Clinical application of pharmacokinetic

and PHARMACODYNAMIC models

Dr V. Billard[1]

Anaesthetists have a special interest in pharmacology compared to other physicians who choose the drugs to prescribe then often follow narrow approved guidelines to choose the doses. Conversely, the anesthetic drugs could be administered in a wide range of doses, and underdosage (awareness, muscle tension or pain) adequate anesthesia, or overdosage (late recovery, side effects) can all be observed inside the approved range of doses.

The art (or the science ?) of the anaesthetist is to choose both the drug and the doses in order to achieve an adequate level of anesthesia as fast as possible and maintain it just as long as necessary.

Pharmacokinetic (PK) and pharmacodynamic (PD) modeling could help to achieve this clinical goals because it splits the relationship between a dose and its effects into successive physiologic steps (dose à concentration à effects). Clinically, this approach is first useful to understand what is happening when giving a dose, and how will change the corresponding effects. Then the PKPD modeling could be used up side down to adjust the dose to a desired level of effect at any time through target controlled delivery systems.

PKPD modeling to understand the dose-effect relationship

PK or PD models are a set of mathematical equations that link a dose of a drug to the corresponding concentration and effects at different times.

Models are usually established by giving a known dose of drug, measuring blood concentration and effects over time and fitting the measured values to a "a priori" chosen model, as 2 or 3 compartments mamillary models for PK or Hill E_max model for PD (figure 1).

The parameters obtained from the fitting are used to predict concentration and effects for other patients at different times, and with different doses, if the model has been demonstrated to be linear ¹. This process assumes that the most important factor in the dose-effect relationship is the drug and not the patient.

This assumption is essential because most of the patients to whom we give anesthesia come only once. There is no way to sample for PK modeling on a first session and call them for surgery on a second one, or to wait for a real steady state to allow the surgeon to start to work.

Fortunately, this basic assumption is often true for anesthetic drugs.

Rational choice of drugs and doses based on PK models and simulation softwares

The first use of PKPD modeling is to describe the fundamental properties of a drug and to compare the drugs to each other in order to choose the best drug for each use.

Reading the PK parameters (figure 1) is tricky and may induce interpretations errors ². As anaesthetists could not all be experts in pharmacology, the PK parameters could be incorporated into simulation softwares that display the time course of predicted concentration when entering the drug and the doses (table 1).
Those softwares are easy to use for all anaesthetists (certified doctor, student or nurse) to understand what they have done by giving a dose to a patient, or what they should do to achieve stable and adequate anesthesia. Most of the softwares use previously published 2 or 3 compartments models together with an additional compartment for effect-site ³ and display the predicted drug concentration in plasma and effect-site for any dose.

Simulation programs are wonderful tools for understanding and teaching. Their clinical relevance could be separated into 3 main areas :

- How to deal with delay and duration of action for short term (induction, short cases)

- How to optimize recovery after long term use.

- How to adjust doses to special patients

Use of PK models for short term procedures

The PK modeling including effect-site compartment shows the time from an iv bolus to its maximal effect (Tmax, figure 1) ⁴. It indicates how long time a bolus should be anticipated from the time the effect is needed. For example, it illustrates why oipioids as remifentanil or alfentanil are suitable for short duration stimuli (fast intubation, endoscopy, bone fracture reduction), just to avoid to achieve the maximal effect when the procedure is over (figure 2).

It also shows the decrease in concentrations after a bolus, suggesting that fentanyl or sufentanil may be given by repeated boluses, whereas alfentanil or remifentanil requires continuous infusion for procedures longer than a few minutes (figure 2).

Finally, simulating the time course of effect-site concentration after a bolus shows how increasing the dose can shorten the onset (since the time to maximal effect is supposed to be constant), but increases the duration of action. This property has been widely used for choosing the doses of muscle relaxants : it resulted in the classical choice of 2 ED₉₅ for intubation, to optimize the balance between a short onset to intubate quickly and a reasonable duration of blockade to avoid post operative residual blockade. But everybody knows that the dose may be doubled or more for emergency intubation.

Preparing recovery after long time administrations

We learnt at medical school that the duration of action was a constant for each anesthetic drug, and was related to the elimination of the drug from the body.

PKPD modeling showed us that, in realty, adequate anesthesia corresponds to a certain concentration, and recovery occurs below another concentration ("MAC awake" or iv equivalent), although substantial amount of drug is still present in the body. The time to go from the maintenance concentration to the recovery concentration depends on how far they were from each other, how fast the drug is cleared from the body, but also how much the drug accumulates in the body, and how long time it was administered.

