F E A T U R EF E A T U R E

Control technology influences modernmedicine through robotic surgery,electrophysiological systems (pace-makers and automatic implantabledefibrillators), life support (ventilatorsand artificial hearts), and image-guided

therapy and surgery. An additional area of medicinesuited for applications of control is clinical pharmacol-ogy, in which mathematical modeling plays a promi-nent role [1], [2]. Although numerous drugs areavailable for treating disease, proper dosing is oftenimprecise, resulting in increased costs, morbidity, andmortality. In this article, we discuss potential applicationsof control technology to clinical pharmacology, specificallythe control of drug dosing [3]. (See “Control Engineeringand Medicine: A Fruitful Collaboration.”)

We begin by considering how dosage guidelines are devel-oped. Drug development begins with animal experimentation.Promising agents are moved to human trials, which begin withhealthy volunteers and progress to patients with the specific dis-ease for which the drug is being developed. Early stages of these tri-als focus on safety, while the final trials usually involve administrationof a placebo and different drug doses to evaluate efficacy. Efficacy isdefined statistically, and aggregate therapeutic effects do not preclude theexistence of individual patients for whom the drug is either not efficacious orcauses side effects. If a therapeutic effect is observed, then the drug may beapproved by the Food and Drug Administration (FDA). In general, the recommendeddosage is the level found to be efficacious in the “average” patient. Herein lies the prob-lem: the “average” patient does not exist.

Paradigms, benefits,and challenges

By James M. Bailey and Wassim M. Haddad

© ARTVILLE

April 2005 350272-1708/05/$20.00©2005IEEE

IEEE Control Systems Magazine

April 200536 IEEE Control Systems Magazine

Substantial variability exists among patients both inthe drug concentration at the locus of the effect (theeffect-site concentration) resulting from a given dose andin the therapeutic efficacy of a given effect site concentra-tion. Frequently, the appropriate dose for a specificpatient is found by trial and error. For example, a physi-cian treating a patient for hypertension typically beginsby prescribing the recommended dose. After observingthe effect of the drug on blood pressure, the doctoradjusts the dose empirically.

A Primer on Clinical Pharmacology

Pharmacokinetic ModelsDrug dosing can be made more precise by using pharma-cokinetic and pharmacodynamic modeling [4]. Pharmacoki-netics is the study of the concentration of drugs in tissueas a function of time and dose schedule. Pharmacodynam-ics is the study of the relationship between drug concen-tration and drug effect. By relating dose to resultant drugconcentration (pharmacokinetics) and concentration toeffect (pharmacodynamics), a model for drug dosing canbe generated.

The distribution of drugs in the body depends ontransport and metabolic processes, many of which arepoorly understood [1], [4]. However, compartmental mod-els, that is, dynamical models based on conservation lawsthat capture the exchange of material between coupledmacroscopic subsystems or compartments, are widelyused to model these processes [2]. Pharmacokinetic com-partmental models typically assume that the body is

comprised of more than one compartment. Within eachcompartment, the drug concentration is assumed to beuniform due to perfect, instantaneous mixing. Transport

The barriers between control engineering and medicineare slowly eroding as it becomes more evident thatcontrol system technology has a great deal to offer

medicine. Our interdisciplinary collaboration is based on thedesire of one author to explore the area of biomedical controlengineering and the interest of the other author in applica-tions of pharmacokinetic and pharmacodynamic modeling.Our collaboration began with e-mail communication, fol-lowed by a period of mutual education. It then progressed tomonthly meetings to discuss and identify control methodolo-gies and paradigms for addressing pharmacological prob-lems. From our own experience, it takes somewhat longer forthe clinician to become comfortable with the vocabulary andconcepts of control than vice versa. While anesthesiologistsuse feedback control every working day, few are familiarwith the mathematical rigor of the control scientist.

Our collaboration is facilitated by the fact that whilepharmacokinetics, the clinician’s research interest, is basedon dynamical systems theory, the level of rigor in pharma-cokinetics is well below that of control science. Thus, therate-limiting step in the collaboration is introducing theclinician to the vocabulary of the control engineer. Thecollaboration is now at a point where the clinician canserve on the dissertation committee of the control engi-neer’s graduate students. Furthermore, the clinician servesa valuable role in the collaboration by giving the controlengineer an idea of which assumptions and approxima-tions are clinically realistic. This exchange is facilitated byhaving the control engineer observe the clinician in theoperating room and the intensive care unit, as we havedone. This interaction is particularly important as theorytransitions to clinical implementation.

Pharmacokinetic Terminology

Pharmacokinetic models are frequently described interms of half lives, compartment volumes, and clear-ance. For instance, most pharmacokinetic papers

report the terminal elimination half life, which is the timerequired for drug concentration to decrease by 50% if all tis-sues were equilibrated with the blood concentration. Thisparameter is useful to clinicians desiring a measure of theduration of drug effect once the drug is discontinued fromchronic use. Similarly, pharmacokinetic papers sometimesreport distribution half lives, the time needed for 50% equili-bration between two compartments when a drug is initiallyconfined to one compartment. Another parameter is clear-ance, a term originally taken from the renal physiology litera-ture. Clearance is defined as the effective volume of tissue orblood cleared of drug per unit time. Elimination clearance isequal to the product of the compartment volume and theelimination rate from the compartment. While drugs are noteliminated in the simplistic fashion implied by the above def-inition, the concept is useful since the constant drug inputrate needed to achieve a given drug concentration is equalthe product of the desired concentration and clearance.

Control Engineering and Medicine: A Fruitful Collaboration

April 2005 37IEEE Control Systems Magazine

to other compartments and elimination from the bodyoccur through metabolic processes. For simplicity, thetransport rate is often assumed to be proportional to drugconcentration. Although the assumption of instantaneousmixing is an idealization, it has little effect on the accuracyof the model as long as we do not try to predict drug con-centrations immediately after the initial drug dose.

In a simple one-compartment model, the body isassumed to consist of a single compartment in whichinstantaneous mixing occurs, followed by elimination. It isusually assumed that elimination is linear, with the rate ofelimination directly proportional to the drug concentrationin the compartment. This model is characterized by twoparameters: the compartmental volume Vd and the elimina-tion rate constant ae. For this simple model, the concentra-tion C (in moles/volume or mass/volume) immediatelyafter a dose of mass D is equal to D/Vd , and the drug issubsequently eliminated at the rate aeC with exponentialdecay. While the behavior of a few drugs can be describedadequately using this model, the model is too simplistic formost drugs. Furthermore, for drugs taken orally, a modelwith two or more compartments is required. One compart-ment represents the gastro-intestinal tract, which receivesthe dose and transfers it to a second compartment. Thesecond compartment represents intravascular blood (bloodwithin arteries or veins) and other organ systems.

A two-compartment mammillary model [2] can also beused for drugs administered intravenously. This modelincludes a central compartment that receives the intra-venous dose with instantaneous mixing and is typically iden-tified with organs such as the heart, brain, liver, and kidney.These organ systems receive a large amount of blood flowper unit mass and, hence, are well mixed with intravascularblood. The drug is then transferred to a peripheral compart-ment comprised of muscle and fat or it is metabolized andeliminated from the body. For most drugs, the enzyme sys-tems responsible for drug metabolism are found in the liveror kidney so that metabolism in the peripheral compartmentcan be ignored. Drug in the peripheral compartment trans-fers back to the central compartment with linear kinetics.

