprolol group died and 13% (n=10) of the placebo group died.
However, when fractional shortening change over time was included as a continuous covariate, it was no longer significantly related to survival when a Cox regression model was used. This discrepancy means that the relationship between fractional shortening change over time and log hazard is not linear but rather could be threshold shaped. Indeed, when the nonparametric model is fitted, the log hazard versus fractional shortening change over time does not appear linear, as shown in Fig 5⇓, nor does it have a clear threshold shape; the slope of the curve is steep in the range of negative values of fractional shortening variation and almost null for positive values. This complex relationship is confirmed with the use of a Cox model that includes the fractional shortening change over time not directly as an explanatory variable but after a transformation aimed at taking into account such a brokenline shape: a negative slope for negative values of fractional shortening changes over time, a horizontal slope for positive values. The test assessing the effect of this transformed covariate on survival using the Cox model is significant (P<.05).
In addition, the relationship between left ventricular fractional shortening change over time and hazard (nonparametric approach) displays an interaction with treatment group with crossing of the curves at a point close to the zero value for such a change (Fig 6⇓). The statistical computation (using a Cox model implementing these effects with the equation formulated in Appendix B) indicates that the hazard rate is significantly increased (P<.04) in the bisoprolol group compared with the placebo group for the patients exhibiting a negative value (worsening) of fractional shortening change over time but is not significantly different between treatment groups for positive values. However, fractional shortening decreased for only a small proportion (18%) of patients in the bisoprolol group, who were therefore at high risk, compared with 49% in the placebo group (P<.02; see previous section). This is consistent with results provided by the standard Cox analysis and illustrated by Fig 4⇑.
Combined Effects of Study Treatment Group, Heart Rate Change Over Time, and Left Ventricular Fractional Shortening Change Over Time on Survival
Calculation of the hazard variance, based on the estimated joint distribution of fractional shortening change over time, heart rate change over time, treatment, and hazard, is given in Table 3⇓. Hazard variance was calculated after stepbystep inclusion of the different studied covariates.
Results of such an analysis show that the most informative covariate (in terms of hazard variance reduction) is the fractional shortening change over time. Given this covariate, hazard variance can be reduced if either heart rate change over time or treatment is known. Finally, given any two of these covariates, hazard variance is still reduced by the third one.
These results are illustrated by the graphic representation of the relationship of the joint effects of covariates on survival (Figs 7⇓ and 8⇓).
Discussion
Hemodynamic data reported herein from the CIBIS trial confirm results of previous randomized studies but suggest in addition potential relationships between bisoprololinduced hemodynamic effects and improved survival.
Bisoprolol significantly reduced heart rate, suggesting that the demonstrated cardiac βadrenergic receptor downregulation in the failing heart^{16} ^{17} is unable to completely offset the presence of increased sympathetic activity. Amplitude of heart rate reduction was significantly related to the baseline value of heart rate in the bisoprolol group, even after adjustment for regression to the mean. Therefore, the baseline value of heart rate could represent one element of identification of heart failure patients who could benefit most from βblocker treatment because our results strongly suggest a relationship between heart rate reduction and prognosis.
Bisoprolol significantly reduced diastolic but not systolic blood pressure. This effect may result from reduction of sympathetic tone secondary to hemodynamic improvement and could be somewhat similar to that observed in hypertensive patients.
Left ventricular function significantly improved in the bisoprolol group compared with the placebo group, as previously demonstrated with other βblockers in heart failure.^{5} ^{6} ^{8} ^{9} In contrast to heart rate reduction, which is rapidly obtained after initial administrations of βblocker treatment, left ventricular function improvement takes several months to fully develop and counteracts the negative inotropic effect that should be derived from blockade of myocardial βadrenergic receptors and therefore can be considered as a net effect.