The effect of the drug and the duration of infusion was illustrated by the concept of Context Sensitive Half Time described almost 10 years ago ⁵. It shows why remifentanil is suitable for long infusion when fast recovery is required (for example for severe COPD or obese patients), alfentanil or sufentanil may be suitable for long infusion when intermediate delay of recovery is possible and fentanyl should be avoided when postoperative ventilation is undesirable. It can also explain why propofol is suitable for maintenance of anesthesia whereas thiopental is not.

For clinical use, context sensitive half time gives an estimates of the time to recovery only when the maintenance concentration is twice the recovery concentration : this is often but not always true. So, the concept of CSHT has been extended : beside the time necessary to decrease the concentration by 50%, the decrement time was defined as the time necessary to decrease the concentration by any percent ⁶.

This parameter, available from the simulation softwares, could be helpful clinically not only to choose a drug but to decide the level of concentration to maintain, and the consequences of this choice on recovery times. This has been nicely described for opioids, specially by Shafer and the Stanford group (figure 3).

- For example, for minor surgery with little postoperative pain, a concentration above the recovery concentration by 20% may be sufficient. In that case, there is no difference between fentanyl, alfentanil or sufentanil regarding recovery time up to 120 minutes of surgery and any opioid can be used (figure 3, top).

- If the surgical stimulus is stronger, opioid concentration around twice the recovery concentration may be chosen, but the physician must be aware that postoperative ventilation may be necessary with fentanyl, and that sufentanil allows faster recovery than alfentanil up to 8 hours of infusion (figure 3, middle).

- When analgesia is the main concern and postoperative sedation is usual as in cardiac surgery, opioid concentrations 4 fold above the recovery concentration can be proposed. Simulation shows that the recovery would be faster using alfentanil than sufentanil for surgery longer than 2 hours, which is not very much used in clinical practice ! (fig 3, bottom).

- Finally, PK calculation of decrement time show the specific behaviour of remifentanil compared to the other opioids : this feature is an advantage when fast recovery is the main concern, but the fast disappearance of all opioid effect must be anticipated when postoperative analgesia is required.

Adjusting to special populations

Another advantage of PK modeling is to show the influence of physiological variables as weight or age on the time course of concentration.

Unfortunately, most of initial PK studies published when the actual intravenous agents have been released did exclude obeses and extremes ages. Some other did a class analysis describing a set of PK parameters for young adults and another set for elderly, but the use of these models is limited because for intermediate values of the physiological variables, interpolation between 2 PK models is necessary.

The most useful models to incorporate in a simulation software express the physiological variables as a covariate of the model as could be fitted by population analysis. ^7-10.

Models including physiological variables are very relevant in clinical practice because they show how much the dose should be modified in special populations of patients to achieve a chosen concentration, as illustrated for remifentanil in figure 5.

Until now, only simple and constant variables (age, weight, gender, lean body mass, … ) are included in simulation software for iv drugs.

Changing variables as cardiac output are included in simulation software for volatile agents as Gasman. For iv drugs, they are only used in physiological models, and rarely used by routine physicians. They will be developed in another communication.

Clinical benefits of PD modeling

PD models described the mathematical relationship between the concentration of a drug and its effect (for example a BIS value at 50), or the probability of effect for binary effects (response to incision, to verbal command,…).

They first induced a change in the minds : anaesthetists stopped to describe the effects of a drug as a function of the dose (since this relationship is changing every second) but started to control the concentration on one hand, and assess the corresponding response in the other hand. Then, they had to decide if the response was adequate, and if it was not, they adjusted the dose not as a final goal but in order to increase or decrease the concentration.

This process pulled up the management of intravenous anesthetics closed to the delivery of volatile agents because nobody cares about the number of milliliters of volatile agent is given, but the dose delivered is adjusted to achieve a chosen end-tidal fraction, then to maintain it if the level of anesthesia is considered adequate.

PD modeling showed clinically important features of concentration-effect relationship:

- The requirements differ with the effect considered. The opioid concentration necessary for intubation is higher than for incision ¹¹ (figure 4) and the concentration of muscle relaxant to block the diaphragm or the larynx is higher than to block the peripheral muscles of the hand ¹². That is why at any time of the anesthesia, doses should be adjusted to achieve the adequate concentration for the current surgical time.

- The potency of a drug could be modified by physiological variables as age : for example, the adequate concentrations are reduced by about 50% in elderly patients for all mu-opioids ¹³, and by 30% for propofol ¹⁴.

- The time to equilibration between blood and effect site (brain) is longer in elderly for remifentanil (figure 5) ¹³ and also for the hemodynamic effects of propofol ¹⁵.

These properties suggest that induction in elderly patients should achieve lower concentration than in younger adults, and achieve it slower to avoid overdosage and side effects.

