A Model Predictive Control Strategy Toward Optimal Structured Treatment Interruptions in Anti-HIV...

1040 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 57, NO. 5, MAY 2010

A Model Predictive Control Strategy TowardOptimal Structured Treatment Interruptions

in Anti-HIV TherapyGabriele Pannocchia!, Marco Laurino, and Alberto Landi, Member, IEEE

Abstract—In this paper, model predictive control (MPC) strate-gies are applied to the control of human immunodeficiency virusinfection, with the final goal of implementing an optimal structuredtreatment interruptions protocol. The MPC algorithms proposedin this paper use a dynamic model recently developed [1] in orderto mimic both transient responses and ultimate behavior, and todescribe accordingly the different effect of commonly used drugsin highly active antiretroviral therapy (HAART). Simulation stud-ies show that the proposed methods achieve the goal of reducingthe drug consumption (thus minimizing the severe side effects ofHAART drugs) while respecting the desired constraints on CD4+cells and free virions concentration. Such promising results areobtained with realistic assumptions of infrequent (possibly noisy)measurements of a subset of model state variables. Furthermore,the control objectives are achieved even in the presence of mismatchbetween the dynamics of true patients and that of the MPC model.

Index Terms—Antiretroviral therapy, human immunodefi-ciency virus control, model predictive control (MPC), therapyoptimization.

I. INTRODUCTION

THE DISEASE control area gained an increasing attentionin the 1960s, until the early 1970s, with the development of

several analytical models; the ambitious goal of researchers wasto apply a mathematical framework for helping medical diag-nostic techniques and new therapeutic protocols. Very often, thestarting point was a linear time-invariant deterministic modelframework. Unfortunately, physiological systems are intrinsi-cally time-variant and highly nonlinear, as well as they showrelevant stochastic behavior. Furthermore, an effective balanceof the model complexity is a difficult task to satisfy: low-ordermodels are usually too simple to be useful, conversely, high-order models are too complex for simulation purposes and havetoo many unknown parameters requiring identification. There-fore, many clinical researchers do not consider a mathemati-cal quantitative approach relevant for a practical progress inmedicine, and “open-loop” control is the norm in drug therapies.

Manuscript received October 30, 2008; revised May 11, 2009, October 20,2009, and November 2, 2009. First published February 17, 2010; current versionpublished April 21, 2010. Asterisk indicates corresponding author.

!G. Pannocchia is with the Dipartimento di Ingegneria Chimica, Chimica In-dustriale e Scienza dei Materiali, University of Pisa, 56122 Pisa, Italy (e-mail:g.pannocchia@ing.unipi.it).

M. Laurino was with the Dipartimento di Sistemi Elettrici e Automazione,University of Pisa, 56125 Pisa, Italy. He is now with the Dipartimentodi Scienze Fisiologiche, University of Pisa, 56126 Pisa, Italy (e-mail:marco.laurino@studenti.ing.unipi.it).

A. Landi is with the Dipartimento di Sistemi Elettrici e Automazione, Uni-versity of Pisa, 56122 Pisa, Italy (e-mail: alberto.landi@dsea.unipi.it).

Digital Object Identifier 10.1109/TBME.2009.2039571

Although we are aware that the adaptation of the parame-ters of a mathematical model to each patient is an extremelyoptimistic goal, we believe that the introduction of a feedbackcontrol loop for testing the effects of the therapy is difficult,but attainable. In the literature, several mathematical modelshave been proposed to describe human immunodeficiency virus(HIV) infection and evolution (see e.g., [2], [3], and referencestherein). In HIV infection, pharmacological therapy offers in-creased life expectation and quality to the patient. Combineddrugs are used to reduce viral replication and to delay the pro-gression of pathology [4]. Highly active antiretroviral therapy(HAART) is a combination therapy that includes the following.

1) Reverse transcriptase inhibitors (RTI), to inhibit reversetranscriptase activity and prevent cell-to-cell transmission.

2) Protease inhibitors (PI), to inhibit the production of viralprotein precursors and to prevent the production of virionsby infected cells.

However, there are some limitations to the effectiveness ofHAART. Infected cells have a short half-life (from days tomonths), but hidden reservoirs of virus contribute to an evenslower disease phase that makes complete eradication of thevirus from the body impossible with current therapies. Becausecontinued administration of drugs is associated with severe sideeffects and the generation of drug resistance, more recent re-search efforts have been directed at finding therapy regimesthat boost HIV-specific immune responses [5], [6]. There aresome clinical data, which suggest that the so-called structuredtreatment interruptions (STI) can boost immunity against HIV,especially, when performed relatively early after infection [7]. Itwas observed that HIV progression was linked to a loss of spe-cific cytotoxic T-lymphocyte (CTL) and that long-term non Pro-gressors (LTNP), i.e., infected patients who do show significantdisease progression for many years without treatment, maintainhigh counts of CTLs, while people with fast disease progres-sion do not. Several studies suggest that STI enhance immuneresponse against HIV. The use of STI is currently a matter ofdebate (see [8]–[14] and references therein), most studies agreethat STI may be successful if therapy is initiated early in HIVinfection, but unsuccessful for people who started therapy later.

