^{1}Respiratory depression,

^{1}although rare, is a concern. Tolerance to the analgesic effect may develop during opioid therapy, even at a very early stage.

^{2}

*via*activation of the

*N*-methyl-d-aspartate (NMDA) receptor.

^{3–5}Spinal cord hyperexcitability is involved in the pathophysiology of acute pain.

^{6}High doses opioids may activate NMDA pain facilitatory processes, which causes hyperalgesia and could enhance postoperative pain.

^{7}Therefore, the NMDA antagonist ketamine may have a role in the treatment of postoperative pain. The concomitant administration of an NMDA antagonist and an opioid may result in a synergistic or additive analgesic effect in animals.

^{5,8–10}This may allow a reduction in the doses of both drugs, which could result in a lower incidence of side effects. Animal studies have shown that NMDA antagonists prevent the development of tolerance to continuous exposure to morphine

^{8}and attenuate and reverse opioid-induced tolerance.

^{9}These data are consistent with clinical investigations showing that adding ketamine to opioids improves postoperative analgesia and reduces side effects.

^{10,11}On the other hand, the results of other studies did not confirm these findings and question the usefulness of adding ketamine to morphine for PCA.

^{12–14}Thus, while basic pain research clearly favors the combination of opioids with NMDA antagonists, the results of clinical research are still equivocal.

*i.e.*, the minimal allowed time between two consecutive boluses) is not known. When two drugs are combined at different concentrations in the PCA solution and different lockout times are investigated, hundreds of combinations of PCA regimens are possible. For example, if 5 different values for each variable are considered, 5

^{3}(= 125) different combinations exist. Therefore, the optimal combination is unlikely to be identified by randomized controlled studies, since a very small proportion of all possible combinations is analyzed.

^{15}we applied a “direct search” model

^{16}for the first time in a clinical investigation to optimize drug combinations for thoracic epidural analgesia after major abdominal surgery. The main advantage of this method is that a limited number of combinations have to be investigated.

^{15–17}

#### Materials and Methods

##### Anesthetic Procedure

##### Postoperative Management

*i.e.*, the minimum time allowed between two boluses) was one of the independent variables of the study and therefore depended on the regimen analyzed (table 1). Patients were instructed to press the PCA button when pain of any intensity at rest or moderate, strong, or very strong pain during mobilization occurred. If adequate analgesia was not obtained after six subsequent bolus requests (counting also demands below the lockout time, whereby no drug was delivered), the PCA bolus was permanently increased by 0.2 ml, to a maximum of 2 ml per bolus.

*i.e.*, lockout time). After each morphine injection, the PCA bolus was increased by 0.2 ml. After the first hour postoperatively, no supplemental analgesia or sedation was administered. One hour after extubation was considered the beginning of the postoperative study period, which included the following 48 h. Thus, data pertaining the first postoperative hour were not used for the optimization model.

*via*nasal probe, was administered to maintain an oxygen saturation of more than 93%. 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.

##### Data Collection

##### Optimization Procedure

^{16}that we previously applied in a clinical study.

^{15}The aim of the procedure was to increase the analgesic effect,

*i.e.*, to minimize the pain score, by sequentially optimizing the combination of morphine concentration in PCA solution, ketamine concentration in PCA solution, and lockout interval. Rules of the procedure included minimum and maximum value of independent variables, their minimum and maximum increase between two subsequent optimization steps (table 3), 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 PCA combination (table 2).

*complex*. Each complex consisted of eight combinations. The rationale for choosing the number of patients in a combination group (

*i.e.*, six) and the number of combinations in a complex (

*i.e.*, eight) is explained in the Appendix.

^{15}(see Appendix). The new combination replaced the one mentioned in point 2. The new combination was studied in a subsequent group of patients.

^{15}at the end of the optimization procedure, we randomly selected two of the three best combinations and retested them on two additional groups of patients (n = 6 for each group). The patient allocation to the groups was randomized.

##### Statistical Analysis

^{16}The method focuses on the trend of the optimization procedure and avoids excessive weight on any individual combination.

^{18}The optimization model to identify new combinations was based on a statistical method that is described in the Appendix.

#### Results

Fig. 1 Image Tools |
Fig. 2 Image Tools |
Fig. 3 Image Tools |

Table 5 Image Tools |
Table 6 Image Tools |

*i.e.*, the study was not discontinued in more than two patients because of the same side effect in any combination. No complication occurred.