State-Space ModelsThe two-compartment mammillary system is described bythe state-space model

x(t) = Ax(t), x(0) = x0, t ≥ 0, (1)

where

A =[−(a21 + a11) a12

a21 −a12

],

x = [x1, x2]T is the state vector representing the masses inthe two compartments; a12 and a21 are the nonnegative

transfer coefficients from compartment 2 to compartment 1and from compartment 1 to compartment 2, respectively;and the nonnegative coefficient a11 is the rate at whichdrug is eliminated from the system through the centralcompartment 1. Entry (2, 2) of A reflects no eliminationfrom compartment 2. An additional parameter is the vol-ume V1 of the central compartment, for a total of four phar-macokinetic parameters. (See “PharmacokineticTerminology.”) Note that with the assumption of instanta-neous mixing, the concentration at t = 0+ after dose D isadministered is D/V1.

The two-compartment mammillary model is useful formany drugs administered intravenously. The basic two-compartment model predicts that the drug concentrationin the central compartment after a bolus (impulsive) initialdose can be described by the sum of two terms decreasingexponentially with time. However, to fit the data, somedrugs require three or more exponential terms, whichmotivates an extension of the two-compartment model [2].Figure 1 shows an n-compartment mammillary model,which includes a central compartment that distributesdrug into the interstitial spaces of several organs and tis-sues of the body.

In most cases, the assumption of linear transfer ismaintained so that the system is modeled by (1), wherex ∈ Rn represents the system compartmental masses orsystem compartmental concentrations and A ∈ Rn×n is acompartmental matrix [2] when x represents compart-mental masses and a Metzler matrix [2] when x represents

Figure 1. The n-compartment mammillary model. The cen-tral compartment, which is the site for drug administration, isgenerally thought to be comprised of the intravascular bloodvolume as well as highly perfused organs such as the heart,brain, kidney, and liver. The central compartment exchangesthe drug with the peripheral compartments comprised ofmuscle, fat, and other organs and tissues of the body, whichare metabolically inert as far as the drug is concerned.

CentralCompartment

2

3 n−1

n

April 200538 IEEE Control Systems Magazine

compartmental concentrations. Hence, (1) describes anonnegative, compartmental dynamical system [2].

Compartmental pharmacokinetic models, especiallymammillary models, are coarse-grained oversimplifica-tions. Consider the injection of a drug into a small periph-eral vein in the hand. The drug is transported by thevenous stream to the right atrium and the right ventricle,binding to blood cells or proteins and mixing with venousstreams as veins coalesce. Large-scale mixing occurs in theright atrium and ventricle, transporting the drug to thelung, where some of the drug can bind to tissue. From thelung, the drug returns to the left atrium and ventricle.There, it is expelled into the aorta for transport to otherinert tissues, where drug binding occurs, and to the liverand kidney, where the drug is metabolized. Modeling thisprocess with a small number of compartments is thus anapproximation.

Drug Action, Effect, and InteractionThe clinical utility of pharmacokinetic models dependsentirely on the time scale of the application. For example,these models work well for determining suitable dosingintervals for drugs administered orally as well as for main-taining appropriate anesthetic concentrations duringsurgery. Using simplified mammillary models, it is possibleto achieve median absolute performance errors (the nor-malized difference between target and measured anestheticconcentrations) of less than 20% when drug concentrationsare sampled every 15 minutes. This level of performance isclinically acceptable since drug concentrations within thisrange generally achieve the desired effect.

We now consider the problem of predicting drug con-centrations during the administration of anesthesia. Anes-thesia is typically initiated by administering a bolus doseof a hypnotic drug intravenously. During the first few min-utes after initiation of intravenous anesthesia by adminis-tration of a bolus dose, mammillary compartmentalmodels fail to accurately predict drug concentrationsbecause of the assumption of instantaneous mixing. Moreelaborate models postulate multiple compartments inseries to approximate the transport of the drug from thesite of injection to the central circulation. Additional paral-lel compartments (similar to mammillary models) accountfor drug distribution to peripheral tissue (muscle and fat).These extensions help to describe drug concentrationimmediately after a bolus dose initiation of anesthesia [5].

While the most commonly used pharmacokinetic mod-els are linear, the underlying processes that determinepharmacokinetic behavior are nonlinear. For example, themolecular processes of drug metabolism are described byMichaelis-Menten kinetics, in which the rate of drugmetabolism is given by VmC /(Km + C ), where Vm is themaximum rate of reaction, Km is the drug concentrationthat achieves 50% of the maximal effect, and C is the drug

concentration. However, large-scale pharmacokineticmodels assume linear drug metabolism or elimination.Similarly, most compartmental models assume that drugdistribution between tissues is linear. However, deliveryof drug to tissue per unit time is equal to the product ofdrug concentration in the blood and regional perfusion,the blood flow to tissue per unit time. Regional perfusionmay be nonlinearly related to the drug concentrationsince anesthetic drugs alter cardiovascular function.

Pharmacokinetic Parameter EstimationData used for pharmacokinetic modeling is collected byadministering a drug to patients, drawing blood samplesat designated times after the initiation of dosing, anddetermining the concentration of the drug as a function oftime (but not in real time). Consequently, most pharmaco-kinetic investigations focus on blood concentrations. Onegoal of this analysis is to derive an expression for the unitdisposition function, the time-dependent blood concentra-tion that results from a unit bolus dose. Assuming linearkinetics, if the unit disposition function fud is known, thenthe resulting blood concentration is given by the convolu-tion integral

C (t) =∫ t

0fud(τ)D(t − τ)dτ, (2)

where D(t) is the dose as a function of time [6].It is not feasible to measure drug concentration in the

tissue at the site of the therapeutic effect. Since drugs aredistributed to the action site by blood flow, the effect siterapidly equilibrates with blood, and it is often assumedthat effect-site concentration and blood concentration areequal. If the equilibration time between the centralintravascular blood volume and the effect site is clinicallyrelevant, then the pharmacokinetic model must be revisedto include an effect-site compartment distinct from thecentral compartment.

Pharmacokinetic model parameters that comprise thesystem matrix A are estimated by fitting models to thedata. There are numerous sources of noise in the data,from assay error to human recording error. Because ofmodel approximation and noise, there is always an offsetbetween the concentration predicted by the model andthe observed data, the prediction error. One method forestimating pharmacokinetic parameters is maximum likeli-hood [7]. This approach assumes a statistical distributionfor the prediction error and then determines the parame-ter values that maximize the likelihood of the observedresults. Suppose we conduct a study in a single patientfrom whom we collect blood samples at ten differentpoints in time after a single intravenous bolus. If weassume that the prediction error for an individual patient

April 2005 39IEEE Control Systems Magazine

(intrapatient error model ) has a normal or Gaussian distri-bution, then the likelihood L of the observed results isgiven by

L =r∏

i=1

1√2πσ 2

e−PE2i /2σ 2

, (3)

with the prediction error PEi of the i th observation givenby PEi = Cpi − Cmi , where Cpi is the i th predicted drugconcentration and Cm i is the i th measured drug concen-tration, σ 2 is the variance of the Gaussian distribution ofprediction errors, and r is the number of observations(measured concentrations).

The likelihood (3) of the observed results is a functionof σ and the pharmacokinetic parameters. By maximizing(3) (or, more commonly, its logarithm) with respect tothe pharmacokinetic parameters and σ , one can estimatethe structural model parameters (the entries of the sys-tem matrix A) and the error model parameters (in thiscase, σ ) that maximize the likelihood of the observedresults. The above example reduces to least squares esti-mation when σ 2 is a constant. Using a more sophisticatederror model, in which σ 2 is proportional to a power of thepredicted concentration, leads to weighted least squaresestimation [7].

There are two distinct approaches to estimating meanpharmacokinetic parameters for a population of patients[8], [9]. In the first approach, models are fitted to datafrom individual patients and the pharmacokinetic parame-ters are then averaged (two-stage analysis) to provide ameasure of the pharmacokinetic parameters for the popu-lation. The second approach, called mixed-effects modeling,is to pool the data from individual patients. In this situa-tion, the prediction error is determined by the stochasticnoise of the experiment as well as by the fact that differentpatients have different pharmacokinetic parameters. The sta-tistical model used to account for the discrepancybetween observed and predicted concentrations must takeinto consideration not only variability between observedand predicted concentrations within the same patient(intrapatient variability), but also variability betweenpatients (interpatient variability). Most commonly, it isassumed that the interpatient variability of pharmacokinet-ic parameters conforms to a log-normal distribution. Thissophisticated method of analysis estimates the meanstructural pharmacokinetic parameters as well as the sta-tistical variability of these elements in the population.