The mechanism of such improvement, however, remains unclear. The CIBIS echocardiography substudy showed that left ventricular fractional shortening improvement was mainly related to left ventricular ESD reduction and was not correlated either to baseline heart rate or to bisoprololinduced bradycardia. However, when both placebo and bisoprolol groups were pooled together, left ventricular fractional shortening change over time was significantly correlated to heart rate reduction. Such results indicate that heart rate reduction per se might play an important role in the development of left ventricular function improvement. The antiischemic effect of such heart rate reduction on these dilated hearts working with depleted energetic stores^{18} could play an important role.
Heart rate reduction could also improve left ventricular function by means of a favorable effect on excitationcontraction coupling. Indeed, the normal positive staircase phenomenon (increase in developed force with rate, or the Bowditch effect) is reversed in the myocardium of patients with primary dilated cardiomyopathy.^{19} In this case, reduction of heart rate should increase contractility and contribute to left ventricular function improvement on βblockade. The capacity of the calciumdependent ATPase of the sarcoplasmic reticulum, which ensures prompt calcium refilling of the sarcoplasmic reticulum and subsequently regulates the level of contractile force, has been shown to be altered in experimental hypertrophy and cardiac failure.^{20} ^{21} ^{22} Thus, βblocker–induced bradycardia could increase contractile force by allowing a longer period of time for calcium refilling of the sarcoplasmic reticulum during the diastolic period. In addition, it has been shown using a cardiomyopathy model that βblocker therapy (with carteolol) can increase sarcoplasmic reticulum Ca^{2+} ATPase activity.^{23} βblocker–induced upregulation of β1adrenergic receptors could represent an additional mechanism of left ventricular function recovery with βblocker therapy^{24} ^{25} but remains greatly limited by βblocker occupancy of βreceptors themselves. In addition, carvedilol, which does not induce upregulation of βadrenergic adrenoceptors, also provides left ventricular function improvement.^{9} ^{26}
Several other mechanisms have been proposed to explain the βblocker–induced improvement in left ventricular function, but it is not yet possible to draw definite conclusions on their potential respective roles. Several reviews have already addressed this question.^{27} ^{28}
Given the potential beneficial effects of heart rate change over time on left ventricular function during βblockade in heart failure, the question of the link and interrelationships between these hemodynamic parameters and survival arises. Using a nonparametric estimation of relationships between survival and covariates, our results show that left ventricular function change over time induces the largest hazard variance reduction. Each of the three factors (heart rate change over time, left ventricular function change over time, and bisoprolol treatment) plays a specific additional role in hazard explanation. In addition, their combination has a synergic impact on survival. The complexity of this question is still evidenced by analysis of the relationship between these covariates and survival. Our analysis disclosed that relationships between log hazard (instantaneous mortality rate) and covariates may not be linear, as hypothesized by the usual Cox model. If the relationship of heart rate change over time with log hazard could be considered somehow linear in the range of 90% of values, the relationship between left ventricular fractional shortening change over time and log hazard is obviously nonlinear, with a steeper slope in the range of the negative changes of fractional shortening. Such nonlinearity explains the discrepancy between Cox analysis of the relationship between survival and left ventricular change over time when this last covariate is entered as a continuous covariate (nonsignificant result) or a discrete variable with separation of the studied population into two groups around the median value (significant result).
In addition, our analysis disclosed an interaction between left ventricular function improvement and treatment group. In patients whose left ventricular function improved, the hazard was significantly lower than in those patients whose left ventricular function did not improve, but the difference between the two treatment groups was not significant, and there was no further significant decrease of hazard with a further increase of left ventricular fractional shortening. In contrast, for those patients without such left ventricular function improvement, the hazard rate appeared higher in the bisoprolol group than in the placebo group, even with heart rate reduction. However, the proportion of patients whose left ventricular function improved was significantly higher (82%) in the bisoprolol group than in the placebo group (51%). These results imply that the most important objective of therapy when prognosis is concerned in heart failure would be the preservation of left ventricular function and the prevention of further deterioration rather than achievement of the greatest improvement in left ventricular function.
