Discrete time state feedback with setpoint control, actual state observer and load estimation for a tumor growth model

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1 th IEEE Internatonal Symposum on Apple Computatonal Intellgence an Informatcs May -4, 06 mşoara, Romana Dscrete tme state feeback wth setpont control, actual state observer an loa estmaton for a tumor growth moel Johanna Sáp *, Dánel Anrás Dreler **, Levente Kovács * * Research an Innovaton Center of Óbua Unversty, Physologcal Controls Group, Óbua Unversty, Buapest, Hungary Emal: {sap.johanna, kovacs.levente}@nk.un-obua.hu ** Department of Control Engneerng an Informaton echnology, Buapest Unversty of echnology an Economcs, Buapest, Hungary Emal: reler@t.bme.hu Abstract Wth the current scentfc knowlege, there s no mecal evce whch can hanle contnuous nfuson cancer therapy. Hence, only a quas contnuous nfuson therapy can be acheve usng screte rug amnstraton (whch can be suffcently frequent). herefore, a screte tme control has to be esgne for a tumor growth moel for real-lfe applcaton. We esgne state feeback control, augmente wth setpont control to follow nonzero reference sgnal, an also augmente wth actual state observer ue to the fact that we are unable to measure all states of the system. In aton, the control system contans loa estmaton as well to nvestgate the effect of sturbance on the nput of the moel. I. INRODUCION Nowaays n cancer treatment there s a we scale of avalable therapes. A recent treatment type s calle as argete Molecular herapes (Ms) whch am to fght rectly aganst specfc, entfe cancer mechansms. A promsng target metho s to nhbt tumor vascularzaton because f we can cease the process of angogeness (new bloo vessel formaton), tumor growth s lmte. hs leas to a new approach where the pont s not to elmnate the whole cancer but to keep t n a controlle steay state. We nvestgate a well-known tumor growth moel uner antangogenc therapy [] an esgne several contnuous tme controllers lke LQ control metho an state observer [-4], flat control [5-7], moern robust control metho [8-0], feeback lnearzaton metho [], an aaptve fuzzy technques []. However, wth the current scentfc knowlege, there s no mecal evce whch can hanle contnuous nfuson cancer therapy [3]; hence we orente n the current research work on screte tme control. he paper s organze as follows. In Secton II, we present the nonlnear moel of tumor growth uner angogenc nhbton, an escrbe a lnear moel whch s acqure by workng pont lnearzaton. Secton III contans the escrpton of the esgn structure nclung state feeback, setpont control, actual state observer an loa estmaton. In Secton IV, we present the smulaton results. he paper ens wth the concluson n Secton V. II. HE APPLIED MODEL OF UMOR GROWH P. Hahnfelt et al. create a ynamc moel for tumor growth uner antangogenc therapy []. In ther eperment mce were njecte wth Lews lung carcnoma cells an they have nvestgate the effect of three fferent angogenc nhbtors (angostatn, enostatn an NP-470). he orgnal moel was analyze an mofe n several stues [4-6]. he most mportant alteraton s the contnuous nfuson therapy [5], where the nput (the nhbtor amnstraton rate) s equal to the concentraton of amnstere nhbtor (serum level of nhbtor). he moel whch takes nto account the contnuous nfuson therapy s the followng secon-orer system: = λ ln () / 3 = b e g () y =, (3) where s the tumor volume (mm 3 ), s the enothelal/ vascular volume (mm 3 ) an g s the concentraton of the amnstere