Analysis based on mixed-effects modeling is powerful fortwo reasons. First, this approach gives the clinician an esti-mate of both the pharmacokinetic parameters and theirvariance. These statistics are important for the cliniciansince no matter how desirable the properties of a drug areon average, extreme variability in these parameters mayindicate that the drug is not safe for clinical use. Second,

mixed-effects modeling may allow a reduction in theamount of data needed from each patient. In a two-stageanalysis, one must have enough data points from eachpatient to estimate the patient’s pharmacokinetic parame-ters. For example, a two-compartment mammillary modelrequires four pharmacokinetic parameters. With mixed-effects modeling, it is possible to estimate these parame-ters for any one patient with four or fewer data points.

Pharmacodynamic ModelsIn contrast to pharmacokinetic modeling, pharmacody-namic modeling is less readily related to molecularprocesses. The molecular mechanism of action of manydrugs is well understood; most drugs act by binding to areceptor on or within target cells [4]. There is a well-devel-oped theory of multiple equilibrium binding of ligands,such as drug molecules, to receptors on larger macromole-cules, such as proteins. In theory, pharmacodynamics,which models the relationship between drug concentra-tion and effect, should follow from models of molecularbinding. However, the physiological effect is an interplay ofnumerous factors, and it is generally not possible to ana-lytically relate the drug effect at the level of the intactorganism to the number of receptors bound by the drug atthe molecular level. Empirical models are thus needed. Itmight be assumed that drug effect is proportional to thedrug concentration at the effect site, but this simple linearmodel is unrealistic since it admits the possibility of limit-less drug effect as drug concentration increases whileignoring saturation effects.

One empirical pharmacodynamic model is given by theHill equation

E = EmaxC γ /(C γ + C γ

50

), (4)

where E is the drug effect, Emax is the maximum drugeffect, C is the drug concentration, C50 is the drug concen-tration associated with 50% of the maximum effect, and γis a dimensionless parameter that determines the steep-ness of the concentration-effect relationship [10]. Thismodel was developed in 1906 to describe a molecularinteraction, the binding of oxygen to hemoglobin. Sincethat time the Hill model has been applied to various phe-nomena that are far removed from explanations at themolecular level. A number of modified versions of thismodel have been employed, including the case in whichthe drug effect is a binary (yes-or-no) variable. An exam-ple of a binary variable is anesthesia, for which the patientis either responsive or not. In this case, Emax = 1, and thepharmacodynamic model becomes

P = C γ /(C γ + C γ

50

),

April 200540 IEEE Control Systems Magazine

where the effect is now the probability P that the patientdoes not respond to some noxious stimuli [11], [12].

In typical pharmacodynamic studies, a drug is adminis-tered and the effect is measured by taking a blood sampleat various points in time to determine the drug concentra-tion at the time of observation of effect. The parameters ofthe pharmacodynamic model (Emax, C50, γ ) can then beestimated by the maximum likelihood or generalized leastsquares methods described above. Obviously, if drug con-centrations in the blood and effect site have not equili-brated, this analysis does not apply.

It should be noted that pharmacodynamic models areinherently nonlinear. This property is in contrast to thelinearity of pharmacokinetic models. However, the inter-play with pharmacodynamics may also lead to nonlinearpharmacokinetics. For example, some intravenous anes-thetics depress cardiac output, the volume of bloodpumped by the heart per unit time. Since the basic trans-port processes that determine pharmacokinetic behaviordepend on blood flow, administration of the drug alters itskinetics. Furthermore, since the pharmacodynamic rela-tionship between drug concentration and depression ofcardiac output is nonlinear, the pharmacokinetics of thedrug are, in reality, also nonlinear.

Clinical Pharmacology and Drug Dosing

Open-Loop Drug DosingIn addition to safety and efficacy, the FDA requires that,prior to the approval of any new drug, manufacturers pro-vide a pharmacokinetic evaluation to establish a basis fordosage guidelines. The concentration that results from drugadministration is determined by the transport processesthat distribute the drugs to various tissues as well as by themetabolic processes that transform the drug. However, thevast majority of drugs are given for chronic conditions;when the time scale of treatment greatly exceeds the timescale of the distributive processes, there will be equilibra-tion of drug through the various tissues.

The pharmacokinetics of most drugs given chronicallyare described by (1), where A is a scalar. Drug calculationsare now greatly simplified. For example, consider an anti-hypertensive drug with a half life (the time needed for thedrug concentration to decrease by 50% after discontinua-tion of administration) of 12 hours. If a dose of 50 mg is effi-cacious in the average patient, then a suitable dosingschedule would be an initial dose of 50 mg with subse-quent dosing of 25 mg every 12 hours. As another example,suppose that a blood concentration of an intravenousanesthetic of 100 µg/ml reliably produces unconscious-ness and the clearance (the effective volume of bloodcleared of drug per unit time) is 150 ml/minute. An infusionrate of 100 µg/ml × 150 ml/min = 15000 µg/min maintainsthe desired blood concentration, although this concentra-

tion is achieved only when distributive processes haveequilibrated. Many of the dosing guidelines recommendedby the manufacturers of drugs are based on simple calcula-tions like these.

There have been attempts to develop more preciseopen-loop control in acute care, especially in anestheticpharmacology. Since the appearance of computers in theoperating room in the 1980s, investigators have developedcomputer-controlled pump systems that continually adjustthe drug infusion rate to achieve and maintain the drugconcentration desired by the clinician [13]–[16]. Thesesystems use the pharmacokinetic model

x(t) = Ax(t) + Bu(t), x(0) = x0, t ≥ 0

to calculate the dose u(t) needed to achieve and maintainthe target drug concentration. To implement thisapproach, it is necessary to know the pharmacokineticparameters that define A and B. In open-loop control, spe-cific pharmacokinetic parameters are not known for theindividual patient. Instead, it is assumed that average para-meter values, taken from the pharmacokinetic literature,are applicable to the individual patient. The dose calculat-ed using these average parameter values is then deliveredby a pump controlled by the computer. Despite the obvi-ous fact that these systems ignore interpatient pharmaco-kinetic variability, studies have demonstrated that drugconcentrations are better maintained in therapeutic rangesby open-loop control than with standard clinical practice.

Closed-Loop Drug DosingWhile initial dosing guidelines are often based on the aver-age patient, the significant interpatient pharmacokineticand pharmacodynamic variability observed for most drugssuggests that precise drug dosing requires closed-loop con-trol. Most patients, especially those treated for chronicdisease, are familiar with this closed-loop process. Thephysician prescribes an initial dose, observes theresponse, and adjusts the dose. Although some physiciansare adept at this process, it is usually time consuming.Also, the efficiency of the process depends on the experi-ence of the clinician since there is not enough data avail-able on many drugs to develop quantitative guidelines.

The process of dose titration can be made more preciseby using mixed-effects pharmacokinetic modeling andpost-hoc Bayesian estimation of individual patient pharma-cokinetic parameters [7]–[8]. Recall that mixed-effectsmodeling provides not only estimates of pharmacokineticparameters, but also their variance within the population.Suppose one or more drug concentrations are measured inan individual patient. Using Bayesian probability princi-ples, the likelihood of a given value of a pharmacokinetic

April 2005 41IEEE Control Systems Magazine

parameter � is proportional to P (C |�)P (�) , whereP (C |�) is the probability of the observed concentration Cas a function of �. Furthermore, P (�) is the a priori prob-ability of a given value of �, which is given by theassumed distribution of � (as noted above, usually log-normal) with the variance of � estimated from the mixed-effects analysis.

