Potential Benefits of Discrete-Time Controllerbased Treatments over Protocol-based Cancer Therapies

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1 Acta Polytechnca Hungarca Vol. 4, No., 207 Potental Benefts of Dscrete-me Controllerbase reatments over Protocol-base Cancer herapes Johanna Sáp, Dánel Anrás Drexler, Levente Kovács Research an Innovaton Center of Óbua Unversty, Physologcal Controls Group, Óbua Unversty, Kscell utca 82, H-032 Buapest, Hungary {sap.johanna, rexler.anel, Abstract: In mecal practce, the effectveness of fghtng cancer s not only etermne by the composton of the use rug, but etermne by the amnstraton metho as well. As a result, havng rugs wth a sutable acton profle s just a promsng begnnng, but wthout approprate elvery methos, the therapy stll can be neffectve. Fnng the optmal bologc ose s an emprcal process n mecal practce; however, usng controllers, an automate optmal amnstraton can be etermne. In ths paper, we evaluate the effectveness of fferent rug elvery protocols; usng n slco smulatons (lke bolus oses, low-ose metronomc regmen an contnuous nfuson therapy). In aton, we compare these results wth screte-tme controller-base treatments contanng state feeback, setpont control, actual state observer an loa estmaton. Keywors: antangogenc therapy; maxmum tolerate ose; bolus ose; low-ose metronomc regmen; contnuous nfuson therapy; optmal bologc ose; screte-tme control; state feeback; setpont control; actual state observer; loa estmaton Introucton. Bomecal Backgroun umor cells can appear n the human boy after a somatc mutaton. As tumor cells prolferate, the number of cells ncrease, an the tumor volume grows. hs growth, however, s lmte snce bloo supply s prove by the nearby capllares, an f the tumor cells grow farther than the ffuson stance (50 μm), nutrton an oxygen access ecrease. In orer to overcome ths problem, tumor cells nee ther own bloo supply. here are two man ways to form new bloo vessels. he formaton of the frst prmtve vascular plexus s calle vasculogeness, whle the formaton of new bloo vessels from the preexstng

2 J. Sáp et al. Potental Benefts of Dscrete-me Controller-base reatments over Protocol-base Cancer herapes mcrovasculature s angogeness []. In the case of tumor growth, angogeness takes place, whch s regulate by pro- an antangogenc factors. he most mportant proangogenc factor s the vascular enothelal growth factor (VEGF) snce t specfcally regulates enothelal prolferaton [2] whch s essental for angogeness. herefore, VEGF nhbton s an mportant therapeutc target [3]; an to control angogeness, ant-vegf agents an other VEGF nhbtors are beng use aroun the worl [4]. However, the best angogenc nhbton amnstraton metho s stll unknown n clncal practce [5], thus an effectve an automatc amnstraton metho s requre..2 Backgroun of the Control Problem We nvestgate a well-known tumor growth moel uner antangogenc therapy [6] an esgne several contnuous-tme controllers lke an LQ control metho an state observer [7-9], flat control [0-2], moern robust control metho [3-5], feeback lnearzaton metho [6] an aaptve fuzzy technques [7]. However, wth the current scentfc knowlege, there s no mecal evce whch can hanle contnuous nfuson cancer therapy [8]; hence we esgne a scretetme control heren. 2 umor Growth Moel P. Hahnfelt et al. create a moel whch escrbes tumor growth uner angogenc nhbton [6]. Assumng that after the njecton, the level of the nhbtor n the bloostream s equal to the amount of the njecte nhbtor, the orgnal thr-orer system was mofe to a secon-orer system: x x x ln x 2 x 2 bx x y x, 2 / 3 x 2 ex 2 g () (2) (3) where the frst state varable (x ) s the tumor volume [mm 3 ], whle the secon state varable (x 2 ) s the volume of the vasculature of the tumor [mm 3 ]. he nput of the moel s the concentraton of the njecte nhbtor (g [mg/kg]). he frst equaton contans the λ parameter whch escrbes the tumor growth rate (/ay). he change of the vasculature volume epens on three effects: a) the tumor can stmulate the alreay exstng capllares to form new bloo vessels by the process of sproutng (parameter b [/ay]), b) enothelal cell eath causes volume loss n vasculature (parameter [/(ay mm 2 ]), c) the amnstraton of antangogenc 2