Among possible control strategies model predictive control(MPC) appears to be suitable for an “optimal” application ofSTI, due to its intrinsic robustness to disturbances and modeluncertainties, and most importantly because of the effectivecapabilities of handling of constraints [e.g., on the viral load(VL)]. Thus, the aim of this paper is the application of an MPCstrategy to therapy against the HIV infection. The first step in an

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PANNOCCHIA et al.: MPC STRATEGY TOWARD OPTIMAL STRUCTURED TREATMENT INTERRUPTIONS 1041

MPC application is the choice of the model. Among the manyavailable models, this paper considers a variant of the Wodarzand Nowak models [2], [7], [15], adding a state variable named“aggressiveness” (or “virulence”). Such variant was proposed in[1] and it was developed in order to mirror the natural evolutionof HIV infection, as qualitatively described in several clinicalstudies and to introduce the impact of therapy effectiveness intoHIV dynamics in a simple way, suitable for use in feedbackcontrol. One last comment is due because simulation resultsmust be coherent with the medical findings; a close cooperationwith clinical researchers expert in HIV therapies was helpful intesting the model [1] and is active for evaluating the effectivenessof the proposed control methods.

The rest of this paper is organized as follows. In Section II,mathematical models for HIV control are reviewed. InSection III, two methods for defining “optimal” drug therapyprotocols based on MPC techniques are presented. In Section IV,simulation results are presented and discussed. In Section V,conclusions are drawn and future perspectives of the researchactivity are sketched.

II. MATHEMATICAL MODELS OF HIV

A. Previous Work

In the literature, several different models have been developedto describe the HIV evolution, and some review articles arealready available [2], [3]. We briefly review two models that arenecessary to understand the model used in this paper and theissues associated with the development of an MPC algorithmproposal.

A five-state model was presented in [7]!""""""#

""""""$

x = ! " dx " !xv

y = !xv " ay " pyz

v = ky " uv

w = cxyw " cqyw " bw

z = cqyw " hz.

In such a model, the first equation represents the dynamics ofthe concentration of healthy CD4+ cells (x); ! represents theconstant rate at which new CD4+ cells are generated and thedeath rate of healthy cells is dx. In the case of active HIV infec-tion, the concentration of healthy cells decreases proportionallyto the product !xv, where ! represents a coefficient depend-ing on various factors, (e.g., the velocity of penetration of virusinto cells, the frequency of encounters between uninfected cellsand free virus). The second equation describes the dynamicsof the concentration of infected CD4+ cells (y); !xv is therate of infection and ay is the death rate of infected cells. Thethird equation describes the concentration of free virions (v)produced by the infected cells at a rate ky and uv is the deathrate of the virions. The fourth and fifth differential equationsdescribe the dynamics of CTLp (w), which are responsible forthe development of immune memory, and CTL effectors (CTLe)(z), which are responsible for killing virus-infected cells withrate pyz. This model can discriminate the trend of infection asa function of the rate of viral replication, if the rate is high,

a successful immune memory cannot establish, conversely, ifthe replication rate is slow, the CTL-mediated immune memoryhelps the patient to “successfully” fight the infection.

Since RTI drugs inhibit the infection of cells, their effectcan be included in model (1) by reducing the value of ! withrespect to the case of untreated patient. In the hypothetical caseof 100% effectiveness of RTI drugs, ! should be zero, but thisis not achievable in practice. Hence, one can replace

! # !(1 " "f) (2)

in which f is the control input (i.e., the drug uptake, normalizedbetween 0 and 1) and " represents the therapy effectiveness,which varies in [0, 1).

The variant to model (1) proposed by Nowak [16] and detailedbelow, is considered in [12]

!"""""#

"""""$

x = ! " dx " !(1 " "f)xyy = !(1 " "f)xy " ay " p1z1y " p2z2yw = c2xyw " c2qyw " bwz1 = c1z1y " h1z1

z2 = c2qyw " h2z2 .

In such a model, a differentiation between helper-independentCTL z1 , and helper-dependent CTL z2 , is considered.

One important feature of both models (1) and (3) is the ex-istence of bifurcation points, and thus, different steady states,which may be stable or unstable depending on the values of themodel parameters. In particular, Zurakowski and Teel [12] con-sider the case in which two stable steady states exist in absenceof therapy (i.e., when f = 0), one describing a progressive in-fection leading to AIDS and one describing the establishmentof a successful and permanent immune response. Under suchexistence hypothesis, [12] proposes using an MPC strategy fordetermining an “optimal” therapy protocol that leads the patienttoward the immune steady state.

B. Model Considered in This Paper

In this paper, we consider the following model, recently pro-posed by Landi et al. [1]:

!""""""""#

""""""""$

x = ! " dx " rxvy = rxv " ay " pyzw = cxyw " cqyw " bwz = cqyw " hzv = k(1 " µP fP )y " uvr = r0(1 " µT fT ).

