^{1}it is likely that concerns about the accuracy of such devices have had a significant role in delaying approval.

^{2}

^{3–6}alfentanil,

^{7–10}sufentanil,

^{11,12}remifentanil,

^{13,14}propofol,

^{15–24}etomidate,

^{25}thiopental,

^{26}midazolam,

^{27,28}dexmedetomidine,

^{29}and lidocaine.

^{30}One striking feature about the accuracy of TCI devices is that it resembles the accuracy observed after simple infusions or bolus drug delivery.

#### Materials and Methods

##### Mathematical Proof

Equation 1 Image Tools |
Equation (Uncited) Image Tools |

*C***are the coefficients of the pharmacokinetic model, and λ**

*i***are the exponents of the pharmacokinetic model. Because these coefficients and exponents describe how the body “disposes” of a unit bolus of drug, we refer to this sum of exponentials as the**

*i***(UDF). Therefore, the UDF in equation 1 is**

*unit disposition function***and**

*time-invariant***pharmacokinetics.**

*dose-invariant***(), with the disposition function using the convolution integral**

*I*Equation 2 Image Tools |
Equation (Uncited) Image Tools |

**(τ) is the input, consisting of one or more bolus injections. For an infusion,**

*I***(τ) consists of a very large number of very small boluses. The integral simply adds up the concentrations from each bolus, after shifting the curves to reflect the timing of the doses (accomplished by the τ and**

*I***− τ). Because any infusion regimen, including TCI, can be reduced to series of infinitely small boluses, equation 2 is a general description of the relation between drug administration,**

*t***(), the pharmacokinetic model,**

*I***(). We can simplify our mathematical notation by letting * refer to the operation of convolution and referring to**

*Cp***(**

*U***):**

*t***, the SD, ς, and the coefficient of variation (CV) in a conventional way. For any vector**

*E***= (**

*X*

*x*_{1}, . . .,

*x***) of length**

*n***,**

*n*Equation 4 Image Tools |
Equation 5 Image Tools |
Equation 6 Image Tools |

**is the average**

*X***. Equation 5 states that the SD is the average squared difference between the average**

*X***and each individual observation. Equation 6 states that the CV is the SD divided by the average. Therefore, equations 4, 5, and 6 are standard statistical definitions, but they are introduced here to help clarify the formal proof that follows.**

*X*##### Theorem 1

**(**

*U***) = {**

*t*

*u***(**

*i***)},**

*t***= 1, . . .,**

*i***be a collection of unit disposition functions in a population of**

*n***patients. We define ρ**

*n***(**

*u***) as the CV of the set of UDFs at time**

*t***. We also define**

*t***= max**

*M***ρ**

*t***(**

*u***),**

*t***, the maximum value of the CV of the UDFs over time.**

*i.e.*

*D***be a series of consecutive boluses of amount**

*j*

*d***given at time**

*j*

*t***,**

*j***= 1, . . .**

*j***. Consistent with equations 2 and 3, we define**

*k***(**

*I***) as the dosing function, which consists of boluses**

*t*

*D***. Let each individual patient receive the same**

*j***(**

*I***).**

*t***(**

*I***), the observed CV in concentration in the population of patients will be less than or equal to**

*t***, the maximum CV after unit bolus injection. Stated formally, we propose that for any**

*M***(**

*I***),**

*t***th bolus in the**

*j***th individual. Let**

*i***be the number of boluses up to time**

*m***. The response at time**

*t***in the**

*t***th individual is the sum of the responses to all preceding boluses,**

*i***is**

*t*Equation 10 Image Tools |
Equation (Uncited) Image Tools |

**th bolus. By definition of**

*j***, the maximum CV after the**

*M***th bolus injection is**

*j*Equation 13 Image Tools |
Equation 15 Image Tools |
Equation (Uncited) Image Tools |

##### Simulations

**,**

*et al.*^{31}we calculated the expected volumes and clearances for a 50-yr-old man with a weight of 70 kg and a height of 170 cm: V

_{1}= 4.27 l, V

_{2}= 20.07 l, V

_{3}= 238 l, Cl

_{1}= 1.64 l/min, Cl

_{2}= 1.36 l/min, Cl

_{3}= 0.84 l/min. We then created a population of 1,000 individuals whose volumes and clearances varied by 30% (log-normally distributed) from the nominal individual.

##### Experimental Data

^{32}We used these data to examine whether the predictions of the mathematical proof and the simulations were observed in experimental data. The CV was calculated using Excel.

#### Results

##### Mathematical Proof

##### Simulations

##### Experimental Data

#### Discussion

^{1}TCI devices have been associated with biologic variability, reflecting the desire of investigators to obtain an accurate prediction of concentration. Although this effort to obtain an accurate prediction is commendable, the accuracy of the prediction is inexorably limited by biologic variability. The transformation of a drug administration profile to a plasma or effect site concentration is done by each individual patient’s body. TCI devices are not responsible for biologic variability. The case has been made that biologic variability prevents safe use of TCI devices.

^{2}This is nonsense.

**, age-related reduction in intercompartmental compartmental clearance) are virtually impossible to incorporate into drug dosing without TCI. As the science of drug pharmacokinetics produces ever more complex models, including physiologically based models and models incorporating complex genetic traits, there will be increasing need for TCI devices to translate these advances into accurate drug administration.**

*e.g.*^{1}

^{33}patients have identical concentrations at all times except at 20 min. At 0, 10, 20, 30, and 40 min, the concentrations are 4 ± 0%, 2 ± 0%, 1 ± 50%, 0.5 ± 0%, and 0.25 ± 0% CV, respectively, as shown in table 1. Now, consider giving two identical boluses of Duzitol to the same population of patients, the first given at time 0 and the next given 10 min later. The resulting concentrations are shown in the right columns in table 1. Because the times of maximum CV for the two boluses do not line up, the maximum CV after two boluses (table 1, far right column) is less than the maximum after a single bolus.

^{34,35}Similarly, the pharmacokinetic software in TCI devices should meet appropriate standards for software development and validation. The software for the Diprifusor has been rigorously tested by regulatory agencies.

^{36,37}Parenthetically, all contemporary infusion systems are controlled by computer software, and the software required to execute the pharmacokinetic model is trivial compared with the software required for the user interface, stepper motor tracking, pump error handling, and other essential functions of infusion devices.

**–response relations. These are simply dose–response relations, but they take advantage of the ability of TCI devices to achieve a pseudo–steady state. In addition, the exact dose (amount of drug over time) specified by the target concentration can be readily calculated. As such, the infusion rates called for at a given target concentration can be compared with the range of doses specified in the drug labeling to demonstrate whether they conform to the package insert.**

*versus*