Comparison of protocol based cancer therapies and discrete controller based treatments in the case of endostatin administration

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1 Comparison of protocol base cancer therapies an iscrete controller base treatments in the case of enostatin aministration Johanna Sápi *, Dániel Anrás Drexler **, Levente Kovács * * Research an Innovation Center of Óbua University, Physiological Controls Group, Óbua University, Buapest, Hungary {sapi.johanna, kovacs.levente}@nik.uni-obua.hu ** Department of Control Engineering an Information echnology, Buapest University of echnology an Economics, Buapest, Hungary rexler@iit.bme.hu Abstract In the meical practice, there are several methos to aminister anti-cancer rugs. A commonly use metho is the intermittent bolus oses (BD) aministration when the patient receives rug on given ays an the therapy has rest perios between the injections. he amount of bolus oses can be the maximum tolerate ose (MD) or less. Anti-cancer rug can be aministere in low oses over prolonge perios without extene rest perios which is calle as low-ose metronomic therapy (LDM). In aition, continuous infusion therapy is applicable within clinical environment, not yet as a portable evice. he major isavantage of these methos is the empiricism associate with etermining the optimal biologic ose (OBD). In orer to solve the problem, we have esigne iscrete-time controllers which realize automate optimal treatments. Keywors cancer therapy, protocol, iscrete-time controller, enostatin, bolus oses, maximum tolerate ose, low-ose metronomic therapy, state feeback, setpoint control, actual state observer, loa estimation I. INRODUCION Newest research stuies on cancer treatments show the importance of specific cancer therapies. he group of argete Molecular herapies (Ms) [, ] contains several ifferent therapeutic methos; however what they have in common is the aim to fight irectly against specific, ientifie cancer mechanisms. Antiangiogenic therapy [3] is a type of M, which acts by inhibiting tumor vascularization. Ceasing the process of angiogenesis (new bloo vessel formation), tumor growth is limite. here are several types of antiangiogenic rugs (also known as angiogenic inhibitors) like bevacizumab [4], angiostatin [5] an enostatin [6]. Delivery of an antiangiogenic rug is not a trivial problem since the biological effectiveness of the treatment highly epens on the applie osage. Using an empirical metho to achieve the biological effective osage, it is not guarantee that the optimal therapy will be foun; however, using close-loop control, optimal solution can be realize [7, 8]. In this paper, we iscuss the possible elivery methos in cancer therapies which use enostatin. Enostatin is able to target neovascular enothelial cells (ECs) an has the potential for antiangiogenic an antitumor activities [9]. However, it has to be taken into account that enostatin has a short half-time [0]. he paper is organize as follows. In Section II, we present the nonlinear moel of tumor growth uner angiogenic inhibition. Section III iscusses first the cancer protocols in the light of the osage problem, an then it presents the simulation results of the protocol base cancer therapies. In Section IV, first the esign aspects of the iscrete time controller with state feeback, setpoint control, actual state observer an loa estimation are summarize, an then the simulation results of the iscrete-time controller base treatments are evaluate. he paper ens with the results an iscussion in Section V. II. HE APPLIED MODEL OF UMOR GROWH P. Hahnfelt et al. create a ynamic moel for tumor growth uner antiangiogenic therapy []. In a simplifie version of this moel, continuous infusion therapy is taken into account [], where the input (the inhibitor aministration rate) is equal to the concentration of aministere inhibitor (serum level of inhibitor) an the system is escribe by the equations: x x = λ x ln x () / 3 x = bx x x ex g () y = x, (3) where x is the tumor volume (mm 3 ), x is the enothelial/ vascular volume (mm 3 ) an g is the concentration of the aministere inhibitor (mg/kg). he moel contains the following parameters: λ is the tumor growth rate (/ay), b is the stimulatory capacity of the tumor to the vasculature (/ay), is the enogenous inhibition of previously generate vasculature (/(ay mm )), e is the antiangiogenic effect of the aministere inhibitor on the tumor vasculature (kg/(ay mg)). Parameter values for the consiere Lewis lung carcinoma an the mice use in the experiment are []: λ = 0.9 /ay, b = 5.85 /ay, = /ay mm. he experiment has shown that the most effective inhibitor was enostatin; therefore, we have applie this /6/$ IEEE SMC_

