The common approach to total intravenous (i.v.) anaesthesia is to use a combination of several drugs, e.g. an opioid and a hypnotic, to ensure an adequate anaesthetic state. Increasing the dose of one drug may result in a decreased requirement of the other drug. This leads to an ambiguity in the selection of the proper ratio of drug doses when administering drug combinations to ensure adequate anaesthesia. There are a number of studies showing that a hypnotic such as propofol reduced the requirement for the opioid (e.g. alfentanil or remifentanil) to reach a certain clinical end-point [^{1–7} ]. However, such observations alone cannot characterize and quantify the nature of the drug-drug interaction, in terms of an additive, or more or less than additive interaction. There are only a few studies that model the type of interaction between a hypnotic and an opioid. Vuyk and colleagues [^{4} ] studied the pharmacodynamic interaction between propofol and alfentanil by observing signs of inadequate anaesthesia in response to different clinical stimuli, such as laryngoscopy, skin incisions, etc. In another paper [^{5} ], the same group investigated the interaction between propofol and alfentanil given for the induction of anaesthesia. Both studies concluded that the interaction between propofol and alfentanil was more than additive. As the electroencephalogram (EEG) is increasingly used in monitoring anaesthesia and in guiding anaesthetic drug dosing, there is a need to quantify the interaction of hypnotics and opioids when EEG derivations are used as a pharmacodynamic end-point. This study was undertaken to characterize and quantify the interaction between propofol and alfentanil on median EEG frequency in patients undergoing abdominal surgery under common clinical routine conditions.

Methods
Participants
With the approval of the appropriate Medical Ethics Committee and after obtaining patients' written informed consent, 20 male/female patients were enrolled in the study. Inclusion criteria were: age 18-60 yr, ASA I-II and scheduled for abdominal surgery either in general surgery or in gynaecology. Exclusion criteria were: concomitant cardiac, pulmonary or renal diseases, excessive consumption of alcohol or excessive smoking, mental disorders, and continued use of opioids, benzodiazepines, antiepileptics and other psychopharmacological drugs. Eligible patients were those fulfilling all inclusion criteria and none of the exclusion criteria.

General anaesthetic procedure
Premedication (midazolam 7.5 mg) was administered orally to the patients in the ward 1 h before induction of anaesthesia. Tracheal intubation was facilitated by rocuronium and anaesthesia was induced and maintained by target-controlled infusions for propofol and alfentanil. The choice of the target concentrations for propofol and alfentanil during the non-study period was left to the discretion of the attending anaesthetist. After the start of the study period, the target of alfentanil was chosen according to the random list and propofol was dosed by a feedback system. Artificial ventilation was adjusted to maintain normocapnia. The electrocardiograph, noninvasive blood pressure and S_{P} O_{2} were monitored at regular intervals.

Study design
The observational period was defined as that after the opening of the peritoneum and before its closure. During this period, each patient received a targetcontrolled infusion of alfentanil using an already published pharmacokinetic parameter [^{8} ]. Three target concentrations (T150, T225, T300) of 150, 225 and 300 ng mL^{−1} were applied to each patient in a random order according to a computer-generated random list. These targets correspond to steady-state maintenance infusion rates of approximately 3, 4.5 and 6 mg h^{−1} in subjects of 75 kg body weight, whereby the dose of 4.5 mg h^{−1} was the maintenance infusion rate recommended by the pharmaceutical manufacturer (Janssen-Cilag GmbH, Neuss, Germany). After the target had been reached, this concentration was maintained for at least 30 min. During the observational period, propofol was added to the alfentanil infusion by an already described closed-loop feedback system [^{9,10} ]. The set point was the range of 1.5-2.5 Hz median EEG frequency. During the last 20 min of each target interval, four blood samples were taken from a cannula inserted into the radial artery of the non-dominant hand for the estimation of the drug concentration of propofol and alfentanil. The mean drug concentrations were used to determine the type of interaction (additive, supra-additive, infraadditive) as detailed in Appendix 1 and to construct the isobole as described in Appendix 2 .