3 Acta Polytechnca Hungarca Vol. 4, No., 207 rug causes volume loss n vasculature as well (parameter e [kg/(ay mg]). In the case of Lews lung carcnoma, an usng enostatn as antangogenc rug, the parameters are the followng []: λ = 0.92 /ay, b = 5.85 /ay, = /ay mm 2, e = 0.66 kg/(ay mg). 3 Protocol-base Cancer herapes 3. Cancer Protocols n the Lght of the Dosage Problem As t was scusse prevously, there s no best way for antangogenc rug amnstraton n clncal practce. here are three man methos whch are use; however, both ones have avantages an savantages. Bolus ose (BD) amnstraton means that the patent receves rug boluses on gven ays, an between the njectons, the treatment has rest peros when there s no rug amnstraton at all. he amount of njecte ose can be the Maxmum olerate Dose (MD) or any lower ose. After an MD njecton, the treatment shoul nclue an extene rest pero n orer to avo averse events. Instea of bolus oses, antcancer rugs can be elvere over prolonge peros usng low-oses, ths therapy s calle as Low-Dose Metronomc (LDM) regmen. Of course, n ths case the rest peros can be shorter; but the real queston s to fn the Optmal Bologc Dose (OBD) whch results n the best therapeutc effcacy. Fnally, n clncal envronment contnuous nfuson therapy s feasble (e.g. usng mnosmotc pumps), but there s no portable evce yet. Clncal experments have shown that low-ose amnstraton therapes have better therapeutc effcacy than bolus ose njectons, an contnuous nfuson therapes have even better results [9]. 3.2 Smulaton Results of the Protocol-base Cancer herapes he effect of bolus ose amnstraton, low-ose metronomc regmen an contnuous nfuson therapy was nvestgate n slco, usng the mofe Hahnfelt-moel escrbe by Eq. ()-(3). he total amnstere nhbtor concentraton s 300 mg/kg, an treatment pero s 5 ays n every smulaton, n orer to get comparable results. Smulatons start from the lethal steay-state of the moel when the ntal value of tumor volume (x (0)) an vascular volume (x 2 (0)) are mm 3. Four fferent scenaros were examne [20] (left se of Fgure ). 3

4 J. Sáp et al. Potental Benefts of Dscrete-me Controller-base reatments over Protocol-base Cancer herapes Fgure Protocol base therapes (treatment pero s 5 ays) a) herapy P: bolus oses wth maxmum tolerate ose (BD MD) herapy: 00 mg/kg bolus njecte for one hour; treatment ays: st, 6 th an 2 th ays; rest peros: 5 ays. otal nhbtor concentraton: 300 mg/kg, steay state tumour volume: 6330 mm 3. b) herapy P2: bolus oses (BD) herapy: 20 mg/kg bolus njecte for one hour; treatment ays: every ay of the therapy; rest peros: 23 hours. otal nhbtor concentraton: 300 mg/kg, steay state tumour volume: 5580 mm 3. c) herapy P3: low-ose metronomc regmen (LDM) herapy: 2.5 mg/kg nfuson amnstere for one ay; treatment ays: st, 4 th,7 th, 0 th, 3 th ays; rest peros: 2 ays. otal nhbtor concentraton: 300 mg/kg, steay state tumour volume: 5660 mm 3. ) herapy P4: contnuous nfuson therapy (cont) herapy: mg/kg/h contnuous nfuson amnstraton urng the whole therapy; wthout rest peros. otal nhbtor concentraton: 300 mg/kg, steay state tumour volume: 5360 mm 3. herapy P (BD MD). he therapy usng bolus oses wth maxmum tolerate ose contans 00 mg/kg boluses whch are njecte for one hour. reatment ays are the st, 6 th an 2 th ays (3 tmes); between these ays, the therapy contans 5 ays long rest peros. herapy P2 (BD). In ths case lower bolus oses are use than the maxmum tolerate ose. 20 mg/kg bolus s njecte for one hour every ay of the therapy (5 tmes). he treatment contans 23 hour rest peros. herapy P3 (LDM). Low-ose metronomc regmen s carre out wth 2.5 mg/kg nfusons whch are amnstere for one ay. reatment ays are the st, 4 th,7 th, 0 th an 3 th ays (5 tmes). he therapy contans 2 ay rest peros. 4

