Systematic Method for Determining Intravenous Drug Treatment Strategies Aiding the Humoral Immune Response 1

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1 Systematic Method for Determining Intravenos Drg reatment Strategies Aiding the Hmoral Immne Response A. Rndell R. DeCarlo V. Balakrishnan H. Hogensch School of lectrical and Compter ngineering School of lectrical and Compter ngineering School of lectrical and Compter ngineering School of Veterinary Medicine Prde University, West Lafayette, IN 4796 Keywords: predator-prey modeling, nonlinear modeling, immne response modeling, intravenos antibiotic treatment, robst control, linear matrix ineqalit ies Abstract: his paper delineates a systematic method for determining "optimal" intravenos drg delivery strategies for patients having illnesses that primarily evoke a hmoral immne response and are treatable by antibiotics. he method derives from a nonlinear, distribted predator -prey model that captres the dominant antigen and antibody interaction. his model is developed from relevant physiology, past predator-prey-type modeling work, available data, and pertinent parameter identification. mbedding this predator-prey model into a larger class of ncertain systems, by a finite dimensional approximation and a transformation to a linear fractional representation, enables the application of robst control based on linear matrix ineqality optimization techniqes. he optimization problem is solved by minimizing an pper bond on a measre of the total drg delivered sbject to patient recovery (stability to healthy eqilibrim state). Specifically, the paper addresses the treatment of Haemophils inflenzae throgh modeling, controller development, and simlations of infected adlt patients sbjected to typical and proposed intravenos antibiotic treatments. hrogh simlations the proposed intravenos drg strategy shortens patient recovery time, lowers peak drg concentrations and decreases the total drg administered when compared to standard antibiotic strategies. I Introdction In the immne response to antigen, the host and antigen are both predator and prey: the antigen preys on the host and the host monts an immne response to attack the antigen. he work herein describes a predator -prey model that captres the dominant immne response to the bacterim, Haemophils inflenzae, bt can be generalized to most antibiotic ssceptible antigens that primarily evoke a hmoral immne response. H. inflenzae is a Gram negative bacterim which can case bacterial sepsis, pnemonia, or meningitis. Becase it spreads by respiratory droplets, day care centers and homes for the elderly are often otbreak centers. Standard treatments to combat infection inclde antibiotics (ampicillin or ceftriaxone for ampicillin resistant strains) or immnization which is only partially sccessfl [3]. Within the context of the predator-prey model, or overall goal is to investigate the feasibility and effectiveness of two antibiotic treatments for H. Inflenzae: (i) a typical intravenos antibiotic treatment, and (ii) an "optimal" intravenos drg strategy. he consistency of the simlated model response with the clinical response provides a partial model validation. he control strategy is obtained by applying convex optimization techniqes based on linear matrix ineqalities (LMIs) [] to redce the total qantity of drg administered to a level below that of a typical intravenos treatment. Becase of the approximations inherent in the LMI problem formlation, the soltion strategies are conservative, bt nevertheless, provide improvement and represent a first step in obtaining a systematic method for determining optimal drg delivery strategies. II Relevant Immnology verview he host's immne system predominantly responds to H. inflenzae with a specific hmoral response prodcing antibodies which bind to the bacteria. hese bond antibodies are bactericidal (kill the bacteria) and opsonic (promote phagocyte englfing) [4] thereby eliminating the bacteria threat to the host. he antibodies (types IgG, IgM, and IgA [4]) are prodced by plasma cells. Plasma cells are differentiated B cells which have been stimlated by active - helper cells, stimlatory signals, and antigen. For hosts with immne deficiencies, immatre, or sppressed immne systems, the bacteria may escape early detection and proliferate to debilitating or lethal levels. AIDS patients with compromised immne systems are particlarly ssceptible to H. inflenzae indced pnemonia [5]. Conseqently, a mathematical model of the immne response to H. inflenzae provides a tool for illness evalation pls analysis and development of disease treatment strategies. his work was spported in part by NR nder contract N

