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Optimal PID Based Computed Torque Control of Tumor Growth Models⁎
Proceedings Proceedings of of the the 3rd 3rd IFAC IFAC Conference Conference on on Advances in Proportional-Integral-Derivative Proceedings of the 3rd IFAC Conference on Control Available online at www.sciencedirect.com Advances in Proportional-Integral-Derivative Proceedings of the 3rd IFAC Conference on Control Ghent, May Advances in Proportional-Integral-Derivative Control Ghent, Belgium, Belgium, May 9-11, 9-11, 2018 2018 Advances in Proportional-Integral-Derivative Control Ghent, Belgium, May 9-11, 2018 Ghent, Belgium, May 9-11, 2018
ScienceDirect
IFAC PapersOnLine 51-4 (2018) 900–905
Optimal PID PID Based Based Computed Computed Torque Torque Optimal Optimal PID Based Computed Torque ⋆⋆ Control of Tumor Models Optimal PID BasedGrowth Computed Torque Control of Tumor Growth Models Control of Tumor Growth Models ⋆⋆ Control of Tumor Growth Models Bence G. Czak´ o, Johanna S´ api, Levente Kov´ acs
Bence G. Czak´ o, Johanna S´ api, Levente Kov´ acs Bence G. Czak´ o, Johanna S´ api, Levente Kov´ acs Bence Controls G. Czak´ o, Johanna S´ api, LeventeInnovation, Kov´ acs Physiological Research Center, Research, Physiological Controls Research Center, Research, Innovation, and and ´ Physiological Controls Center, Research, Innovation, and Service Center, Obuda University Budapest, Hungary (email: ´ Research Service Center, Obuda University Budapest, Hungary (email: Physiological Controls Center, Research, Innovation, ´ Research [email protected], Center, Obuda University Budapest, Hungary (email:and {kovacs.levente, ´ {kovacs.levente, [email protected], Center, Obuda University Budapest, Hungary (email: [email protected], {kovacs.levente, sapi.johanna}@nik.uni-obuda.hu) sapi.johanna}@nik.uni-obuda.hu) [email protected], {kovacs.levente, sapi.johanna}@nik.uni-obuda.hu) sapi.johanna}@nik.uni-obuda.hu) Abstract: Abstract: In In the the past past few few decades decades cancer cancer research research has has delivered delivered several several new new treatment treatment options, options, Abstract: Inbe thehighly past few decadesthus cancer research has delivered several new practice. treatmentHowever, options, of which can expensive reducing its applicability in medical of which can be highly expensive thus reducing its applicability in medical practice. However, Abstract: thehighly past few decadescan cancer research has delivered several new treatmentHowever, options, of which can be expensive thus reducing applicability in medical practice. advances inIncontrol control engineering tackle thisits issue by the the use use of an an appropriate optimal advances in engineering can tackle this issue by of appropriate optimal of which can be highly expensive thus reducing its applicability in medical practice. However, advances inIn control engineering can Torque tackle Control this issue by the use PID of an appropriate optimal controller. this paper a Computed (CTC) based controller was designed controller. this paper a Computed Control (CTC) based controller was designed advances inIn engineering can Torque tackle this provide issue by the use PID of an appropriate optimal controller. In control this tumor paper a Computed Control (CTC) based PID controller was designed for growth model which an administration protocol for for the the Hahnfeldt Hahnfeldt tumor growth modelTorque which can can provide an optimal optimal administration protocol for controller. In this paper a Computed Torque Control (CTC) based PID controller was designed for theindividual Hahnfeldtpatient. tumor growth model which can providemodel an optimal administration protocol for every The paper contains the system in conjunction with the detailed every individual patient. The paper contains the system model in conjunction with the detailed for theindividual Hahnfeldt tumor growth model which can provide an optimal administration protocol for every patient. The paper contains the system in with the detailed design steps of the the controller. The control strategy was model tested byconjunction numerical simulations which design steps of controller. The control strategy was tested by numerical simulations which every individual patient. The paper contains the system model in conjunction with the detailed design control strategy tested by numerical simulations which can be be steps foundofat atthe thecontroller. end of of the theThe paper together with was the conclusions. conclusions. can found the end paper together with the design control strategy tested by numerical simulations which can be steps foundofatthe thecontroller. end of theThe paper together with was the conclusions. © 2018, IFAC (International Automaticwith Control) Hosting by Elsevier Ltd. All rights reserved. can be found at the end ofFederation the paperoftogether the conclusions. Keywords: Computed Computed Torque Torque Control, Control, PID PID Control, Control, LQR LQR tuning, tuning, Tumor Tumor Growth Growth Model, Model, Keywords: Keywords: Computed Torque Control, PID Control, LQR tuning, Tumor Growth Model, Hahnfeldt Model Model Hahnfeldt Keywords: Computed Torque Control, PID Control, LQR tuning, Tumor Growth Model, Hahnfeldt Model Hahnfeldt Model 1. In 1. INTRODUCTION INTRODUCTION In the the recent recent years years biomedical biomedical control control has has become become a a flourflour1. INTRODUCTION In the discipline recent years biomedical control has become a flourishing in engineering. Mathematical models are ishing discipline in engineering. Mathematical models are 1. are INTRODUCTION In the discipline recent years biomedical control has become a flourishing in engineering. Mathematical models are employed on physiological processes which can lead to Cancerous diseases one of the most serious illnesses on physiological processes which can lead to Cancerous diseases are one of the most serious illnesses employed ishing discipline in engineering. Mathematical models employed on physiological processes whichIonescu can lead to effective individualized treatment solutions, Ionescu et are al. Cancerous diseaseswhich are one of the most seriousrates. illnesses of modern society cause high mortality Aceffective individualized treatment solutions, et al. of modern society which cause high mortality rates. Acemployed on physiological processes which can lead to Cancerous diseaseswhich areal.one of the most seriousrates. illnesses effective individualized treatment solutions, Ionescu et al. (2011) and Ionescu et al. (2017) for example, and curing of modern society cause high mortality According to Malvezzi et (2017), approximately 1 373 500 (2011) and Ionescu et al. (2017) for example, and curing cording to Malvezzi et al. (2017), approximately 1 373 500 effective individualized treatment solutions, Ionescu et al. of modern society which cause high mortality rates. Ionescu et al. (2017) for From example, and curing cancer and is an as aa control engicording to Malvezzi et al.to (2017), approximately 373 Ac500 (2011) people in due cancer which people died died in 2017 2017 et due to cancerapproximately which is is a a 3% 3%11 increase increase is not not an exception exception as well. well. control engi(2011) and Ionescu et al. (2017) for From example, and can curing cording to to Malvezzi al.to (2017), 373 500 cancer cancer is not an exception as well. From a control engineering perspective the tumor regulation problem be people died in2012. 2017These due cancer whichcan is abe3% increase compared high numbers attributed neering perspective the tumor regulation problem can be compared to 2012. These high numbers can be attributed cancer isperspective notananoptimal exception as well. Fromaims a control engipeople diedtoin 2017These duestrategies tohigh cancer which is abe3% increase neering the tumor regulation problem can be solved using using controller which to decrease decrease compared 2012. numbers can attributed to inefficient treatment such as chemotherapy or solved an optimal controller which aims to to inefficient treatment strategies such as chemotherapy or neering perspective the tumor regulation problem can be compared to 2012. These high numbers can be attributed solved using an optimal controller which aims to decrease the size of the tumor in the shortest manner while minito inefficient treatment strategies suchimpact as chemotherapy or the size of the tumor in the shortest manner while miniradiotherapy which often often has adverse adverse on the the health health radiotherapy which has impact on solved using an optimal controller which aims to decrease to inefficient treatment strategies such as chemotherapy or the sizethe of magnitude the tumor of in input the shortest manner while minisignal the radiotherapy of of the the patient. patient.which often has adverse impact on the health mizing mizing of input signal i.e. i.e. the concentration concentration sizethe of magnitude the tumor inorder the shortest manner while miniradiotherapy mizing the magnitude of input signal i.e. the concentration of the medication. In to design such a of the patient.which often has adverse impact on the health the the the medication. Inoforder to design such a controller controller Researchers have developed developed a a number number of of new new methodmethod- of mizing magnitude input signal i.e. the concentration of the patient. of the medication. In order to design such a controller Researchers have an appropriate appropriate model model should should be be used used which which describes describes the an the Researchers have developed a number of new methodologies from which targeted molecular therapies (TMT) of the medication. In order to design such a controller an appropriate model should be used which describes the ologies from which targeted molecular therapies (TMT) tumor growth under anti-angiogenic inhibition. An imporResearchers have developed a number of new methodtumor growth under anti-angiogenic inhibition. An imporologies from which targeted molecular therapies (TMT) offers promising promising results (Charlton and Spicer Spicer (2016)). In an appropriate model should be usedinhibition. whichintroduced describes the tumor growth under anti-angiogenic An important growth model is the Hahnfeldt model by offers results (Charlton and (2016)). In ologies from which targeted molecular therapies (TMT) growth model isanti-angiogenic the Hahnfeldt inhibition. model introduced by offers promising results (Charlton and Spicer (2016)). In tant particular anti-angiogenic treatment has been a significant tumor growth under An important growth model is the Hahnfeldt model introduced by Hanhfeldt et al. (1999). While the model seems simple at particular anti-angiogenic treatment has been a significant Hanhfeldt et al. (1999). While the model seems simple at offers promising results (Charlton and Spicer (2016)). In particular anti-angiogenic treatment has been a significant advancement which which targets targets special special tumor tumor growth growth mechamecha- Hanhfeldt tant glance, growth is theWhile Hahnfeldt model introduced et model al.has (1999). the model seems simple by at advancement first it severe nonlinearities which should be first glance, it has severe nonlinearities which should be particular anti-angiogenic treatment has been a significant advancement which targets special tumor growth mechanisms. In theory the side effects are mild compared to conHanhfeldt et al. (1999). While the model seems simple at first glance, it has severe nonlinearities which should be nisms. In theory the side effects are mild compared