EEG analysis and drug dosing
The electroencephalogram was recorded by the processed EEG monitor A-1000^{®} (Aspect Medical Systems, Inc., Natick, MA, USA) using frontal leads. The raw EEG, digitized at a rate of 128 Hz, was obtained from the serial port of the A-1000^{®} . After artefact rejection and filtering, the data between 0.5 and 32 Hz were used to determine the median EEG frequency, which has been used in the past for feedback-controlled dosing of i.v. anaesthetics [^{9,10} ]. EEG analysis and drug infusions of alfentanil and propofol were performed by the computer program ‘ivfeed4.0’, which was developed by the authors. It can analyse two EEG channels and drive two pumps simultaneously. Each pump can be used in three different modes: manual, target-controlled and feedback mode. In the manual mode, the infusion pump is set to constant rate infusions. In the targetcontrolled mode, the pump is used as target-controlled infusion device and in the feedback mode the pump is used within the feedback system. ‘ivfeed4.0’ merges several pieces of software which had been developed at our institution and used in the past for other studies. We used the Braun Perfursor fm^{®} (Braun, Melsungen, Germany) infusion pump. The software ran on a notebook Tecra 500 cdt^{®} (Toshiba, Inc., Tokyo, Japan) with Microsoft Windows 95^{®} operating system.

Blood samples and assays
During the last 20 min of each alfentanil-target period, four arterial blood samples were drawn. Samples were collected in heparinized tubes and stored at 4°C for up to 2 h. After centrifugation, the plasma was stored at −20°C pending analysis.

Plasma propofol analysis was performed using high-pressure liquid chromatography with electrochemical detection [^{11} ]. The lower detection limit for propofol was 1 ng mL^{−1} with inter- and intra-assay variability rates of 1.7 and 7.7%, respectively.

Alfentanil was determined in human plasma by gas chromatography using a phosphorous-nitrogen detector (PND). It was extracted by subsequent extraction procedures with varying pHs. Saturated saline solution (200 μL) with sodium hydroxide 0.5 mol were added to 500 μL plasma and extracted for 30 min with heptan 2 mL. The organic phase was acidified with 200 μL 1 mol HCl and extracted for 30 min. To the aqueous phase 150 μL 5 mol sodium hydroxide solution was added and vortexed with heptane 50 μL for 1 min. The organic phase was injected into the gas chromatograph in splitless mode at injector temperature of 280°C. The PND detector had a temperature of 320°C. The starting temperature of the oven was 210°C and it was heated to 290°C and held for 16 min. Helium was used as carrier gas. The retention time for alfentanil was 16.4 min. The limit of quantitation was 10 ng mL^{−1} with intra- and interassay variability rates of <10%.

Data analysis and statistics
In each patient for which blood concentrations of propofol and alfentanil could be determined during the three phases, T1, T2 and T3, there were three data points q^{1} , q^{2} and q^{3} of alfentanil-propofol concentrations in the alfentanil-propofol concentration plane. The relative distance rd of point q_{2} from the line connecting q_{1} and q_{3} was used as a measure for the type of interaction (see Appendix 1 ). An rd > 0 would indicate a more than additive interaction; rd < 0 a less than additive interaction; rd = 0 an additive interaction. On the basis of the data of Vuyk and colleagues [^{5} ], we formulated the null hypothesis HO: rd = 0 versus the alternative hypothesis H1: rd > 0. Under the assumption that the rd are normally distributed, we used the t -test to test the hypothesis. The rd were tested for normality using the Shapiro-Wilk's test. The typical median prediction error for blood concentrations was in the order: 25-30%. We therefore decided to consider an rd ≥ 20% as clinically important. Given a significance level of α = 5% and a test power of 80% to detect rd = 20%, and assuming a standard deviation for rd = 25%, at least 12 patients were needed. Assuming that one-third of all included patients will not successfully complete all three targets, we decided to enrol 20 patients in the study. For the modelling of the isobole we used least-square fitting to draw appropriate Bernstein spline functions as isoboles in the alfentanil-propofol concentration plane (see Appendix 2 ). The F -test was applied to select the appropriate number of parameters for the Bernstein splines.