By determining the mode of P (C |�)P (�) with respectto � (the value of � at which P (C |�)P (�) is maximized),one can derive a maximum likelihood estimate of � for thespecific patient. The patient-specific parameter estimatecan be used to calculate the dose needed to achieve agiven drug concentration [17]. However, because of phar-macodynamic variability, more precise control of drug con-centration does not necessarily lead to better control ofdrug effect.

The process of titrating drug dose to the desired effectmay be acceptable for chronic outpatient therapy; yet, inthe acute care environment, such as the operating room orthe intensive care unit, this process is often either danger-ously slow or imprecise. Feedback control of drug effect, incontrast with drug concentration, has much to offer mod-ern medicine. The remainder of this article focuses ondrugs used in the acute care setting.

To implement closed-loop control in an acute careenvironment, real-time measurement of drug effect isrequired. Early attempts at closed-loop control focusedon regulating variables that are conveniently measured.By their very nature, cardiovascular and central nervoussystem functions are critical in the acute care environ-ment. Thus, technologies have evolved for their measure-ment. The primary applications of closed-loop drugadministration are hemodynamic management and con-trol of consciousness. Next, we review closed-loop con-trol of the cardiovascular system, which illustratesproblems inherent in the application of control technolo-gy to physiological function.

Closed-Loop Controlof Cardiovascular FunctionA major side effect of cardiac surgery is that patients canbecome hypertensive [18], requiring treatment to preventcardiac dysfunction, pulmonary edema, myocardialischemia, stroke, and bleeding from fragile sutures.Although drugs are available for treating post-operativehypertension, titrating these drugs to regulate blood pres-sure is often difficult. Underdosing leaves the patienthypertensive, whereas overdosing can reduce the bloodpressure to levels associated with shock. Since the late1970s, there has been interest in developing controllers foradministering sodium nitroprusside (SNP), a commonlyused and potent antihypertensive.

The problems encountered in hemodynamic control areenlightening. Initial attempts used simple nonadaptive

methods such as proportional-derivative (PD) or propor-tional-integral-derivative (PID) controllers, which assume alinear relationship between infusion rate and effect [19],[20]. However, while the drug concentration is given by theconvolution of the infusion rate and a transfer function asin (2), the relationship between effect and infusion rate isnonlinear as in (4). Also, a significant challenge to thedesign of a blood-pressure controller is the time delaybetween drug administration and the clinical effect, whichcan lead to system oscillations. Although early blood pres-sure controllers included time delays in the system model,the delays were assumed to be the same for each patient[19], [20].

While early controllers were successful in some patients,these techniques have not been widely implemented due tononlinear patient response and differences in drug sensitivi-ty among patients. Interpatient variability, as well as apatient’s sensitivity to drugs, motivated the development ofsingle-model and multiple-model adaptive controllers [21],[22]. Single-model adaptive controllers are based on on-lineestimation of system parameters using minimum varianceor least squares methodologies. These controllers performpoorly due to large amplitude transients [21].

Multiple-model adaptive controllers represent the sys-tem by means of a finite number of models. For eachmodel, there is a separate controller. The probability thatthe system is represented by each of the different modelsis calculated from the relative offsets of the systemresponse and the response predicted by each model. Theoutput of the controller is the probability-weighted sum ofthe outputs from each model [23], [24]. Multiple-modeladaptive controllers have proven to be more satisfactorythan single-model adaptive controllers and fixed-gain con-trollers [24]. Subsequent refinements to blood pressurecontrol include both single-model, reference adaptive con-trol [25], which appears promising in simulations, andneural network-based methods [26]. There is also interestin optimal control since SNP has toxic side effects whenthe dose is too high [27].

These investigations into controlling blood pressurereveal the challenges inherent to biological systems: non-linearity, interpatient variability (system uncertainty), andtime delay. Despite the refinements of closed-loop bloodpressure controllers, such controllers are seldom usedclinically. Although blood pressure control is important,cardiovascular function involves several other importantvariables, all of which are interrelated [18]. The intensivecare unit clinician must ensure not only that blood pres-sure is within appropriate limits, but also that cardiac out-put (the amount of blood pumped by the heart perminute) is acceptable and that the heart rate is withinreasonable limits. Mean arterial blood pressure is propor-tional to cardiac output, with the proportionality constantdenoting the systemic vascular resistance, in analogy with

April 200542 IEEE Control Systems Magazine

Ohm’s law. Cardiac output is equal to the product of heartrate and stroke volume, the volume of blood pumped witheach beat of the heart. Stroke volume, in turn, is a functionof contractility (the intrinsic strength of the cardiac con-traction), preload (the volume of blood in the heart at thebeginning of the contraction), and afterload (the imped-ance to ejection by the heart).

The intensive care unit clinician must balance all ofthese variables. Inotropic agent drugs, or drugs thatincrease the strength of contraction of the heart, also havevariable effects on heart rate and afterload. There are alsovasopressor drugs, which increase afterload, and vasodila-tor drugs, which decrease afterload. Finally, stroke volumecan be improved by giving the patient intravenous fluidsand increasing preload. However, too much fluid can poten-tially be deleterious by impairing pulmonary function asfluid builds up in the lungs. The fact that closed-loop con-trol of blood pressure has not been widely adopted by clini-cians is not surprising when one considers the complexinterrelationships among hemodynamic variables. Futureapplications of control theory in the form of adaptive androbust optimal controllers that oversee the administrationof multiple drugs (inotropes, vasopressors, and vasodila-tors) and fluids will be a major advance in critical care med-icine. Preliminary investigation of the control of multiplehemodynamic drugs has already begun [28], [29].

Closed-Loop Control of Anesthesia

Automated Anesthesia: Inhalational and IntravenousAnesthesia involves several components: analgesia, lack ofreflex response (such as increased blood pressure or heartrate) to surgical stimulus; areflexia, lack of movement(which simplifies the task of the surgeon); and hypnosis, orlack of consciousness. Closed-loop control of anesthesiamay be implemented either by controlling these compo-nents of anesthesia or, more simply, by controlling theanesthetic concentration and assuming that the appropri-ate concentration will lead to the desired effect.

Real-time spectroscopic methods for measuring theconcentration of inhaled anesthetic agent in end-tidal (thatis, exhaled) gases are now routinely available in most oper-ating rooms. End-tidal anesthetic gas concentration is areasonable surrogate for arterial blood anesthetic concen-tration [30]. End-tidal anesthetic agent concentrations canbe measured in real time, which facilitates closed-loop con-trol of end-tidal anesthetic concentration. However, anes-thetic concentration cannot be equated with anestheticeffect. More recently, real-time processed electroencephalo-graph (EEG) measurement has offered the possibility ofclosed-loop control of anesthetic effect. It has been knownfor decades that the induction of anesthesia causeschanges in the EEG [31]. In the last decade, there has been

substantial progress in developing processed EEG moni-tors that measure the depth of anesthesia to provide per-formance variables for closed-loop controllers [31].

Inhaled anesthetic agents have been the mainstay of clin-ical practice since the first delivery of anesthesia. A funda-mental characteristic of every inhaled anesthetic agent is its“MAC” value, minimum (in suppressing the response topainful stimuli) alveolar (alveoli are the functional units ofthe lung) concentration, which is associated with a 50%probability of patient movement in response to surgicalstimulus [28]. By maintaining end-tidal concentrations wellabove MAC, the practitioner is relatively assured of hypno-sis. The ready availability of spectroscopic systems for mea-suring end-tidal anesthetic concentration in real time hasled several investigators to develop closed-loop controllers.