to conhandled with care. advancement which targets special tumor growth mechahandled withitcare. nisms. In theory thewhich side effects are mild compared to conventional protocols renders the method attractive to first glance, has severe nonlinearities which should be with care. ventional protocols which renders the method attractive to handled linear nisms. theory thewhich side effects are mild compared to concontrol ventional protocols renders the method attractive to Several medicalInprofessionals professionals (Harris (2003)). handled with care. Several linear control methods methods were were proposed proposed by by various various medical (Harris (2003)). ventional protocols which renders the method attractive to Several linear control methods were proposed by various authors in order to control the volume of the tumor. In medical professionals (Harris (2003)). authors in order to control the volume of the tumor. In Besides its its promising (Harris features, anti-angiogenic treatment treatment Several linear control methods were proposed by various medical professionals (2003)). authors in order to control the volume of the tumor. In S´ a pi et al. (2015) the authors investigated several linear Besides promising features, anti-angiogenic S´ a pi et al. (2015) the authors investigated several linear Besides its promising features, anti-angiogenic treatment has numerous numerous disadvantages. Based on on Jayson Jayson et al. al. authors inincluding order tothe control theinvestigated volume of the tumor. In S´ a pi et al. (2015) authors several linear strategies pole placement and LQR design, while has disadvantages. Based et poleauthors placement and LQRseveral design, linear while Besides its drug promising features, anti-angiogenic treatment has numerous disadvantages. Based on cancer Jayson et al. strategies (2016), the has no effect on particular illnesses, S´ api et et al.including (2015) the investigated strategies including pole placement and LQR design, while (2016), the drug has no effect on particular cancer illnesses, Kovacs al. (2013) analyzed modern robust control poset including al. (2013)pole analyzed modern robust controlwhile poshas numerous disadvantages. Based on cancer Jayson etital. (2016), the has no effect particular illnesses, prostate or drug pancreatic canceronfor for example. However, is Kovacs strategies placement and LQR design, Kovacs etAlthough al. (2013)these analyzed modern robust control possibilities. controllers provided significantly prostate or pancreatic cancer example. However, it is sibilities. Although these controllers provided significantly (2016), the drug has no effect on particular cancer illnesses, prostate or noting pancreatic for example. However, it is Kovacs also worth worth thatcancer nowadays the protocol protocol is vastly vastly etAlthough al. (2013) analyzed modern robust control they posthese provided significantly better results compared to existing protocols, also noting that nowadays the is better results compared tocontrollers existing medical medical protocols, they prostate or anoting pancreatic for example. However, ital. is sibilities. also worth thatcancer nowadays thepractice protocol is vastly applied as combined treatment in (Ilic et sibilities. Although these controllers provided significantly results compared to existing medical protocols, they are exploiting the which could applied as anoting combined treatment in practice (Ilic et al. better also worth that perspective nowadays in the protocol is should vastly are not notresults exploiting the nonlinearities nonlinearities which protocols, could improve improve applied as a combined treatment practice (Ilic et al. better (2016)). From biological future research compared to existing medical they are not exploiting the nonlinearities whichNonlinear could improve (2016)). as From biological perspective future research should the overall performance of the treatment. techthe overall performance of the treatment. Nonlinear techapplied a combined treatment in practice (Ilic et al. (2016)). From biological perspectivehowever future research should scrutinise predictive biomarkers, from an engiare not exploiting the nonlinearities which could improve overall treatment. Nonlinear techniques were investigated in Czak´ oo et or scrutiniseFrom predictive biomarkers, however from anshould engi- the wereperformance investigated of in the Czak´ et al. al. (2017) (2017) or Drexler Drexler (2016)). biological perspective future scrutinise predictive biomarkers, however from an engineering standpoint standpoint a different different problem will research be discussed in niques the overall performance of the treatment. Nonlinear techniques were investigated in Czak´ o et al. (2017) or Drexler et al. (2017a) such as nonlinear model predictive control neering a problem will be discussed in al. (2017a) such as nonlinear model predictive control scrutinise predictive biomarkers, however from an engineering standpoint a different problem will be discussed in et this paper that is the treatment often infers high medical niques were investigated in Czak´ o et al. (2017) or Drexler et al. (2017a) such as nonlinear model predictive control (NMPC), robust robust fixed fixed point point transformation transformation (RFPT) (RFPT) based based this paper that is the treatment often infers high medical neering standpoint a different problem will behigh discussed in (NMPC), this paper that should is the treatment often infers medical expenses which be in order be applied al. (2017a) such as point nonlinear model predictive control (NMPC), robust fixed transformation (RFPT) based expenses which should be addressed addressed in infers order to to be medical applied et control and exact linearization in order to decrease the control and exact linearization in order to decrease the this paper that is the treatment often high expenses widely. which should be addressed in order to be applied (NMPC), robust fixed point transformation (RFPT) based control and exact linearization in order to decrease the total inhibitor concentration while improving robustness widely. total inhibitor concentration while improving robustness expenses which should be addressed in order to be applied widely. control and exact linearization in order to decrease the total inhibitor concentration while improving robustness of the controlled system. ⋆ of theinhibitor controlledconcentration system. widely. ⋆ This This project project has has received received funding funding from from the the European European Research Research total while improving robustness of the controlled system. ⋆ This project Council (ERC) under the Horizon 2020 research received funding Unions from the European In this paper, different Council (ERC) has under the European European Unions Horizon 2020 Research research of ⋆ This project In the thiscontrolled paper, aa system. different nonlinear nonlinear strategy strategy is is proposed proposed has received funding from the European Research and innovation programme (grant agreement No 679681). B.G. Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement No 679681). B.G. In this paper, a different nonlinear strategy isshares proposed which is not computationally expensive and the which is not computationally expensive and shares the Council (ERC) under the European Unions Horizon 2020 research Czak was supported by the UNKP-17-2/I. New National Excellence and innovation programme (grant agreement No 679681). B.G. In this ispaper, a of different nonlinear strategy isshares proposed Czak was supported by the UNKP-17-2/I. New National Excellence which not computationally expensive and the important traits the other controllers. Section 2 gives and innovation programme (grant agreement No 679681). B.G. important traits of the other controllers. Section 2 gives Program of the Ministry Ministry of Human Human Capacities. Czak wasof supported by the UNKP-17-2/I. New National Excellence which is not computationally expensive and shares the Program the of Capacities. important traits of the other controllers. Section 2 gives Czak wasof supported by the UNKP-17-2/I. New National Excellence Program the Ministry of Human Capacities. important traits of the other controllers. Section 2 gives Program of the Ministry of Human Capacities.
2405-8963 © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright © 2018 900 Copyright 2018 IFAC IFAC 900 Control. Peer review© under responsibility of International Federation of Automatic Copyright © 2018 IFAC 900 10.1016/j.ifacol.2018.06.109 Copyright © 2018 IFAC 900
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a brief description on the considered tumor growth model, while Section 3 and 4 presents the control concept in the context of RFPT control design. In Section 5 the proposed algorithm is validated by numerical simulations followed by conclusions and further research possibilities in Section 6.
901
Therefore, the main goal of the control algorithm is to govern the system from an arbitrary initial state (which is less or equal than the maximal value without inhibition) to a safe steady state tumor volume, preferably smaller than 10 mm3 . 3. THE FEEDBACK LINEARIZATION APPROACH
2. THE SIMPLIFIED TUMOR GROWTH MODEL In order to create a stabilizing controller, a proper tumor growth model under anti-angiogenic inhibition is indispensable. The basic model is (still) considered the work of Hanhfeldt et al. (1999) although some phenomena is not covered by it. As a result, several models were proposed by various authors including Drexler et al. (2017b) and Csercsik et al. (2017) in order to include recent pathophysiological advancements in the process model, however in this paper we are still considering the original Hahnfeldt model as the main idea is to demonstrate the applicability of the introduced control methodology in comparison with other approaches used on the same model. Based on Sapi et al. (2013) a simplification of the original Hahnfeldt model can be carried out which has the following form: (
x1 x˙1 = −λx1 ln x2 2/3
)
x˙2 = bx1 − dx1 x2 − ex2 g(t) y = x1
In order to steer the states to the desired regime, an appropriate controller should be designed. Besides, the controller should minimize the control effort so that expenses are smaller compared to current medical protocols. In this paper, a feedback linearization-based controller is utilized, with and LQR-based tuning rule. Suppose that the first equation of 1 can be expressed as: x ¨1 = −x˙ 1 λln
(
x1 x2
)
− λx˙ 1 + λ
x1 x˙ 2 x2
(2)
If x˙ 2 is substituted into (2) one can obtain the following form: (
x1 x ¨1 = −x˙ 1 λln x2
)
− λx˙ 1 + bλ
5 x21 − λdx13 − eλx1 g(t) (3) x2
By choosing a suitable input signal, x ¨1 can be linearized. Therefore, g(t) can be determined as: (1)
where x1 denotes the tumor volume (mm3 ) representing the output of the model as well, x2 is the volume of the tumor vasculature (mm3 ), λ is the growth parameter of the tumor (1/day), b is the angiogenic factor (1/day), d describes the cellular blocking mechanisms of the vasculature (1/day · mm2 ), e is the inhibition of the vasculature by the drug (kg/day · mg), and g(t) is the concentration of the administered inhibitor (mg/kg) considered the input of the model. One should be aware, that the model has a singularity at x1 = 0, x2 = 0 which implies that the tumor can not be eradicated. However, the main goal of the research is to tame the cancer by decreasing the size of the tumor to a point where it does not pose any threat to the patient health. Simulation parameters were chosen according to S´ api et al. (2015) which are presented in Table 1.