nhbtor (mg/kg). he moel contans the followng parameters: λ s the tumor growth rate (/ay), b s the stmulatory capacty of the tumor to the vasculature (/ay), s the enogenous nhbton of prevously generate vasculature (/(ay mm )), e s the antangogenc effect of the amnstere nhbtor on the tumor vasculature (kg/(ay mg)). Parameter values for the consere Lews lung carcnoma an the mce use n the eperment are []: λ = 0.9 /ay, b = 5.85 /ay, = /ay mm. he eperment has shown that the most effectve nhbtor was enostatn; therefore, we have apple ths antangogenc rug n controller esgn (e enostatn = 0.66 kg/(ay mg)). From equaton () t s clear that the system s n steaystate when tumor an vascular volumes are equal. umor growth wthout antangogenc therapy leas to hgh steay-state tumor volume (.734ˇ0 4 mm 3 ) an t represents the lethal steay-state case /6/$ IEEE

2 J. Sáp et al. Dscrete me State Feeback wth Setpont Control, Actual State Observer an Loa Estmaton for a umor Growth Moel Snce state feeback control esgn requres lnear moel, we have apple operatng pont lnearzaton n the g 0 = 0 workng pont. he matrces of the lnear moel are: λ log λ A = 3 b 3 0 B = e C = D = [ 0] [] 0 λ 3 (4) (5) (6) (7) III. CONROLLER DESIGN In the case of screte tme controller esgn, the close loop contans a DAC (gtal to analog converter) rght before the tumor moel n the feeforwar branch, whch s moele by a zero-orer hol. In the feeback branch, rght after the moel, an ADC (analog to gtal converter) can be foun (Fg.). he whole controller structure was esgne for the lnearze an scretze tumor growth moel; however, the smulatons were carre out wth the orgnal nonlnear contnuous moel. Controller esgn was eecute n Matlab (R009b). A. Samplng tme, observablty an controllablty of the lnearze screte moel Samplng tme ( ) was chosen to fulfll the contons of Shannon theorem for every sgnal of the accelerate system. akng nto account a screte tme moel whch can be represente by the followng state space an output equatons: + = A + Bu (8) y = C, (9) the controllablty an observablty matrces are as follows: n M = [ B A B... A B ] (0) C C CA M O =, ()... n CA where n s the menson of the state varables. o fulfll the contons of controllablty an observablty, M C an M O have to be full rank (rank M C = n = m, rank M O = n = m ). he matrces are full rank for every nonzero operatng pont; thus, the system s controllable an observable. B. State feeback he general state feeback law s gven by : u =, () K where K can be calculate base on pole placement or LQ control metho. he state equaton of the close loop system usng state feeback s: + = ( A B K ). (3) In the case of pole placement, the feeback matr K can be etermne by the Ackermann s formula,.e. K PP = en M C ϕclose ( A ), (4) where e n s the nth unt vector an φ close (A) s the characterstc polynomal of the close loop evaluate at the matr A. he poles of the close loop are etermne by acceleratng the poles of the lnear contnuous moel an then usng Z-transform metho (pole zero mappng): e s z =, (5) where z s the screte tme pole, s s the contnuous tme pole. LQ control metho ams to mnmze the tumor volume ( ) usng the least possble control sgnal. he screte tme cost functon whch has to be mnmze wth the constrant (8) s the followng: J ( u) = { Q + u Ru }, (6) = where Q an R are postve efnte weghtng matrces. We chose to mnmze the square of the output ( = y ), thus the Q weghtng matr s: Q = C C. (7) he feeback matr K for the screte tme LQ problem can be calculate by the