The second major clinical interest of PD modeling is to describe and quantify interactions between drugs. The most relevant interaction is anesthesia is the synergism between hypnotics (intravenous or volatile) and opioids. It could be displayed as a 3 dimensional surface model where x and y are drugs concentrations and z is effect ¹⁶.

As for the concentration-effect relationship, interactions differ with the effect considered :

- Opioids reduce only moderately the concentration of hypnotic necessary to loose consciousness (30 to 50%) (figure 6, top) ¹⁷.

- They reduce markedly (60-85%) the concentration of hypnotic needed to block motor response to stimulations

- The maximal synergism is observed when considering the hemodynamic response to noxious stimuli (up to 90%reduction) ¹⁸.

- However, synergism also occurs on side effects as hypotension ¹⁹ or respiratory depression.

So, for every surgical event and every type of response, several combinations of concentrations can all provide adequate anesthesia (figure 6). Whereas the shape of the interaction curve is always the same, the values depends on both drugs combined, and the anaesthetist can choose his strategy according to several criteria :

- To maintain non moving patient and hemodynamic stability, any point of the curve is OK. Below the curve, the patient may show signs of light anesthesia, and above overdosage.

- If the noxious stimulation occurs or change rapidly, it may be interesting to shift the balanced anesthesia to the faster reacting drug : to the opioid side (right part of the curve) with remifentanil of alfentanil, or to the hypnotic side (left part of the curve) with propofol, desflurane or sevoflurane.

- If fast recovery and discharge are the main concern, the balance should be shift to the drug having the faster elimination.

This property has been nicely illustrated by Vuyk & col. who performed simulations of anesthesia for gynecologic surgery and recovery with opioids and propofol (figure 6) ²⁰ :

- when fentanyl was used, as its decrement time was much more longer than the one of propofol, the fastest recovery was obtained with an excess of hypnotic.

- with alfentanil or sufentanil who have a decrement time similar to propofol, fastest recovery was achieved for a balanced combination.

- with remifentanil (fast decrement time compared to propofol whatever the duration of infusion), the fastest recovery was obtained in excess of opioid.

In summary, PD modeling of interactions can help the anesthetist to choose the optimal strategy of drug combination after having answered 2 questions :

- what is my main concern for the coming up surgical time?

- How do all the drugs I chose perform to achieve this goal?

PKPD modeling to adjust the doses to a desired concentration or effect

Three clinical applications of PKPD modeling have been described in anesthesia to control the doses of drug given. The first one, target controlled infusion, is already widely used in routine practice (at least for 1 drug…). The others as bayesian adjustment and closed loops, are still limited to prototypes and research protocols but could come to the routine practice of anesthesia in the next future if the regulatory requirements are performed.

Target controlled infusion

As PD models describe the adequate concentrations of all drugs for every surgical time, the anaesthetist could choose a concentration (or effect) to achieve and maintain and ask to a dedicated software to calculate the corresponding doses : this idea is the basis of target controlled infusion ("TCI" or "AIVOC" in french).

Described more than 30 years ago ²¹, this technique has been developed during the last 20 years through several prototypes in Germany ^22;23, Netherland ²⁴ and USA ^25-28.

Then, TCI has moved to the routine practice for propofol with the development by Zeneca of a dedicated medical device, the Diprifusor™ which was CE approved in 1996 ^29;30.

To day, there is no need to define TCI : several general reviews have described it ^31-33 and most of the anaesthetists in Europe either practice TCI or at least have been told about it.

Propofol TCI has been widely used in routine : from minor procedures ³⁴, to spinal ³⁵ or to cardiac surgery ³⁶ where it has been very helpful for early extubation ³⁷. It has also been proposed for sedation associated with local or regional anesthesia ³⁸, as well as for endoscopy or fiberoptic intubation when spontaneous ventilation must be maintained.

TCI has shown several clinical benefits, due to a better adjustment of the drug effect over time.

- For propofol, TCI reduced the incidence of movements during surgery ^39;40, and decreases the number of human interventions ^40;41. It didn't save drug ^39;40;42, but shortened the time to discharge from recovery room ⁴¹and decreased the incidence of PONV compared to manual infusion ⁴². It has been described as easy to use with a minimal training, and most of the anaesthetists who tried it for studies would continue to use it afterwards ³⁹.

- For opioids, it improved hemodynamic stability because the opioid concentration could be titrated at any surgical time ^25;43;44.

- TCI has also been proposed in ICU for propofol and for midazolam and reduces the interindividual variability of sedation score ⁴⁵.

TCI could target either the plasma (as the Diprifusor™), or the effect site, or the effect itself when it is well defined as for muscle relaxants ⁴⁶.