5 (x ) x21 3 1 + λx x˙ 1 λln x − bλ + λdx +u ˙ 1 1 x 2 2 g(t) = − eλx1
(4)
where u is an auxiliary control input. Note that now x ¨1 = u which is linear. Equation (4) can be further simplified to: 5 ( 1) x21 3 λ(x˙ 1 (ln x x2 + 1) − b x2 + dx1 ) + u g(t) = − eλx1
(5)
In order to govern the volume of the tumor one should consider defining the error between the desired and actual states, namely: e = x1 − xd1
(6)
where xd1 is the desired tumor volume which is determined by the control objective. Differentiating the error two times leads to:
Table 1. Simulation parameters Parameter
Value
λ b d e
0.192 5.85 0.00873 0.66
e˙ = x˙ 1 − x˙ d1
e¨ = x ¨1 − x ¨d1
(7)
One can see that the second equation contains the linear term x ¨1 which produces:
With these parameters and constant zero inhibitor administration, the final value of the state variables are x1 = x2 = 1.734 · 104 mm3 ; hence, these values will be used as initial conditions. 901
¨d1 = u − x ¨d1 = u ˆ e¨ = x ¨1 − x
(8)
If one defines the error vector as e = [e e] ˙ T the following linear system can be obtained:
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[ ] e˙ e¨
[
=
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0 1 0 0
][ ] e e˙
+
[ ] 0 1
u ˆ
xd1 , x˙ d1
e, e˙ −
which is equivalent to: (10)
Assume that a static feedback is defined in the form of u ˆ = −Ke, where K is constructed so that it minimizes the functional: ∫
Kinematic block
u
∞
eT Qe + µˆ u2 dt
(11)
0
In (11), µ > 1 penalize the control effort (note that in u, but R is now general the second term should be u ˆT Rˆ just a constant term), and Q is the error weighting matrix as follows:
Q=
[
ξ1 0 0 ξ2
]
(12)
where ξ1 and ξ2 are responsible for the tracking precision. Upon possessing A, B, Q, R, one can calculate the gain matrix K analytically by K = R−1 B T P , where P is the solution of the Ricatti equation AT P + P A − P BR−1 B T P + Q = 0. In this case, the gain matrix K is just a vector, precisely K = [k1 k2 ]. Substituting u ˆ = −Ke into (8) one can calculate u as follows: u=x ¨1 + u ˆ=x ¨1 − k2 e˙ − k1 e
(13)
The last step is to obtain x2 in (5). In this case, x2 can be calculated from the desired tumor volume as follows: x2 = xd1 exp−1
(
x˙ d1 −λxd1
)
(14)
Using these equations, a stable controller can be utilized. In each control cycle the error between the desired and actual tumor volume is computed altogether with its first derivative so that e can be obtained by (6) and (7). In the next step, the value of u is determined by (13) based on preliminary calculation of the gain vector K. By using (14), the appropriate value of x2 can be carried out which then substituted into (5) in conjunction with u results in the control signal g(t) that is applied to the plant. In the next section, a slightly robust version of the controller is presented which uses the RFPT based control technique. In theory this augmentation improves the performance of the controller; however, there is no tuning technique of the RFPT method which results in an optimal control sequence, therefore it can not be utilized by itself.
r
Deform function
x1 , x˙ 1 Delay
ˆ e˙ = Ae + B u
J=
Delay
x ¨d1
(9)
r, xn1 , x˙ n1
Inverse model
g(t)
Plant
x ¨1
Fig. 1. A schematic depiction of the control loop. mode (SM) controller, called the kinematic block. This SM controller is then connected to a deformation block that can properly manipulate the corresponding state variables which then applied on the inverse model results in a proper control input. A more detailed explanation of the method can be found in Tar et al. (2012). The original idea was applied in Czak´o et al. (2017) nevertheless it does not impose any penalty on the input signal. In this paper a slightly modified version of the RFPT controller was designed which only uses the deformation block altogether with the feedback linearization approach. This entails that the kinematic block is replaced by (13), where the gains are determined by the LQR tuning and not the original operator which is:
(
d + Λ)n+1 eint = 0 dt
(15)
where eint is the integral of the error, Λ is a controller parameter and n is the order of the control task. Note that in this case, the controller does not require the integral of the error term. Using the fact that x ¨1 = u, based on the inverse model (5), a deformation is applied to the system. In this SISO case, the deform function can be defined as: def
G(r|u) = (r + K)[1 + B tanh(A[f (r) − u])] − K G(u∗ |u) = u∗ if f (u∗ ) = u G(−K|u) = −K if r = −K
(16)
which then is iterated as rn+1 = G (rn |u). In (16), A, B and K are the control parameters, f (r) is the response of the system for the deformed input r, u∗ is the fixed point of the equation, and the role of r is to maintain the iteration. In each control cycle, the value of r is used to compute x2 by (14) that is substituted into (5) in conjunction with u. The missing x1 and x˙ 1 can be computed from r by integrating it two times. Note that if r = u∗ the input of the deform block equals with the output, so that u∗ = x ¨1 , which justifies the integration. It is also worth mentioning that the iteration entails two time delays which coincide with the step size of the simulation. On Fig. 1 a simple sketch of the control loop is presented in order to facilitate the understanding of the RFPT algorithm. For the desired trajectory, multiple prescriptions were considered. First, a heuristic tanh() function based trajectory is proposed based on Czak´o et al. (2017), which has the following general form:
4. AUGMENTED ROBUST FIXED POINT TRANSFORMATION BASED CONTROLLER The idea of the RFPT method originates from Tar et al. (2009) in which the underlying idea is to construct an inverse model of the system that is connected to a sliding 902
xd1 (t) = (−tanh(ct) + 1)(xs1 − xf1 ) + xf1
(17)
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where c > 0 is a scaling constant, xs1 is the initial value of the tumor volume and 0 < xf1 ≤ 10 denotes its final value. Differentiating the expression above twice leads to: x˙ d1 (t) = c(xs1 − xf1 )sech2 (ct)
x ¨d1 (t) = −2c2 (xf1 − xs1 )sech2 (ct)tanh(ct)
(18)
The problem with this prescription is that at t = 0 the first derivative is not zero which may cause high initial dosage levels. A remedy to this issue could be provided by an exponential function based trajectory which is defined according to Rymansaib et al. (2013) as: xd1 (t) = exp((−ct)3 )(xs1 − xf1 ) + xf1
(19)
903
ficiently. On Fig. 3. one can see, that the error for the initial control parameters was high at the beginning, but reduces to zero in finite time. If one increase the value of ξ1 , more accurate tracking can be obtained. On Fig. 4. the administration protocol can be viewed. It is notable that because the trajectory prescription has non zero derivative at time t = 0, there is a jump at the beginning of the administration. It should also be clear, that negative input is not possible (one can not remove inhibitor from the patient) which means that a saturation had to be employed to the system that limits the input signal magnitude between 0 mg/kg and 30 mg/kg. The purpose of the upper bound is to avoid high dosage profiles which could jeopardize the health of the patient. The reduction of both tumor and vasculature volume can be seen on Fig. 2. 104
and their first and second derivatives are:
Vasculature volume Tumor volume
3
3 3
= −3c exp(−c t
)(xf1
−
xs1 )t(−2
3 3
+ 3c t )
(20)
Observing the above expressions one can easily deduce that at t = 0 the derivatives are both zero which solves the problem. A third set point prescription was also defined as a constant xd1 (t) = xf1 with zero derivatives. 5. NUMERICAL SIMULATION Several simulations were conducted in order to measure the qualitative behaviour of the proposed control algorithms. The model parameters were indicated in Table 1. before, with the corresponding initial values of the tumor and its vasculature of x1 = x2 = 1.734 · 104 mm3 . Therefore, the value xs1 = 1.734 · 104 mm3 was assigned to the prescriptions in conjunction with xf1 = 1 mm3 . The scaling constant of the tanh() and exponential case were both c = 0.1. By these choices, the tumor volume reduces to a safe level in 30 days both cases. The initial controller parameters can be seen in Table 2 which was determined on the basis of numerical simulations. In order to tune the controller, the RFPT part must be adjusted first with the original operator (15) in conjunction with Λ = 1 so that the tracking error is minimal. After that the LQR parameters can be set so that they fulfill the treatment criteria. The simulation time was 100 days, and a continuous therapy was assumed. One should note, that continuous treatment is not likely to be possible because a proper feedback is not available, however it can be employed to investigate the basic properties of the controllers. Table 2. Initial controller parameters for the tanh() tracking
1.5
1
0.5
0 0
10
20
30
40
50
60
70
80
90
100
Time (days)
Fig. 2. Reduction of the volumes by using tanh() prescription. Error
Tumor volume error (mm 3 )
x ¨d1 (t)
150
100
50
0 0
10
20
30
40
50
60
70
80
90
100
Time (days)
Fig. 3. Error of the tumor volume by using tanh() prescription. 30 Inhibitor concentration 20
Concentration (mg/kg)
x˙ d1 (t) = 3c3 exp(−c3 t3 )(xf1 − xs1 )t2
Volumes (mm 3 )
2
10 0 -10 -20 -30 -40
Parameter
Value
ξ1 ξ2 µ K A B
107
0
10
20
30
40
50
60
70
80
90
100
Time (days)
1 10 7 · 1010 10−11 −1
Fig. 4. Inhibitor profile by using tanh() prescription.