5 Acta Polytechnca Hungarca Vol. 4, No., 207 herapy P4 (cont). Contnuous nfuson therapy s carre out wth mg/kg/h contnuous nfuson urng the whole treatment, wthout rest peros. he rght se of Fgure epcts the outputs of the tumor growth moel, usng herapy P - herapy P4 as nputs. Smlarly to the clncal expermental results, smulatons show that the less effectve therapy s the bolus oses wth maxmum tolerate ose (BD MD). umor volume reucton s not effectve (steay state tumor volume s 6330 mm 3 ), an bese ths, se-effects can occur an qualty of lfe (QoL) of the patent ecreases ue to the therapy. Lower bolus oses (BD) cause contnuous slght reucton of the tumor volume, however ths s not sgnfcant (steay state tumor volume s 5580 mm 3 ). Another savantage of ths metho s the resultng hgh frequency oscllaton-lke characterstcs of the vascular volume. Low-ose metronomc amnstraton (LDM) has smlar results as BD n terms of tumor volume reucton (steay state tumor volume s 5660 mm 3 ) an oscllaton-lke characterstcs of the vascular volume; however, the oscllaton frequency an ampltue are lower whch can be more tolerable for the patent. he most effectve treatment s the contnuous nfuson therapy (cont) snce t results n the lower steay state tumor volume (5360 mm 3 ) an the change of the vascular volume s a smooth curve. In aton, ue to the extremely low osage, contnuous nfuson therapy has vrtually no se-effects. 4 Dscrete-me Controller-base reatments he mofe Hahnfelt-moel escrbes a nonlnear system, whch has to be lnearze ue to controller esgn aspects. We apple operatng pont lnearzaton n the g 0 = 0 operatng pont. he resultng LI (lnear tme nvarant) system usng state space representaton s x Ax Bu y Cx Du, where the matrces are (4) (5) x log A x2 2 b x 3 x2 3 x x x (6) 5

6 J. Sáp et al. Potental Benefts of Dscrete-me Controller-base reatments over Protocol-base Cancer herapes 0 B ex C D (7) (8) (9) 4. Dscrete-me Controller Desgn wth State Feeback, Setpont Control, Actual State Observer an Loa Estmaton akng nto account a feasble screte-tme system, the state space equatons are x A x B u y Cx. he controllablty an observablty matrces of the screte-tme system are M M C O [ B C CA... CA A B n,... A n B ] where n s the menson of the state varables. Snce for every nonzero operatng pont, the matrces are full rank, the system s controllable an observable n every operatng pont. In orer to fn optmal solutons, we use the LQ control metho as state feeback to mnmze the tumor volume (x ) usng the lowest possble control sgnal. he screte-tme cost functon contanng the postve efnte Q an R weghtng matrces s x Qx u Ru. (0) () (2) (3) J( u) (4) As our am was to mnmze the square of the output (x 2 = y 2 ), the Q matrx s the followng: Q C C. (5) 6

