_{2}agonists.

^{1–6}These drugs are variously combined to maximize analgesia and minimize side effects, such as motor block, hypotension, sedation, nausea, vomiting, pruritus, and respiratory depression.

^{2,6–8}Different combinations have been compared in randomized controlled studies.

^{2,5–7,9–12}However, when combining drugs at different concentrations, hundreds or thousands of combinations are possible. For example, if five different values for three drugs are considered, 5

^{3}= 125 different combinations exist. Clearly, it is impossible to analyze all possible combinations by randomized controlled trials. The choice of the combinations that are compared in such trials is mostly made arbitrarily. As a result, the chosen combinations might be far away from the region of the optimal combinations.

^{13}we applied a model

^{14}to optimize drug combination for thoracic epidural analgesia after major abdominal surgery. Later, we improved the method and applied it to two additional studies: one on intravenous patient-controlled analgesia

^{15}and the current one. Initially, eight combinations were studied. On the ground of the results observed, further steps were made by investigating new combinations until no further improvement was shown in three consecutive steps. Using this method, optimization may be performed by investigating a limited number of combinations.

^{13–16}

^{17}Therefore, the results obtained from the aforementioned study on thoracic epidural anagesia

^{13}cannot be applied to lumbar epidural analgesia. Motor block prevents patient’s mobilization, which is important to avoid postoperative complications, such as thromboembolism. Moreover, it may mask the occurrence of severe complications that lead to permanent neurologic sequel, such as epidural hematoma. Adding clonidine to bupivacaine and fentanyl may reduce the concentration of local anesthetic required to obtain pain relief, thereby reducing the incidence of motor block. However, this practice may also increase side effects. For example, the interaction of clonidine with bupivacaine may increase the incidence of hypotension.

^{18}Therefore, the right balance of the drug components in the epidural solution is still unclear.

#### Materials and Methods

##### Anesthetic Procedure

^{−1}· h

^{−1}during the operation.

##### Postoperative Management

*via*nasal probe was administered to maintain an oxygen saturation of more than 93%. Lactated Ringer’s solution, 2,500 ml/24 h, was infused. If the urine output was less than 100 ml/2 h, 500 ml in addition was infused. Blood loss was replaced using the same criteria as in the intraoperative phase. Systolic blood pressure, heart rate, and respiratory rate were monitored and recorded every 2 h during the first 6 h postoperatively and then every 4 h.

*i.e.*, a score of 0 at rest and 2 during mobilization) at 15 ml/h, 5–20 ml carbonated lidocaine, 2% wt/vol, was injected in 5-ml increments. If no segmental hypoalgesia to cold or a unilateral block involving the side contralateral to the site of operation resulted, failure to provide pain relief was attributed to factors other than the epidural combination (

*e.g.*, catheter placement or anatomical factors). These patients were excluded from the analysis. If segmental hypoalgesia on the side of surgery or bilaterally was observed, failure to provide adequate analgesia was attributed to the combination investigated. The epidural combination investigated was discontinued, and the patient remained in the analysis; only data collected before the injection of lidocaine were considered.

^{19}(table 1). Motor block was defined as a score greater than 0 or muscle weakness that prevented the patient from lifting the operated leg by 20–30 cm. Motor block was accepted during the first 6 h after the end of the operation, because during this time it may have resulted from the intraoperative administration of epidural local anesthetics. Thereafter, in the presence of motor block, the epidural infusion was interrupted until motor function was restored and then restarted at a rate of 2 ml/h lower than previously.

##### Data Collection

##### Optimization Procedure

^{14}that we previously applied in a clinical study.

^{13}The aim of the procedure was to increase the analgesic effect,

*i.e.*, to minimize the pain score, by sequentially optimizing the combinations of bupivacaine, fentanyl, and clonidine in the epidural solution. Rules of the procedure included minimum and maximum values of drug concentrations in the epidural solution, their minimum and maximum increases between two subsequent optimization steps (table 2), and constraints. Constraint of the search procedure was an unacceptable incidence of side effects. A combination violated a constraint when the study had to be discontinued because of the same side effect in three patients who received that combination, according to the criteria for discontinuing the epidural combination (table 1).

*complex*. Each complex consisted of eight combinations.

^{13,15}whereby the optimal m and n were identified by simulation procedure (see appendix).