The earliest anesthesia controllers use fixed-gain PIDcontrollers [32], [33], which assume that all patients arethe same. In contrast, the adaptive model-based con-trollers in [34] and [35] rely on least squares methods toestimate the patient-specific system parameters. In animalstudies [23], the adaptive controllers have performedmore effectively than the fixed-gain controllers. However,the adaptive controllers have not been widely adoptedclinically since control of anesthetic concentration doesnot translate into control of anesthetic effect due to inter-patient pharmacodynamic variability.

EEG-Based ControlClosed-loop control of anesthesia requires a monitor ofanesthetic effect, specifically consciousness, which hasbeen an elusive challenge for anesthesiologists. The EEG,which measures electrical activity in the brain, has beenan obvious candidate. In particular, neurophysiologistshave observed that the EEG of an anesthetized patientcontains slower waves with higher amplitudes. However,the EEG is comprised of multiple time series and multiplespectra and, while anesthesia induces characteristicchanges in the EEG, it is not clear which, if any, charac-teristic of the EEG best reflects the anesthetic state.

EEG-based closed-loop control of anesthesia was firstproposed in [36]. Subsequently, a closed-loop, model-based adaptive controller was developed and clinicallytested in [37], delivering intravenous anesthesia using themedian frequency of the EEG power spectrum as the regu-lated variable. The model used in [37] assumes a two-com-partment pharmacokinetic model for which the drugconcentration C (t) after a single bolus dose is given by

C (t) = Ae−αt + Be−βt, t ≥ 0,

where A, B, α, and β are patient-specific pharmacokineticparameters. It is also assumed that the median EEG fre-quency E is related to the drug concentration by the modi-fied Hill equation

April 2005 43IEEE Control Systems Magazine

E = E0 − Emax[C γ /

(C γ + C γ

50

)], (5)

where E0 is the baseline signal, Emax is the maximumdecrease in signal with increasing drug concentration, C50 isthe drug concentration associated with 50% of the maxi-mum effect, and the parameter γ describes the steepness ofthe concentration-effect curve. From (5) it can be seen thatthe drug effect is a function of the pharmacokinetic parame-ters A, B, α, and β as well as the pharmacodynamic parame-ters E0, Emax, C50, and γ . If these parameters are known, itis straightforward to calculate the dose regimen needed toachieve the target EEG signal. However, these parametersare not known for individual patients, and interpatient vari-ability may be significant. Estimates for the coefficients ofvariability for some parameters are as high as 100%.

The algorithm in [37] assumes that the pharmacody-namic parameters E0, Emax, C50, and γ as well as the phar-macokinetic parameters α and β are equal to the meanvalues reported in prior studies. Using the mean values ofthe pharmacokinetic parameters A and B from prior stud-ies as starting values, estimates of these parameters arerefined by analyzing the difference �E between the targetand observed EEG signal. Linearizing �E with respect to Aand B yields

�E = (∂E/∂ A)δ A + (∂E/∂ B)δB, (6)

where δ A and δB represent the updates to the values of Aand B in the adaptive control algorithm. In conjunctionwith minimizing (δ A)2 + (δB)2, (6) is used to estimate δ Aand δB. This algorithm is only partially adaptive in that Aand B are the only parameters of the model that areupdated. This algorithm was implemented for the intra-venous anesthetic agents methohexital and propofol butdid not appear to offer great advantage over standard man-ual control [37], [38]. The observed performance mighthave been due to the approximations of the algorithm orthe deficiencies of the median EEG frequency as a measureof the depth of anesthesia.

Bispectral Index-Based ControlSince the work of [37], alternative EEG measures of depthof anesthesia have been developed. Possibly the mostnotable of these measures is the bispectral index or BIS[39], [40]. The BIS is a single-composite EEG measure,which appears to be closely related to the level of con-sciousness (see Figure 2). The BIS signal is related to drugconcentration by the empirical relationship

BIS(ceff) = BIS0

(1 − cγ

eff

cγ

eff + ECγ

50

), (7)

where BIS 0 denotes the baseline (awake state) value,which, by convention, is typically assigned a value of 100;ceff is the drug concentration in µg/ml in the effect-sitecompartment (brain); EC50 is the concentration at halfmaximal effect and represents the patient’s sensitivity tothe drug; and γ determines the degree of nonlinearity in(7). Here, the effect-site compartment is introduced toaccount for finite equilibration time between the centralcompartment concentration and the central nervous sys-tem concentration [41].

In [42], closed-loop control is used to deliver the intra-venous anesthetic propofol based on a model-based adap-tive algorithm with the BIS as the regulated variable. Thealgorithm in [42] is based on a pharmacokinetic modelthat predicts the drug concentration as a function of infu-sion rate and time and uses a pharmacodynamic modelthat relates the BIS signal to concentration.

In contrast to [37], it is assumed in [42] that the phar-macokinetic parameters are always correct and that dif-ferences in individual patient response are due topharmacodynamic variability. Moreover, the approach of[42] predicts the anesthetic concentration using the phar-macokinetic model and then constructs a BIS-concentra-tion curve using both the observed BIS during inductionand the predicted propofol concentration. During eachtime epoch, the difference between the target BIS signaland the observed BIS signal is used to update the phar-macodynamic parameters relating concentration and BISsignal for the individual patient. However, this algorithmdoes not update the pharmacokinetic parameters.

The results in [42] demonstrate excellent performanceas measured by the difference between the target andobserved BIS signals. However, as pointed out in [43], the

Figure 2. Bispectral index (BIS) monitor. A sensor is placedon the patient’s forehead to measure electrical activity in thebrain and monitor the patient’s level of consciousness.

CO

UR

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SY

OF

AS

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April 200544 IEEE Control Systems Magazine

performance of the model-based adaptive controller mayreflect the fact that the patient was not fully stressed. In[42], a high dose of the opioid remifentanil, a neurotrans-mitter inhibitor resulting in significant analgesic effect, wasadministered in conjunction with propofol. Consequently,central nervous system excitation due to surgical stimuluswas blunted and, thus, the need to adjust the propofoldose as surgical stimulus varied was diminished. It isunknown whether the control system would have beeneffective in the absence of deep narcotization.

In contrast to the model-based adaptive controllers in[37], [38], and [42], a PID controller using the BIS signal asthe variable to control the infusion of propofol is consid-ered in [44]. The median absolute performance error, themedian value of the absolute value of �E/Etarget , wasgood (8.0%), although in three of the ten patients, oscilla-tions of the BIS signal around the setpoint were observed,and anesthesia was deemed clinically inadequate in one ofthe ten patients. The same system was used in [45], withan auditory-evoked potential based on somatosensoryinformation provided by auditory stimulation generatingoscillations within the EEG signal as the regulated perfor-mance variable.

Intravenous propofol anesthesia was delivered in [44]by means of a fuzzy logic closed-loop controller that usesboth auditory (constant frequency, constant amplitude sig-nal delivered by earphones) evoked responses and cardio-vascular responses as the regulated variables. This systemhas had only minimal clinical testing [46]. More recently,[47] considers model-based controllers for inhalation anes-thetic agents that attempt to control the BIS signal or meanarterial blood pressure, while keeping end-tidal anestheticconcentrations within prespecified limits.

Nonlinear Adaptive and Neuro AdaptiveControl for General AnesthesiaBecause of the uncertainties in the pharmacokinetic andpharmacodynamic parameters due to interpatient vari-ability, we have developed adaptive controllers that canbe implemented using the processed EEG as a perfor-mance variable (see Figure 3). Using compartmental

models, a Lyapunov-based direct adaptive controlframework developed in [48] and [49] guarantees partialasymptotic setpoint stability of the closed-loop system(asymptotic setpoint stability with respect to part of theclosed-loop system states associated with the physio-logical state variables). Furthermore, the remainingstates associated with the adaptive controller gains arebounded. The adaptive controllers, which are construct-ed without requiring knowledge of the system pharma-cokinetic and pharmacodynamic parameters, provide anonnegative control input for stabilization with respectto a given setpoint in the nonnegative orthant.