formula: K LQ ( R + B PB ) B PA, = (8) where P s the soluton of the Dscrete Control Algebrac Rcatt Equaton (DARE): P = A PA A PB R + B PB B PA (9) ( )( ) ( ) Q + C. Setpont control he soluton of the state equaton n steay state s characterze by the equvalence of the prevous an actual states: = A + B u (0) y = C (). akng nto account a constant reference sgnal, two matrces (N an N u ) are neee to eten the control structure for setpont control. For zero steay state error on the output, the followng equatons nee to be satsfe: = N r () y = r u = N ur. (4) Conserng on the one han (), () an (3) equatons, one can epress: CN = I m, (5) where m y = m r = m u = m. On the other han, substtutng () an (4) nto (0) leas to the: (3)

3 th IEEE Internatonal Symposum on Apple Computatonal Intellgence an Informatcs May -4, 06 mşoara, Romana ( A I ) N + B N =. (6) u 0 nm Fgure. Block agram of the screte tme control contanng state feeback, setpont control, actual state observer an loa estmaton. Fnally, equaton (5) an (6) can be wrtten n matr equaton form that epresses the vector whch contans the requre N an N u matrces: N N u A I = C D. Actual state observer B 0 0 I nm m. (7) In screte tme, an actual state observer can be use to estmate the non-measurable state varables. Let us conser that the matr M o A s full rank,.e. the screte tme system s observable wth an actual observer. In ths case we can choose an actual state observer whch s escrbe by the fference equaton: Fˆ + Gy Hu. (8) ˆ = + Let ~ be the error of estmaton: ~ = ˆ, (9) then ~ ( ˆ = F ) + ( B GCB H ) u + (30) + ( A GCA F ). In orer to assure ~ 0, we choose the followng parameters for the actual state observer: H = B GCB (3) ~ = ~ F. (33) Fnally, gan G can be calculate usng the Ackermann s formula by substtutng A :=A an B :=A C nto (4) an usng the prescrbe poles of the observer to polynomal n (4). efne the characterstc F = A GCA (3) E. Loa estmaton We assume that the sturbance s reuce to the nput of the system (loa change) an has a constant value. Consequently the fferental equaton of the sturbance s: + =. (34) Etenng the system wth the state varable of the sturbance ( ) an usng the notaton ~ = ( ),, the state equaton becomes: + A B B = + u + 0 I 0 ~ ~ ~ = A ~ + B u + y = [ C 0] y = C ~~. (35.a) (35.b) (36.a) (36.b) he state feeback an the setpont control s esgne for the orgnal system; however, the actual state observer has to calculate not only the estmaton of the state varables ( ˆ ), but the estmaton of the sturbance ( ˆ ) as well. herefore, actual state observer was esgne for the etene system [7] whose fference equaton s: ˆ ˆ ~ ~ ~ = F + Gy + Hu. (37) ˆ ˆ Fg. epcts the whole close-loop control system contanng the controller an the nonlnear system. We place saturaton between the tumor moel an the controller. he control nput has a lower lmt n orer to eclue negatve nputs, snce they have no physologcal meanng; an an upper lmt because too hgh nput coul be angerous n bologcal systems. IV. SIMULAION RESULS Smulatons were eecute n Smulnk Smulaton peros were 50 ays n all cases an enostatn was use as angogenc nhbtor. Intal value of tumor volume an enothelal volume was the steay state volume wthout control nput (.734ˇ0 4 mm 3 ). Acceleraton of the actual state observer was a o = 5. Control strateges were evaluate base on three crtera: () the total concentraton of the amnstere nhbtor urng the treatment (mg/kg), () the steay state 3