The simple LQR controller was scrutinised first. Simulations showed that it could track the tanh() signal ef903
The next simulations targeted the constant reference case. Here, the desired volume was xd1 (t) = xf1 = 1 mm3 and the derivatives were both zero. Multiple simulations showed, that the best results can be obtained if one sets ξ1 = ξ2 = 1 altogether with µ ∈ [100; 1000]. By varying µ different
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Simulations showed, that the augmented RFPT controller has not proven to be useful with parameters included in Table 2. The tracking error grew significantly, while the inhibitor profile became inconsistent which can be seen on Fig. 7. It was not possible to remove the negative values with a saturation as well, because it destabilized the system. On Fig. 8. one can see that compared to the tanh() case, the error differs considerably and the corresponding reduction is shown on Fig. 9. The poor performance of the controller can be attributed to the structure of the kinematic block. Compared to the original prescription (15), the LQR based PID controller has slower convergence rate which results in higher tracking error and initial oscillations in the control signal. Unfortunately this behaviour can not be alleviated by varying the control parameters of the LQR or RFPT functions which means that the PD type structure of the kinematic block is inadequate for the RFPT controller. Vasculature volume Tumor volume
Volumes (mm 3 )
15000
10000
Concentration (mg/kg)
30 Inhibitor concentration 25
20
15
10
0
10
20
30
40
50
60
70
80
90
100
Time (days)
Fig. 6. Inhibitor profile by using constant set point tracking.
50 Inhibitor concentration
Concentration (mg/kg)
In the case of the exponential trajectory, the results were unsatisfying. While the controller could track the trajectory with minimal error, the inhibitor dosage profile was unacceptable due to high concentration levels, hence this type of trajectory is omitted.
prescription, a fully customized treatment plan can be constructed.
40 30 20 10 0 -10 -20 0
10
20
30
40
50
60
70
80
90
100
Time (days)
Fig. 7. Inhibitor profile in the case of the augmented RFPT controller. Error
Tumor volume error (mm 3 )
settling times and inhibitor profiles can be achieved, which entails that larger µ values leads to slower settling time and lower dosage profile. On Fig. 5. one can view the reduction of the tumor and vasculature volumes. It has similar characteristic as the tanh() case, but it reaches a safe level considerably faster. The corresponding inhibitor profile can be seen on Fig. 6. It should be noted that under a 100 days period the volume only reached 1.79 mm3 , which is not identical to the desired prescription. However, one should consider that under 10 mm3 tumor volume the treatment is successful, and by increasing the simulation time the error term vanishes.
500 400 300 200 100 0
5000
0
10
20
30
40
50
60
70
80
90
100
Time (days) 0 0
10
20
30
40
50
60
70
80
90
Fig. 8. Error of the tumor volume in the case of the augmented RFPT controller.
100
Time (days)
Fig. 5. Reduction of the volumes by using constant set point tracking.
104 Vasculature volume Tumor volume
Compared to other controllers, the augmented RFPT controller obviously did not perform well, however the PID controller in the constant reference case has many promising features. By varying parameter µ different control strategies can be created which may have a longer time span but uses lower dosages. This is also true for the tanh() case, where by changing the value of parameter c in the trajectory prescription leads to the same effect as before. In addition the huge similarity between the constant and tanh() inhibitor profiles implies some connection between both methods, which means that by using the tanh() 904
Volumes (mm 3 )
2
1.5
1
0.5
0 0
10
20
30
40
50
60
70
80
90
100
Time (days)
Fig. 9. Reduction of the volumes in the case of the augmented RFPT controller.
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6. CONCLUSION In this paper a control engineering based approach was presented in order to lower the medical expenses of the antiangiogenic TMT treatment. A feedback linearizationbased controller was designed, which based on the Hahnfeldt model could track various tumor volume prescriptions including set point tracking as well. A modified approach was presented as well which in theory could improve the performance of the PID controller. Simulations showed however that this augmented RFPT controller was not operated as expected. In order to be applicable, several other features of the PID controller has to be investigated. Robustness for example rises many questions regard to model parameters and measurement disturbances. Since system parameters are highly unlikely to be an accurate representation of the reality, the parameter robustness of the system is essential in order to be employed in every day practice. Discrete time control also has to be simulated due to the lack of continuous measurement, and has to compensate for the time intervals between inhibitor dosages. These effects altogether can cause the system to be unstable which leads to ineffective treatment. REFERENCES Charlton, P. and Spicer, J. (2016). Targeted therapy in cancer. Medicine, 44(1), 34–38. Csercsik, D., S´api, J., G¨onczy, T., and Kov´ acs, L. (2017). Bi-compartmental modelling of tumor and supporting vasculature growth dynamics for cancer treatment optimization purpose. In 57th IEEE Conference on Decision and Control (CDC 2017), 4698–4702. Czak´o, B.G., S´api, J., and Kov´ acs, L. (2017). Modelbased optimal control method for cancer treatment using model predictive control and robust fixed point method. In 21st IEEE International Conference on Intelligent Engineering Systems (INES 2017), 271–276. Drexler, D., J.Sapi, and Kovacs, L. (2017a). Positive control of minimal model of tumor growth with bevacizumab treatment. In 12th IEEE Conference on Industrial Electronics and Applications (ICIEA 2017), 2081 – 2084. Drexler, D.A., Sapi, J., and Kovacs, L. (2017b). A minimal model of tumor growth with angiogenic inhibition using bevacizumab. In 15th IEEE International Symposium on Applied Machine Intelligence and Informatics (SAMI 2017), 185–190. Hanhfeldt, P., Panigrahy, D., Folkman, J., and Hlatky, L. (1999). Tumor development under angiogenic signaling: a dynamical theory of tumor growth, treatment response, and postvascular dormancy. Cancer Research, 59(19), 4770–4775. Harris, A. (2003). Angiogenesis as a new target for cancer control. European Journal of Cancer Supplements, 1(2), 1–12. Ilic, I., Jankovic, S., and Ilic, M. (2016). Bevacizumab combined with chemotherapy improves survival for patients with metastatic colorectal cancer: Evidence frommeta analysis. PLoS ONE, 11(8), e0161912. Ionescu, C., Keyser, R.D., Sabatier, J., Oustaloup, A., and Levron, F. (2011). Low frequency constant-phase behavior in the respiratory impedance. Biomedical Signal Processing and Control, 6(2), 197–208. 905
905
Ionescu, C., Lopes, A., Copot, D., Machado, J., and Bates, J. (2017). The role of fractional calculus in modeling biological phenomena: A review. Communications in Nonlinear Science and Numerical Simulation, 51, 141– 159. Jayson, G.C., Kerbel, R., Ellis, L.M., and Harris, A.L. (2016). Antiangiogenic therapy in oncology: current status and future directions. The Lancet, 388, 518–529. Kovacs, L., Szeles, A., Drexler, D., Sapi, J., and Harmati, I. (2013). Model-based angiogenic inhibition of tumor growth using modern robust control method. Computer Methods and Programs in Biomedicine, 114(3), 98–110. Malvezzi, M., Carioli, G., Bertuccio, P., Boffetta, P., Levi, F., Vecchia, C.L., and Negri, E. (2017). European cancer mortality predictions for the year 2017, with focus on lung cancer. Annals of Oncology, 28, 1117–1123. Rymansaib, Z., Iravani, P., and Sahinkaya, M.N. (2013). Exponential trajectory generation for point to point motions. In IEEE/ASME International Conference on Advanced Intelligent Mechatronics, 906 – 911. S´api, J., Drexler, D.A., Harmati, I., S´api, Z., and Kov´ acs, L. (2015). Qualitative analysis of tumor growth model under antiangiogenic therapy - choosing the effective operating point and design parameters for controller design. Optimal Control Applications and Methods, 37(5), 848–866. Sapi, J., Drexler, D.A., and Kovacs, L. (2013). Parameter optimization of h infinity controller designed for tumor growth in the light of physiological aspects. In 14th IEEE International Symposium on Computational Intelligence and Informatics (CINTI 2013), 19 – 24. Tar, J.K., N´adai, L., and Rudas, I.J. (2012). System and Control Theory with Especial Emphasis on Nonlinear Systems. Typotex. Tar, J., Bit´o, J., N´adai, L., and Machado, J.T. (2009). Robust Fixed Point Transformations in adaptive control using local basin of attraction. Acta Polytechnica Hungarica, 6(1), 21–37.