Results
Type of interaction
In 17 patients we could administer all three target alfentanil concentrations and propofol via the feedback system. In the other three enrolled patients we did not have the opportunity to measure all three target points necessary. The patients' anthropometric data are given in Table 1 . Table 2 shows the rd of point T225 from the line defined by T150 and T300 (see Appendix 1 ). The Shapiro-Wilk's test for normality yields a Type I error of P > 0.8. Applying the one-tailed t-test gives a Type I error of P > 0.3. Thus, we cannot reject the null hypothesis rd = 0. On the basis of the given data, rd = 0.15 could have been detected with a power >90%.

Table 1: Anthropometric data of patients who completed the study.

Table 2: Relative distance of rd of the concentration point at T225 from the line connecting the two points achieved at T150 and T300 (Appendix 2).

Modelling the isobole
To construct an isobole we used the spline technique as described in Appendix 2 . Figure 1a shows the additive isobole and the isoboles using three control points together with the measured drug concentrations. Each datum point represents the mean of four propofol concentrations and four alfentanil concentrations. Table 3 gives the sums of squares (SS) and the degrees of freedom (DF) together with F values for the comparison of the goodness of fit for the two versus three-parameter fits and for the three versus four-parameter fits and the corresponding critical values. Figure 1a identifies some rather high propofol concentrations that might be suspected as outliers. These values belong consistently to Patient 2. One of these values is greater than the third quartile plus three times interquartile range of the measured blood concentrations and may be considered a definite outlier. On omitting the data from Patient 2 as outliers, Table 4 gives the adjusted values corresponding to Table 3 , and Figure 1b shows the corresponding fits. Figure 1c shows the group means (±SEM) of the alfentanil and propofol concentrations for each of the three targets. The product-moment correlation coefficients r for the data shown in Figures 1a–c are r = −0.31, −0.47 and < −0.99, respectively. Table 5 gives the concentrations c_{A0} and c_{P0} , denoting the intersections of the line of additivity with the alfentanil and propofol axis, and their standard deviations for all subjects and omitting Patient 2.

Figure 1.: (a) Measured drug concentrations (○) and fitted Bernstein spline functions, ___: Two-parameter model; ___: three-parameter model; ___: four-parameter model; ___: envelope of the 95% confidence region for the line of the two parameter model. (b) Data points from patient 2 are the three points with the highest propofol concentrations. They can be considered as outliers and are omitted here. (c) Mean (±SEM) alfentanil and propofol concentrations (●) for the three targets (T150, T225, T300) for the data set shown inFigure 1b.

Table 3: Weighted sum of squares (SS) for fitting a Bernstein spline function with two, three and four parameters to the measured concentrations.

Table 4: Weighted sum of squares (SS) for fitting a Bernstein spline function with two, three and four parameters to the measured concentrations with the exception of Patient 2, who was considered as outlier.

Table 5: Fitted concentrations CP0 and CA0 as determined from the line of additivity, which are required to maintain a median EEG frequency between 1.5 and 2.5 Hz if each drug is given solely.