#### Discussion

##### Clinical Aspects

^{10,11}However, other investigations on major abdominal surgery did not confirm this finding.

^{12,13}In the current study, the application of the optimization model produced an increase in the morphine and ketamine concentrations in the PCA solutions analyzed (fig. 3). According to the principle of the optimization method employed, new combinations identified by the stepwise procedure contain more ketamine than the previous ones if the average ketamine concentration of the “good” combinations is higher than the average ketamine concentration of the “bad” combinations (fig. 2 top; see Appendix for details). Therefore, the increase in ketamine concentration in the PCA solutions analyzed during the four optimization steps is indirect evidence that very low concentrations of ketamine in the PCA solution have a clinically detectable analgesic effect when combined with morphine. The lockout interval displayed minimal changes (fig. 3). This suggests that the lockout interval, in the range investigated, might have been within the optimal area. Within this range, the lockout interval is probably less sensitive than changes in the drug concentrations in the PCA solution. Furthermore, because of the relatively slow onset of the drugs studied, differences in lockout time may not have an impact on the pain score over a fairly broad range.

##### Methodological Aspects

^{15–17}We developed a more effective method to identify the new combinations at each step of the optimization. Based on a simulation procedure that used the data collected in a previous study,

^{15}we minimized the number of patients per combination and optimized the number of combinations in each complex, assuming that the interindividual variability in the two studies is similar (see Appendix). In this way, we rendered the procedure more efficient by reducing the number of patients investigated. In fact, the final combinations were identified by investigating 12 out of several hundreds of possible combinations after only five steps enrolling only 102 patients.

^{15}First, as it stands in its original version,

^{16}the algorithm does not provide guidelines to choose the parameters m (

*e.g.*, number of combinations per complex) and n (

*e.g.*, number of patients per combination). Choosing excessively low values of m and n may reduce the time necessary to test a complex but does not necessarily reduce the number of optimization steps required to reach the final solution. In fact, with few patients testing each combination and few combinations in the complex, the correct search direction may be deviated by measurements coming from outlying patients. As a result, more steps would be required to head back to the optimal point. On the other hand, high values for m and n may provide correct search direction but at the cost of an expensive or even unfeasible study. We addressed this point by defining the optimal values for m and n (see Appendix).

^{15}This means that in a complex of, for example, eight combinations, four would be “good” and four would be “bad.” Partitioning the complexes into the categories “good” and “bad” to compute the next combination makes sense if we believe that combinations can be naturally clustered into two groups. However, given that the assumption is true, defining the clusters by 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 very 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. We needed a more rational algorithm to define clusters (see Appendix).

*i.e.*, away from the “bad” combinations. Lower values of α cause small changes, requiring more steps to reach the end point. On the other hand, large values of α may result in missing the optimum and possibly end up in toxic range. The optimal value of α remains undetermined, and the exact choice of α is inevitably somewhat arbitrary. The optimal α value is likely to depend on the type of experiment. For example, whenever severe toxicity is anticipated, low α values should be chosen, which was not the case of our investigation. The actual value chosen for α can be defended if the procedure converges to the end point in a limited number of iterations, without overshooting its target too often. Based on experience from previous studies,

^{15,16}we chose a value of α of 1.3 for the current study.

^{19}adequately characterized biologic response surfaces show almost always one single optimum. In this study, we assumed that that was the case. In spite of aforementioned improvements in the model, the study does not provide guarantee that the best combinations are really the best ones among all possible combinations. In this sense, the term

*optimization*must be taken with caution. It indicates the process of sequentially improving the end point, rather than the assumption that the best combination has been identified with certainty.

^{10,11}On the other hand, the recent publication of negative studies

^{12–14}questions the usefulness of adding ketamine to morphine for PCA. However, none of these studies used one of the combinations that were included in the final set of the current study. Because of the contradictory results of clinical research, there is still a place for additional randomized controlled trials comparing the combination with morphine alone. In this case, the current study provides indication on the combination that should be used as a comparison with morphine alone. In our opinion, the scientific based approach presented here should be preferred to the purely empiric criteria usually employed to select the combinations analyzed in randomized controlled trials.