4 J. Sáp et al. Dscrete me State Feeback wth Setpont Control, Actual State Observer an Loa Estmaton for a umor Growth Moel nhbtor concentraton at the en of the treatment (mg/kg), () the steay state tumor volume at the en of the treatment (mm 3 ). We foun that operatng pont ( 0 ) has a etermnatve effect on the control snce the approprate operatng pont can be chosen only from a narrow range. Fgure 3. Effect of R weghtng matr on the evaluaton crtera (LQ control, operatng pont: 00 mm 3 ; saturaton: 5 mg/kg; reference sgnal: 5 mm 3 ; sturbance: 0%). Fgure. Effect of operatng pont on the evaluaton crtera (LQ control, saturaton: 5 mg/kg; R weghtng matr: 0; reference sgnal: 5 mm 3 ; sturbance: 0%). If 0 < 00 mm 3 or 0 > 50 mm 3, the control s not effectve. For 0 < 00 mm 3, the control nput before the saturaton (u accorng to Fg. ) s unusable because ts ampltue s hgh an t oscllates wth hgh frequency; for 0 > 50 mm 3, the 50 ays smulaton pero s not enough to reach the steay state. If 00 mm 3 < 0 < 50 mm 3, the control strategy s effectve an the smallest amount of nhbtor s neee n the case of 0 = 00 mm 3 operatng pont (Fg. ). If the gan of state feeback s calculate base on LQ control metho, the most mportant parameter to be chosen s the R weghtng matr. heoretcally large R attempts to mnmze the nput, whle small R allows hgh nputs; however, we foun that larger R values resulte n larger total nhbtor concentraton (Fg. 3). hs behavor can be eplane by the effect of saturaton. Smlarly to the operatng pont, R also has a range n whch case the control strategy s effectve ( < R < 00). Fgure 4. Effect of saturaton on the evaluaton crtera (LQ control, operatng pont: 00 mm 3 ; R weghtng matr: 0; reference sgnal: 5 mm 3 ; sturbance: 0%). he effect of saturaton was foun to be very smlar to the contnuous tme state feeback [4]. Increasng the saturaton lmt, the total concentraton of the amnstere nhbtor also ncreases, whlst the steay state nhbtor concentraton slghtly ecreases an the steay state tumor volume remans the same value (Fg. 4). It means that lower saturaton value s not only approprate ue to physologcal aspects (less se effects) an economc conseratons (better cost-effectveness), but also because of engneerng pont of vew. In real-lfe applcablty, a key queston s the prescrbe value of reference sgnal. Evently, small steay state tumor volume has smaller cytotoc an other harmful physologcal effect than larger ones; however, the problem s more complcate. In cancer treatment, we have to take nto account cost-effectveness aspect as well whch can be accomplshe n the followng way. 4

5 th IEEE Internatonal Symposum on Apple Computatonal Intellgence an Informatcs May -4, 06 mşoara, Romana Fgure 5. Effect of reference sgnal on the evaluaton crtera (LQ control, operatng pont: 00 mm 3 ; saturaton: 5 mg/kg; R weghtng matr: 0; sturbance: 0%). saturaton. One can see that the corresponng total nhbtor concentraton values vary n a large range (95 mg/kg an 555 mg/kg for -0% an +0%, respectvely). In aton, the steay state nhbtor concentraton also vares n a relatvely we range (0.3 mg/kg an 7.5 mg/kg for -0% an +0%, respectvely). Fnally, f the gan of state feeback s calculate base on pole placement, the acceleraton (a) of the poles has to be set as a parameter. If a <, the 50 ays smulaton pero s not enough to reach the steay state. Otherwse, f a > 5, the system s over-accelerate. Wthn ths range, the hgher the acceleraton, the lower the total nhbtor concentraton becomes (Fg. 7). Fg. 8 shows the reference, nput an output sgnals of the tumor growth moel n