^{13}(see appendix). The new combination replaced the one mentioned in point 2. The new combination was studied in a subsequent group of patients.

^{13,15}at the end of the optimization procedure, we randomly selected two of them to be retested on two additional groups of patients (n = 6/group). Patient allocation to the groups was randomized.

##### Statistical Analysis

^{14}Therefore, it cannot be concluded that any combination is statistically significantly different from other ones. Rather, the method focuses on the trend of the optimization procedure and avoids excessive weight on any individual combination.

^{14}The identified optimal combinations must be investigated in further randomized controlled trials. This practice reproduces the one of the other two clinical studies in which this model was applied.

^{13,15}

#### Results

Fig. 1 Image Tools |
Fig. 2 Image Tools |

Fig. 3 Image Tools |
Table 7 Image Tools |

*i.e.*, the study was discontinued in three patients because of motor block in combinations I (complex 2) and L (complex 4). At these points, the regression procedure could successfully bring the search out of the toxic range back to the therapeutic range at complexes 3 and 5, respectively (see appendix, Logistic Regression). The incidence of other side effects remained low and constant (fig. 3).

Table 5 Image Tools |
Table 6 Image Tools |

#### Discussion

^{13,15}To our knowledge, the regression model to deal with occurrence of side effects had not been used in human studies. During the study period, an initial decrease in pain score (fig. 2) and a continuous decrease in incidence of insufficient analgesia (fig. 3) were observed. A low incidence of motor block and other side effects was achieved at the end of the direct search procedure (fig. 3 and table 7).

##### Clinical Aspects

^{13,15}

^{6}In the previous study, applying the direct search model to combinations of the same three drugs for thoracic epidural anesthesia identified a combination of 9 mg/h bupivacaine, 21 μg/h fentanyl, and 5 μg/h clonidine as one of the best three combinations.

^{13}In the other two best combinations, clonidine was not present in the epidural solution. In a clinical trial on thoracic epidural, the addition of higher doses of clonidine (20 μg/h) to bupivacaine and fentanyl improved the quality of postoperative analgesia but induced hypotension that increased vasopressor requirement.

^{20}It is possible that clonidine in a low concentration is a useful additive for lumbar epidural postoperative analgesia in that it allows a reduction in bupivacaine concentration and therefore reduces the incidence of motor block. This is probably less important in thoracic epidural analgesia, in which motor block due to administration of local anesthetic is unlikely to occur. However, the nature of the direct search method does not allow the conclusion that adding clonidine to bupivacaine and fentanyl is advantageous. Rather, the study identifies combinations that must be compared to a bupivacaine–fentanyl mixture in a randomized controlled trial.

*e.g.*, monitoring. In this respect, optimization procedures remain potentially useful but probably underused methodologies in medicine.

##### Methodologic Aspects

^{15}and in the current one. This study was completed later than a patient-controlled analgesia study because of longer patient recruiting. The method not used before, even in the patient-controlled analgesia study, is the regression model (see appendix, Logistic Regression). None of the combinations of morphine and ketamine for patient-controlled analgesia violated a constraint, so there was no need to use logistic regression in that study.

*Via*logistic regression, we calculated the probability that each of the three variables investigated (concentrations of bupivacaine, fentanyl, and clonidine) has led to the specific side effect. After identifying bupivacaine as responsible for motor block, we recalculated the new combination that would bring the search out of the toxic range: Bupivacaine concentration was recalculated as the average concentration among the good combinations of the complex that violated a constraint, whereas fentanyl and clonidine concentrations remained the same. The method proved successful in identifying combinations that did not violate the constraint

*motor block*in subsequent steps.

^{13}and intravenous patient-controlled analgesia,

^{15}most good combinations were not present in the initial complex but resulted from the optimization procedure.

^{13}(see appendix). We cannot say whether these m and n values would be optimal also for investigations other then the one that we performed.

^{14,21}at each optimization step, the new combination is calculated by partitioning the complex in two halves,

*i.e.*, good and bad clusters—combinations. Using that principle in our case, four combinations would be good and four would be bad. However, cutting the ranked list at its half is purely arbitrary. For example, the worst combination of the good subgroup and the best combination of the bad subgroup could be characterized by similar and clinically indistinguishable pain scores. In this case, it would be more logical and more productive for the optimization procedure if these two combinations belonged to the same cluster, either the good or the bad one. In this study, we used a more rational algorithm to define clusters. In brief, to avoid the limitation of using the average pain score, we considered the distribution of pain to allocate combinations into good and bad groups. For example, two combinations whose average pain scores differ from each other markedly may still belong to the same cluster if the distributions of pain scores among the patients are such that they significantly overlap.