ScienceDirect
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Optimal PID PID Based Based Computed Computed Torque Torque Optimal Optimal PID Based Computed Torque ⋆⋆ Control of Tumor Models Optimal PID BasedGrowth Computed Torque Control of Tumor Growth Models Control of Tumor Growth Models ⋆⋆ Control of Tumor Growth Models Bence G. Czak´ o, Johanna S´ api, Levente Kov´ acs
Bence G. Czak´ o, Johanna S´ api, Levente Kov´ acs Bence G. Czak´ o, Johanna S´ api, Levente Kov´ acs Bence Controls G. Czak´ o, Johanna S´ api, LeventeInnovation, Kov´ acs Physiological Research Center, Research, Physiological Controls Research Center, Research, Innovation, and and ´ Physiological Controls Center, Research, Innovation, and Service Center, Obuda University Budapest, Hungary (email: ´ Research Service Center, Obuda University Budapest, Hungary (email: Physiological Controls Center, Research, Innovation, ´ Research [email protected], Center, Obuda University Budapest, Hungary (email:and {kovacs.levente, ´ {kovacs.levente, [email protected], Center, Obuda University Budapest, Hungary (email: [email protected], {kovacs.levente, sapi.johanna}@nik.uni-obuda.hu) sapi.johanna}@nik.uni-obuda.hu) [email protected], {kovacs.levente, sapi.johanna}@nik.uni-obuda.hu) sapi.johanna}@nik.uni-obuda.hu) Abstract: Abstract: In In the the past past few few decades decades cancer cancer research research has has delivered delivered several several new new treatment treatment options, options, Abstract: Inbe thehighly past few decadesthus cancer research has delivered several new practice. treatmentHowever, options, of which can expensive reducing its applicability in medical of which can be highly expensive thus reducing its applicability in medical practice. However, Abstract: thehighly past few decadescan cancer research has delivered several new treatmentHowever, options, of which can be expensive thus reducing applicability in medical practice. advances inIncontrol control engineering tackle thisits issue by the the use use of an an appropriate optimal advances in engineering can tackle this issue by of appropriate optimal of which can be highly expensive thus reducing its applicability in medical practice. However, advances inIn control engineering can Torque tackle Control this issue by the use PID of an appropriate optimal controller. this paper a Computed (CTC) based controller was designed controller. this paper a Computed Control (CTC) based controller was designed advances inIn engineering can Torque tackle this provide issue by the use PID of an appropriate optimal controller. In control this tumor paper a Computed Control (CTC) based PID controller was designed for growth model which an administration protocol for for the the Hahnfeldt Hahnfeldt tumor growth modelTorque which can can provide an optimal optimal administration protocol for controller. In this paper a Computed Torque Control (CTC) based PID controller was designed for theindividual Hahnfeldtpatient. tumor growth model which can providemodel an optimal administration protocol for every The paper contains the system in conjunction with the detailed every individual patient. The paper contains the system model in conjunction with the detailed for theindividual Hahnfeldt tumor growth model which can provide an optimal administration protocol for every patient. The paper contains the system in with the detailed design steps of the the controller. The control strategy was model tested byconjunction numerical simulations which design steps of controller. The control strategy was tested by numerical simulations which every individual patient. The paper contains the system model in conjunction with the detailed design control strategy tested by numerical simulations which can be be steps foundofat atthe thecontroller. end of of the theThe paper together with was the conclusions. conclusions. can found the end paper together with the design control strategy tested by numerical simulations which can be steps foundofatthe thecontroller. end of theThe paper together with was the conclusions. © 2018, IFAC (International Automaticwith Control) Hosting by Elsevier Ltd. All rights reserved. can be found at the end ofFederation the paperoftogether the conclusions. Keywords: Computed Computed Torque Torque Control, Control, PID PID Control, Control, LQR LQR tuning, tuning, Tumor Tumor Growth Growth Model, Model, Keywords: Keywords: Computed Torque Control, PID Control, LQR tuning, Tumor Growth Model, Hahnfeldt Model Model Hahnfeldt Keywords: Computed Torque Control, PID Control, LQR tuning, Tumor Growth Model, Hahnfeldt Model Hahnfeldt Model 1. In 1. INTRODUCTION INTRODUCTION In the the recent recent years years biomedical biomedical control control has has become become a a flourflour1. INTRODUCTION In the discipline recent years biomedical control has become a flourishing in engineering. Mathematical models are ishing discipline in engineering. Mathematical models are 1. are INTRODUCTION In the discipline recent years biomedical control has become a flourishing in engineering. Mathematical models are employed on physiological processes which can lead to Cancerous diseases one of the most serious illnesses on physiological processes which can lead to Cancerous diseases are one of the most serious illnesses employed ishing discipline in engineering. Mathematical models employed on physiological processes whichIonescu can lead to effective individualized treatment solutions, Ionescu et are al. Cancerous diseaseswhich are one of the most seriousrates. illnesses of modern society cause high mortality Aceffective individualized treatment solutions, et al. of modern society which cause high mortality rates. Acemployed on physiological processes which can lead to Cancerous diseaseswhich areal.one of the most seriousrates. illnesses effective individualized treatment solutions, Ionescu et al. (2011) and Ionescu et al. (2017) for example, and curing of modern society cause high mortality According to Malvezzi et (2017), approximately 1 373 500 (2011) and Ionescu et al. (2017) for example, and curing cording to Malvezzi et al. (2017), approximately 1 373 500 effective individualized treatment solutions, Ionescu et al. of modern society which cause high mortality rates. Ionescu et al. (2017) for From example, and curing cancer and is an as aa control engicording to Malvezzi et al.to (2017), approximately 373 Ac500 (2011) people in due cancer which people died died in 2017 2017 et due to cancerapproximately which is is a a 3% 3%11 increase increase is not not an exception exception as well. well. control engi(2011) and Ionescu et al. (2017) for From example, and can curing cording to to Malvezzi al.to (2017), 373 500 cancer cancer is not an exception as well. From a control engineering perspective the tumor regulation problem be people died in2012. 2017These due cancer whichcan is abe3% increase compared high numbers attributed neering perspective the tumor regulation problem can be compared to 2012. These high numbers can be attributed cancer isperspective notananoptimal exception as well. Fromaims a control engipeople diedtoin 2017These duestrategies tohigh cancer which is abe3% increase neering the tumor regulation problem can be solved using using controller which to decrease decrease compared 2012. numbers can attributed to inefficient treatment such as chemotherapy or solved an optimal controller which aims to to inefficient treatment strategies such as chemotherapy or neering perspective the tumor regulation problem can be compared to 2012. These high numbers can be attributed solved using an optimal controller which aims to decrease the size of the tumor in the shortest manner while minito inefficient treatment strategies suchimpact as chemotherapy or the size of the tumor in the shortest manner while miniradiotherapy which often often has adverse adverse on the the health health radiotherapy which has impact on solved using an optimal controller which aims to decrease to inefficient treatment strategies such as chemotherapy or the sizethe of magnitude the tumor of in input the shortest manner while minisignal the radiotherapy of of the the patient. patient.which often has adverse impact on the health mizing mizing of input signal i.e. i.e. the concentration concentration sizethe of magnitude the tumor inorder the shortest manner while miniradiotherapy mizing the magnitude of input signal i.e. the concentration of the medication. In to design such a of the patient.which often has adverse impact on the health the the the medication. Inoforder to design such a controller controller Researchers have developed developed a a number number of of new new methodmethod- of mizing magnitude input signal i.e. the concentration of the patient. of the medication. In order to design such a controller Researchers have an appropriate appropriate model model should should be be used used which which describes describes the an the Researchers have developed a number of new methodologies from which targeted molecular therapies (TMT) of the medication. In order to design such a controller an appropriate model should be used which describes the ologies from which targeted molecular therapies (TMT) tumor growth under anti-angiogenic inhibition. An imporResearchers have developed a number of new methodtumor growth under anti-angiogenic inhibition. An imporologies from which targeted molecular therapies (TMT) offers promising promising results (Charlton and Spicer Spicer (2016)). In an appropriate model should be usedinhibition. whichintroduced describes the tumor growth under anti-angiogenic An important growth model is the Hahnfeldt model by offers results (Charlton and (2016)). In ologies from which targeted molecular therapies (TMT) growth model isanti-angiogenic the Hahnfeldt inhibition. model introduced by offers promising results (Charlton and Spicer (2016)). In tant particular anti-angiogenic treatment has been a significant tumor growth under An important growth model is the Hahnfeldt model introduced by Hanhfeldt et al. (1999). While the model seems simple at particular anti-angiogenic treatment has been a significant Hanhfeldt et al. (1999). While the model seems simple at offers promising results (Charlton and Spicer (2016)). In particular anti-angiogenic treatment has been a significant advancement which which targets targets special special tumor tumor growth growth mechamecha- Hanhfeldt tant glance, growth is theWhile Hahnfeldt model introduced et model al.has (1999). the model seems simple by at advancement first it severe nonlinearities which should be first glance, it has severe nonlinearities which should be particular anti-angiogenic treatment has been a significant advancement which targets special tumor growth mechanisms. In theory the side effects are mild compared to conHanhfeldt et al. (1999). While the model seems simple at first glance, it has severe nonlinearities which should be nisms. In theory the side effects are mild compared to conhandled with care. advancement which targets special tumor growth mechahandled withitcare. nisms. In theory thewhich side effects are mild compared to conventional protocols renders the method attractive to first glance, has severe nonlinearities which should be with care. ventional protocols which renders the method