Discussion
Both the relative distance test and the isobole technique come consistently to the conclusion that with our data there is no indication of a more than additive interaction. This is in agreement with recent data [^{12} ] concerning the type of interaction between propofol and sufentanil at loss of consciousness. However, it is in contrast to conclusions from Vuyk and colleagues [^{4–7} ] who reported a more than additive type of interaction between propofol and alfentanil at loss of consciousness and loss of the eyelash reflex. In addition, these workers reported a negative concentration of alfentanil (EC_{50Alf} ) associated with a 50% probability of no response. This group argued that ‘this suggests that alfentanil is not capable of the induction of loss of the eyelash reflex and loss of consciousness in the absence of propofol …’. However, this speculation is contradicted by evidence of induction of loss of consciousness with fentanyl and alfentanil in unpremedicated healthy male patients, which was reported by Scott and colleagues [^{13} ] in 1985. As detailed in Appendix 2 , there are several mathematical objections in using the methodology applied by Vuyk's group. We therefore applied the methodology presented here to reanalyse the data given in Table 2 by Vuyk and colleagues [^{5} ]. Table 6 gives in correspondence to Tables 3 and 4 the SS, DF and F value when comparing the two-parameter with the three-parameter fit, and the three-parameter with the four-parameter fit, for both loss of the eyelash reflex and loss of corneal reflex. The F -test clearly prefers an additive interaction. This is confirmed in Figure 2 where the data for loss of the eyelash reflex are plotted and the various fits shown. Given the line of additivity, one would estimate the alfentanil concentration at the loss of the eyelash reflex to be somewhere between 600 and 800 ng mL^{−1} . This is in fair agreement with the early findings of Scott and colleagues [^{13} ] who found loss of consciousness within ≤2 min during a constant rate infusion of alfentanil 1.5 mg min^{−1} at measured plasma concentrations in the order of between 600 and 1000 ng mL^{−1} . As loss of the eyelash reflex usually precedes or accompanies a loss of consciousness, we believe that the alfentanil concentrations we found in reanalysing Vuyk and colleagues' data - 690 ng mL^{−1} for loss of the eyelash reflex and 730 ng mL^{−1} for loss of consciousness - are strongly supported by Scott and colleagues [^{13} ].

Table 6: The method to construct isoboles described in the present paper was applied to the data of Table 2 by Vuyk and colleagues [5].

Figure 2.: Measured data of Vuyk and colleagues at the loss of the eyelash reflex (○) and fitted curves according to Vuyk and recalculated using Bernstein spline functions. ⋄: Fit according to Vuyk and colleagues; ____: two-parameter model; ____: three-parameter model; ____: four-parameter model.

It has been argued that the entire isobole is needed to determine correctly the type of drug-drug interaction [^{14} ]. However, the fact that the type of interaction is, in general, a local phenomenon - which may depend on which portion of the isobole is studied - has to be considered. There are examples in the literature [^{15} ] where the isobole shows supraadditivity in one portion, additivity in another portion and antagonistic properties in a third portion. To avoid confusion with an imprecise terminology, it should be made clear whether the type of interaction is considered to hold globally (for the entire isobole) or locally (for only a certain portion of the isobole). Given these definitions of ‘global’ and ‘local’ interaction, as detailed in Appendix 2 , this paper gives only evidence for the local additivity of the interaction of propofol and alfentanil. Any smooth function can be locally approximated by a straight line and it has been pointed out that studying isoboles in a small concentration range might result falsely in an additive isobole [^{15} ]. However, our data cover concentration ranges of 800 ng mL^{−1} (56-858 ng mL^{−1} ) for alfentanil and 11.4 μg mL^{−1} (0.7-12.1 μg mL^{−1} ) for propofol, which we believe cover a clinically relevant range of concentrations. If we had also considered alfentanil concentrations equal to zero, we might have found data that could have suggested locally, at small alfentanil concentrations, a more than additive type of interaction. However, this remains rather speculative, for concentration ranges such as those reported in this paper the evidence for an additive type of interaction is twofold.

A secondary aspect of our results is that the measured concentrations of alfentanil are on average about 70% higher than the target set point [^{16} ] (median prediction error = 0.67, median absolute prediction error = 0.69). This may have several causes, such as an inappropriate choice of pharmacokinetic model or drug-drug interaction, that are, however, not the focus of this study and need to be studied separately.

This brings the discussion to the point where it is necessary to ask for the clinical implications of our findings. An isobole as such identifies only equipotent drug combinations with respect to the chosen end-point. In this sense, any point on the isobole is equivalent to all others. Additional criteria are needed to select a specific point of the isobole. These might be considerations about the postoperative period or the intraoperative treatment of pain. The definition of pain given by the International Association for the Study of Pain (IASP) implies that an unconscious person cannot have pain although he/she may well experience nociception. Given the isobole of this study, alfentanil can be freely exchanged for propofol, and vice versa, without jeopardizing the major goals of general anaesthesia - loss of consciousness and painlessness, which is implied by unconsciousness - if it is believed that the chosen end-point for the isobole (median EEG frequency = 2 Hz) reflects an hypnotic effect associated with unconsciousness. A rationale to choose an appropriate dose of opioid could lie in the consideration of the amount of nociceptive stimulation reaching the central nervous system. One may speculate that the neuroplastic cascade of events [^{17,18} ], which can be induced and triggered by an insufficient protection of nociceptive input to the central nervous system, may have great effects for the postoperative well-being of surgical patients. Research in the field of neuroplasticity and its implications for clinical anaesthesia has been started [^{19–21} ] and may help to identify the rationale of dose selection for intraoperative delivery of opioids.