*a posteriori*rescaled in the experimentally significant range 0–2. (For further details on the rescaling and computing methods, refer to the appendix.)

^{13,14}

*optimization*must be taken with caution. It is used as a description of a stepwise process of improving the endpoint, rather than the indicator of a certain identification of the best combination.

#### Conclusions

^{22}

##### Appendix

##### The Direct Search Procedure

^{15}In the current study, combinations of bupivacaine, fentanyl, and clonidine concentration in the epidural solution were optimized. In a previous study,

^{15}we used the direct search method by Berenbaum

^{14}to optimize combinations of bupivacaine, fentanyl, clonidine and infusion rate for postoperative epidural analgesia. We will further refer to the algorithm adopted in this early study as

*method 1*. For the current study and a recently published study,

^{17}we developed and tested an improved direct search method, to which we will refer as

*method 2*. This method does address the major considerations of method 1 mentioned in the Discussion.

^{15}Method 2 was designed to improve the efficiency of the clinical investigation because it aims at using the minimum number of patients that is required to reach the endpoint.

##### Method 1

_{1}, c

_{2}, …, c

_{m}}. In our case, the variables in each combination are bupivacaine, fentanyl, and clonidine concentrations in the epidural solution. Precisely, c

_{i}= (b

_{i}, f

_{i}, cl

_{i}), where b

_{i}, f

_{i}, and cl

_{i}are the bupivacaine, fentanyl, and clonidine concentrations in that particular combination (i = 1, …, m). Let us define as n

_{i}as the number of patients testing combination i. According to the study protocol, this number was constant (n

_{i}= n) for all combinations. Let us denote as PS

_{ij}as the pain score reported by patient j when testing the combination i. PS

_{ij}represents the average pain score reported by patient j in the 48-h study period. The pain score of the combination i is defined as the average pain score across the patients who tested the combination:

_{1}, c

_{2}, …, c

_{m}} according to their average pain score from the lowest to the highest. Namely,

Equation (Uncited) Image Tools |
Equation (Uncited) Image Tools |

_{b}would be (0.6 + 0.9 + 1.0 + 1.0)/4 = 0.9. B

_{b}is calculated for bupivacaine in the same fashion, by considering the bupivacaine concentrations of the four bad combinations. The same procedure is applied to fentanyl and clonidine concentrations.

##### Method 2

##### Choosing m and n.

^{15}we concluded that the optimal values in the optimization algorithm to be used are m = 8 and n = 6. In that study, a direct search procedure was applied to combinations of bupivacaine, fentanyl, and clonidine and infusion rate to minimize the pain score and the side effects. By using data of the previous study to modify the search procedure of the current one, we implicitly assume that the interindividual variabilities among the subjects with regard to the drugs used in the two studies are comparable and the optimal m and n values are the same. The data from the previous study was used uniquely to determine the optimal values of m and n.

^{2}

_{1}, δ

^{2}

_{2}, δ

^{2}

_{3}, and δ

^{2}

_{4}the variabilities of the four independent variables in the previous study (1 = bupivacaine, 2 = fentanyl, 3 = clonidine, 4 = infusion rate). As an example, δ

^{2}

_{3}is plotted in figure 5 as a function of the number of patients per combinations (n). All the variabilities except δ

^{2}

_{4}decrease with increasing n and m. A significant reduction in variability is obtained for n = 6 and m = 8. Higher values for both n and m do not result in significant improvements. Based on these data, we included six patients per combination and eight combinations per complex in the current study.

##### Partitioning the Complexes.

_{max}; g(h) is the pain score value associated with pain class h, and N

_{ih}is the number of individuals having received combination i falling into pain class h. The number of subjects investigated per combination is constant (n

_{i}= n), so we can define PS

_{i,t}as

*total*.

Equation 1 Image Tools |
Equation 6 Image Tools |

_{i}as defined in equation 6 does coincide with PS

_{i}defined in equation 1: In equation 1, the sum is over the different patients, whereas in equation 6, the sum is over the different pain classes.