the case of LQ control. One can see that the parameters have sgnfcant effect on the nput sgnal. If the nput has large over- an unershoot, t wll cause larger total nhbtor concentraton on the one han, an on the other han t may result n se-effects for the patent. Fgure 6. Effect of sturbance on the evaluaton crtera (LQ control, operatng pont: 00 mm 3 ; saturaton: 5 mg/kg; R weghtng matr: 0; reference sgnal: 5 mm 3 ). Fgure 7. Effect of acceleraton on the evaluaton crtera (pole placement, operatng pont: 00 mm 3 ; saturaton: 5 mg/kg; reference sgnal: 5 mm 3 ; sturbance: 0%). One has to efne oncologcally homogenous groups whch contan a set of cases (e.g. a set of steay state tumor volumes) whch have nearly the same physologcal effect to the host organsm. Wthn a certan homogenous group, the most cost-effectve treatment shoul be chosen. If we can count the mm 3 < r < 0 mm 3 reference sgnal range as an oncologcally homogenous group, the selecton crteron shoul be the r whch results n the smallest total nhbtor concentraton, vz. r = 0 mm 3 (Fg. 5). he effect of sturbance () s also not neglgble. A possble type of sturbance whch s reuce to the nput of the system can be the error cause by the non-precse tumor volume measurement [3]. Fg. 6 shows the effect of the sturbance when s [-0%,+0%] of the V. CONCLUSION We esgne state feeback control, augmente wth setpont control to follow nonzero reference sgnal. State feeback was realze usng both pole placement an LQ control metho. he control structure s also augmente wth an actual state observer an loa estmaton. We eamne the effect of several parameters on the control such as operatng pont, R weghtng matr, saturaton, reference sgnal, sturbance an acceleraton. We foun that a current set of parameters can be chosen n orer to reuce total nhbtor concentraton an avo se-effect as much as possble. Further work wll focus on the nvestgaton of LPVbase moelng [8]. REFERENCES 5

6 J. Sáp et al. Dscrete me State Feeback wth Setpont Control, Actual State Observer an Loa Estmaton for a umor Growth Moel [] P. Hahnfelt, D. Pangrahy, J. Folkman, an L. Hlatky, umor evelopment uner angogenc sgnalng: A ynamcal theory of tumor growth, treatment response, an postvascular ormancy, Cancer research, vol. 59, pp , 999. [] D. A. Dreler, L. Kovács, J. Sáp, I. Harmat, an Z. Benyó, Moel-base analyss an synthess of tumor growth uner angogenc nhbton: a case stuy. IFAC WC 0 8th Worl Congress of the Internatonal Feeraton of Automatc Control, pp , August 0, Mlano, Italy. [3] J. Sáp, D. A. Dreler, I. Harmat, Z. Sáp, an L. Kovács, Lnear state-feeback control synthess of tumor growth control n antangogenc therapy, SAMI 0 0th IEEE Internatonal Symposum on Apple Machne Intellgence an Informatcs, pp , January 0, Herlany, Slovaka. [4] J. Sáp, D. A. Dreler, I. Harmat, Z. Sáp, an L. Kovács, Qualtatve analyss of tumor growth moel uner antangogenc therapy choosng the effectve operatng pont an esgn parameters for controller esgn, Optmal Control Applcatons an Methos, Artcle frst publshe onlne: 9 SEP 05, DOI: 0.00/oca.96. [5] D. A. Dreler, J. Sáp, A. Szeles, I. Harmat, A. Kovács, an L. Kovács, Flat control of tumor growth wth angogenc nhbton, SACI 0 6th IEEE Internatonal Symposum on Apple Computatonal Intellgence an Informatcs, pp , May 0, msoara, Romana. [6] D. A. Dreler, J. Sáp, A. Szeles, I. Harmat, L. Kovács, Comparson of Path rackng Flat Control an Workng Pont Lnearzaton Base Set Pont Control of umor Growth wth Angogenc Inhbton, Scentfc Bulletn of