2 III Model Development he dominant disease dynamics for H. inflenzae are modeled by the predator-prey characteristics of bacteria and antibody serm concentrations. his new predator-prey model evolved from that of Rai, Kmar, Pandé (RKP) [] into a model qite similar to Bell s model [4]. nhancements to and differences from Bell's model inclde a time delay on the antibody prodction rate, an pper limit on antigen concentration, and the effects of antibiotics. r new model has an antigen rate eqation db( t) and an antibody rate eqation ( A( t) A eq ) B( t) = a B(t) w d + B( t) + A( t) A eq bb (t) αb(t) (t) da( t) A( t τ) B( t τ) = ρ r + A( t τ) + B( t τ) ( A( t) Aeq ) B( t) w d + B ( t ) + A( t) A eq (3.a) A( t) * + ( t τ ) A a (A(t) A eq ) (3.b) where + (t-τ) = for t τ and for t < τ, and B(t) and A(t) represent antigen (bacteria) and antibody concentrations in the blood respectively. (he bacteria concentration is measred in colony forming nits (cf) per ml and the antibodies are stated in µg per ml.) 3. he Antigen Rate qation Bacterial growth, modeled by a B(t), with appropriate choice of a, acconts for the rapid proliferation of H. inflenzae (which in vitro is x 4 cfml to 6.5x 7 cfml in a matter of hors [6]) as well as the growth inhibiting effects of nonspecific serm components sch as acte phase proteins and collectins [7,8]. ther bacterial growth limiting factors are the specific immne system response, resorce limitations, and antibiotics. he specific immne system response prodces antibodies which are bactericidal and opsonic for H. Inflenzae, i.e. antigen removal by antigen-antibody complex formation and phagocytosis. he removal rate is assmed to be proportional to the concentration of bond bacteria ( A( t) Aeq ) B( t) approximated by the fraction: d + B ( t ) + A( t) A eq derived from a mass balance eqation with the Law of Mass Action [4]. he parameter, w, is the rate of removal of these bond bacteria. he parameter d is related to antibody avidity and immnity levels (i.e. antibody concentrations which prevent infection); is a constant acconting for mltivalent antigens and nit conversions. Resorce availability, competition for limited resorces within the host, limits growth for extremely large bacteria concentrations and is accommodated within the model by a self-interaction term []: - bb (t). Neglecting the effects of antibiotics and antibodies, this self-interaction term cases bacterial growth to cease as bacteria concentrations approach a b. his follows from the fact that B = [a - bb(t)]b(t) is negative for B(t) > a b and positive for B(t) < a b. he term -αb(t) (t) represents bacterial growth inhibition de to antibiotics where (t) is the drg concentration in the blood. he prodct term, B(t) (t), reflects the necessity of drg-antigen interaction for halting antigen growth; α is a parameter dependent on the drg's minimm inhibitory concentration (MIC); however, α mst be lowered to accont for bacteria inaccessible to the blood. qation 3.a differs from the Bell eqation in the last two terms which limit growth for extremely large bacteria concentrations, -bb (t), and incorporate antibiotics, αb(t) (t). 3. he Antibody Rate qation Prodction of antibodies for H. inflenzae occrs after specific -helper and B cells are activated by stimlatory signals and antigen contact. his is assmed to be proportional to the concentration of bond antibodies as A t B t represented by ρ ( τ) ( τ). his term is r + A( t τ) + B( t τ) derives its form from the Law of Mass action and the associated parameters ρ and r correspond to antibody prodction rates, the host sensitivity to this particlar antigen, and the immne system activation capability. he delay terms, A(t-τ) and B(t-τ), (a modification to the Bell model) compensate for the activation, proliferation, and differentiation times of the -helper and B cells. o simlate satration antibody concentration levels [4], the antibody prodction term is mltiplied by [-A(t)A * ] so that as A(t) approaches the pper limit A *, antibody prodction ceases. he finite effective life span of an antibody is incorporated by - a (A(t)-A eq ) where a is the inverse of the antibody half-life and A eq represents the eqilibrim level of naive B cell membrane srface immnogloblins available for stimlation at the onset of an immne response. As bond bacteria is removed from the host, the antibodies bond to the bacteria are also removed. his physiological effect is ( A( t) Aeq ) B( t) acconted for by w d B t which + ( ) + A( t) A eq