Targetting the effect-site slows down the time to achieve the target compared to a plasma target, but minimizes the time to achieve a chosen level of effect ⁴⁷.

It is necessarily associated with an initial overdosage of the plasma concentration that may theoretically be detrimental, but this risk has never been clinically demonstrated, and could likely be avoided by careful titration in fragile patients.

Most of the simulation programs mentioned above (table 1) can pilot a syringe pump in a TCI mode, and can target either the plasma or the effect site. However, all are dissociated systems (a computer + a cable + a syringe pump) and are not approved for clinical use out of research with IRB approval. To be CE approved for TCI, a medical device will need to be submitted to a risks and safety analysis, and the recommended target concentration need to be added to the regulatory recommendations ³⁰.

Based on the numerous clinical benefits demonstrated with TCI, the approval of such TCI devices are really an clinical emergency need, specially for opioids or midazolam, but would be also useful for propofol in special populations as children or to target the effect site.

Bayesian adaptation

Both PK and PD responses for an individual differ more or less from the model established on another population. When the population PKPD model is coupled with a quantitative assessment of concentration or effect in one individual, the model could be adjusted to this individual : it is the bayesian approach.

Bayesian techniques are well known for antibiotics and antineoplasic agents to improve PK models. But those drugs are given in repeated doses over longer periods of time, and could be adjusted to plasma assays that needs several hours to be performed. Conversely, adjustment of anesthesia could not wait for a classical drug assay since the result would show up far after the patient recovery or discharge ! Only one study described the theoretical benefit of bayesian approach to improve the PK model of alfentanil, but it was retrospective ⁴⁸.

So, bayesian technique could be clinically helpful for improving PK models only if associated with fast assays, available in a few minutes.

Another use of bayesian approach in anesthesia would be to measure the effect and adjust the whole PK and PD population model according to this measurement.

This could be done in real time for muscle relaxants ^46;49 and for hypnotics using EEG parameters. The bayesian adjustment could be manual as in Stanpump or included in an automatic closed loop.

Closed loop control

When the effect could be measured and the measure automatically transmitted to the infusion controller software, the dose (or the target concentration) could be adjusted by iterations to minimize the difference between desired and observed effect, realizing a closed-loop.

The adjustment could be done directly on the dose, using simple controllers as in industrial processes (Proportional, Proportional-Derivative, Proportional-Integral-Derivative) or fuzzy logic controllers ⁴⁹. However, this technique needs repeated measurements during the whole procedure, and loss of the signal could result in crazy dose adjustments.

The second way to build a closed loop system is to adjust not the dose but the PK or the PD model according to the measured value. This has been described for hypnotics using, as a quantitative effect, spectral analysis of EEG ⁵⁰, BIS ⁵¹ or auditory evoked potentials ⁵² . In all studies, quite stable anesthesia could be maintained. It has also been proposed for muscle relaxants ⁴⁶. Theoretically, this approach is more robust than a closed loop adjusting the dose, because it can reduce the inter individual variability (main cause of PD variability) and adjust the model to each patient using very few measures. Then, even if the measured effect is lost, the model is still appropriate for this patient, and the anesthesia should remain stable.

However, the influence of the opioid and the surgical stimulation on the EEG parameters and subsequently on the adjustment of the model are uneasy to describe and should be further studied before a routine use in clinical conditions.

Conclusion

At the stage of research, PKPD modeling is an essential tool to understand step by step the general behaviour of anesthetic drugs in the body, and to determine the covariates who are relevant and those who are not.

At the bedsite, PKPD models could be used through simulation softwares to display predicted concentration and effect for a rational choice of the drugs and the doses.

They give to the anesthetist the opportunity to think directly in terms of concentrations and concentration–effect relationship, to optimize the onset and predict the recovery and could dramatically improve the stability of anesthesia through target controlled delivery devices.

Table 1 : Some of the simulation softwares available. All could be used either in dose units or in TCI mode ; in TCI all of them can target the plasma or the effect site, and some can target the EEG effect (Rugloop, next version) or the level of neuromuscular blockade (Stanpump).

Name	Author and request address	Hardware	Drugs included and specific features
Stanpump	SL Shafer http://pkpd.icon.palo-alto.med.va.gov	PC (DOS) or MAC	Most of iv, run 1 pump Several models / drug Bayesian for NMBA
Stelpump	J Coetzee http://pkpd.icon.palo-alto.med.va.gov	PC (DOS)	Most of iv, run 2 pumps Few models / drug
Rugloop	M Struys http://allserv.rug.ac.be/~mstruys	PC (Windows)	Most of iv Several models / drug + data management (Datex AS3, BIS, Anemon)
Tivatrainer	F Engbers http://www.eurosiva.org ou webmaster@eurosiva.org