In conclusion, the interaction of alfentanil and propofol on the median EEG frequency is additive and for the rational selection of opioid dosing in i.v. anaesthesia a search has to be made for criteria other than EEG-derived parameters.

Acknowledgement
This work was supported in part by the Bayerische Forschungsstiftung (Grant No. 261/98).

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Appendix 1: Testing the type of interaction

Given the three measured pairs of concentration q_{i} = (c_{Ai} ,c_{Pi} ), where i = 1, 2, 3, that can be visualized as points in the propofol-alfentanil concentration plane, a straight line can be drawn through the points q_{1} and q_{3} (line of additivity) (Fig. A1 ). If the point q_{2} is ‘far enough’ away from this line, it could be said that there is a more than additive interaction where q_{2} is nearer to the origin, and less than additive if q_{2} is further from the origin than the line of additivity. However, the measure for the distance from the line of additivity should be independent of the units in which the concentrations or doses are expressed. It is therefore reasonable to use the depicted sum of ratios rd = Δc_{A} /c_{A0} + Δc_{P} /c_{P0} as a relative measure of this distance. One may regard the expression rd as the sum of proportions by which the two drug concentrations are decreased or increased as a result of their interaction in order to produce a given effect as compared with an additive interaction.

Figure A1.: The distance of point q2 from the line connecting q1 with q3 can be used as a measure for the deviation from additivity. As this measure should be independent of the units in which the concentrations are expressed, the partial distances are scaled with cA and cP, respectively.

Denoting (with a and b) the slope and intercept of the line connecting the intercepts c_{P0} and c_{A0} yields c_{P0} = b and c_{A0} = -b/a, which, when inserted into the expression for rd, yields rd = (Δc_{P} - aΔ_{cA} )/b. If a and b are expressed in terms of the co-ordinates c_{Ai} , c_{Pi} , where i = 1, 2, 3, and mindful that for the intermediate point q_{x} the relation q_{Px} = aq_{Ax} + b holds, after elementary rearrangements, the equation becomes:

For example, if q_{1} = (0, 1), q_{2} = (1/4, 1/4) and q_{3} = (1,0) then rd = 0.50. Thus, there is a 50% saving in concentration due to the interaction compared with the line of additivity that would have predicted q_{2} = (1/2, 1/2). The hypothesis of an additive, or supra-additive, or infra-additive interaction could be formulated as rd = 0, >0 and <0, respectively.

Appendix 2

Modelling of an isobole
To characterize a drug-drug interaction, the sum of concentration ratios for SR is investigated:

where the pair (c_{A} , c_{P} ) is the concentration of A and P given simultaneously to achieve the desired effect, and (C_{A0} , 0) and (0, C_{P0} ) are the concentrations needed to achieve the effect when A respectively P are given alone. If the ratio SR equates to 1, then the interaction is said to be additive, if SR < 1 the interaction is said to be more than additive, and if SR is > 1 the interaction is said to be less than additive.

An often-used specific approach to quantitate drug-drug interaction is to use the formula [^{5} ]

The r term is used to model the interaction and r itself is a measure of the strength of interaction. The parameters C_{A0} , C_{P0} and r have to be estimated from the measured concentration pairs (c_{Ai} , c_{Pi} ), where i = 1, …, n. If r < 0, then the sum of ratios

may be > 1 indicating less than additive interaction; if r > 0 then the sum of ratios is < 1 so indicating more than additive interaction. To estimate the parameters C_{AO} ,C_{P0} and r, equation (1) is solved for either c_{A} or c_{P} .