7 Acta Polytechnca Hungarca Vol. 4, No., 207 he sought K feeback matrx of the screte-tme LQ problem can be foun usng the P soluton of the Dscrete Control Algebrac Rcatt Equaton (DARE): R B PB B PA A PA A PB R B PB B PA Q. K (6) P For setpont control, we assume that the reference sgnal s constant. he control structure s neee to be extene by two matrces (N x an N u ) n orer to use nonzero reference sgnal. he values of these matrces can be calculate as follows: N N x u A I C B 0 0 I nxm m, where n s the menson of the state varables, whle m s the menson of the nputs (an outputs). As the vascular volume s non-measurable, we esgne an actual state observer to estmate ths state varable. We have verfe that the matrx M o A s full rank, thus the screte-tme system s observable wth an actual observer escrbe by the followng fference equaton: x Fxˆ Gy Hu. (9) ˆ he F, H an G parameter matrces of the observer can be calculate as follows: F A GCA G H B GCB e M A, A C A, n c F where φ F (A ) refers to the characterstc polynomal of the matrx F evaluate at the matrx A. Assumng that a sturbance reuce to the nput of the system can occur (loa change), we esgne loa estmaton as well. he system was extene by the sturbance moele as a constant state-varable that as up to the nput of the orgnal moel. he state feeback an the setpont control were esgne for the orgnal system; however, the actual state observer was esgne for the extene system. As a consequence, the fference equaton of the state observer s xˆ xˆ ~ xˆ F xˆ ~ Gy ~ Hu where xˆ s the estmaton of the sturbance., (7) (8) (20) (2) (22) (23) 7

8 J. Sáp et al. Potental Benefts of Dscrete-me Controller-base reatments over Protocol-base Cancer herapes Fgure 2 epcts the whole block agram of the close-loop screte-tme control system contanng state feeback, setpont control, actual state observer an loa estmaton. Please note that saturaton s use before the nput of the tumor moel n orer to avo negatve or too hgh nput values ue to physologcal aspects. Fgure 2 Block agram of the screte-tme control contanng state feeback, setpont control, actual state observer an loa estmaton 4.2 Smulaton Results of the Dscrete-me Controller-base reatments Usng screte-tme controller, the treatment can contan bolus oses, low-ose metronomc parts an contnuous peros as well. In orer to get comparable results wth the protocol base therapes, the treatment pero was chosen to be 5 ays. Parameters of the screte-tme controllers were chosen accorng to [2]. he operatng pont of the lnearzaton s x = x 2 =0 mm 3, the R weghtng matrx use n the esgn of the LQ control s. In orer to get steay state tumor volumes close the protocol base cancer therapes values, the reference sgnal s 3000 mm 3. Snce protocol base cancer therapes o not have sturbance, the sturbance s 0% n the case of screte-tme controllers. hree fferent scenaros were examne n the lght of the saturaton level (left se of Fgure 3). 8

9 Acta Polytechnca Hungarca Vol. 4, No., 207 Fgure 3 Dscrete-tme controller base therapes (treatment pero s 5 ays) Parameters: operatng pont: 0 mm 3 ; R: ; reference sgnal: 3000 mm 3 ; sturbance: 0%. a) herapy C: saturaton = 00 mg/kg otal nhbtor concentraton: 38 mg/kg, steay state tumour volume: 9870 mm 3. b) herapy C2: saturaton = 20 mg/kg otal nhbtor concentraton: 56 mg/kg, steay state tumour volume: 360 mm 3. c) herapy C3: saturaton = 2.5 mg/kg otal nhbtor concentraton: 30 mg/kg, steay state tumour volume: 353 mm 3. herapy C (sat = 00). he saturaton level was chosen to be the maxmum tolerate ose (herapy P). he control sgnal mostly contans MD boluses; the amnstere boluses have lower ampltue only n a few cases. he treatment contans 3 rest peros, the longer one s approxmately 6.5 ays an t appears n the mle of the therapy. herapy C2 (sat = 20). hs therapy has the same saturaton level as herapy P2 (BD). Due to the lower saturaton level compare to herapy C, ths treatment has shorter rest peros; however, the characterstcs of the treatments are smlar. In the begnnng of the therapy, bolus oses follow each other frequently for approxmately 7.5 ays, an n some cases the boluses are smaller than the level of saturaton. herapy C3 (sat = 2.5). Fnally, the saturaton level was chosen to be equal to the nput of the contnuous nfuson therapy (herapy P3). he resultng treatment contans only one rest pero n the very begnnng of the treatment. After that a contnuous amnstraton can be obtane for approxmately 8 ays, whch s 9