attractive to handled linear nisms. theory thewhich side effects are mild compared to concontrol ventional protocols renders the method attractive to Several medicalInprofessionals professionals (Harris (2003)). handled with care. Several linear control methods methods were were proposed proposed by by various various medical (Harris (2003)). ventional protocols which renders the method attractive to Several linear control methods were proposed by various authors in order to control the volume of the tumor. In medical professionals (Harris (2003)). authors in order to control the volume of the tumor. In Besides its its promising (Harris features, anti-angiogenic treatment treatment Several linear control methods were proposed by various medical professionals (2003)). authors in order to control the volume of the tumor. In S´ a pi et al. (2015) the authors investigated several linear Besides promising features, anti-angiogenic S´ a pi et al. (2015) the authors investigated several linear Besides its promising features, anti-angiogenic treatment has numerous numerous disadvantages. Based on on Jayson Jayson et al. al. authors inincluding order tothe control theinvestigated volume of the tumor. In S´ a pi et al. (2015) authors several linear strategies pole placement and LQR design, while has disadvantages. Based et poleauthors placement and LQRseveral design, linear while Besides its drug promising features, anti-angiogenic treatment has numerous disadvantages. Based on cancer Jayson et al. strategies (2016), the has no effect on particular illnesses, S´ api et et al.including (2015) the investigated strategies including pole placement and LQR design, while (2016), the drug has no effect on particular cancer illnesses, Kovacs al. (2013) analyzed modern robust control poset including al. (2013)pole analyzed modern robust controlwhile poshas numerous disadvantages. Based on cancer Jayson etital. (2016), the has no effect particular illnesses, prostate or drug pancreatic canceronfor for example. However, is Kovacs strategies placement and LQR design, Kovacs etAlthough al. (2013)these analyzed modern robust control possibilities. controllers provided significantly prostate or pancreatic cancer example. However, it is sibilities. Although these controllers provided significantly (2016), the drug has no effect on particular cancer illnesses, prostate or noting pancreatic for example. However, it is Kovacs also worth worth thatcancer nowadays the protocol protocol is vastly vastly etAlthough al. (2013) analyzed modern robust control they posthese provided significantly better results compared to existing protocols, also noting that nowadays the is better results compared tocontrollers existing medical medical protocols, they prostate or anoting pancreatic for example. However, ital. is sibilities. also worth thatcancer nowadays thepractice protocol is vastly applied as combined treatment in (Ilic et sibilities. Although these controllers provided significantly results compared to existing medical protocols, they are exploiting the which could applied as anoting combined treatment in practice (Ilic et al. better also worth that perspective nowadays in the protocol is should vastly are not notresults exploiting the nonlinearities nonlinearities which protocols, could improve improve applied as a combined treatment practice (Ilic et al. better (2016)). From biological future research compared to existing medical they are not exploiting the nonlinearities whichNonlinear could improve (2016)). as From biological perspective future research should the overall performance of the treatment. techthe overall performance of the treatment. Nonlinear techapplied a combined treatment in practice (Ilic et al. (2016)). From biological perspectivehowever future research should scrutinise predictive biomarkers, from an engiare not exploiting the nonlinearities which could improve overall treatment. Nonlinear techniques were investigated in Czak´ oo et or scrutiniseFrom predictive biomarkers, however from anshould engi- the wereperformance investigated of in the Czak´ et al. al. (2017) (2017) or Drexler Drexler (2016)). biological perspective future scrutinise predictive biomarkers, however from an engineering standpoint standpoint a different different problem will research be discussed in niques the overall performance of the treatment. Nonlinear techniques were investigated in Czak´ o et al. (2017) or Drexler et al. (2017a) such as nonlinear model predictive control neering a problem will be discussed in al. (2017a) such as nonlinear model predictive control scrutinise predictive biomarkers, however from an engineering standpoint a different problem will be discussed in et this paper that is the treatment often infers high medical niques were investigated in Czak´ o et al. (2017) or Drexler et al. (2017a) such as nonlinear model predictive control (NMPC), robust robust fixed fixed point point transformation transformation (RFPT) (RFPT) based based this paper that is the treatment often infers high medical neering standpoint a different problem will behigh discussed in (NMPC), this paper that should is the treatment often infers medical expenses which be in order be applied al. (2017a) such as point nonlinear model predictive control (NMPC), robust fixed transformation (RFPT) based expenses which should be addressed addressed in infers order to to be medical applied et control and exact linearization in order to decrease the control and exact linearization in order to decrease the this paper that is the treatment often high expenses widely. which should be addressed in order to be applied (NMPC), robust fixed point transformation (RFPT) based control and exact linearization in order to decrease the total inhibitor concentration while improving robustness widely. total inhibitor concentration while improving robustness expenses which should be addressed in order to be applied widely. control and exact linearization in order to decrease the total inhibitor concentration while improving robustness of the controlled system. ⋆ of theinhibitor controlledconcentration system. widely. ⋆ This This project project has has received received funding funding from from the the European European Research Research total while improving robustness of the controlled system. ⋆ This project Council (ERC) under the Horizon 2020 research received funding Unions from the European In this paper, different Council (ERC) has under the European European Unions Horizon 2020 Research research of ⋆ This project In the thiscontrolled paper, aa system. different nonlinear nonlinear strategy strategy is is proposed proposed has received funding from the European Research and innovation programme (grant agreement No 679681). B.G. Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement No 679681). B.G. In this paper, a different nonlinear strategy isshares proposed which is not computationally expensive and the which is not computationally expensive and shares the Council (ERC) under the European Unions Horizon 2020 research Czak was supported by the UNKP-17-2/I. New National Excellence and innovation programme (grant agreement No 679681). B.G. In this ispaper, a of different nonlinear strategy isshares proposed Czak was supported by the UNKP-17-2/I. New National Excellence which not computationally expensive and the important traits the other controllers. Section 2 gives and innovation programme (grant agreement No 679681). B.G. important traits of the other controllers. Section 2 gives Program of the Ministry Ministry of Human Human Capacities. Czak wasof supported by the UNKP-17-2/I. New National Excellence which is not computationally expensive and shares the Program the of Capacities. important traits of the other controllers. Section 2 gives Czak wasof supported by the UNKP-17-2/I. New National Excellence Program the Ministry of Human Capacities. important traits of the other controllers. Section 2 gives Program of the Ministry of Human Capacities.
2405-8963 © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright © 2018 900 Copyright 2018 IFAC IFAC 900 Control. Peer review© under responsibility of International Federation of Automatic Copyright © 2018 IFAC 900 10.1016/j.ifacol.2018.06.109 Copyright © 2018 IFAC 900
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a brief description on the considered tumor growth model, while Section 3 and 4 presents the control concept in the context of RFPT control design. In Section 5 the proposed algorithm is validated by numerical simulations followed by conclusions and further research possibilities in Section 6.
901
Therefore, the main goal of the control algorithm is to govern the system from an arbitrary initial state (which is less or equal than the maximal value without inhibition) to a safe steady state tumor volume, preferably smaller than 10 mm3 . 3. THE FEEDBACK LINEARIZATION APPROACH
2. THE SIMPLIFIED TUMOR GROWTH MODEL In order to create a stabilizing controller, a proper tumor growth model under anti-angiogenic inhibition is indispensable. The basic model is (still) considered the work of Hanhfeldt et al. (1999) although some phenomena is not covered by it. As a result, several models were proposed by various authors including Drexler et al. (2017b) and Csercsik et al. (2017) in order to include recent pathophysiological advancements in the process model, however in this paper we are still considering the original Hahnfeldt model as the main idea is to demonstrate the applicability of the introduced control methodology in comparison with other approaches used on the same model. Based on Sapi et al. (2013) a simplification of the original Hahnfeldt model can be carried out which has the following form: (
x1 x˙1 = −λx1 ln x2 2/3
)
x˙2 = bx1 − dx1 x2 − ex2 g(t) y = x1
In order to steer the states to the desired regime, an appropriate controller should be designed. Besides, the controller should minimize the control effort so that expenses are smaller compared to current medical protocols. In this paper, a feedback linearization-based controller is utilized, with and LQR-based tuning rule. Suppose that the first equation of 1 can be expressed as: x ¨1 = −x˙ 1 λln
(
x1 x2
)
− λx˙ 1 + λ
x1 x˙ 2 x2
(2)
If x˙ 2 is substituted into (2) one can obtain the following form: (
x1 x ¨1 = −x˙ 1 λln x2
)
− λx˙ 1 + bλ
5 x21 − λdx13 − eλx1 g(t) (3) x2
By choosing a suitable input signal, x ¨1 can be linearized. Therefore, g(t) can be determined as: (1)
where x1 denotes the tumor volume (mm3 ) representing the output of the model as well, x2 is the volume of the tumor vasculature (mm3 ), λ is the growth parameter of the tumor (1/day), b is the angiogenic factor (1/day), d describes the cellular blocking mechanisms of the vasculature (1/day · mm2 ), e is the inhibition of the vasculature by the drug (kg/day · mg), and g(t) is the concentration of the administered inhibitor (mg/kg) considered the input of the model. One should be aware, that the model has a singularity at x1 = 0, x2 = 0 which implies that the tumor can not be eradicated. However, the main goal of the research is to tame the cancer by decreasing the size of the tumor to a point where it does not pose any threat to the patient health. Simulation parameters were chosen according to S´ api et al. (2015) which are presented in Table 1.