3 coincides (as expected) with the antigen -antibody complex removal rate in 3.a except in the parameter,, which adjsts for the effects of mltivalent antigens and nit conversions. his activity is not explicitly represented in the original Bell model since withot the delay it can be accommodated only by lowering the antibody prodction rate appropriately. IV Drg herapy as Control o establish a baseline for later comparison of recovery time and total drg administered, we evalate the standard intravenos drg treatment s ing or model for an AIDS patient with H. inflenzae indced pnemonia. 3.3 Simlation of Model raditional model validation is difficlt withot an abndance of experimental data. he available experimental biological data for adlt infections with H. Inflenzae was tilized to find model parameters. he model parameters, a, a, b,, α, A *, A eq, and τ, were chosen to be consistent with known physiological qantities and the remaining parameters, w, d, ρ, and r, were fond by meeting the few immnological observations of a typical adlt response for the bacterial and antibody crves as discssed below. Using the parameter vales and initial conditions listed in Appendix I, a typical H. inflenzae infection for a healthy adlt was simlated. Figre 3. shows a bacteria concentration that peaks at 34 cfml on abot day five. his represents a typical adlt response given that children not immne to H. inflenzae exhibit mch higher bacterial levels on the order of cfml []. Figre 3. also indicates antibody prodction for five to seven days plateaing at day 7 in agreement with typical hmoral immne responses []. For weeks after the H. inflenzae infection, the reslting antibody concentration is 53 µgml, well within the expected concentration range after one month of 35 to 8 µgml [4,3]. Sch a simplified model (antigen and antibody responses only) from eqation 3. can only be expected to captre dominant aspects of the hmoral immne response. As a validation exercise, simlations rn with minor variations in the model parameters gave similar reasonable bacteria and antibody responses. As a frther independent validation, we examined the initial bacteria response to varios initial conditions on antibody concentrations. r model indicates that for A() <. µgml the bacteria concentration increases (infection occrs) while for initial concentrations greater than. µgml the bacteria concentration decays which is consistent with literatre [4, 3] that sggests bacterial growth is inhibited when circlating H. Inflenzae specific antibody concentrations exceed the immnity level estimated between.6 µgml to µgml. Illnesses which elicit a dominant hmoral immne response have similar antigen-antibody dynamics since the nderlying physiology is the same. his sggests that the strctre of or predator prey model will captre the dynamics of other illnesses similar to H. inflenzae with appropriate parameter vales for antigen growth rate, a, resorce availability, b, antibody-antigen avidity, d and w, and antibiotic efficacy, α. 4. Standard Antibiotic Intravenos reatment Ampicillin, an antibiotic commonly sed to treat H. inflenzae, inhibits bacteria proliferation by interfering in the bacterial cell wall synthesis process [3]. It is administered every 6 hors intravenosly with dosages ranging from 5mg to 3 g over to 4 days for adlts [9,6]. he dominant pharmacokinetics of ampicillin, inclding its half - life ( to.8 hors), dosage level and schedle, specify the drg concentration in the blood, (t). With a day regimen administering 3 g of ampicillin for times a day and assming a half-life of.5 hors: 4* + ( t) ( t 6k) 5exp ( t 6k) 5. k= assming that the 3 g of ampicillin will distribte throghot the 6 ml of blood in a short period of time. 4. valation of Standard Antibiotic Intravenos reatment he rapid growth of H. inflenzae reqires drg treatment for patients with sppressed immne systems. An adlt AIDS patient can contract pnemonia from exposre to the H. inflenzae bacteria [5]. We model an AIDS patient with a sppressed immne system by making the antibody prodction rate, ρ, small compared to the normal healthy parameter. (Simlation shown in Figre 4..) With bacterial levels exceeding cfml (greater than sixteen times that of a healthy adlt patient), this patient is extremely ill and wold probably not srvive more than a few days ntreated. Since an AIDS patient has a severely depressed immne system, it is reasonable to assme that the administered