It differs from (1) in the introduction of the new state variabler, an index of the aggressiveness of the virus, which substitutesthe constant ! of (1). According to the new equation describingthe r-state dynamics, r increases linearly with time in the caseof an untreated HIV-infected individual, with a growth rate thatdepends on the constant r0 (a higher r0 value indicates a highervirulence growth rate). In the model presented here, the increaseof virulence with time is assumed to be linear, this hypothesisis consistent with the simulation results obtained in the case ofLTNPs patients. The coefficients µT and µP represent the drug

TABLE IMODEL PARAMETERS AND VALUES IN [1] (IF DIFFERENT)

effectiveness weights for specific external inputs fT and fP ,which are the RTI and PI normalized drug uptakes.

In some existing models, the effects of RTI and PI drugshave been aggregated (also for time-scale separation reasons),but in several works they are treated separately [14], [17]–[19].PI drugs reduce the rate of virus production, and this action isconsidered in model (4) by modifying the rate coefficient k ofproduction of infected cells in the dynamical equation

v = k(1 " µP fP )y " uv (5)

where the term µP fP reduces the rate k of virions production,according to the PI drug effectiveness µP and dose fP . The ef-fect of PI drugs is to reduce the production of infectious virionsfrom already infected cells and to interfere with posttranslationalviral particle assembly, previously released infectious virus de-cays, but continues to infect cells. The effect of RTI drugs ismodeled by reducing the infection rate, and in this way, block-ing the infection of CD4+ cells by free virus. Hence, in model(4) the RTI drugs have an effect on virulence because their mainrole is halting cellular infection [20].

It is important to point out that the considered model does notexhibit any stable (immune) steady state, since we will assumethat the model parameters (see Table I) are such that

(µT fT ) < 1 (6)

and hence, r > 0, i.e., the virulence never becomes a constant.The inclusion of virulence as a new state variable in earlierWodarz and Nowak models represents the main result of [1].The idea of infection rate written as a dynamic state is alreadypresent in other papers [17], [21], [22] in which the infectionrate is supposed to be a function of VL or time-dependent, butin [1] it slowly increases with time to simulate the long-termuncontrollability of the HIV disease (at present, the possibilityof eradicating completely the virus has not been demonstrated).Today, a temporary condition of nonprogressor equilibrium ispossible due to HAART treatment, but inexorably the diseaseprogression slowly goes on to arrive at the final stage whenthe population of CD4+ cells is close to zero and the virusgrows unbounded. For this reason, a slow increase of virulencedescribes well a real clinical progression of the HIV disease,and simulation results obtained with model (4) are very similarto real courses of disease. Furthermore, it is important to remarkthat other studies such as that in [3] discussed that models, whichadmit stable steady states in absence of therapy do not describeaccurately well the long-term evolution of HIV.

III. MPC ALGORITHMS

A. Introduction

MPC refers to control strategies in which at each decisiontime point, a process model—either based on first-principleequations, such as mass/energy balances, equilibrium equations,etc., or derived from data—is used to compute an optimal “fu-ture” control sequence over a (finite) prediction horizon andan associated sequence of states starting from the current statevalue [23]. The control sequence is chosen in a way that anappropriate cost function, typically comprising a measure ofthe deviation of the future state sequence from reference targetvalues and a measure of the control effort, is minimized whilestate and control constraints are fulfilled. Usually, only the firstcontrol input is injected into the system to be controlled, andat the subsequent decision time point the overall optimizationprocess is repeated, computing new control and state sequencesstarting from the new current state value, possibly updated fromfeedback information (measures of output), if available. Morespecifically, feedback from measurements can be incorporatedby updating the model prediction with a correction term thattakes into account the difference between the predicted and themeasured outputs [23]–[25].

MPC algorithms typically achieve superior performance withrespect to other control strategies when manipulated and con-trolled variables have constraints to meet (possibly with differ-ent priority). Moreover, MPC algorithms can efficiently handlemultivariable systems and do not require that the number of con-trolled and manipulated variables be equal. Thus, they use allmanipulated variables to achieve optimization goals, such as: 1)keeping the key variables close to their targets; 2) maintaining allcontrolled variables within limits; and 3) minimizing the controleffort while respecting constraints on the manipulated variables.

MPC is a control technology widely used in many areas,especially in the process industries [24], for systems with alarge number of controlled and manipulated variables, whichinteract significantly. In general, feedback control technologies,and MPC in particular, have started to gain significant attentionin the biomedical area [26], [27]. Typical biomedical applica-tions of control methods include the glucose–insulin system indiabetics [28]–[32], anesthesia [26], [33]–[35], anticoagulantadministration [36], [37], and HIV [12]–[14], [38]–[40].

In general, any kind of (linear or nonlinear) model, cost func-tion and constraints can be used, although it is often desirableto use linear models, linear constraints, and quadratic (or linear)cost functions. In this way, the associated optimal control prob-lem is a quadratic program (QP) or a linear program (LP) forwhich reliable and efficient solvers exist. Indeed, one importantpoint that needs to be taken into account when considering MPCas possible control strategy is, its computational cost (i.e., theCPU time, often higher than that of other simpler control algo-rithms) compared to the allowed decision time. This is an activearea of current research in the predictive control and optimiza-tion research community aimed at reducing the complexity ofthe online calculations [41]–[44], and potentially extending therange of applicability of MPC to systems with shorter samplingtimes.