The fact that prognosis may worsen under βblockade when left ventricular function does not improve, as suggested by our findings, can be interpreted as a failure of βblocker treatment to improve left ventricular function in some patients with heart failure and might imply progressive treatment withdrawal. In these patients, the negative inotropic effect of βblockade is therefore not counterbalanced by the expected left ventricular function improvement secondary to blockade of the deleterious effects of sympathetic stimulation. The proportion of such patients without βblocker–induced left ventricular function improvement is, however, relatively low (18%) and lower than in the placebo group (49%).
Such results will have to be confirmed because the accuracy of the nonparametric approach that we used is based on the total number of events, which was relatively low in the 160 patients included in the echocardiographic substudy. In addition, the survival study was performed on patients still alive after a 5month period of treatment, which excluded the most severely ill patients who died very early during the followup period in the CIBIS trial.
Our results show, however, that relationships between βblockade, heart rate reduction, and left ventricular function improvement are complex, interact, and cannot be reduced to a simple succession of causal events. It appears from our findings that the most important goal of therapy in heart failure where prognosis is concerned is more the preservation of left ventricular function and prevention of deterioration than the achievement of the greatest left ventricular function improvement. However, preventive effects of βblocker treatment on sudden death related to severe arrhythmias may also interfere with the overall effect on prognosis. In the CIBIS study, no particular reduction of sudden death was found with bisoprolol. In the recently published US carvedilol trials,^{10} carvedilol showed a significant trend in reducing sudden death, although the reduction was inferior to that obtained for deaths related to progressive heart failure. However, such results concerning mortality cannot be considered demonstrative because mortality was not the primary end point of these trials. Still, some βblockers may improve survival independently of their effect on left ventricular function, that is, by reducing sudden death. This remains to be studied further.
Future highly powered trials with βblockers in chronic heart failure should provide the opportunity to more precisely define potential differences between compounds according to their pharmacological profile and to confirm the relationships between βblocker–induced shortterm cardiac effects, especially on heart rate and left ventricular function, that have been found in CIBIS with bisoprolol and that could enable the identification of patients who will experience longterm survival improvement in response to βblocker treatment in heart failure.
Selected Abbreviations and Acronyms
CIBIS  =  Cardiac Insufficiency Bisoprolol Study 
EDD  =  enddiastolic diameter 
ESD  =  endsystolic diameter 
NYHA  =  New York Heart Association 
RR  =  risk ratio 
Appendix AA
Let us consider z = (z^{1},…,z^{p}) the pdimensional vector of covariates. Hazard estimation is based on iterative estimation of its two components, the baseline hazard λ_{0} and a predictor function f(z) expressing the effect of covariate z. The entire hazard λ is the product of λ_{0}(t) and exp[f(z)]. Estimation of λ_{0} is done on similar bases as Breslow’s estimation for the Cox model.^{29} The remainder of the appendix concerns estimation of f.
The model belongs to the class of random effect models. This means that it involves random parameters whose distribution is to be estimated. In our model, two random parameters are considered: H (the log hazard) and Q (the covariate vector).
We assume that each patient i has an individual realization (h,q) of the two random variables (H,Q); h and q cannot be measured directly (they are unobservable) but are related to observation of the censored survival time (t_{i},δ_{i}), where δ_{i} is the censoring indicator (δ_{i}=1 if i died, δ_{i}=0 if not), and of the measured covariate z_{i}.
(t_{i}, δ_{i}) is related to h through the conditional density of the survival time, which expresses the probability (or probability density) for a patient with hazard h to die or be censored at time t_{i}, that is if patient i actually died or if patient i was censored. Thus, a general expression is To take into account prognosis factors without focusing on the relationship between them and hazard (in other words, taking into account unspecified nuisance factors), the specification of the model is generalized according to where, for instance, x_{i} could be an individual score for the nuisance factors, derived from the regression parameters of a previous parametric estimation, by use of the Cox model.