treatment wold be the most aggressive strategy available (3 g dosages every 6 hors.) Applying this standard intravenos drg treatment to or model of an adlt AIDS patient reslts in antigen elimination (less than one cf6ml) after 9.5 days as shown in Figre 4.. his recovery reqired a total administration of 4 g of ampicillin which is reasonable. V Mathematical Formlation of the Control Problem and Soltion By applying nmerical LMI optimization techniqes we generate an alternative intravenos drg strategy that redces the total qantity of drg administered to a level below that of the standard intravenos treatment. he model of eqations 3., mst be adapted to the application of LMI techniqes since it is nonlinear and distribted (de to the delay). o obtain a finite dimensional model we approximate 3

4 the delay with a finite-dimensional linear time -invariant system. he nonlinearities are addressed by taking advantage of their rational natre and transforming the nonlinear model to a linear-fractional representation which consists of a linear system with a feedback loop containing mltiple copies of the state variables. he state variables appearing in the feedback loop can be artificially regarded as "strctred, bonded ncertainties" and standard LMI techniqes from robst control can be employed. In effect, this embeds the nonlinear system in a family of ncertain systems. Controllers are generated that are garanteed to work for all members of this family, which in trn are garanteed to work for or specific nonlinear model. r objective is to determine a strategy for drg delivery that minimizes the total drg administered sbject to the constraint that the patient recovers; this is accomplished by minimizing an pper bond for the drg - antigen interaction. 5. xpressing Model in Sitable Form for Application of LMI echniqes A 4 th order Bessel filter with linear phase [5] is sed to obtain a finite dimensional approximation of the time delay: dx = A d x (t) + B d h(t) y(t) = x (t) h(t-τ) where x (t) R 4 and y(t) approximates the delayed inpt h(t). (he Bessel filter was chosen to be forth order so as not to increase the dimension and complexity of the reslting LMIs beyond what is reasonable to formlate and solve.) Incorporating this model for the delay in eqations 3. reslts in a model of the form: dx = f(x)+ B g(x) (5.) where x = [B A x x x 3 x 4 ] and f(x) and g(x) are rational fnctions. We reqire the finite dimensional model to approximately match the inptotpt relationship (antigen infection doseantigen and antibody responses) of the original nonlinear differential delay model (3.). his reqires adjsting the parameters associated with the ant ibody prodction rate, ρ and r, in eqation 5.. Figre 5. compares the reslting approximate finite dimensional model and the original model. Appendix II lists the explicit eqations of 5. with the parameters for the Bessel filter and the adjsted vales for ρ and r. With the sbstittion of = g(x), the rational form of f(x) can be sed to obtain an eqivalent linear fractional representation (LFR) of 5. sing techniqes from l Ghaoi and Scorletti [] and Balakrishnan and Boyd [6]. he reslting LFR is given by: dx = A mx + BpP + B q = C q x + D qp p (5.) p = (x)q (x) = diag(b(t), B(t), B(t), B(t), A(t) - A eq, A(t) - A eq, A(t) - A eq, A(t) - A eq ) where x = [B (A-A eq ) x x x 3 x 4 ]. In effect, we have an approximate model of or system which consists of a linear part with mltiple copies of the state x(t) appearing in the feedback loop (see Figre 5.). By bonding the norm of the feedback, (x) σ, the LFR defines a class of ncertain systems. r original nonlinear model is a member of this class. Models sch as these rotinely appear in robst control [, ] and motivate or se of robst control techniqes. Note that system 5. has an eqilibrim state at x = which represents a patient with no infection. Appendix III states the associated LFR matrices. (t) p(t) LI (x) q(t) Figre 5. Linear Fractional Representation x(t) 5. LMI based Control Synthesis here are three steps to generate a controller sing LMI techniqes: (i) the selection of a Lyapnov fnction, V(x) = x Px with symmetric P >, and controller architectre, (t) = K x + K p, (ii) the formlation of the conditions on the Lyapnov fnction and controller parameters that garantee properties desired of the closed loop system, and (iii) a nmerical search to solve the reslting LMIs from which we can find nmerical vales for P, K and K sbject to (i) and (ii). he objective is to generate a drg delivery strategy that minimizes total drg dosage sbject to the patient's recovery. Stability to the zero eqilibrim level assres patient recovery, garanteed when the derivative of the Lyapnov fnction is negative, dv < for x. ne measre for qantifying drg dosage is a bond on the total drg-antigen interaction as represented by ( B) and thereby limit the total drg delivered. Under the assmption that the linear portion of the c ontrol (K x) dominates the controller effort (jstified since the bacterial growth rate term in eqation 3.a is linear and is the only positive rate term), we have: 4