In this section, two MPC algorithms are presented to de-termine the “optimal” doses of RTI and PI drugs over a finite-prediction horizon, and their simulated performance is evaluatedand compared in the next section. In the first variant, the algo-rithm considers that in each sampling period (defined later on),the drugs can assume any value from zero to its maximum value(1), whereas the second algorithm constrains the drugs either tobe 0 or 1, thus leading to an STI protocol.

Compared to the MPC method proposed in [12], the MPCalgorithms developed in this paper differ in a number ofrelevant issues. The method proposed in [12] assumes to mea-sure all state variables every week, whereas we assume to mea-sure only CD4+ concentration and VL every month (or evenless frequently and possibly irregularly). Furthermore, whilethe method in [12] does not use any correction strategy to copewith model errors, we use as a state augmentation approach thatallows us to keep the controlled variables within the desiredranges in the presence of model errors. As discussed later inSection III-B, the strategy is based on fictitious disturbances(with integrating dynamics) that are estimated from availablemeasurements, and such an approach is that typically used inMPC to achieve offset-free control, i.e., to track desired sig-nals without a permanent bias in the presence of model errorsas extensively described in [25]. However, the most importantdifference between the MPC methods proposed in this paperand that in [12] is related to the internal model. The model usedin [12] admits that an immune steady state in absence of therapyexists, and hence, the goal of the MPC algorithm is to adjust thetherapy to reach that “safe” steady state. Afterward, the therapyis permanently stopped. In this paper, instead the internal model(4) does not admit a stable steady state, because, in general, itis not possible to zero the derivative of the last state variable r.This consideration is particularly important in the definition ofan MPC algorithm because the usual approach of defining thestage cost in deviation from a steady state is not applicable here.Furthermore, as will be shown in Section IV, the model used inthis paper makes impossible for the therapy to be permanentlysuspended, and this in agreement with the fact that no definitecure for HIV is currently possible. Finally, from an optimizationpoint of view the stage cost in [12] is quadratic for the state de-viation, linear for the drug uptake, and no output constraints areconsidered. In our paper, instead, we use a linear stage cost thatincludes the drug uptake and the output constraint violations.

B. Preliminary Definitions and Augmented Model

In order to give a formal description of the proposed MPCalgorithms, some preliminary definitions are necessary. Let !be the sampling time, we denote with X(i) the state of thesystem at time ti = i!, i.e.,

X(i) = [x(ti) y(ti) w(ti) z(ti) v(ti) r(ti) ]T

= [x(i) y(i) w(i) z(i) v(i) r(i) ]T . (7)

Next, let U(i) = [fP (i) fT (i)]T denote the vector of PI andRTI drugs taken in the time interval [i!, (i + 1)!], and rewrite

the system model (4) in the integrated form

X(i + 1) = F (X(i), U(i)) (8)

where F (·) is a vector-value function obtained by numericalintegration of (4) over the sampling time !, assuming a constantdrug uptake during the sampling time.

In this paper, we assume that only two state variables, theconcentration of healthy CD4+ cells (x) and the concentra-tion of free virions (v in logarithmic scale), are measured atregular intervals (every M sampling times, namely at discretetimes i = 0,M, 2M, 3M, . . .). We denote with Y (i) the vectorof such measured values (if available at time i), and with Y (i)the corresponding vector of model state values, i.e.,

Y (i) = [xmeas(i) log vmeas(i) ]T ,

Y (i) = [x(i) log v(i) ]T = H(X(i)) (9)

in which H(·) is the vector-value function that selects the firstand fifth components of X(i) (and apply the logarithmic trans-formation for the virus concentration). Next, we define the dis-turbance vector variable as follows:

D(i) =%

Y (i) " Y (i), if i $ {0,M, 2M, . . .}D(i " 1), otherwise.

Notice that the vector D(i) $ R2 is equal to the error betweenthe (last available) measured output variables and the corre-sponding values predicted by the model. Also notice that thisapproach can still be applied when the output variables are mea-sured irregularly, by updating D(i) at any time in which Y (i)is available. Such term D(i) will be used in the MPC opti-mization to evaluate an updated trajectory for cost computationand enforcement of constraints on concentration of CD4+ cellsand free virions. We notice that this approach corresponds toaugmenting the system (8) as follows:

X(i + 1) = F (X(i), U(i))D(i + 1) = D(i)Y (i) = H(X(i)) + D(i)

and using a deadbeat observer for estimating the augmentedstate from the measured output. Such an augmented system isusually referred to as output disturbance model because an in-tegrated disturbance is added to the output of the system. Therole of such disturbance is to capture the effect of all sources ofpermanent discrepancy between the patient’s and the model’sresponses, such as actual disturbances with nonzero mean (andsufficiently slow dynamics) and general model errors (e.g., in-correct parameters’ values). Such an approach is commonlyused in industrial applications of MPC to enforce offset-freecontrol of the output variables. Clearly, due to the use of anobserver, it is possible that the initial disturbance estimates arenot particularly accurate, but as more and more output mea-surements become available, the estimation quality improves.The interested reader is referred to [24], [25] for more completediscussion on offset-free MPC algorithms.