##### Appendix

##### The Direct Search Procedure

^{15}In the current study, combinations of morphine concentration in the PCA solution, ketamine concentration, and lockout interval were optimized. In a previous study,

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

^{16}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, we developed and tested an improved direct search method, to which we will refer as method 2. This method does address the three major drawbacks of method 1 mentioned in the section “Discussion.”

^{15}Method 2 was designed to improve the efficiency of the clinical investigation, since it aims at using the minimum number of patients that is required to reach the end point. For the sake of simplicity, we assume that morphine concentration in the PCA solution, ketamine concentration in the PCA solution, and lockout interval are the independent variables in both methods.

##### Method 1

_{1}, c

_{2}, …, c

_{m}}. In our case, the variables in each combination are lockout time and morphine and ketamine concentrations in the PCA solution. Precisely, ci = (m

_{i}, k

_{i}, l

_{i}), where m

_{i}, k

_{i}, and l

_{i}are, respectively, the morphine concentration, the ketamine concentration, and the lockout time investigated in that particular combination (i = 1, …, m). Each combination is tested on n subjects. Let us denote as PS

_{ij}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 UA3 Image Tools |
Equation UA4 Image Tools |

_{k}would be (0.7 + 0.7 + 1.0 + 0.8)/4 = 0.8. B

_{k}is calculated for ketamine in the same fashion, by considering the ketamine concentrations of the 4 “bad” combinations. The same procedure is applied to morphine concentration and lockout interval.

##### Method 2

##### Choosing m and n.

^{15}we could conclude 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, clonidine, and infusion rate to minimize the pain score and the side effects. By using the data of the previous study to modify the search procedure of the current one, we implicitly assume that the interindividual variability among the subjects with regard to the drugs used in the two studies is 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 4 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.

##### Partitioning the Complexes.

Equation A6 Image Tools |
Equation A1 Image Tools |
Table 7 Image Tools |

_{ih}is the number of individuals with average pain score h receiving combination i. Note that PS

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

_{i}defined in equation 1. We normalized the pain scores by constructing class intervals (table 7). Despite the scaling of the pain scores, we did not modify the indices in all equations of this appendix. Indeed, neither the mathematical treatise nor the decision process about optimization steps depends on the scale adopted for the pain scores.

_{A}= (2 × 0.2 + 1 × 0.6 + 2 × 1.0 + 0 × 1.4 + 1 × 1.8)/6 = 0.8

_{ih}is a random variable, then PS

_{i}is also random. If the number of subjects investigated per combination n is constant, we could consider equivalently:MATH

_{i,t}have not been published.

^{18}However, by defining as n

_{i}the number of patients reporting an average pain score in the class i, the density of PS

_{i,t}can be given as where

*P*(N

_{0}= n

_{0}, …, N

_{k}= n

_{k}) denotes the probability that the number of patients falling into the first pain class N

_{0}is n

_{0}, the number of patients falling into pain class N

_{1}is n

_{1}, and so on.

_{i,t}assumes the value h for every possible h. Since PS

_{i,t}is a discrete variable, this probability can be calculated by summing the probability of all possible distribution of patients across the pain score, such that PS

_{i,t}is equal to h. Mathematically, the set of such combinations can be expressed as MATH

Equation A8 Image Tools |
Equation UA10 Image Tools |

_{j}that a patient testing the combination will rate pain with a score j. To do this, we can use the maximum likelihood estimates:MATH

_{0}, …, n

_{k}}. Two examples are represented in figure 5.

_{1}, c

_{2}, …, c

_{m}} according to their average pain score as in equation 2, let us define the two groups with the index q such that MATH is maximized. The above probability can be calculated as

Equation A15 Image Tools |
Equation A14 Image Tools |

_{E}< PS

_{A}, PS

_{A}< PS

_{F}, PS

_{F}< PS

_{G}, PS

_{G}< PS

_{D}, PS

_{D}< PS

_{C}, PS

_{C}< PS

_{B}, PS

_{B}< PS

_{H}) in equation 15 according to equation 14, and we chose the highest probability as the cutting point for the partitioning of the complexes. The highest probability occurs between the combination F and G with

*P*(PS

_{F}< PS

_{G}= 0.73;fig. 5).

##### Computing the New Combination.

Equation A11 Image Tools |
Equation A12 Image Tools |

*i.e.*, unacceptable incidence of side effects). Therefore, it was not necessary to apply the regression model proposed by Berenbaum

^{16}to perform a step back from the toxicity into the therapeutic response surface. Cited Here...