5 # % % % % % $ % "! " ' & -. ( + 5 ( B) = [ K x] K x where = g(x) and g(x) = B. o minimize the total drg delivered we propose to minimize an pper bond of or objective [ K x] K x. If the Lyapnov fnction, V(x) = x Px with P = P >, satisfies dv < x K K x (5.3) QA m + A Q + B Y + Y B + B B + B RB + B R B *) * QC + B RD + B D R B Y m p p p p q p qp qp QC q + B RD + B D p qp qp B R Y, DqpDqp σ I < (5.6) for every state trajectory of the system 5. sch that (x) is contained within the ball of radis σ, then the system is well posed and stable. By integrating both sides of the ineqality 5.3, x() Px() x() Px() V() > [ K x] K x and x() Px() provides an pper bond for the objective fnction since V() for all. A physiological interpretation of this spposition asserts: for small infections (B(t) < σ, A(t) A eq < σ), the patient will recover since dv < and the total drg delivered will be less than x() Px(). A sfficient condition for 5.3 which incorporates the restriction of the norm of (x) to σ is dv σ p Sp + q Sq < x K K x (5.4) where S satisfies the following conditions: S = S >, (x)s = S (x) for every (x) of form 5.. Since S commtes with (x), S exploits or knowledge of the diagonal strctre of (x) []. Sbstitting for dv sing eqations 5. with (t) = K x + K p, condition 5.4 holds if the following matrix ineqality is satisfied: m A P + PA + C SC + K B P + PB K + K K m q q ( p q qp ) PB + PB K + C SD p q qp PB + C SD + PB K D SD σ S qp qp <.(5.5) he set of P, K, K, and S sch that ineqality 5.5 holds is non-convex; this precldes a soltion by standard LMI techniqes. However, with the following change of variables: Q = P -, Y = K Q, = (σ S) -, R = K, we obtain the eqivalent matrix ineqality which is linear in the new variables (Q, Y,, and R). Minimizing x() Q - x() minimizes the pper bond of or objective, and is a so-called eigenvale problem [, p. ]: sbject to 43 4 γ x( ) minimize γ x( ) 6 > and ineqality 5.6. (5.7) Q nce the minimization is complete the "optimal" control (t) = K x + K p can be recovered from K = YQ - and K = R -. Note γ is x() Px() which is approximately x() Px() V() since V() for large. 5.3 Soltion to LMIs It is now well-recognized that LMI optimization is qite efficient, and ths is ideally sited for nmerical linear control system design. Briefly, LMI problems are convex optimization problems with a finite nmber of variables []. herefore, they can be solved in polynomial-time, and every stationary point is a global minimizer. Moreover, we can immediately write down necessary and sfficient optimality conditions, and there is a well-developed dality theory. From a practical standpoint, there are effective and powerfl algorithms for the soltion of LMI problems, i.e. algorithms that rapidly compte the global optimm with non-heristic stopping criteria [8,, ]. Several recently developed software packages for LMI optimization are now available [9, ]. We sed the LMI toolbox from MatLab [9] to solve the LMI problem 5.7 for Q,, Y and R. In or case, there exists a linear control satisfying the LMIs (with R = ). A soltion to the LMIs for a nonlinear controller (R ) allows s to frther redce the control energy. he reslting stabilizing controller is (t) = K x + K p where K = YQ - and K = R -. We noted, after solving nmeros LMI problems each with a different set of model parameters, that K and K have the strctre: K = [κ κ κ 3 κ 4 ] K = [ κ 5 κ 5 κ 6 κ 5 ]. 5