2 antiangiogenic rug in controller esign (the value of parameter e for enostatin is e enostatin = 0.66 kg/(ay mg)). III. PROOCOL BASED CANCER HERAPIES A. Cancer protocols in the light of the osage problem Methos of rug aministration fall into four categories: a) bolus oses (BD), b) bolus oses with maximum tolerate ose (BD MD), c) low-ose metronomic (LDM) regimen, an ) continuous infusion therapy. In this subsection, we raw a istinction between these treatments; in aition, we lay emphasis on the possible elivery oses in the case of enostatin treatment. Using intermittent bolus oses aministration, the patient receives rug on given ays an the therapy has rest perios between the injections. he injecte amount of boluses can be the maximum tolerate ose (MD) or lower ose. he length of the rest perios epens on the amount of boluses (e.g. after MD a longer rest perio is require). he isavantage of this metho is that it involves regrowth of tumor cells, an in several cases the growth of selecte clones will be resistant to the therapy [3]. o avoi the averse events of bolus oses, anti-cancer rug can be aministere in low oses over prolonge perios without extene rest perios which is calle as low-ose metronomic therapy [4]. Avantages of LDM are its antitumor efficacy an reuce acute toxicities; however the major isavantage is the empiricism associate with etermining the optimal biologic ose (OBD) [5]. his is the most important problem oncologists are face with when they are trying to translate LDM into the clinical application; however, accoring to our previous research, this can be solve by a close-loop control [6]. Low-ose aily aministration of antiangiogenic rugs was foun to be more effective than high-oses with rest perios. In [6], the effect of low-ose bevacizumab treatment on mice an human aenocarcinoma was examine. Article [7] makes a case for low-ose enostatin treatment on human lung cancer. In this experiment, 3 mg/kg enostatin was aministere intravenously aily for 0 ays. he results len support to the view that enostatin normalizes the structure an function of tumor vasculature. he importance of tumor vasculature normalization lies on the further elivery of chemotherapeutic rugs. Low-ose antiangiogenic therapy creates the possibility of further efficient therapeutic agent use (if any) or assists the more effective irect antitumor effect of the angiogenic inhibitor [8]. Finally, continuous infusion therapy is applicable within clinical environment, but not yet as a portable evice. However, prolonge elivery of antiangiogenic rugs may be carrie out by mini-osmotic pumps or cell encapsulation systems [9, 9]. In the stuy [0], enostatin was microencapsulate into poly(lactic-coglycolic aci) (PLGA) microspheres to treat Lewis lung cancer. hey foun that in vivo releasing enostatin was biologically active an significantly inhibite the migration of enothelial cells. Several stuies examine the antitumor effect of enostatin as a function of elivery (for instance in the case of pancreatic cancer [0, ]). Continuous aministration (using micro-osmotic pumps) was foun to be more effective (97% inhibition of tumor growth) than aily bolus oses (66% inhibition of tumor growth), using the same osage (0mg/kg/ay). o sum up, there is a lot of supporting evience from the meical experiments showing that low-ose aministration of antiangiogenic rugs is more effective than high-ose elivery. Continuous rug aministration may have even better efficacy; however, long-term continuous therapy raises technical questions which are not solve yet. herefore we raw attention to the iscrete-time control as an alternative an promising solution for antiangiogenic rug elivery. B. Simulation results of the protocol base cancer therapies We simulate the effect of the ifferent meical protocols on the tumor growth moel escribe by ()-(3). he therapies were esigne to be comparable; therefore the total aministere