5 (x ) x21 3 1 + λx x˙ 1 λln x − bλ + λdx +u ˙ 1 1 x 2 2 g(t) = − eλx1
(4)
where u is an auxiliary control input. Note that now x ¨1 = u which is linear. Equation (4) can be further simplified to: 5 ( 1) x21 3 λ(x˙ 1 (ln x x2 + 1) − b x2 + dx1 ) + u g(t) = − eλx1
(5)
In order to govern the volume of the tumor one should consider defining the error between the desired and actual states, namely: e = x1 − xd1
(6)
where xd1 is the desired tumor volume which is determined by the control objective. Differentiating the error two times leads to:
Table 1. Simulation parameters Parameter
Value
λ b d e
0.192 5.85 0.00873 0.66
e˙ = x˙ 1 − x˙ d1
e¨ = x ¨1 − x ¨d1
(7)
One can see that the second equation contains the linear term x ¨1 which produces:
With these parameters and constant zero inhibitor administration, the final value of the state variables are x1 = x2 = 1.734 · 104 mm3 ; hence, these values will be used as initial conditions. 901
¨d1 = u − x ¨d1 = u ˆ e¨ = x ¨1 − x
(8)
If one defines the error vector as e = [e e] ˙ T the following linear system can be obtained:
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[ ] e˙ e¨
[
=
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0 1 0 0
][ ] e e˙
+
[ ] 0 1
u ˆ
xd1 , x˙ d1
e, e˙ −
which is equivalent to: (10)
Assume that a static feedback is defined in the form of u ˆ = −Ke, where K is constructed so that it minimizes the functional: ∫
Kinematic block
u
∞
eT Qe + µˆ u2 dt
(11)
0
In (11), µ > 1 penalize the control effort (note that in u, but R is now general the second term should be u ˆT Rˆ just a constant term), and Q is the error weighting matrix as follows:
Q=
[
ξ1 0 0 ξ2
]
(12)
where ξ1 and ξ2 are responsible for the tracking precision. Upon possessing A, B, Q, R, one can calculate the gain matrix K analytically by K = R−1 B T P , where P is the solution of the Ricatti equation AT P + P A − P BR−1 B T P + Q = 0. In this case, the gain matrix K is just a vector, precisely K = [k1 k2 ]. Substituting u ˆ = −Ke into (8) one can calculate u as follows: u=x ¨1 + u ˆ=x ¨1 − k2 e˙ − k1 e
(13)
The last step is to obtain x2 in (5). In this case, x2 can be calculated from the desired tumor volume as follows: x2 = xd1 exp−1
(
x˙ d1 −λxd1
)
(14)
Using these equations, a stable controller can be utilized. In each control cycle the error between the desired and actual tumor volume is computed altogether with its first derivative so that e can be obtained by (6) and (7). In the next step, the value of u is determined by (13) based on preliminary calculation of the gain vector K. By using (14), the appropriate value of x2 can be carried out which then substituted into (5) in conjunction with u results in the control signal g(t) that is applied to the plant. In the next section, a slightly robust version of the controller is presented which uses the RFPT based control technique. In theory this augmentation improves the performance of the controller; however, there is no tuning technique of the RFPT method which results in an optimal control sequence, therefore it can not be utilized by itself.
r
Deform function
x1 , x˙ 1 Delay
ˆ e˙ = Ae + B u
J=
Delay
x ¨d1
(9)
r, xn1 , x˙ n1
Inverse model
g(t)
Plant
x ¨1
Fig. 1. A schematic depiction of the control loop. mode (SM) controller, called the kinematic block. This SM controller is then connected to a deformation block that can properly manipulate the corresponding state variables which then applied on the inverse model results in a proper control input. A more detailed explanation of the method can be found in Tar et al. (2012). The original idea was applied in Czak´o et al. (2017) nevertheless it does not impose any penalty on the input signal. In this paper a slightly modified version of the RFPT controller was designed which only uses the deformation block altogether with the feedback linearization approach. This entails that the kinematic block is replaced by (13), where the gains are determined by the LQR tuning and not the original operator which is:
(
d + Λ)n+1 eint = 0 dt
(15)
where eint is the integral of the error, Λ is a controller parameter and n is the order of the control task. Note that in this case, the controller does not require the integral of the error term. Using the fact that x ¨1 = u, based on the inverse model (5), a deformation is applied to the system. In this SISO case, the deform function can be defined as: def
G(r|u) = (r + K)[1 + B tanh(A[f (r) − u])] − K G(u∗ |u) = u∗ if f (u∗ ) = u G(−K|u) = −K if r = −K
(16)
which then is iterated as rn+1 = G (rn |u). In (16), A, B and K are the control parameters, f (r) is the response of the system for the deformed input r, u∗ is the fixed point of the equation, and the role of r is to maintain the iteration. In each control cycle, the value of r is used to compute x2 by (14) that is substituted into (5) in conjunction with u. The missing x1 and x˙ 1 can be computed from r by integrating it two times. Note that if r = u∗ the input of the deform block equals with the output, so that u∗ = x ¨1 , which justifies the integration. It is also worth mentioning that the iteration entails two time delays which coincide with the step size of the simulation. On Fig. 1 a simple sketch of the control loop is presented in order to facilitate the understanding of the RFPT algorithm. For the desired trajectory, multiple prescriptions were considered. First, a heuristic tanh() function based trajectory is proposed based on Czak´o et al. (2017), which has the following general form:
4. AUGMENTED ROBUST FIXED POINT TRANSFORMATION BASED CONTROLLER The idea of the RFPT method originates from Tar et al. (2009) in which the underlying idea is to construct an inverse model of the system that is connected to a sliding 902
xd1 (t) = (−tanh(ct) + 1)(xs1 − xf1 ) + xf1
(17)
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where c > 0 is a scaling constant, xs1 is the initial value of the tumor volume and 0 < xf1 ≤ 10 denotes its final value. Differentiating the expression above twice leads to: x˙ d1 (t) = c(xs1 − xf1 )sech2 (ct)
x ¨d1 (t) = −2c2 (xf1 − xs1 )sech2 (ct)tanh(ct)
(18)
The problem with this prescription is that at t = 0 the first derivative is not zero which may cause high initial dosage levels. A remedy to this issue could be provided by an exponential function based trajectory which is defined according to Rymansaib et al. (2013) as: xd1 (t) = exp((−ct)3 )(xs1 − xf1 ) + xf1
(19)
903
ficiently. On Fig. 3. one can see, that the error for the initial control parameters was high at the beginning, but reduces to zero in finite time. If one increase the value of ξ1 , more accurate tracking can be obtained. On Fig. 4. the administration protocol can be viewed. It is notable that because the trajectory prescription has non zero derivative at time t = 0, there is a jump at the beginning of the administration. It should also be clear, that negative input is not possible (one can not remove inhibitor from the patient) which means that a saturation had to be employed to the system that limits the input signal magnitude between 0 mg/kg and 30 mg/kg. The purpose of the upper bound is to avoid high dosage profiles which could jeopardize the health of the patient. The reduction of both tumor and vasculature volume can be seen on Fig. 2. 104
and their first and second derivatives are:
Vasculature volume Tumor volume
3
3 3
= −3c exp(−c t
)(xf1
−
xs1 )t(−2
3 3
+ 3c t )
(20)
Observing the above expressions one can easily deduce that at t = 0 the derivatives are both zero which solves the problem. A third set point prescription was also defined as a constant xd1 (t) = xf1 with zero derivatives. 5. NUMERICAL SIMULATION Several simulations were conducted in order to measure the qualitative behaviour of the proposed control algorithms. The model parameters were indicated in Table 1. before, with the corresponding initial values of the tumor and its vasculature of x1 = x2 = 1.734 · 104 mm3 . Therefore, the value xs1 = 1.734 · 104 mm3 was assigned to the prescriptions in conjunction with xf1 = 1 mm3 . The scaling constant of the tanh() and exponential case were both c = 0.1. By these choices, the tumor volume reduces to a safe level in 30 days both cases. The initial controller parameters can be seen in Table 2 which was determined on the basis of numerical simulations. In order to tune the controller, the RFPT part must be adjusted first with the original operator (15) in conjunction with Λ = 1 so that the tracking error is minimal. After that the LQR parameters can be set so that they fulfill the treatment criteria. The simulation time was 100 days, and a continuous therapy was assumed. One should note, that continuous treatment is not likely to be possible because a proper feedback is not available, however it can be employed to investigate the basic properties of the controllers. Table 2. Initial controller parameters for the tanh() tracking
1.5
1
0.5
0 0
10
20
30
40
50
60
70
80
90
100
Time (days)
Fig. 2. Reduction of the volumes by using tanh() prescription. Error
Tumor volume error (mm 3 )
x ¨d1 (t)
150
100
50
0 0
10
20
30
40
50
60
70
80
90
100
Time (days)
Fig. 3. Error of the tumor volume by using tanh() prescription. 30 Inhibitor concentration 20
Concentration (mg/kg)
x˙ d1 (t) = 3c3 exp(−c3 t3 )(xf1 − xs1 )t2
Volumes (mm 3 )
2
10 0 -10 -20 -30 -40
Parameter
Value
ξ1 ξ2 µ K A B
107
0
10
20
30
40
50
60
70
80
90
100
Time (days)
1 10 7 · 1010 10−11 −1
Fig. 4. Inhibitor profile by using tanh() prescription.