C. MPC Algorithm 1: A Continuous Dosage Approach

With the previous definitions, it is now possible to state thefollowing finite-horizon optimal control problem (FHOCP) tobe solved at each discrete time i:

minU i ,E i

i+N "1&

[cP cT ]U(j) + qxex(j + 1) + qv ev (j + 1)

(12a)subject to

X(i) given, X(j + 1) = F (X(j), U(j)), j = i, . . . , i+ N " 1

(12b)'

(% U(j) %

(, j = i, . . . , i + N " 1 (12c)

x(j) + D1(i) & x " ex(j), j = i + 1, . . . , i + N (12d)

log v(j) + D2(i) % log v + ev (j), j = i + 1, . . . , i + N

ex(j) & 0, j = i + 1, . . . , i + N (12f)

ev (j) & 0, j = i + 1, . . . , i + N (12g)

[1 " 1] U(j) = 0, j = i, . . . , i + N " 1 (12h)

in which N is an integer (control horizon), cP , cT , qx , qv arenonnegative scalar weights, and the decision variables are asfollows:

Ui = [U(i)T U(i + 1)T · · · U(i + N " 1)T ]T (13)

Ei = [ ex(i + 1) ev (i + 1) · · · ex(i + N) ev (i + N) ]T

in which Ei is the vector of slack variables. The value x rep-resents the lower bound imposed on CD4+ cells concentration,whereas v is the upper bound imposed on the concentrationof free virions. Consequently, ex(j) and ev (j) represent theviolations (if any) of such constraints over the trajectory.

The rationale behind the FHOCP (12) is the following: itaims at finding the sequence of future PI and RTI drugs (overa horizon of N sampling times) that (possibly) keeps CD4+cells concentration above its lower bound and the concentrationof free virions below its upper bound, while minimizing theoverall dosage (according to the weights cP and cT ). In moredetail, (12b) is the model evolution, (12c) are the constraints on(normalized) PI and RTI doses, (12d)–(12e) are the constraintson the prediction of CD4+ cells and virions concentration (up-dated with the disturbance term computed using the last avail-able measures), and (12f)–(12g) ensure that the slack variablesare zero if the predicted concentrations of CD4+ cells and freevirions satisfy the corresponding bounds. The constraint (12h)enforces PI and RTI drugs doses to be equal and this is addedboth for practical reasons of simplicity of the computed optimalprotocol, and most importantly, because of the known effectivetherapeutic action of combined drugs. It is important to point outthat in (12), the output constraints are posed as soft constraints toensure feasibility of the control problem [45], i.e., violations of

the bounds on CD4+ concentration and VL are allowed but theamount of such possible violations is penalized in the objectivefunction so that the optimizer tries to avoid (or reduce) them.Finally, it must be noted that all terms in the (linear) objectivefunction (12a) are nonnegative.

For open-loop analysis, one can consider the entire optimalsequence of PI and RTI drugs Ui , whereas in closed-loop op-eration only the first component U(i) is actually injected intothe system. At the successive decision time, the FHOCP (12)is solved again using the new current state vector and the newdisturbance vector (updated if measurements of CD4+ cells andfree virions concentration are available). Notice that even if nomeasurements are available (hence, the current state and distur-bance are the values estimated at the previous decision time), theFHOCP (12) is solved because its prediction time window hasmoved one sample time forward. Thus, compared to the drugsequence computed at the previous sample time, the current oneis based on a slightly more farsighted prediction.

D. MPC Algorithm 2: A Discrete Dosage Approach

From a therapeutic point of view, it may be unsafe to adminis-trate drugs at a dose less than maximum because virus mutationsmay occur (see e.g., [12] and references therein for a more ex-haustive discussion on this point). Therefore, standard HAARTprotocols require persistent drugs uptake at maximum value.However, a number of clinical and theoretical studies attemptedSTI protocols in which periods of therapy at maximum dosagesare alternated with periods of treatment suspension [9]–[14],[40]. The reasons for these attempts can be found in severalside effects of HAART, such as serious hepatic damages andthe high cost of the therapy, but also in the evidence that appro-priate suspension periods may enhance the immune response ofthe patient.

In this section, the optimal control problem of MPC 1 ismodified to obtain an STI protocol approach by restricting the PIand RTI doses to take values of either 0 or 1. Thus, the modifiedFHOCP, which defines algorithm MPC 2, is as follows:

minU i ,E i

i+N "1&

[cP cT ]U(j) + qxex(j + 1) + qv ev (j + 1)

(15a)subject to (12b), (12d), (12e), (12f), (12g), and

[1 0]U(j) = [0 1]U(j) $ {0, 1}, j = i, . . . , i + N " 1.(15b)

Notice that also in (15), it is considered that PI and RTI drugsare always taken (or suspended) together.

It is important to point out that from optimization point ofview the FHOCP (15) is a mixed-integer nonlinear program(MINLP), and therefore, it would require appropriate softwareto be solved. However, given the structure of the problem andthe fact that the computational time is irrelevant in the presentapplication (see Section IV), the approach considered in thispaper is to enumerate all possible sequences of drug doses andto evaluate for each one its cost function value. Although po-tentially inefficient, this approach depicted in Fig. 1 suffices for

Fig. 1. Tree diagram describing the dose sequences considered in MPC 2.

the purpose of this study. A similar full enumeration approachis, for instance, considered in [12].