inhibitor concentration was 300 mg/kg in all cases. reatment perio was 5 ays in every case. In the case of bolus oses with maximum tolerate ose (BD MD), 00 mg/kg bolus was aministere once a ay in the st, 6th an th ays of the treatment (3 times). he length of the rug infusion was chosen to be one hour. he rest perios were 5 ays. When bolus oses (BD) were use in lower osage, 0 mg/kg bolus was elivere once a ay every ay of the treatment (5 times). he length of the rug infusion was chosen to be one hour, similarly to the BD MD therapy. Consequently, the rest perios were 3 hours. Using low-ose metronomic regimen (LDM),.5 mg/kg infusion was elivere for one ay in the st, 4th, 7th, 0th an 3th ays of the treatment (5 times). he rest perios were ays. In the case of continuous infusion therapy, mg/kg/h continuous infusion was use uring the whole treatment, without rest perios. Fig. shows the results of the simulations. he less effective treatment was BD MD, since this therapy reuce the tumor volume in the least amount (steay state tumor volume is 6330 mm 3 ). his is consistent with the above mentione clinical experimental results. In aition, maximum tolerate oses are not just therapeutically ineffective, but may cause serious averse events for the patient, an therefore eroe the quality of life. he effectiveness of BD an LDM therapies was similar. Using aily bolus oses, the steay state tumor volume is 5580 mm 3, while low-ose metronomic therapy resulte in 5660 mm 3 steay state tumor volume. Despite the similar results relating to steay state tumor volume, the aspect of physiological loa has to be taken into account as well. For a long-term treatment, low-ose metronomic regimen can be more tolerable, an as it can be seen from Fig., ue to the smaller rug osage, vascular volume changes more slowly an to a lesser extent in LDM therapy in contrast to BD treatment. he simulation results of continuous infusion therapy (marke as cont in Fig. ) len support to the clinical experimental results, since the lowest steay state tumor volume (5360 mm 3 ) was achieve using this treatment. Other avantages of this metho are the smooth change SMC_

3 Fig.. Protocol base therapies. reatment perio: 5 ays. a) Bolus oses with maximum tolerate ose (BD MD) herapy: 00 mg/kg bolus once a ay in the st, 6th an th ays of the treatment; rest perios: 5 ays. otal inhibitor concentration: 300 mg/kg, steay state tumor volume: 6330 mm 3. b) Bolus oses (BD) herapy: 0 mg/kg bolus once a ay every ay of the treatment; rest perios: 3 hours. otal inhibitor concentration: 300 mg/kg, steay state tumor volume: 5580 mm 3. c) Low-ose metronomic regimen (LDM) herapy:.5 mg/kg infusion for one ay in the st, 4th, 7th, 0th, 3th ays of the treatment; rest perios: ays. otal inhibitor concentration: 300 mg/kg, steay state tumor volume: 5660 mm 3. ) Continuous infusion therapy (cont) herapy: mg/kg/h continuous infusion therapy uring the whole treatment; no rest perios. otal inhibitor concentration: 300 mg/kg, steay state tumor volume: 5360 mm 3. of vascular volume uring the treatment on the one han, an minimal or no sie effects on the other han. However, we woul like to emphasize once more that continuous long-term therapy is not yet available nowaays. IV. DISCREE IME CONROLLER BASED REAMENS A. Discrete time controller with state feeback, setpoint control, actual state observer an loa estimation Detaile escription of the iscrete-time controller esign can be foun in [7]. Here we give a concise summary of the esign aspects of the iscrete time controller with state feeback, setpoint control, actual state observer an loa estimation. aking into account that state feeback control esign requires linear moel, we applie operating point linearization in the g 0 = 0 working point. he matrices of the linear moel with system matrix A, input matrix B (i.e. the right-han sie of its ifferential equation is x = Ax + Bu), an output efine as y = Cx + Du are x λ log λ A = x 3 b x x 3 0 B = ex C = D = [ 0] [] 0 x λ x x 3 (4) (5) (6) (7) he whole controller structure was esigne for the linearize an iscretize tumor growth moel; however the simulations were carrie out on the original nonlinear continuous moel. Sampling time was chosen to fulfill the conitions of Shannon theorem for every signal of the accelerate (close-loop) system. he continuous-time linear moel was transforme to a iscretize moel that has equivalent step response in the sampling times, thus the ifference equation of the iscrete-time moel is SMC_