z_{i} is related to q through the conditional density of the measurement error made on the covariate Q, which expresses the probability density for a patient with covariate q to be measured at the level z_{i}. If the error of each covariate component z^{j} is supposed to be independent and gaussian, with zero mean and ς_{j}^{2} variance, then Assuming that (t_{i},δ_{i}) and z_{i} are independent, conditional on h and q, then the unconditional density of the triplet (t_{i},δ_{i,}z_{i}), if μ is the joint probability distribution of (H,Q), is Estimation of μ is based on the maximumlikelihood principle. The optimal solution μ̂ has been proved to be a discrete distribution.^{30} In our case, μ̂ is a n*uplet of atoms denoted as (α_{i}, r_{i}, q_{i})_{i}_{=1…}_{n*}, with n*<n.
Thanks to this estimated joint distribution, we can easily calculate conditional distributions, then conditional expectations. In particular, for a given level of covariate z, for instance, z_{0}, the conditional distribution of H is characterized by the n*uplet [α′_{i}(z_{0}),r_{i}]_{i=1…}_{n*}
where and the estimated conditional expectation given z_{0} is This expectation expresses the value of the log hazard for an individual with covariate z_{0}. When calculated for successive values of z, it corresponds to the evolution of the log hazard against z, so that we proposed it as an estimate of function f:
Appendix AB
The Cox regression model specified on the basis of the nonparametric results was defined by the following equation: where 1_{tt} equals 1 if the patient belongs to the bisoprolol group, 2 otherwise; 1_{FS} equals 1 if fractional shortening (FS) change is negative, 0 otherwise; and 1_{int} equals 0 if fractional shortening did not increase, 2 if fractional shortening increased and patient belongs to the placebo group, and 1 otherwise.
Numerical results, expressed per unit of change in fractional shortening (±SE), were as follows:
Acknowledgments
The CIBIS study was sponsored by E. Merck (Darmstadt) Co. The list of all CIBIS committees and investigators has been published in the initial publication.^{1} The authors thank the following investigators for their active participation in the echocardiographic substudy. Belgium: J. Carpentier and W. Van Mieghen. Finland: L. Hämäläinen. France: M. Baudet, J.P. Leroux, D. Bleinc, J.P. Bouhour, A. Hage, P. Charbonnier, P. Fournier, P. Cornaert, P. Dambrine, F. Labaki, A. Gabriel, J.P. Dove, S. Gaba, L. Ferrière, M. Galinier, J.Y. Ketelers, W. Mouawad, M. Komajda, D. Thomas, M. Richaud, D. Matina, P. Morelon, J.C. Quiret, G. Jarry, C. Thery, P. Asseman, and S. Witchitz. Germany: V. Hombach, W. Haerer, M. Hetzel, and P. Limburg. Holland: R. Ciampricotti and J.A. Henneman. Italy: R. Fogari, L.J. Placer, V. Vallé, J. Lupon, P. Puddu, and L. De Biase. Portugal: S. Gomes. Spain: E. Roig.
 Received February 13, 1997.
 Revision received May 2, 1997.
 Accepted May 12, 1997.
 Copyright © 1997 by American Heart Association
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 Prognostic Value of BisoprololInduced Hemodynamic Effects in Heart Failure During the Cardiac Insufficiency BIsoprolol Study (CIBIS)P. Lechat, S. Escolano, J. L. Golmard, H. Lardoux, S. Witchitz, J. A. Henneman, B. Maisch, M. Hetzel, P. Jaillon, J.P. Boissel and A. MalletCirculation. 1997;96:21972205, originally published October 7, 1997http://dx.doi.org/10.1161/01.CIR.96.7.2197
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 Prognostic Value of BisoprololInduced Hemodynamic Effects in Heart Failure During the Cardiac Insufficiency BIsoprolol Study (CIBIS)P. Lechat, S. Escolano, J. L. Golmard, H. Lardoux, S. Witchitz, J. A. Henneman, B. Maisch, M. Hetzel, P. Jaillon, J.P. Boissel and A. MalletCirculation. 1997;96:21972205, originally published October 7, 1997http://dx.doi.org/10.1161/01.CIR.96.7.2197
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