the Poltehnca Unversty of msoara, ransactons on Automatc Control an Computer Scence, vol. 57 (7):(), pp. 3 0, 0. [7] A. Szeles, D. A. Dreler, J. Sáp, I. Harmat, an L. Kovács, Stuy of Moern Control Methoologes Apple to umor Growth uner Angogenc Inhbton, IFAC WC 04 9th Worl Congress of the Internatonal Feeraton of Automatc Control, pp , August 04, Cape own, South Afrca. [8] A. Szeles, J. Sáp, D. A. Dreler, I. Harmat, Z. Sáp, an L. Kovács, Moel-base angogenc nhbton of tumor growth usng moern robust control metho, IFAC BMS 0 8th IFAC Symposum on Bologcal an Mecal Systems, pp. 3 8, August 0, Buapest, Hungary. [9] J. Sáp, D. A. Dreler, L. Kovács, Parameter optmzaton of H controller esgne for tumor growth n the lght of physologcal aspects, CINI 03 4th IEEE Internatonal Symposum on Computatonal Intellgence an Informatcs, pp. 9 4, November 03, Buapest, Hungary. [0] L. Kovács, A. Szeles, J. Sáp, D. A. Dreler, I. Ruas, I. Harmat, an Z. Sáp, Moel-base angogenc nhbton of tumor growth usng moern robust control metho, Computer Methos an Programs n Bomecne, vol. 4, pp. 98 0, 04. [] A. Szeles, D. A. Dreler, J. Sáp, I. Harmat, Z. Sáp, an L. Kovács, Moel-base Angogenc Inhbton of umor Growth usng Feeback Lnearzaton, CDC 03 5n IEEE Conference on Decson an Control, pp , December 03, Florence, Italy. [] A. Szeles, D. A. Dreler, J. Sáp, I. Harmat, an L. Kovács, Moel-base Angogenc Inhbton of umor Growth usng Aaptve Fuzzy echnques, Peroca Polytechnca: Electrcal Engneerng an Computer Scence, vol. 58:(), pp. 9 36, 04. [3] J. Sáp, L. Kovács, D.A. Dreler, P. Kocss, D. Gajár, Z. Sáp, umor Volume Estmaton an Quas-Contnuous Amnstraton for Most Effectve Bevaczumab herapy, Plos One, vol. 0:(), Paper e p, 05. [4] A. 'Onofro, an P. Cerra, A b-parametrc moel for the tumour angogeness an antangogeness therapy, Mathematcal an Computer Moellng, vol. 49, pp , 009. [5] U. Lezewcz, an H. Schätler, A synthess of optmal controls for a moel of tumor growth uner angogenc nhbtors, CDC th IEEE Conference on Decson an Control, an the European Control Conference, pp , December 005, Sevlla, Span. [6] A. 'Onofro, A. Ganolf, an A. Rocca, he ynamcs of tumour-vasculature nteracton suggests low-ose, tme-ense antangogenc scheulng, Cell Prolferaton, vol. 4, pp , 009. [7] B. Lantos, heory an Desgn of Control Systems I-II (n Hungaran), Akaéma Kaó, Buapest, 005. [8] Gy. Egner, J. K. ar, I. Ruas, an L. Kovács, LPV-base qualty nterpretatons on moelng an control of abetes, Acta Polytechnca Hungarca, vol. 3(), pp. 7 90, 06. Fgure 8. Reference, nput an output sgnals of the tumor growth moel n the case of LQ control. a) Parameters: operatng pont: 0 mm 3 ; saturaton: 3 mg/kg; R: ; reference sgnal: 0 mm 3 ; sturbance: 0%. 6

7 th IEEE Internatonal Symposum on Apple Computatonal Intellgence an Informatcs May -4, 06 mşoara, Romana otal nhbtor concentraton: 506 mg/kg, steay state nhbtor concentraton: 7.5 mg/kg, steay state tumor volume: 0 mm 3. b) Parameters: operatng pont: 0 mm 3 ; saturaton: 3 mg/kg; R: ; reference sgnal: 5 mm 3 ; sturbance: 0%. otal nhbtor concentraton: 550 mg/kg, steay state nhbtor concentraton: 7.5 mg/kg, steay state tumor volume: 5 mm 3. c) Parameters: operatng pont: 00 mm 3 ; saturaton: 5 mg/kg; R: 0; reference sgnal: 5 mm 3 ; sturbance: 0%. otal nhbtor concentraton: 580 mg/kg, steay state nhbtor concentraton: 7.3 mg/kg, steay state tumor volume: 5 mm 3. 7

8 J. Sáp et al. Dscrete me State Feeback wth Setpont Control, Actual State Observer an Loa Estmaton for a umor Growth Moel 8