4 Fig.. Block iagram of the iscrete time control containing state feeback, setpoint control, actual state observer an loa estimation. x. k = A xk + Bu (7) + k We investigate the observability an controllability of the linearize iscrete moel, an we foun that the matrices are full rank for every nonzero operating point, thus the system is controllable an observable. State feeback control problem was solve by two ifferent control methos. In the case of pole placement, the feeback matrix K was etermine by the Ackermann s formula, i.e. KPP = en MC ( A, B ) ϕclose( A ), (8) where e n is the nth unit vector, φ close (A ) is the characteristic polynomial of the close loop evaluate at the matrix A an M c is the controllability matrix evaluate as n MC( A, B ) = ( B A B A B ). (9) he feeback matrix K was also calculate for the iscrete time LQ problem using the formula K LQ ( R + B PB ) B PA, = (0) where P is the solution of the Discrete-time Algebraic Ricatti Equation (DARE) P = A PA A PB R + B PB B PA () ( )( ) ( ) Q + where Q = C C. For setpoint control, two matrices (N x an N u ) are neee to exten the control structure (provie that the reference signal is piecewise continuous): N x A I B 0 nxm =. 0 () N u C I m Actual state observer was esigne to estimate the nonmeasurable state variables. It is escribe by the ifference equation x ˆ ˆ i = Fxi + Gyi + Hui, (3) where F = A GCA (4) H = B GCB (5) G = ( e M ( A, A C ) ϕ ( A ), n c F (6) where φ F (A ) refers to the characteristic polynomial of the matrix F evaluate at the matrix A. We also esigne loa estimation; we assume that the isturbance is reuce to the input of the system (loa change) an it is piecewise constant, so the iscrete-time moel was extene by the loa moele as a constant state-variable that as up to the input of the original moel. he state feeback an the setpoint control was esigne for the original system; however, the actual state observer was esigne for the extene system, whose ifference equation is xˆ ˆ i xi ~ ~ ~ = F + Gyi + Hu. (7) i ˆ ˆ x i x i Fig. shows the whole close-loop control system containing the controller an the nonlinear system. We place saturation between the tumor moel an the controller; in aition, control input has a lower an an upper limit ue to physiological consierations. B. Simulation results of the iscrete-time controller base treatments Discrete controller base therapies are automatic treatments which can contain bolus oses an metronomic perios as inputs. he whole treatment perios are the same as in protocol base cancer therapies, i.e. 5 ays. Each iscrete controller has the following parameters (which were chosen accoring to [7]). he operating point, where the original nonlinear system was linearize, is 0 mm 3. Controllers use LQ optimal control as a state feeback metho. he value of R weighting matrix is. he reference signal is 3000 mm 3 to get steay state tumor volume results close to the protocol base cancer therapies values. he isturbance was chosen to be 0%, since protocol base cancer therapies o not have isturbance either in the simulations. he controllers iffer only in the chosen saturation level. Fig. 3 shows the results of the simulations. Using the maximum tolerate ose as the saturation level (00 mg/kg, note that this is the same value as it was at the protocol base therapies BD MD case), there are more than one rest perios, an these perios are not locate at equal intervals. he therapy begins with a maximum bolus ose, an after that, a half ay rest perio follows. he next phase takes 3 ays; uring this perio, maximum bolus oses follow each other in about 5-0 hours. After that an approximately 6.5-ay rest perio follows. his SMC_