In [50] and [51], we present a neural network adaptivecontrol framework that accounts for combined interpa-tient pharmacokinetic and pharmacodynamic variability.In particular, we develop a framework for adaptive set-point regulation of nonlinear uncertain compartmental sys-tems. The formulation in [50] and [51] addresses adaptiveoutput feedback controllers for nonlinear compartmentalsystems with unmodeled dynamics of unknown order. Italso guarantees ultimate boundedness of the error signalscorresponding to the physical system states as well as theneural network weighting gains.

Nonlinear Adaptive Control for General Anesthesia

Pharmacokinetic Model for PropofolAlmost all anesthetics are myocardial depressants, mean-ing that they decrease the strength of the contraction ofthe heart and lower cardiac output. As a consequence,decreased cardiac output slows down the transfer ofblood from the central compartments comprising theheart, brain, kidney, and liver to the peripheral compart-ments of muscle and fat. In addition, decreased cardiacoutput can increase drug concentrations in the centralcompartments, compounding side effects. This instabilitycan lead to overdosing that, at the very least, can delayrecovery from anesthesia and, in the worst case, canresult in respiratory and cardiovascular collapse. Alterna-tively, underdosing can cause psychological trauma from

awareness and pain during surgery.Control of drug effect is clinically

important since overdosing orunderdosing incur risks for thepatient. To illustrate adaptive con-trol for general anesthesia, we con-sider a hypothetical model for theintravenous anesthetic propofol. Thepharmacokinetics of propofol aredescribed by the three-compartmentmodel [49], [52] shown in Figure 4,where x1 denotes the mass of drug inthe central compartment, which is

Figure 3. Adaptive closed-loop control for drug administration. Active control canimprove the medical care of patients requiring anesthesia or sedation in the oper-ating room or intensive care unit.

EEG

System

BISTarget

Controller

BIS Score

−+

April 2005 45IEEE Control Systems Magazine

the site for drug administration andincludes the intravascular blood vol-ume as well as highly perfused organs(organs with high ratios of bloodflow to weight such as the heart,brain, kidney, and liver), whichreceive a large fraction of the car-diac output. The remainder of thedrug in the body is assumed toreside in two peripheral compart-ments: one identified with muscleand one with fat. The masses inthese compartments are denoted byx2 and x3, respectively. These com-partments receive less than 20% ofthe cardiac output.

A mass balance for the three-statecompartmental model is of the form

x1(t) = −[a11(c(t)) + a21(c(t)) + a31(c(t))]x1(t)

+ a12(c(t))x2(t) + a13(c(t))x3(t) + u(t),

x1(0) = x10, t ≥ 0,

x2(t) =a21(c(t))x1(t) − a12(c(t))x2(t), x2(0) = x20,

x3(t) =a31(c(t))x1(t) − a13(c(t))x3(t), x3(0) = x30,

where c(t) = x1(t)/Vc, Vc is the volume of the central com-partment (about 15 l for a 70-kg patient), aij(c) for i �= j isthe rate of transfer of drug from the j th compartment tothe i th compartment, a11(c) is the rate of drug metabolismand elimination (metabolism typically occurs in the liver),and u(t) is the infusion rate of the anesthetic drug propofolinto the central compartment. The transfer coefficients areassumed to be functions of the drug concentration c sinceit is well known that the pharmacokinetics of propofol areinfluenced by cardiac output [53]. In turn, cardiac outputis influenced by propofol plasma concentrations, both dueto venodilation (pooling of blood in dilated veins) [54] andmyocardial depression [55].

Experimental data indicate that the transfer coefficientsaij are nonincreasing functions of the propofol concentra-tion [54], [55]. The most widely used empirical models forpharmacodynamic concentration-effect relationships aremodifications of the Hill equation (4). Applying the Hillequation to the relationship between transfer coefficientsand drug concentration implies that

aij(c) = AijQij(c), Qij(c) = Q0Cαij

50,ij

/(C

αij

50,ij + cαij)

,

where, for distinct i, j ∈ {1, 2, 3}, C50,ij is the drug concen-tration associated with a 50% decrease in the transfer coef-

ficient, αij is a parameter that determines the steepness ofthe concentration-effect relationship, and Aij are positiveconstants. Note that αij and Aij are functions of i and j,meaning that there are distinct Hill equations for eachtransfer coefficient. Furthermore, since for many drugs therate of metabolism a11(c) is proportional to the rate ofdrug transport to the liver, we assume that a11(c) is pro-portional to the cardiac output so that a11(c) = A11Q11(c).

To illustrate the adaptive control of propofol, we assumefor simplicity that C50,ij and αij are independent of i and j.Also, since decreases in cardiac output are observed atclinically utilized propofol concentrations, we arbitrarilyassign C50 a value of 4 µg/ml, which is in the mid-range ofclinically utilized values. We also set α = 3, which is typi-cal for ligand-receptor binding (see the discussion in [56]).The nonnegative transfer and loss coefficients A12, A21,A13, A31, and A11, and the parameters α > 1, C50 > 0, andQ0 > 0 are uncertain due to patient gender, weight, pre-existing disease, age, and concomitant medication.

Pharmacodynamics and the Effect-Site CompartmentAlthough propofol concentration in the blood is correlatedwith lack of responsiveness [57], the concentration cannotbe measured in real time during surgery. Since we aremore interested in drug effect (depth of hypnosis) thandrug concentration, we consider a model involving phar-macokinetics and pharmacodynamics for controlling con-sciousness. We use an EEG signal, specifically the BISsignal, to access the effect of anesthetic compounds on thebrain [40], [58], [59]. Furthermore, we utilize the modifiedHill equation (7) to model the relationship between the BISsignal and the effect-site concentration.

The effect-site compartment concentration is related tothe concentration in the central compartment by the first-order model

Figure 4. Pharmacokinetic model for drug distribution during anesthesia. Thecentral compartment, which is the site for drug administration, is comprised of theintravascular blood volume as well as the highly perfused organs. The two periph-eral compartments comprised of muscle and fat receive a small portion of the car-diac output.

u ≡ Continuous Infusion

IntravascularBlood

Fat

a11(c)x1 ≡ Elimination (Liver, Kidney)

a12(c)x2

a21(c)x1

a31(c)x1

a13(c)x3

Muscle

April 200546 IEEE Control Systems Magazine

ceff(t) = aeff(c(t) − ceff(t)), ceff(0) = c(0), t ≥ 0,

where aeff in min−1 is a time constant. In reality, the effect-site compartment equilibrates with the central compart-ment in a few minutes. The parameters aeff, EC50, and γ are

determined by data fitting and varyfrom patient to patient. BIS index val-ues of 0 and 100 correspond, respec-tively, to an isoelectric EEG signal (nocerebral electrical activity) and anEEG signal of a fully conscious patient.The range 40 to 60 indicates a moder-ate hypnotic state [58]. Figure 5shows the combined pharmacokinet-ic/pharmacodynamic model for thedistribution of propofol.