For instance, this expression is sometimes used in non-linear least square fitting by minimizing:

This approach is inadequate for a number of reasons. First, it considers the measured concentrations c_{Ai} as an independent variable and c_{Pi} as the dependent variable. However, both variables are random variables with corresponding errors of estimation. Instead of minimizing the sum of the squared ordinate distances the sum of squared distances of the data points from the isobole should be minimized. This is shown in Figure A2 . The second disadvantage of minimizing expression (2) is that the expression has a number of singularities at

Figure A2.: The left-hand side shows the distances to be minimized by fitting if the concentration of A is considered as an independent variable and the concentration of P as a dependent variable. The right-hand side shows the minimization procedure if both co-ordinates of the data points pi are considered as random variables.

That is for each measurement c_{Ai} there is an entire line in the n-dimensional hyperplane defined by expression (2) where the hyperplane is infinite, namely at C_{A0} = -c_{Ai} r. This generates a manifold of local minima.

The following example should exemplify how misleading this method of analysis might be. Consider the following fictitious data points: {(200, 10.0), (250, 8.0), (300, 7.7), (350, 6.9), (400, 6.65), (450, 6.5)}. These data are shown in Figure A3 . When viewing the data one would estimate an additive, or a more than additive, interaction. Analysing the data by the fitting procedure as described above for the additive (r = 0 fixed) and non-additive interactions yields the following parameters: degrees of freedom (DF) and sum of squares (SS) (Table A1 ).

Figure A3.: Sample data of a fictitious drug interaction that looks like being additive or supra-additive.

Table A1 This results in F = 27, which exceeds the critical level F_{1,3} (0.05) = 10.1. Hence, the non-additive interaction model with r < 0 has to be chosen, concluding that the interaction is less than additive, in contrast to what was expected from visual inspection of the data. The equation of the isobole for this case reads

If one considers the limit of this expression for cA to infinity one yields

which might lead to the conclusion that the effect being studied might never be achieved with drug A alone, because, even at infinitely high concentrations of A, a concentration of 6.07 of P to achieve the effect is needed. Figure A4 shows again the data points and two isoboles for a additive and non-additive interaction. Obviously there is a singularity at c_{A} = −60/(−0.42) = 143. One might reasonably assume that such an isobole does not occur naturally. This demonstrates how the selection of a certain model might add structure to the data, which cannot be recovered by the data alone.

Figure A4.: Fits to the sample data ofFigure A3usingexpression (2) of Appendix 2as the objective function (thin line) and usingexpression (4) of Appendix 2(bold line). At cA = 143 there is a singularity where the function jumps from - ∞ to + ∞.

To avoid such singularities it is reasonable not to describe the isobole in the c_{A} ,c_{P} plane by a function c_{P} = f(c_{A} ) but to use a parametric description of the curve

c_{A} = c_{A} (t), c_{P} = c_{P} (t), t = 0, …, 1

whereby the parametric representation might be chosen such that

c_{A} (0) = 0, c_{P} (0) = C_{P0} ; c_{A} (1) = C_{A0} , c_{P} (1) = 0

That is, as t varies from 0 to 1, the isobole starts at the upper left corner (0, C_{P0} ) and descends to the lower right corner (C_{A0} , 0) (Fig. A5 ).

Of course the functions c_{A} (t) and c_{P} (t) will depend on some parameters θk, k = 1, …, m

c_{A} (t) = c_{A} (t; θ_{1} , …, θ_{m} ); c_{P} (t) = cp(t; θ1, …, θ_{m} )

or in short hand notation:

c_{A} (t) = c_{A} (t; θ); c_{P} (t) = c_{P} (t; θ)

Now given the set of measured concentration pairs (c_{Ai} , c_{Pi} ), where i = 1, …, n, Figure A2 suggests that one minimize the following expression with respect to θ_{k} , k = 1, …, m:

where w_{Ai} and w_{Pi} are some weighing factors, and ti is that value of 0 ≤ t ≤ 1 which minimizes the distance of the observation (c_{Ai} , c_{Pi} ) from the isobole given as a parametric curve (c_{A} (t,θ), c_{P} (t,θ)). That is the minimization procedure is a two-step procedure: for a given parameter set θ find, for each value of i:

This defines implicitly a set of t_{i} , where i = 1, …, n, as a function of the set of θ, t_{i} = t_{i} (θ). These implicit functions have to be inserted into expression (3) leading to the minimization problem

To solve the minimization problem, an assumption has to be made about the form of the functions c_{A} (t; θ), c_{P} (t; θ). A widely accepted approach to describe curves or surfaces in a model-independent way is to construct spline functions. There are several ways of defining spline functions. For the specific problem of fitting isoboles we used the so-called Bernstein splines or polynomials [^{22} ], which are defined as follows.

Given N + 1 data points p_{0} , …, P_{N} in the plane the corresponding Bernstein spline of order 1 BS^{1} (t) is given by:

As BS^{1} (t = 0) = p_{0} and BS^{1} (t = 1) = p_{l} , the curve starts at P_{0} and ends at pl, i.e. the starting point and end-point lie on the curve, while the intermediate points p_{1} , …, P_{1-1} may lie outside the curve.

The first-order Bernstein spline is just a straight line connecting the points p_{0} and p_{1} .

BS^{1} (t) = p_{o} (1 − t) + p_{1} t

Let p_{0} = (0, C_{P0} ) and p1 = (C_{A0} , 0). If we denote the abscissa and ordinate components of the Bernstein spline by (c_{A} (t), c_{P} (t)), then we have:

c_{A} (t) = C_{A0} t; c_{P} (t) = C_{P0} (1 − t)

The second-order Bernstein spline BS^{2} (t) can describe more complex isoboles:

BS^{2} (t) = p_{o} (1 − t)^{2} + 2p_{1} (1 − t)t + p_{2} t^{2}

Given the co-ordinates of pi, where i = 0, …, 2, as P_{0} = (0, C_{P0} ), p_{1} = (C_{A1} , C_{P1} ) and p_{2} = (C_{A0} , 0), the components of the Bernstein spline are:

c_{A} (t) = 2C_{A1} (1 − t)t + C_{A0} t^{2} ;

c_{P} (t) = C_{P0} (1 − t)^{2} + 2C_{P1} (1 − t)t

Figure A5 shows an isobole constructed from a Bernstein spline of order 2 with points p_{0} , p_{1} and p_{2} .

Figure A5.: Bernstein spline function defined by three control points p0, p1 and p2. Bernstein splines have the property that the first and the last control point lie on the curve, whereby control points in between might lie somewhere. The depicted curve could be an example for a supra-additive isobole.

With each point added to the Bernstein spline, two additional parameters must be estimated. In the present study we investigated the following three Bernstein spline functions with two, three and four parameters as given by their control points:

Two control points, two parameters (C_{P0} , C_{A0} ): P_{0} = (0, C_{P0} ), p_{1} = (C_{A0} , 0)
Three control points, three parameters (C_{P0} , C_{A0} , ρ): P_{0} = (0, C_{P0} ), p_{1} = (ρC_{A0} , ρC_{P0} ), p_{2} = (C_{A0} , 0)
Three control points, four parameters (C_{P0} , C_{A0} , c_{P1} , c_{A1} ), P_{0} = (0,C_{P0} ), p_{1} = (c_{A1} , c_{P1} ), P_{2} =(C_{A0} , 0)
Local and global types of interaction
To specify the meaning of a local and a global type of interaction, consider the parametric representation of the isobole of 2 drugs ‘1’ and ‘2’ by the function pair (c1(t), c2(t)), which goes from (c_{1} (0) = 0, c_{2} (0)) to (c_{1} (1), c_{2} (1) = 0) as the variable t goes from 0 to 1. Given the sum of ratios SR(t) = c_{1} (t)/c_{1} (1) + c_{2} (t)/c_{2} (0), the type of interaction at location t may be defined as additive, supra- or infra-additive if SR(t) = 1 or <1 or >1, respectively. If SR(t) = 1 for all 0 ≤ t ≤ 1, then the interaction is said to be globally additive with corresponding definitions for global supra-additivity and global infra-additivity.