6 7 8 xpressing (t) = K x + K p within the context of or model as a combination of static linear and nonlinear feedback terms pls a dynamic linear feedback component yields: ( A( t) Aeq ) B( t) (t) = κ B(t) + κ 5 d + B ( t ) + A( t) A eq κ 6 B(t) + κ x + κ 3 x 3 + κ 4 x 4. (5.8) his stabilizing controller was generated with a local artificial restriction of the state variables in (x) to a ball of radis σ. Sigma was chosen to be sfficiently large, σ =, so that a controller existed. If the initial conditions of the states are within the ball (B() < σ and A() A eq < σ), the controller maintains the states within the ball and stabilizes the system to the zero bacteria eqilibrim (B = and A = A eq ), garanteeing patient recovery. he ratio R of the L -norm of the dynamic linear feedback terms of the controller (κ x + κ 3 x 3 + κ 4 x 4 ) to the L -norm of the static linear and nonlinear terms approximates - for or particlar model parameters. his negligible inflence permits s to ignore these terms (by setting κ = κ 3 = κ 4 = ). It is easy to check that the original model maintains stability abot the eqilibrim with the trncated control. Sfficient conditions on the controller parameters to eliminate the antigen for all infection levels are fond by examining the bacteria rate eqation. P rovided the controller parameters satisfy κ > a α, κ, κ 3, κ 4, κ 5 < dα, and κ 6 < bα, this controller forces db ( t ) negative, for all non negative vales of A(t) and B(t), driving B(t) towards zero and ensring patient recovery from serios infections. For H. inflenzae the controller parameters meet these conditions and garantee complete bacterial elimination from the host. herefore we have recovery for all non-negative vales of the state variables otside the ball of radis σ. he nonlinear control terms have the same strctre as the antigen-antibody complex removal term and the selfinteraction term on the antigen. hese terms redce the reqired drg concentration as the antibody concentration increases or as the self-interaction antigen rate term increases. he conservative control of 5.8 is the primary bacterial removal sorce ntil the antibody concentration, A(t), reaches inhibitory levels at which point the drg contribtion to bacterial removal is redced and eventally eliminated as A(t) becomes sfficiently large. In sch cases, the LMI derived controller, (t), goes negative nnecessarily eliminating excess antibody by theoretically introdcing controlled amonts of antigen for complex formation. Physiologically, (t) is lower bonded by zero. Incorporation of the constraint (t) in the LMI problem formlation is an area of ftre research. In keeping with physiology, (t) for all simlations. VI Simlations and Immnological Interpretation of Reslts Since mathematical models merely reflect the dominant characteristics of a system, any simlation mst be accompanied by some mathematical interpretation of the physiology, and mathematical assmptions to accommodate nmodeled physiology. For simlation prposes we assme that bacterial levels less than or eqal to one cf in the host (B(t) 6 cfml) corresponds to sccessfl bacterial elimination. We simlate the patient response sing the Gear algorithm in Simlink to nmerically solve the model of eqations 3.. System parameters and initial conditio ns can be fond in Appendices I. he desired drg concentration, (t), is implemented as ( t) = if ( t) < ( t) otherwise. B( t) + ε (6.) his conversion rle from the drg-antigen interaction controller, (t) of eqation 5.8 (recall κ = κ 3 = κ 4 = ), to (t) helps avoid nmerical complications when B(t) is very small by sing a small (less than 6 cfml) positive reglarization constant ε. Physiologically, once B(t) < 6 cfml, the drg dosages may be discontined ( (t) = ). Figre 6. shows the responses of a healthy adlt and an immne sppressed AIDS patient with treatment immediately following infection by cf. In both cases the bacterial removal is completed within.5 days sing a continos constant drg concentration less than 38 µgml tailing off after one and a half days. ven thogh patients are not administered an intravenos drg delivery system pon immediate exposre to the bacteria, the simlations provide validation of the effectiveness of the control strategy. Figre 6. profiles or proposed drg strategy, applied for days after bacterial infection, which sccessflly eliminates the bacteria with a peak drg concentration less than 38 µgml. For the healthy patient, the drg concentration decays rapidly after only one-half day since the antibody concentration has grown large enogh to effectively eliminate the bacteria naided by antibiotics. For an AIDS patient, with a sppressed immne system, the simlation indicates an expected longer treatment dration. Applying or drg strategy at day to a critically ill AIDS patient (bacteria concentration approaching 4 cfml) reqires a continos drg delivery as shown in Figre 6.3. Comparison with the standard ampicillin techniqe (Figre 4. and 6.4) demonstrates a shorter recovery time (8 vs. 9.5 days) and a lower peak drg concentration (38 µgml vs. 5 µgml). In addition, or proposed treatment reqired 6% less drg administration 6