IV. SIMULATION STUDIES

A. Model Response Analysis and Performance Indicators

Table I reports values of the model parameters, most of whichwere taken from [1]. An extensive parametric study conductedshowed that among all model parameters, the constants b and hare the ones, which affect mostly the state variable response. Asa matter of fact, by changing these two parameters, the modelresponse changes from that typical of LTNP patients to thattypical of fast progressor (FP) patients [1]. A complete identifi-ability study is beyond the scope of this paper because most ofthe parameters are meant to be considered as fixed, i.e., patientindependent. Nonetheless, by using the algebraic and geometricidentifiability tools discussed in [46] and [47], it is possible toshow that the two main (patient dependent) parameters b andh, are identifiable from measurements of CD4+ concentrationand VL. In order to better understand the behavior of the modelconsidered in this paper, we simulate the response of three refer-ence “patients”, obtained by varying the main model parametersb and h as follows:

1) Patient A: b = 0.01 (1/day), h = 0.08 (1/day);2) Patient B: b = 0.007 (1/day), h = 0.06 (1/day);3) Patient C: b = 0.006 (1/day), h = 0.05 (1/day).For all patients, we use the following drug effectiveness pa-

rameters: µT = 0.9 and µP = 0.85. In Fig. 2, the response of thethree reference patients is depicted for a period of three years,both for the case of no treatment and for the case of full HAARTstarted two months after the infection event. As expected whenuntreated, patient A shows the fastest progression of the HIVinfection, whereas patient C shows the slowest progression, andpatient B shows an intermediate behavior. Notice that all threepatients can be regarded as FPs. We also note that in the caseof full HAART initiated after two months from the infectionevent, a significant increase of CD4+ concentration occurs cou-pled with a reduction of the VL. It can be noticed that the VLshows faster dynamics than CD4+ concentration, and further-more, during the first one–two months from infection, a naturalimmunological rebound of CD4+ concentration and reductionof VL occur even without therapy. As discussed in more detailby Landi et al. [1], the model responses are in agreement with

Fig. 2. Model responses for three reference patients, untreated and underHAART started after two months from infection over a period of three years.

TABLE IIPERFORMANCE EVALUATION OVER THREE YEARS

clinical data, also with regard to CTLp and CTLe variables (notshown).

In order to assess the achieved performance, we use threeindicators: 1) the overall drug consumption (expressed in termsof weeks of full HAART); 2) the average CD4+ concentra-tion; and 3) the average VL.1 The top part of Table II reportsthe performance indicators obtained for the three reference pa-tients without therapy and with full HAART. It is important tonotice that for all patients, full HAART allows one to obtaina significant reduction of the VL and a clear raise in CD4+concentration. These indicators establish reference values forperformance evaluation of the two MPC algorithms discussedin this paper.

B. Implementation of MPC Algorithms

We implement two therapy protocols based on the MPC al-gorithms described in Section III. Both MPC algorithms use the

1VL is averaged in a logarithmic scale, i.e., VL = 101T

0log v dt

TABLE IIIPARAMETERS OF MPC ALGORITHMS

model of reference patient B, and all MPC parameters are shownin Table III. The dynamic evolution of the “real patient” is sim-ulated using the three reference models, and measurements ofCD4+ concentration and of the VL are assumed available to thecontrol algorithm every four weeks.

It is important to notice the MPC horizon is 8 weeks, butonly the first component of the optimal input sequence is actu-ally applied. Hence, the MPC algorithms compute the therapyevery week, and therapy is held constant within each week.Furthermore, it is important to point out that the model correc-tion term D(i) is updated every four weeks, whenever feedbackmeasurements become available.

For MPC 1, the solution of (12) is obtained at each samplingtime, using the MATLAB (version 2006b) nonlinear constrainedminimization routine fmincon, while a tailored code was writ-ten for the solution of the mixed-integer optimization problemof MPC 2 using the full enumeration approach discussed ear-lier. In both case, the solution time is in order of 30 s (on aPentium-M 1.86 GHz PC), thus irrelevant with respect to theMPC sampling time (one week).

C. Nominal Simulation Results

We first analyze the performance of the MPC-based ther-apy optimization algorithms for the nominal case in which weassume that: 1) the patient response follows that of referencepatient B; and 2) measurements of CD4+ concentration and VLare noise free. Notice that in such a situation the model pre-diction and the actual measurements coincide, thus leading toa correction term D(i) always equal to zero. Unless differentlystated, the therapy starts eight weeks after infection.

We present in Fig. 3, the closed-loop response of CD4+ con-centration, VL, and drug dosage over a period of three years(156 weeks) from infection. As expected, both MPC algorithmsaccomplish the goal of satisfying the constraints on CD4+ con-centration and on the VL (with possible minor violations) whilereducing the drug dosage with respect to sustained HAARTtherapy at maximum dosage. Clearly, this goal is obtained byMPC 1 through a significant dosage reduction, and by MPC2 through an induced STI protocol (comprising 109 weeks onand 39 weeks off). Since MPC 2 only considers on/off ther-apy periods, in order to avoid large violations of the VL upperbound, the algorithm computes a stronger dosage than MPC 1.Thus, the average VL achieved by MPC 2 is significantly be-

Fig. 3. Closed-loop response of reference patient B using MPC 1 or MPC 2.

low the upper bound and the average CD4+ concentration isslightly higher than the corresponding value obtained by MPC1. Furthermore, from Table II, we can notice that despite the20% reduction in drug consumption achieved with MPC 2,the average CD4+ concentration is 897 cells/mm3 and the VLis 45.1 copies/mL, and these values are similar to those ob-tained using full HAART (926 cells/mm3 and 32.5 copies/mL,respectively).