SimulationIn the following simulation involvingthe infusion of the anesthetic drugpropofol, we set EC50 = 5.6 µg/ml,γ = 2.39, and BIS0 = 100, so that theBIS signal is shown in Figure 6. Thedesired target BIS value BIStarget is setat 50. Furthermore, we assume thatthe effect-site compartment equili-brates instantaneously with the cen-tral compartment; that is, we assumethat ceff(t) = c(t) for all t ≥ 0. Theadaptive feedback controller for the

linearized pharmacodynamic model of (7 ) derived in [49]is given by

u1(t) = max{0, u1(t)}, (8)

where

u1(t) = −k1(t)(BIS(ceff(t)) − BIStarget

) + φ1(t), (9)

and k1(t) and φ1(t) are scalars for t ≥ 0. The update lawsk1(t) and φ1(t) are given by

k1(t) ={

0, if u1(t) ≤ 0,

−qBIS1(BIS(ceff(t)) − BIStarget)2, otherwise,

(10)

φ1(t) =

0, if φ1(t) = 0 and BIS(ceff(t))> BIStarget, or if u1(t) ≤ 0,

qBIS1(BIS(ceff(t)) otherwise,

−BIStarget),

(11)

where k1(0) ≤ 0, φ1(0) ≥ 0, and qBIS1 and qBIS1 are pos-itive constants. Theorem 3.1 of [49] guarantees that

Figure 5. Pharmacokinetic/pharmacodynamic model with effect-site (brain)compartment. The effect-site compartment is introduced to account for finite equili-bration time between the central compartment concentration and the central ner-vous system concentration.

u ≡ Continuous Infusion

IntravascularBlood

Muscle Fat

a11(c)x1 ≡ Elimination (Liver, Kidney)

a12(c)x2

a21(c)x1

a31(c)x1

a13(c)x3

aeffc

Effect SiteCompartment

ceff

BIS Index

Pharmacodynamics

Figure 6. BIS index (electroencephalogram indicator) ver-sus effect-site concentration. BIS index values of 0 and 100correspond, respectively, to an isoelectric EEG signal and anEEG signal of a fully conscious patient, while the range 40 to60 indicates a moderate hypnotic state.

Target BIS

EC50 = 5.6 [µg/ml]0

10

20

30

40

50

60

70

80

90

100

BIS

Inde

x [S

core

]

0 2 4 6 8 10 12 14 16 18 20

Effect Site Concentration [µg/ml]

April 2005 47IEEE Control Systems Magazine

BIS(ceff(t)) → BIS target as t → ∞ for all nonnegative valuesof the pharmacokinetic transfer and loss coefficientsA12, A21, A13, A31, A11 as well as for all nonnegative coeffi-cients α, C50, and Q0.

For simulation, we assume Vc = (0.228 l/kg)(M kg),where M = 70 kg is the mass of the patient, A21Q0 = 0.112min−1 , A12Q0 = 0.055 min−1 , A31Q0 = 0.0419 min−1 ,A13Q0 = 0.0033 min−1 , A11Q0 = 0.119 min−1 , α = 3, and

C50 = 4µg/ml [52]. Furthermore, to illustrate the adaptivecontroller, we switch the pharmacodynamic parametersEC50 and γ , respectively, from 5.6 µg/ml and 2.39 to 7.2 µg/ml and 3.39 at t = 15 min and back to 5.6 µg/mland 2.39 at t = 30 min. With qBIS1 = 1 × 10−6 g/min2 ,qBIS1 = 1 × 10−3 g/min2 , and initial conditionsx(0) = [0, 0, 0]T g, k1(0) = 0 g/min, and φ1(0) = 0.01 g/min,Figure 7 shows the mass of propofol in each of the three com-partments versus time. Figure 8 shows the BIS index and thecontrol signal (propofol infusion rate) versus time. Finally, Fig-ure 9 shows the adaptive gain history versus time.

Unlike previous algorithms for closed-loop control ofanesthesia [37], [42], the adaptive controller (8)–(11)does not require knowledge of the pharmacokinetic andpharmacodynamic parameters. However, the adaptivecontroller (8)–(11) does not account for time delays dueto equilibration between the central circulation and theeffect-site compartment or due to the proprietary signal-averaging algorithm within the BIS monitor. The adaptivecontroller also ignores measurement noise. Extensiveclinical testing is needed to access the significance ofthese assumptions and approximations. Since there isoften a substantial delay between observed changes inpatient status and a change in the BIS signal, other mea-sures of depth of anesthesia may be required [60].

Clinical TrialsWe have begun clinical studies of the adaptive controller(8)–(11) at the Northeast Georgia Medical Center. In ini-tial clinical testing, we implemented (8)–(11) using a Dell

Figure 7. Compartmental masses versus time. The pharma-codynamic parameters are switched from their nominal val-ues at t = 15 min and back at t = 30 min.

0 5 10 15 20 25 30 35 400

20

40

60

80

100

120

Com

part

men

tal M

asse

s [m

g]

Time [min]

x1(t )x2(t )x3(t )

Figure 8. BIS index versus time and control signal (infusionrate) versus time achieved by the adaptive controller. Theadaptive controller does not require knowledge of the phar-macokinetic and pharmacodynamic system parameters.

0 5 10 15 20 25 30 35 400

20

40

60

80

100

BIS

Inde

x [S

core

]

Time [min]

0 5 10 15 20 25 30 35 400

50

100

150

Con

trol

Sig

nal [

mg/

min

]

Time [min]

Figure 9. Adaptive gain history versus time. Although thedynamic gains can assume negative values, the physical sys-tem states and the control signal are guaranteed to remainnonnegative.

0 5 10 15 20 25 30 35 40−3.5

−3−2.5

−2−1.5

−1−0.5

0

Ada

ptiv

e G

ain

k[m

g/m

in]

Time [min]

0 5 10 15 20 25 30 35 400

5

10

15

20

25

Ada

ptiv

e G

ain

φ[m

g/m

in]

Time [min]

April 200548 IEEE Control Systems Magazine

Latitude C610 laptop computer with a Pentium (R) IIIprocessor running under Windows XP, an Aspect A 2000BIS monitor (rev 3.23), and a Harvard PHD 2000 program-mable research pump. The BIS monitor sends a datastream that is updated every 5 s. This data stream con-tains the BIS signal as well as other parameters such asdate, time, signal quality indicator, raw EEG information,and electromyographic data. The data are sent to the seri-al port of the laptop computer.

The infusion rate u(t) is calculated using a forward Eulermethod to update the adaptive gains k1(t) and φ1(t) every0.5 s, using the BIS signal. The infusion rate is communicat-ed to the infusion pump using a 9,600 bpm, eight data bits, two stop bits, and zero parity protocol withthe aid of a USB serial port adaptor. An updated infusionrate is sent to the pump at 1-s intervals. Pharmacokineticsimulations predict that a pump update every 5 s or less isadequate in the context of the algorithm under evaluation.An update interval of 1 s was selected in anticipation thatfuture algorithms might benefit from the faster update rate.

The adaptive control algorithm was programmedin Java, an object-oriented programming language cho-sen for its multiplatform portability tools for rapid proto-typing. The program is organized into five modules,which include bisloader, bislogger, controller, pumplog-ger, and pumploader. Bisloader and bislogger handle com-munication between the BIS monitor and the computer,while pumploader and pumplogger manage the Harvardpump apparatus. The module bisloader finds the serialport that receives the BIS signal by using the Java classCommPort Identifier; it then invokes bislogger. Bisloggeruses the Java class SerialPort EventListener to read the sig-

nal and uses the class StringTokenizer to parse the BIS sig-nal from the input stream. The infusion rate is calculatedby the controller module. Finally, pumploader opens theserial port communication to the pump and establishesthe communication protocol, while pumplogger deliversthe infusion rate to the pump.

The protocol for clinical evaluation of the system wasapproved by the Institutional Review Board of NortheastGeorgia Medical Center. Patients are enrolled after provid-ing informed consent. Our protocol excludes patientsrequiring emergency surgery, pediatric patients, hemody-namically unstable patients, and patients for whom weanticipate difficult airway management. Otherwise, all elec-tive surgical patients who can provide informed consent arecandidates. Preoperative management, including adminis-tration of anti-anxiolytic drugs, is left to the discretion of theattending anesthesiologist. Propofol is delivered using theBIS-computer-pump system with a target value of 55. In addi-tion to propofol, all patients receive infusions of eithersufentanil or fentanyl with loading doses of 2 µg/ml or 0.25µg/ml as well as continuous infusions of 2 (µg/ml)/hr or 0.25(µg/ml)/hr, respectively, to provide analgesia. To ensurepatient safety, an independent anesthesia provider observesthe progress of the study and can terminate the study if itappears that the patient’s safety is being jeopardized byeither overdosing or underdosing of propofol.