7 (6.8 g vs. 4 g) where an initial.88 g dosage followed by a continos constant drg delivery rate of.55 ghr for 8 days approximates (t). he LMIs, as stated in problem 5.7, do not explicitly garantee robstness to any model parameter variations. However, the reslting controller stabilizes a class of ncertain systems containing or nonlinear model, this provides some robstness to parameter variations that maintain the system within this class. With the controller from eqation 5.8, we observed robstness to variations in model parameters representing antibody affinity or prodction rates and somewhat robst to bacterial growth rates and antibiotic efficacy. Simlations (not shown) indicated stability for small deviations of these parameters from their nominal vales as wold be expected from the sfficient conditions stated in section 5.3 for a stabilizing controller. o explicitly inclde robstness to certain parameter variations in the controller design one mst agment (x) to inclde parameter ncertainties: specifically (x) wold become (x, δa, δw, δρ, etc.) and wold change from an 8 to an 8+n diagonal matrix (here n = nmber of ncertain parameters). his is a standard techniqe [] which we have chosen not to prse since it attaches an additi onal objective to or goal of minimizing total drg administered and frther complicates or procedre withot providing any additional insight. "ptimal" drg treatments for other illnesses similar to H. inflenzae whose dominant dynamics are captred by or predator-prey model, can be obtained from the nmerical soltion of the same eigenvale problem of 5.7 evalated with appropriate model parameter vale modifications. Similar reslts indicating continos constant drg delivery ntil antibody levels exceed inhibitory levels are expected. VII Conclsions Althogh at the crrent time it is not possible to monitor the bacteria and antibody concentrations continosly, the control techniqes presented in this paper provide a benchmark against which other strategies can be compared and indicate that an alternative continos dosage strategy is preferable to the crrently administered one of periodic dosages. In particlar, simlation reslts sggest that a constant drg concentration may be advantageos over the standard strategy de to a decrease in peak drg concentrations, decrease in treatment dration, and a decrease in total drg administered. Patients with severely sppressed immne systems reqire a longer treatment dration since their antibodies do not contribte significantly to the elimination of bacteria. De to the conservative natre of the LMI soltions, the drg levels proposed are robst to variations in the antibody affinity or prodction rates and somewhat robst to bacterial growth rates and antibiotic efficacy. xplicit robstness to parameter variations in the controller design was not prsed bt wold only reqire a few modifications []. he crrent problem formlation within the LMI framework does not flly tilize the strctre of or original nonlinear eqations. Physiologically, better tilization of the ability of the immne system to respond will allow s to frther redce the total drg delivered. Alternate techniqes which take fller advantage of the nonlinear ities (the effects of the antibodies binding to the antigen) are crrently nder investigation. VIII References [] Rai, V., V. Kmar, and L. Pandé, "A New Prey -Predator Model," I ransactions on Systems, Man and Cybernetics, Vol., No, pp. 6-63, Jan.Feb. 99. [] Boyd, S., L. l Ghaoi,. Feron, and V. Balakrishnan, Linear Matrix Ineqalities in System and Control heory, Stdies in Applied Mathematics SIAM, Vol. 5, Philadelphia, PA, 994. [3] Hoeprich, P., M. Jordan, and A. Ronald, Infectios Diseases, a reatise of Infectios Processes, Philadelphia: J.B. Lippincott Company, 5th edition, 994. [4] Schreiber, J., V. Barrs, K. Cates, and G. Siber, "Fnctional Characterization of Hman IgG, IgM, and IgA Antibody Directed to the Capsle of Haemophils inflenzae ype b," Jornal of Infectios Diseases, Vol. 53, No, pp. 8-6, Jan [5] Johnson, D., Introdction to Filter heory, New Jersey: Prentice-Hall, Inc., pp , 976. [6] Langford, P., and. Moxon, "Growth of Haemophils inflenzae ype b in the Presence of Bovine Aortal ndothelial Cells," Jornal of General Microbiology, Vol. 37, pp , 99. [7] Holmskov, U., R. Malhotra, R. Sim, and J. Jensenis, "Collectins: Collagenos C-type Lectins of the Innate Immne Defense System," Immnology oday, Vol. 5, No, pp , 994. [8] Steel, D., and W. Whitehead, "he Major Acte Phase Reactants: C-reactive Protein, Serm Amyloid P Component and Serm Amyloid A Protein," Immnology oday, Vol. 5, No, pp. 8-88, 994. [9] Drg Information Handbook, ed. Charles Lacy, L. Armstrong, R. Lipsy, and L. Lance, nd dition, Cleveland: Lexi-comp Inc.,

