This research focuses on modeling the diffusion process that occurs in the human body when an analgesic drug is taken up, by using fractional-order impedance models (FOIMs). We discuss the measurement of a suitable feedback signal that can be used in a model-based control strategy. With this knowledge an early dawn concept of a pain sensor is presented. The major challenges that are encountered during this development consist of identification of the patient model, validation of the pain sensor and validation of the effect of the analgesic drug

Chevalier, Amélie

Copot, Dana

De Keyser, Robain

Ionescu, Clara-Mihaela

English

Amelie Chevalier

Dana Copot

Clara M. Ionescu

Robin De Keyser

Crossref

Fractional order impedance models as rising tools for quantification of unconscious analgesia

Ghent University Academic Bibliography

Fractional Order Impedance Models as Rising Tools for Quantificationof Unconscious AnalgesiaAme´lie Chevalier1, Dana Copot1, Clara M. Ionescu1 and Robin De Keyser1Abstract—This research focuses on modeling the diffusionprocess that occurs in the human body when an analgesicdrug is taken up, by using fractional-order impedance models(FOIMs). We discuss the measurement of a suitable feedbacksignal that can be used in a model-based control strategy. Withthis knowledge an early dawn concept of a pain sensor ispresented. The major challenges that are encountered duringthis development consist of identification of the patient model,validation of the pain sensor and validation of the effect of theanalgesic drug.Index Terms—Analgesia, pain relief level, non-invasive painsensor, model-based predictive control (MPC), fractional-orderimpedance model (FOIM)I. INTRODUCTIONThe last few decades, modern medicine has success-fully been influenced by advanced control technologiesresulting in applications such as robotic surgery, electro-physiological system life support and image-guided therapyand surgery [1]. An interesting application of control inmedicine is clinical pharmacology and in particular thecontrol of general anesthesia during surgery and in theintensive care unit (ICU). Monitoring and controlling thedepth of anesthesia for surgical patients poses interestingchallenges to the control engineer [2] as it is a multi-variable interaction process that has captured the attention ofengineers and clinicians already decades ago [3]. The firstdesigns were expert systems that advised the anesthesiologistupon optimal drug infusion rates during clinical trials [4].Control of anesthesia has a manifold of challenges, withmulti-variable characteristics, different dynamics dependingon anesthetic substances and stability problems [5].General anesthesia, where the patient is completely un-conscious, has the aim of ensuring sleep, amnesia, lossof pain, relaxation of skeletal muscles and loss of controlof reflexes of the autonomic nervous system. It consistsof three components acting simultaneously on the patient’svital signs: hypnosis (ensuring sleep and amnesia), analgesia(ensures loss of pain) and neuromuscular blockade (relaxesthe skeletal muscles and the motor reflexes). Hypnosis isrelatively well-characterized and is in standard clinical prac-tice monitored by sensors based on electroencephalogram(EEG) data. Neuromuscular blockade immobilizes the pa-tient during surgical procedures or intensive care and is also1The authors are with the Department of Electrical Energy, Systemsand Automation, Ghent University, Technologiepark 913, 9052 Gent,Belgium, e-mail: {Amelie.Chevalier, Dana.Copot, ClaraMihaela.Ionescu,Robain.DeKeyser}@UGent.bea relatively well-characterized process with standard sensors,such as motion sensors, available. By contrast, analgesia isfar from being well-characterized and no sensor is availablefor measuring the pain relief levels that a patient experiencesduring general anesthesia.The advantage of automated closed loop control of anes-thesia is that it gives a continuous drug delivery, contraryto intermittent control which is nowadays standard practice.A continuous drug delivery ensures that there is no under-or over-dose of hypnotic or analgesic drugs that could resultin patients that feel pain during surgery but are unable tomove. Erroneous feedback information, biased either by thepresence of artifacts (e.g. eye movement, leg movement,coughing, sneezing, choking, shivering) either by patientmodel mismatch, is one of the major problems for thecontrol algorithms [6]. As result the quality of the measuredsignals decreases, leading to the need of complex numericalfiltering techniques. The latter require longer computationtimes, hence introducing artificial time delays which varyfrom one time instant to another, dependent on the signalquality. If not dealt-with appropriately, such varying time-delays are a source of poor feedback control. Advancedcontrol techniques such as model-based predictive control(MPC) can deal successfully with these variable time delays,nonlinearities, input and output constraints [7].The research presented in this paper merges classicalcontrol theory with the young promising field of fractional-order modeling to measure pain relief levels in an uncon-scious patient and initiate the development of a biosensorfor analgesia levels. Few pioneering attempts to measure theanalgesic component of general anesthesia have shown thatcurrent state-of-art is unable to deliver suitable signals andmodels for optimal regulation. The result is then a high riskof drug over- or under-dosing and unwanted post-operativelyeffects, leading to increased hospitalization and health-carecosts for both society and patient [8]. We propose to employa mathematical tool called fractional-order impedance model(FOIM) to model the pharmacological diffusion process thattakes place when the human body takes up an analgesic drugsuch as remifentanil. These models can be used in a MPCcontext to control the depth of analgesia in the unconsciouspatient.The paper is structured as follows: in section II, wedescribe analgesia and the coinciding diffusion process.Section III discusses the control method that will be usedin combination with the proposed analgesia sensor and the!"