10 J. Sáp et al. Potental Benefts of Dscrete-me Controller-base reatments over Protocol-base Cancer herapes followe by a phase where bolus oses follow each other frequently (the ampltue of these boluses s the saturaton level n every case). he rght se of Fgure 3 epcts the outputs of the tumor growth moel, usng herapy C - herapy C3 as nputs. Usng herapy C, the total nhbtor concentraton s 38 mg/kg, whch s the hghest total rug amnstraton among the screte-tme controller base therapes. he acheve steay state tumor volume s 9870 mm 3 (snce at the en of the treatment, an unershoot can be observe). he total nhbtor concentraton n herapy C2 s 56 mg/kg, whch s substantally lower n comparson wth herapy C; however the steay state tumor volume s comparable (360 mm 3 ). Fnally, herapy C3 has resulte n the lowest total nhbtor concentraton (30 mg/kg), an the acheve steay state tumor volume s 353 mm 3 n ths case. Conclusons he effcacy of the therapes are compare an evaluate base on the acheve total nhbtor concentratons an steay state tumor volumes (Fgure 4). Durng the protocol base therapes, the same amount of nhbtor was amnstere n total. As a consequence, the comparson s qute trval: the smaller the steay state tumor volume, the better the therapy. Bolus oses wth maxmum tolerate ose (BD MD) s the less effectve treatment; bolus oses wth lower boluses (BD) an low-ose metronomc regmen (LDM) are better; however, the best metho s the contnuous nfuson therapy (cont) from the protocol base therapes. Nevertheless, screte-tme controller base therapes show better performance regarless of the saturaton value. he choce between these therapes epens on the mecal preferences an constrants. Havng a patent who can tolerate MD, an knowng that the am s the fastest tumor reucton, we have to choose 00 mg/kg saturaton (sat = 00). If we woul lke to fn a trae-off soluton, 20 mg/kg saturaton (sat = 20) s the most approprate choce. However, f slower tumor reucton s esre an/or patent oes not tolerate the nhbtor well, our choce s the 2.5 mg/kg saturaton level (sat = 20). In fact, n most of the cases the output of the tumor growth moel oes not reach the steay state at the en of the smulaton; however, as we woul lke to express the effectveness of the control n terms of tumor reucton, we use the steay state for the fnal state of the nvestgate control an we specfy ts value. 20

11 Acta Polytechnca Hungarca Vol. 4, No., 207 Fgure 4 Comparson of the therapes as functons of total nhbtor concentraton an steay state tumour volume Protocol base therapes: bolus oses wth maxmum tolerate ose (BD MD), bolus oses (BD), low-ose metronomc regmen (LDM), contnuous nfuson therapy (cont). Dscrete controller base Acknowlegement therapes: saturaton: 00 mg/kg (sat = 00), saturaton: 20 mg/kg (sat = 20), saturaton: 2.5 mg/kg (sat = 2.5). hs project has receve funng from the European Research Councl (ERC) uner the European Unon s Horzon 2020 research an nnovaton programme (grant agreement No 67968). References [] J. H. Dstler, A. Hrth, M. Kurowska-Stolarska, R. E. Gay, S. Gay, O. Dstler, Angogenc an Angostatc Factors n the Molecular Control of Angogeness, he Quarterly Journal of Nuclear Mecne, Vol. 47(3), pp. 49-6, 2003 [2] N. Ferrara, Vascular Enothelal Growth Factor an the Regulaton of Angogeness, Recent Prog Horm Res, Vol. 55, pp. 5-35, scusson 35-36, 2000 [3] A. L. Harrs, Angogeness as a New arget for Cancer Control, European Journal of Cancer Supplements, Vol., pp. -2,

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