8 C D = B G F [] l Ghaoi, L., and G. Scorletti, "Control of Rational Systems sing Linear-Fractional Representations and Linear Matrix Ineqalities," Atomatica, Vol. 3, No 9, Sept [] Welch, D. R., Clinical Microbiology Laboratories, University Hospitals, klahoma City, Personal Commniqé, May 996. [] Roitt, I., Brostoff, J., and D. Male, Immnology, Chicago: Mosby, 3rd d., 993. [3] Griswold, W., A. Lcas, J. Bastian, and G. Garcia, "Fnctional Affinity of Antibody to the Haemophils inflenzae ype b Polysaccharide," he Jornal of Infectios Diseases, Vol. 59, No 6, pp , Jne 989. [4] Bell, G. I., " Predator-Prey qations Simlating an Immne Response", Mathematical BioSciences, Vol. 6, pp. 9-34, 973. [5] Conn's Crrent herapy, ed. Robert. Rakel, Philadelphia: W. B. Sanders Company, 996. [6] Gahart, B., Intravenos Medications, 9th dition, Boston: Mosby Year Book, 993. [7] Hiriart-Urrty J-B. and C. Lemaréchal, Convex Analysis and Minimization Algorithms I & II, Grndlehren der Mathematischen Wissenschaften, Vol. 35,36, Springer - Verlag, 993. [8] Nesterov, Y., and A. Nemirovsky, Interior-point Polynomial Methods in Convex Programming, Stdies in Applied Mathematics SIAM, Vol. 3, Philadelphia, PA, 994. [9] LMI Control oolbox, he MathWorks, Inc., 995. [] l Ghaoi, L., F. Delebecqe, and R. Nikokhah, LMIL: A User-friendly Interface for LMI ptimization. NSAINRIA, 995. Software available via anonymos FP from ftp.inria.fr, nder directory pbelghaoilmitool. [] S. Boyd and C. Barratt, Linear Controller Design: Limits of Performance, Prentice-Hall, 99. []. Mangasarian, Nonlinear Programming, McGraw- Hill, 969. Reprinted in 994 in SIAM Classics in Applied Mathematics series. [3] A. Packard, Gain Schedling via Linear Fractional ransformations, System and Control Letters, Vol., pp. 79-9, 994. [5] P. Apkarian, and P. Gahinet, A convex characterization of Gain-Schedled H Controllers, I ransactions on Atomatic Control, Vol. 4, No. 5, pp , 995. [6] V. Balakrishnan and S. Boyd, Global ptimization in Control System Analysis and Design, Control and Dynamic Systems: Advances in heory and Applicati ons, ed. C.. Leondes, New York, New York: Academic Press, Vol. 53, 99. Appendix I: Model Parameters a =.9hr, b =.5x -5 mlcfhr, a =.54x -3 hr, = 4.x 3 cfµg, w =.37hr, d =.58 µgml, ρ = 8x, (AIDS 8x 4 )hr, r = 8x -5 µgml, α =.3 mlµghr, A * = 5µgml, A eq =.455x -7 µgml, τ = 48 hr, A() = A eq, B() =.67 cfml, K = [ ], K = [-8.47, 8.47, -.7,, 8.47,,,, ] Appendix II: Parameters for Finite Dimensional Delay Approximation ρ =.4x (AIDS.5x 8 )hr, r =. (AIDS.) µgml, A d is a standard companion with bottom row [-.978x -5, -9.49x -4, -.95, -.83], B d = [.978x -5 ] B (t) same as 3.a :9; < A( t) A (t) = x + ( t τ ) a * A (A(t) A eq ) w( A( t) Aeq ) B( t) d + B( t) + A( t) A ( eq ) h(t) =?>@ A ρa( t) B( t) r + A( t) + B( t) Appendix III: Matrices for Linear Fractional Representation A m = a A eq a * A ρb4 a a a a ( r + Aeq ) B = [-α ] 4d 3d d d [4] W. M. L and K. Zho and J.C. Doyle, Stabilization of LF systems, Proceedings of the 3 th CDC Conference, pp 39-44, 99. 8

9 H J J J J J J IJ M N Q P L K R S U B p = C q = w w b w w w w * A ρ ρ ρ b4 b4 b4 d ( r + A eq ) ( r + A eq ) r + A eq D qp = d d d ( r + A eq ) ( r + Aeq ) d d d r + Aeq r + Aeq V 9

10 5 Figre 3.: Haemophils Inflenzae Infection of Healthy Patient B(t) (cfml) and A(t) (microgramml) 5 5 Bacteria Antibodies Bacteria Concentration (cfml) 4 3 Antibody Concentration (microgramml) x -4 Figre 4.: Haemophils Inflenzae Infection of AIDS Patient 5 5

11 4 Figre 4.: valation of Standard Intravenos reatment Bacteria x -6 Antibody 4 Drg Figre 5.: Comparison of Finite Dimensional Model and Delay Model 5 Bacteria (cfml) 5 5 original delay model finite dimensional model Antibody (microgramml) 5 5 finite dimensional model original delay model 4 6 8

12 Bacteria (cfml) Antibody (microgramml) Drg (microgramml) Figre 6.: Healthy and AIDS Patients with Drg Initiated Immediately x.5 5 Healthy & AIDS Healthy AIDS Healthy & AIDS Bacteria (cfml) Antibody (microgramml) Drg (microgramml) Figre 6.: Healthy and AIDS Patients with Drg Initiated at Day AIDS Healthy & AIDS Healthy Healthy AIDS 4 6 8

13 Bacteria (cfml) Drg (microgramml) 4 Antibody (microgramml) Figre 6.3: AIDS Patient with Drg Initiated at Day x x -3 Figre 6.4: Comparison of AIDS Patient Recovery Bacteria Concentration (cfml) LMI Derived reatment Standard reatment