To date, we have performed 11 clinical trials. For the firstclinical trial, Figures 10 and 11 show the controlled BIS indexand the control signal versus time. These results are typicalof all 11 clinical trials. We have consistently observed anovershoot at the induction of anesthesia with the BIS signaldropping below the target. We suspect that the initial target

Figure 10. Controlled BIS index versus time for the first clin-ical trial. The oscillations are due to measurement noise inthe BIS signal.

0 2 4 6 8 10 12 14 16 180

10

20

30

40

50

60

70

80

90

100

Time [min]

BIS

Inde

x [S

core

]

Figure 11. Infusion rate versus time for the first clinicaltrial. No initial propofol dose is administered by the anesthe-siologist. Induction of anesthesia is delivered by the adaptivecontroller.

0 2 4 6 8 10 12 14 16 18−10

0

10

20

30

40

50

Con

trol

Sig

nal [

mg/

min

]

Time [min]

April 2005 49IEEE Control Systems Magazine

overshoot stems from the assumption of a linear pharmaco-dynamic model, which is an approximation. Also, signal aver-aging time delays within the BIS monitor may contribute tothis overshoot. We are preparing clinical trials of an exten-sion of the adaptive controller (8)–(11) that addresses non-linear pharmacodynamics and time delays [61], [62].

Challenges and Opportunitiesin Pharmacological ControlClosed-loop control for clinical pharmacology is in itsinfancy, with numerous challenges and opportunitiesahead. An implicit assumption of the control frameworksdiscussed in this article is that the control law is imple-mented without regard to actuator amplitude and rate con-straints. In pharmacological applications, drug infusionrates vary from patient to patient. To avoid overdosing, itis vital that the infusion rate does not exceed the patient-specific threshold values. As a consequence, actuator con-straints, or infusion pump rate constraints, need to beimplemented in drug delivery systems [62].

Another important issue for future research is measure-ment noise. In particular, EEG signals can have as much as10% variation due to noise. For example, the BIS signal maybe corrupted by electromyographic noise, or signals ema-nating from muscle rather than the central nervous sys-tem. Although electromyographic noise can be minimizedby muscle paralysis, there are other sources of measure-ment noise, such as electrocautery, that need to beaccounted for in the control design processes.

In pharmacokinetic and pharmacodynamic models, theassumption of instantaneous mixing between compart-ments is not valid. For example, if a bolus of drug is inject-ed, there is a time lag before the drug is detected in theextracellular and intercellular space of an organ [2]. Phaselag due to mixing times can be approximated by includingadditional compartments in series. To describe the distrib-ution of pharmacological agents in the human body, infor-mation on the past system states can be modeled by delaydynamical systems [63]. This extension necessitates thedevelopment of adaptive control algorithms for compart-mental systems with unknown time delays [62], [64].

ConclusionsControl system technology has a great deal to offer phar-macology, anesthesia, and critical care medicine. Criticalcare patients, whether undergoing surgery or recoveringin intensive care units, require drug administration to reg-ulate physiological variables such as blood pressure, car-diac output, heart rate, and degree of consciousness. Therate of infusion of each administered drug is critical,requiring constant monitoring and frequent adjustments.Open-loop control by clinical personnel can be tedious,imprecise, time consuming, and sometimes of poor quality.Alternatively, closed-loop control can potentially achieve

desirable system performance in the face of the highlyuncertain and hostile environment of surgery and theintensive care unit. Since robust and adaptive controllersachieve system performance without excessive relianceon system models, robust and adaptive closed-loop con-trol may potentially improve the quality of medical care.

Closed-loop control for clinical pharmacology can signif-icantly advance our understanding of the effects of pharma-cological agents and anesthetics, as well as advance thestate-of-the-art in drug delivery systems. In addition todelivering sedation to critically ill patients in an acute careenvironment, potential applications of closed-loop controlinclude glucose, heart rate, and blood pressure regulation.Payoffs will arise from improvements in medical care,health care, reliability of drug dosing equipment, andreduced health-care costs.

AcknowledgmentThis research was supported in part by the Air ForceOffice of Scientific Research under Grant F49620-03-1-0178.

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[51] T. Hayakawa, W.M. Haddad, N. Hovakimyan, and J.M. Bailey,“Neural network adaptive dynamic output feedback control for nonlin-ear nonnegative systems using tapped delay memory units,” in Proc.Amer. Control Conf., Boston, MA, July 2004, pp. 4505–4510.

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James M. Bailey received the B.S. degree from DavidsonCollege in 1969, the Ph.D. in chemistry (physical) from theUniversity of North Carolina at Chapel Hill in 1973, and theM.D. degree from Southern Illinois University School ofMedicine in 1982. He was a Helen Hay Whitney Fellow at theCalifornia Institute of Technology from 1973–1975 and assis-

tant professor of chemistry and biochemistry at SouthernIllinois University from 1975–1979. After receiving his M.D.degree, he completed a residency in anesthesiology andthen a fellowship in cardiac anesthesiology at the EmoryUniversity School of Medicine affiliated hospitals. From1986–2002, he was an assistant professor of anesthesiologyand then associate professor of anesthesiology at Emory,where he also served as director of the critical care service.In September 2002, he moved his clinical practice to North-east Georgia Medical Center in Gainesville, Georgia, asdirector of cardiac anesthesia and consultant in criticalcare medicine. He remains affiliated with the Emory Univer-sity School of Medicine Department of Anesthesiology as aclinical associate professor. He is board certified in anes-thesiology, critical care medicine, and transesophagealechocardiography. His research interests are focused onpharmacokinetic and pharmacodynamic modeling of anes-thetic and vasoactive drugs and, more recently, applica-tions of control theory in medicine. He is the author orcoauthor of 98 journal articles, conference publications,and book chapters.

Wassim M. Haddad (wm.haddad@aerospace.gatech.edu)received the B.S., M.S., and Ph.D. degrees in mechanicalengineering from the Florida Institute of Technology, Mel-bourne, in 1983, 1984, and 1987, respectively, with special-ization in dynamical systems and control. From 1987 to1994, he was a consultant for the Structural Controls Groupof the Government Aerospace Systems Division, Harris Cor-poration, Melbourne, Florida. In 1988, he joined the facultyof the Mechanical and Aerospace Engineering Departmentat the Florida Institute of Technology, where he foundedand developed the systems and control option within thegraduate program. Since 1994, he has been a member of thefaculty in the School of Aerospace Engineering at the Geor-gia Institute of Technology, where he is a professor. Hisresearch contributions in linear and nonlinear dynamicalsystems and control are documented in over 400 archivaljournal and conference publications. His recent research isconcentrated on nonlinear robust and adaptive control,nonlinear dynamical system theory, large-scale systems,hierarchical nonlinear switching control, hybrid and impul-sive control for nonlinear systems, system thermodynamics,thermodynamic modeling of mechanical and aerospace sys-tems, nonlinear analysis and control for biological and phys-iological systems, and active control for clinicalpharmacology. He is an NSF Presidential Faculty Fellow, amember of the Academy of Nonlinear Sciences, and a coau-thor of Hierarchical Nonlinear Switching Control Design withApplications to Propulsion Systems (Springer-Verlag, 2000)and Thermodynamics: A Dynamical Systems Approach(Princeton University Press, 2005). He can be contacted atthe School of Aerospace Engineering, Georgia Institute ofTechnology, Atlanta, GA 30332-0150 USA.

April 2005 51IEEE Control Systems Magazine