5 Fig. 3. Discrete controller base therapies. reatment perio: 5 ays. Parameters: operating point: 0 mm 3 ; R: ; reference signal: 3000 mm 3 ; isturbance: 0%. a) Saturation: 00 mg/kg otal inhibitor concentration: 38 mg/kg, steay state tumor volume: 9870 mm 3. b) Saturation: 0 mg/kg otal inhibitor concentration: 56 mg/kg, steay state tumor volume: 360 mm 3. c) Saturation:.5 mg/kg otal inhibitor concentration: 30 mg/kg, steay state tumor volume: 353 mm 3. ens with a 0 mg/kg bolus ose; an then a -ay perio occurs again with maximum bolus oses following each other in about 5-0 hours. he therapy ens with a.5-ay rest perio. he total inhibitor concentration is 38 mg/kg, an the achieve steay state tumor volume is 9870 mm 3. As it can be seen from Fig. 3, at the en of the 5th ay, the output value shows an unershoot; this is the explanation of the great ifference from the setpoint. In the secon case, the saturation was chosen to be 0 mg/kg (similarly to the protocol base therapies BD case). his treatment also has three rest perios; however, these rest perios are shorter compare to the maximum tolerate ose saturation case. In turn, the characteristics of the treatments are really similar: after the very first bolus ose, the shortest rest perio occurs first, which is followe by the longest bolus ose perio, an then the longest rest perio appears, which is followe by a shorter bolus ose perio, an finally, the secon longest rest perio comes. he bolus oses are not equal to the saturation value in every case (there are lower boluses). he total inhibitor concentration is 56 mg/kg, an the gaine steay state tumor volume is 360 mm 3. Since the input boluses (an hence the over-an unershoots) are not as high as the MD, the steay state tumor volume approximates the setpoint better. he thir iscrete controller has.5 mg/kg saturation limit (which is the same value as it was at the protocol base therapies LDM case). Having regar to the low possible value of the input, to reach the setpoint, the control contains only one short rest perio at the beginning of the treatment (approximately half a ay). After that a 7.5-ay prolonge perio comes where the input keeps the value of saturation. he treatment ens with a 6.5-ay long perio where maximum bolus oses follow each other in about 5 hours. he total inhibitor concentration uring the treatment is 30 mg/kg, an the achieve steay state tumor volume is 353 mm 3. V. RESULS AND DISCUSSION Comparison of the therapies as a function of total inhibitor concentration an steay state tumor volume can be seen in Fig. 4. Since the protocol base therapies were esigne to be comparable (the total aministere inhibitor concentration was 300 mg/kg in all cases), the efficacy orer can be easily set up. Bolus oses therapy with maximum tolerate ose (BD MD) is the less effective in terms of tumor volume reuction. Bolus oses therapy with lower ose (BD) an low-ose metronomic regimen (LDM) has similar results, an continuous infusion therapy reuces the tumor volume in the greatest extent. However, it is trivial from Fig. 4, that even the best protocol base therapy cannot approach the effectiveness of iscrete controller base therapies, since these SMC_

6 Fig. 4. Comparison of the therapies as a function of total inhibitor concentration an steay state tumor volume. Protocol base therapies: bolus oses with maximum tolerate ose (BD MD), bolus oses (BD), low-ose metronomic regimen (LDM), continuous infusion therapy (cont). Discrete controller base therapies: saturation: 00 mg/kg (sat = 00), saturation: 0 mg/kg (sat = 0), saturation:.5 mg/kg (sat =.5). automate treatments achieve consierably lower steay state tumor volumes while using less than half total inhibitor concentration. he choice between the automate treatments epens on the current therapeutic goal. If the goal is to reuce the tumor volume as fast as possible, an the patient tolerates MD boluses, the 00 mg/kg saturation is the most appropriate way. he 0 mg/kg saturation level is a trae-off choice: the physiological loa of the patient is less; however, the gaine steay state tumor volume is higher compare to the BD MD therapy. aking into account the above mentione vasculature normalization aspect an the minimal sie effects;.5 mg/kg saturation seems to be the best therapeutic choice in most cases. VI. ACKNOWLEDGEMEN he research was supporte by the Research an Innovation Center of Óbua University an by the European Research Council Starting Grant ERC-StG REFERENCES [] DE. Gerber, argete therapies: a new generation of cancer treatments, Am Fam Physician, vol. 77(3), pp. 3-39, 008. [] A. Bauino, argete Cancer herapy: he Next Generation of Cancer reatment, Curr Drug Discov echnol, vol. (), pp. 3-0, 05. [3] Y. 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