The simple LQR controller was scrutinised first. Simulations showed that it could track the tanh() signal ef903
The next simulations targeted the constant reference case. Here, the desired volume was xd1 (t) = xf1 = 1 mm3 and the derivatives were both zero. Multiple simulations showed, that the best results can be obtained if one sets ξ1 = ξ2 = 1 altogether with µ ∈ [100; 1000]. By varying µ different
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Simulations showed, that the augmented RFPT controller has not proven to be useful with parameters included in Table 2. The tracking error grew significantly, while the inhibitor profile became inconsistent which can be seen on Fig. 7. It was not possible to remove the negative values with a saturation as well, because it destabilized the system. On Fig. 8. one can see that compared to the tanh() case, the error differs considerably and the corresponding reduction is shown on Fig. 9. The poor performance of the controller can be attributed to the structure of the kinematic block. Compared to the original prescription (15), the LQR based PID controller has slower convergence rate which results in higher tracking error and initial oscillations in the control signal. Unfortunately this behaviour can not be alleviated by varying the control parameters of the LQR or RFPT functions which means that the PD type structure of the kinematic block is inadequate for the RFPT controller. Vasculature volume Tumor volume
Volumes (mm 3 )
15000
10000
Concentration (mg/kg)
30 Inhibitor concentration 25
20
15
10
0
10
20
30
40
50
60
70
80
90
100
Time (days)
Fig. 6. Inhibitor profile by using constant set point tracking.
50 Inhibitor concentration
Concentration (mg/kg)
In the case of the exponential trajectory, the results were unsatisfying. While the controller could track the trajectory with minimal error, the inhibitor dosage profile was unacceptable due to high concentration levels, hence this type of trajectory is omitted.
prescription, a fully customized treatment plan can be constructed.
40 30 20 10 0 -10 -20 0
10
20
30
40
50
60
70
80
90
100
Time (days)
Fig. 7. Inhibitor profile in the case of the augmented RFPT controller. Error
Tumor volume error (mm 3 )
settling times and inhibitor profiles can be achieved, which entails that larger µ values leads to slower settling time and lower dosage profile. On Fig. 5. one can view the reduction of the tumor and vasculature volumes. It has similar characteristic as the tanh() case, but it reaches a safe level considerably faster. The corresponding inhibitor profile can be seen on Fig. 6. It should be noted that under a 100 days period the volume only reached 1.79 mm3 , which is not identical to the desired prescription. However, one should consider that under 10 mm3 tumor volume the treatment is successful, and by increasing the simulation time the error term vanishes.
500 400 300 200 100 0
5000
0
10
20
30
40
50
60
70
80
90
100
Time (days) 0 0
10
20
30
40
50
60
70
80
90
Fig. 8. Error of the tumor volume in the case of the augmented RFPT controller.
100
Time (days)
Fig. 5. Reduction of the volumes by using constant set point tracking.
104 Vasculature volume Tumor volume
Compared to other controllers, the augmented RFPT controller obviously did not perform well, however the PID controller in the constant reference case has many promising features. By varying parameter µ different control strategies can be created which may have a longer time span but uses lower dosages. This is also true for the tanh() case, where by changing the value of parameter c in the trajectory prescription leads to the same effect as before. In addition the huge similarity between the constant and tanh() inhibitor profiles implies some connection between both methods, which means that by using the tanh() 904
Volumes (mm 3 )
2
1.5
1
0.5
0 0
10
20
30
40
50
60
70
80
90
100
Time (days)
Fig. 9. Reduction of the volumes in the case of the augmented RFPT controller.
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6. CONCLUSION In this paper a control engineering based approach was presented in order to lower the medical expenses of the antiangiogenic TMT treatment. A feedback linearizationbased controller was designed, which based on the Hahnfeldt model could track various tumor volume prescriptions including set point tracking as well. A modified approach was presented as well which in theory could improve the performance of the PID controller. Simulations showed however that this augmented RFPT controller was not operated as expected. In order to be applicable, several other features of the PID controller has to be investigated. Robustness for example rises many questions regard to model parameters and measurement disturbances. Since system parameters are highly unlikely to be an accurate representation of the reality, the parameter robustness of the system is essential in order to be employed in every day practice. Discrete time control also has to be simulated due to the lack of continuous measurement, and has to compensate for the time intervals between inhibitor dosages. These effects altogether can cause the system to be unstable which leads to ineffective treatment. REFERENCES Charlton, P. and Spicer, J. (2016). Targeted therapy in cancer. Medicine, 44(1), 34–38. Csercsik, D., S´api, J., G¨onczy, T., and Kov´ acs, L. (2017). Bi-compartmental modelling of tumor and supporting vasculature growth dynamics for cancer treatment optimization purpose. In 57th IEEE Conference on Decision and Control (CDC 2017), 4698–4702. Czak´o, B.G., S´api, J., and Kov´ acs, L. (2017). Modelbased optimal control method for cancer treatment using model predictive control and robust fixed point method. In 21st IEEE International Conference on Intelligent Engineering Systems (INES 2017), 271–276. Drexler, D., J.Sapi, and Kovacs, L. (2017a). Positive control of minimal model of tumor growth with bevacizumab treatment. In 12th IEEE Conference on Industrial Electronics and Applications (ICIEA 2017), 2081 – 2084. Drexler, D.A., Sapi, J., and Kovacs, L. (2017b). A minimal model of tumor growth with angiogenic inhibition using bevacizumab. In 15th IEEE International Symposium on Applied Machine Intelligence and Informatics (SAMI 2017), 185–190. Hanhfeldt, P., Panigrahy, D., Folkman, J., and Hlatky, L. (1999). Tumor development under angiogenic signaling: a dynamical theory of tumor growth, treatment response, and postvascular dormancy. Cancer Research, 59(19), 4770–4775. Harris, A. (2003). Angiogenesis as a new target for cancer control. European Journal of Cancer Supplements, 1(2), 1–12. Ilic, I., Jankovic, S., and Ilic, M. (2016). Bevacizumab combined with chemotherapy improves survival for patients with metastatic colorectal cancer: Evidence frommeta analysis. PLoS ONE, 11(8), e0161912. Ionescu, C., Keyser, R.D., Sabatier, J., Oustaloup, A., and Levron, F. (2011). Low frequency constant-phase behavior in the respiratory impedance. Biomedical Signal Processing and Control, 6(2), 197–208. 905
905
Ionescu, C., Lopes, A., Copot, D., Machado, J., and Bates, J. (2017). The role of fractional calculus in modeling biological phenomena: A review. Communications in Nonlinear Science and Numerical Simulation, 51, 141– 159. Jayson, G.C., Kerbel, R., Ellis, L.M., and Harris, A.L. (2016). Antiangiogenic therapy in oncology: current status and future directions. The Lancet, 388, 518–529. Kovacs, L., Szeles, A., Drexler, D., Sapi, J., and Harmati, I. (2013). Model-based angiogenic inhibition of tumor growth using modern robust control method. Computer Methods and Programs in Biomedicine, 114(3), 98–110. Malvezzi, M., Carioli, G., Bertuccio, P., Boffetta, P., Levi, F., Vecchia, C.L., and Negri, E. (2017). European cancer mortality predictions for the year 2017, with focus on lung cancer. Annals of Oncology, 28, 1117–1123. Rymansaib, Z., Iravani, P., and Sahinkaya, M.N. (2013). Exponential trajectory generation for point to point motions. In IEEE/ASME International Conference on Advanced Intelligent Mechatronics, 906 – 911. S´api, J., Drexler, D.A., Harmati, I., S´api, Z., and Kov´ acs, L. (2015). Qualitative analysis of tumor growth model under antiangiogenic therapy - choosing the effective operating point and design parameters for controller design. Optimal Control Applications and Methods, 37(5), 848–866. Sapi, J., Drexler, D.A., and Kovacs, L. (2013). Parameter optimization of h infinity controller designed for tumor growth in the light of physiological aspects. In 14th IEEE International Symposium on Computational Intelligence and Informatics (CINTI 2013), 19 – 24. Tar, J.K., N´adai, L., and Rudas, I.J. (2012). System and Control Theory with Especial Emphasis on Nonlinear Systems. Typotex. Tar, J., Bit´o, J., N´adai, L., and Machado, J.T. (2009). Robust Fixed Point Transformations in adaptive control using local basin of attraction. Acta Polytechnica Hungarica, 6(1), 21–37.