Next, we analyze how the optimal therapy changes dependingon its starting time. We present in Fig. 4, the closed-loop simula-tion results obtained using the MPC algorithm, when therapy isstarted four months or one month after infection. Clearly, whentherapy is started earlier, a higher overall drug consumption isrequired in order to face the initial period of acute infection, ashighlighted in Table II. As a consequence, the average CD4+concentration is slightly higher and the average VL is lowerwhen the therapy is started earlier. It must be noticed that start-ing the therapy one month after the infection or two monthsafter infection (as in the nominal case depicted in Fig. 3) doesnot result in relevant differences. As a matter of fact, in the rightplot of Fig. 4, we can notice that the MPC algorithms computea zero drug dosage during weeks 5, 6, and 7, given that the VLis well below its upper bound due to the natural immunologicalresponse, whereas the drugs cannot affect the slow rebound ofCD4+ concentration, in this early stage.

D. Robustness to Measurement Noise

We now present closed-loop simulation results obtained inthe presence of measurement noise. More specifically, we stillassume that the patient evolution is described by the model ofreference patient B, but measurements of CD4+ cells concen-tration and VL (available to the MPC algorithms every fourweeks) are affected by random noise with a noise-to-signal ra-tio of 10%. The simulation results are depicted in Fig. 5 over aperiod of three years from infection, while the achieved perfor-mance indicators can be found in Table II. It can be noticed thatMPC 2 appears less sensitive to the presence of measurement

Fig. 4. Closed-loop response of reference patient B when therapy is started(left plots) four months and (right plots) one month after the infectionoccurrence.

Fig. 5. Closed-loop response of patient B with noisy CD4+ and VL measure-ments (10% noise-to-signal ratio) using MPC 1 or MPC 2. Circles indicate themeasurement time points.

noise than MPC 1. This slightly different behavior can be ex-plained by observing that MPC 1 keeps the VL close to its upperbound, and hence, it must react more aggressively to violations,which are likely to occur when measurement noise is present.It is important to remark that if necessary, we can reduce sen-sitivity to noise by a suitable filtering method. For instance, we

Fig. 6. Closed-loop responses of patient A (left plots) and of patient C (rightplots), using MPC 1 or MPC 2 algorithms based on patient B model.

could modify (10) as follows:

D(i) =

'#1 00 #2

((Y (i) " Y (i)), if i $ {0,M, 2M, . . .}

D(i " 1), otherwise(16)

in which #1 and #2 are scalars chosen in the interval [0, 1] toweigh the relative importance between model trust (values closeto 0) and measurement trust (values close to 1).

E. Robustness to Model Error

Next, we consider closed-loop simulation results obtainedwith the MPC algorithms when the patient response is signifi-cantly different from that of the model used by the control al-gorithms. More specifically, we report in Fig. 6, the simulationresults obtained when the patient response is that of referencepatient A (left plots) or that of reference patient C (right plots),but the MPC algorithms are still based on the model of refer-ence patient B. The obtained results in terms of performanceindicators are reported in Table II. It is interesting to notice thatdespite the fact that the MPC algorithms are using an incor-rect model, the output feedback strategy implemented is able tocompensate for model errors and satisfy the desired constraintson CD4+ concentration and VL. We can also observe that forreference patient A, an STI protocol does not seem indicated, asonly one week of therapy suspension is computed. On the otherhand, when applied to patient C, MPC 2 determines a protocol

with 89 weeks of therapy suspension and still achieves an aver-age CD4+ concentration of 821 cell/mm3 and an average VLof 85 copies/mL. These results are in agreement with the factthat patient A shows a much faster disease progression, whereaspatient C displays the slowest progression of HIV.

It is useful to remark that we ensured limited constraint vio-lations, despite the use of the incorrect internal model, by onlyadding an estimated disturbances term to the output predictionsof CD4+ concentration and VL. Such disturbances estimatelumps, the differences between the actual patient’s (patient Aor C) response and the internal model (patient B) used by theMPC algorithm. However, for future clinical applications, if thissimple strategy would not prove sufficiently effective, we mayenvisage an adaptation scheme in which the first available mea-surements are used to update the main parameters of the internalmodel (b and h), while subsequent measurements are treated asusual to update the disturbance term D(i).