Equation (Uncited) Image Tools |
Equation (Uncited) Image Tools |
Equation (Uncited) Image Tools |

Equation (Uncited) Image Tools |

_{i}and PS

_{i,t}as follows:

_{ih}is a random variable, PS

_{i}is also random. The probability that the number of patients testing combination i falling into the first pain class (N

_{i,1}) is equal to n

_{i,1}, that the number of patients falling into the second pain class (N

_{i,2}) is equal to n

_{i,2}, and so on, P(N

_{i,1}= n

_{i,1}, …, N

_{i,hmax}= n

_{i,hmax}) is described by a multinomial distribution

^{22}and can therefore be expressed as

_{i,h}are the probabilities that a patient testing the combination i will fall into pain class h. Each patient might have a different probability to fall into a specific pain class h having received a certain combination i. However, because no

*a priori*knowledge is available about the individual probabilities of every single patient, we can assume them to be equal and estimate them using the maximum likelihood method

Equation 7 Image Tools |
Equation (Uncited) Image Tools |

_{i,t}, as defined in equation 7, is a discrete variable; therefore, its probability density function can be calculated by summing the probability of all possible distributions of patients across the pain score classes, such that PS

_{i,t}is equal to r:

_{i,r}is defined as the set of all possible patient distributions into pain classes resulting for combination i in a PS

_{i,t}of value r:

_{min}= g(1) · n

_{i}to r

_{max}= g(h

_{max}) · n

_{i}with a step size of Δr = 0.4. Let us denote as k (from 1 to k

_{max}) the index running from r

_{min}to r

_{max}.

Equation 8 Image Tools |
Equation 9 Image Tools |
Equation 10 Image Tools |

_{i,t}, assumes the value r for every possible r, from a given realization {n

_{i,1}, …, n

_{i,hmax}}.

Equation (Uncited) Image Tools |
Equation (Uncited) Image Tools |

_{1}, c

_{2}, …, c

_{m}} by increasing pain scores as in equation 2. Let us define the two groups

_{q,t}) lower than the lowest pain score in the group B (PS

_{q + 1,t}),

*i.e.*,

_{q+1,t}> PS

_{q,t}) is maximum.

_{A,t}< PS

_{F,t}, PS

_{F,t}< PS

_{H,t}, PS

_{H,t}< PS

_{G,t}, PS

_{G,t}< PS

_{E,t}, PS

_{E,t}< PS

_{B,t}, PS

_{B,t}< PS

_{C,t}, PS

_{C,t}< PS

_{D,t}) according to equation 12, and we chose the highest probability as the cutting point for the partitioning of the complexes. The highest probability occurs between the combination G and E with P(PS

_{G,t}< PS

_{E,t}) = 0.97.

*i.e.*, violate the toxicity constraint). That happened in our study when combination I of complex 2 and combination L of complex 4 were tested. In these two cases, it was necessary to apply the regression model proposed by Berenbaum

^{14}to perform a step back from the toxicity into the therapeutic response surface.

##### Logistic Regression

Equation 13 Image Tools |
Equation (Uncited) Image Tools |

*P*is the probability of side effect, x

_{1}, x

_{2}, and x

_{3}are the concentrations of bupivacaine, fentanyl, and clonidine, respectively, and α

_{1}, α

_{2}, and α

_{3}are the corresponding regression coefficients. By explicitating the regressors in equation 13, we obtain

_{1}, α

_{2}, and α

_{3}can be calculated

*via*linear regression by replacing x

_{1}, x

_{2}, and x

_{3}with the concentrations of bupivacaine, fentanyl, and clonidine in each of the combinations investigated. For each combination,

*P*can be substituted with an estimate of the probability of side effects π:

_{1}, α

_{2}, and α

_{3}by using combinations A–I and the occurrence of motor block as reported in table 7. Only bupivacaine proved to be a statistically significant regressor (α

_{1}= 1.0 with significance

*P*= 0.01, α

_{2}= α

_{3}= 0.0), which is consistent with the notion that only bupivacaine, among the drugs investigated, produces motor block. Therefore, we averaged bupivacaine concentration in the good combinations for identifying the new combination in the optimization procedure. Cited Here...