Finally, it is worth noticing that we tested the robust perfor-mance of both MPC algorithms with “extreme” patients, i.e.,when the parameters are at the lower or upper bound of theirrange. Even in such cases the control algorithms performed well,in the sense that the therapeutic goals were met, but the resultsare omitted for the sake of space and because they are somehowobvious. For instance, if we consider a patient with parametersb = 0.001 and h = 0.03, besides an initial period with frequentinterruptions, MPC 2 computes no therapy. This is in perfectagreement with the fact that model (4) with such values for band h describes the dynamics of an LTNP patient. Similarly, ifwe consider a patient with parameters b = 0.015 and h = 0.3,the outcome is a sustained therapy with no interruptions.

F. Further Comments and Possible Extensions

It can be noticed that MPC 1 achieves the desired controlobjectives with a significant reduction of drug consumption.Indeed, it is obvious that the overall drug consumption usingMPC 1 is lower than that using MPC 2, because the former onecan modulate the therapy in the [0, 1] range whereas the latterone can only implement either 0 or 1. However, as remarkedpreviously, drug resistance issues suggest to avoid drug mod-ulation, and thus, an STI approach as that computed by MPC2 is to be preferred from a clinical point of view. From theseresults, it is clear that the STI protocols computed by MPC 2vary significantly when applied to different reference patients.Thus, while patient A requires almost continuous therapy (147weeks on/1 week off), patient B can be controlled with a moder-ate STI protocol (109 weeks on/39 weeks off), and patient C canbe controlled with much more frequent interruptions (59 weekson/89 weeks off).

A number of extensions and modifications to the proposedalgorithms can be made. First of all, the MPC algorithms canbe implemented using models different from the one consid-ered in this paper. In particular, we remark that the proposedMPC formulation does not require the existence of a stable (im-mune) steady state (as e.g., that in [12]). Nonetheless, if theproposed MPC algorithms are implemented with such kinds ofmodels, the closed-loop response of such models would show

convergence toward the (safe) steady state similar to [12]. Asecond consideration can be made with regard to the modelcorrection strategy based on feedback measurements. The ap-proach considered here (using an integrated output disturbancemodel) is rather simple, yet very effective. Alternative solutionsthat could be explored, include the use of more general (stateand output) disturbance models coupled with appropriate ob-servers (e.g., moving horizon estimators), parameter updatingstrategies. From a practical point of view, the most importantparameters of the model considered in this paper appear to be band h. Currently, model parameter estimation from clinical datais under investigation. Other possible modifications can be maderegarding the objective function, e.g., by choosing it in quadraticform. From our experience, the linear objective function consid-ered in this paper seems appropriate and effective, especially, forMPC 2. A quadratic objective function could avoid (or reduce)the more pronounced sensitivity to noise of MPC 1.

V. CONCLUSION

In this paper, we presented two novel MPC algorithms for thedetermination of “optimal” therapy protocols for controllingHIV infection. The first algorithm computes at each samplingtime (of one week), the optimal drug dose sequence over ahorizon (of eight weeks) that respects the desired therapeuticconstraints on CD4+ concentration and VL, while minimizingthe drug consumption. The second algorithm instead restrictsthe weekly dose either to be zero or one (maximum dose),thus resulting in an “optimal” STI protocol. Differently fromexisting MPC algorithms for HIV control, we assume to measureonly two state variables (i.e., CD4+ concentration and VL),and an infrequent (possibly irregular) measurement scheme isconsidered. In such a (realistic) framework, simulation resultsshow that the proposed MPC approach is able to control HIVeffectively, despite the possible presence of measurement noiseand relevant model errors. This is due to an appropriate outputfeedback updating strategy that was implemented in this paper.

Future works will be devoted to address a number of impor-tant issues toward a possible clinical application of the proposedstrategies. First of all, we are currently testing the consideredmodel against available clinical data, in order to evaluate thepossibility of model parameters’ identification. Other signifi-cant works might include the development of models of HIVinfection, including the issues of drug resistance and viral mu-tation, as well as the implementation of specific techniques fordetection of such occurrences.

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Gabriele Pannocchia received the M.S. and Ph.D.degrees in chemical engineering from the Universityof Pisa, Pisa, Italy, in 1998 and 2002, respectively.

He was a Visiting Associate with the University ofWisconsin, Madison, in 2000 through 2001 and 2008,and a Postdoctoral Fellow with the Department ofChemical Engineering, University of Pisa, where hehas been an Assistant Professor since 2006. His cur-rent research interests include model predictive con-trol, systems, process simulation and optimization,numerical optimization, and biomedical applications

of automatic control algorithms.

Marco Laurino received the B.S. and M.S. degreesin biomedical engineering from the University ofPisa, Pisa, Italy, in 2007 and 2009, respectively, wherehe is currently working toward the Ph.D. degree inneuroscience.

His current research interests include biomedi-cal systems modeling, nervous and immune systems,and model predictive control applied to biomedicalsystems.

Alberto Landi (M’95) received the M.S. degree inelectrical engineering from the University of Genova,Liguria, Italy, and the Ph.D. degree in electrical en-gineering from the University of Pisa, Pisa, Italy, in1986 and 1991, respectively.

He is currently a Full Professor of automatic con-trol with the University of Pisa, where he teachescourses of fundamentals of automatic control, phys-iological control systems, and process control. Hehas authored or coauthored more than 60 publishedtechnical papers. His research interests include math-

ematical models of physiological systems, sleep slow oscillations, and nonlinearidentification and control.

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