#$%!#&'%()*+'),,-.)-.%/0.1),).2)%0.%/0.',03%4%56'07-'+0.%8(9:;<3-'-.+-&=/>-.+-?%/,)')?%@,))2)?%A6.)%!B=!C?%!"#$DEC=#=FEDD="DDB=EG#$GH$#I""%J!"#$%K999 !"L!"E!"C!"Dtransducer (electrochemical, optical, magnetic, etc.) convertsthe bio-recognition event into a measurable signal, the signalprocessing part converts the signal into a readable form.B. A prediction model for MPCMPC is a model-based control strategy. Standard modelsinclude step response models, transfer function models andlinear state space models, however, these models do notsuffice in modeling the dynamics of the diffusion processesthat occur in the human body.In the past many attempts to model the diffusion pro-cess have been made. Fractional Order Impedance Models(FOIMs) have been shown to well-characterize these diffu-sion processes [15], which in essence take place ubiquitouslyin our body. It is therefore natural to choose these tools indetecting, understanding and characterizing the process ofpain reception at the level of nociceptors.In medicine, the field of fractional-order calculus hasbarely been explored. However, this research field promisesto serve a whole range of applications with a large impacton the progress of science and welfare. The last decadeshave shown an increased interest in the research communityto employ parametric model structures of fractional-orderfor analyzing nonlinear biological systems. The conceptof fractional-order (FO) – or non-integer order – systemsrefers to those dynamical systems whose model structurecontains arbitrary order derivatives and/or integrals [16]. Thefractional-order derivatives and integrals are tools of theFractional Calculus theory. The dynamical systems whosemodel can be approximated in a natural way using FOterms, exhibit specific features: viscoelasticity, diffusion andfractal structure. From previous work [17], we know nowthat the multiple scale adaptation of neurons is consistentwith fractional order differentiation, such that the neuron’sfiring rate is a fractional derivative of slowly varying stimulusparameters [18]. The findings of scale-free fluctuations inthe activity of neurons and synapses have been used toillustrate the existence of multiple time-scale dynamics inneurons and synapses. Additionally, there has been shownthat phase-locking phenomena can be explained by thepresence of fractal electrical neuronal networks, which leadto a fractional-order impedance model of the neural network[19]. However, the theoretical concepts of fractals, chaos andmulti-scale analysis have not yet been employed in the fieldof anesthesia, where the electrical activity of the brain isaltered by the effects of hypnotic (propofol) and analgesic(remifentanil) drugs.Another option to model the diffusion processes in thehuman body is to use compartmental models in combinationwith fractional-order derivatives [20]. Three compartmentsare used in this diffusion model: blood, muscle and fat.Principles of Fractional CalculusThe fractional calculus is a generalization of integrationand derivation to non-integer (fractional) order operators. Atfirst, we generalize the differential and integral operators intoone fundamental operator Dnt (n the order of the operation)which is known as fractional calculus.Several definitions of this operator have been proposed.All of them generalize the standard differential–integraloperator in two main groups: (a) they become the standarddifferential–integral operator of any order when n is aninteger; (b) the Laplace transform of the operator Dnt is sn(provided zero initial conditions), and hence the frequencycharacteristic of this operator is (jω)n. The latter is veryappealing for the design of parametric modelling and controlalgorithms by using specifications in the frequency domain.A fundamental Dnt operator, a generalization of integraland differential operators (differintegration operator), is in-troduced as follows:Dnt =dndtn, n > 01, n = 0∫ t0(dα)−n, n < 0 (1)where n is the fractional order and dα is a derivativefunction. Since this paper will focus on the frequency-domain approach for FO derivatives and integrals, we shallnot introduce the complex mathematics for time domainanalysis. The Laplace transform for integral and derivativeorder n are, respectively:L{D−nt f(t)}= s−nF (s) (2)L {Dnt f(t)} = snF (s) (3)where F (s) = L {f(t)} and s is the Laplace complex vari-able. The Fourier transform can be obtained by replacing sby jω in the Laplace transform and the equivalent frequency-domain expressions are:1(jω)n=1ωn(cosnpi2− j sinnpi2)(4)(jω)n = ωn(cosnpi2+ j sinnpi2)(5)Thus, the modulus and the argument of the FO terms aregiven by:Modulus(dB) = 20 log∣∣(jω)∓n∣∣ = ∓20n log |ω| (6)Phase(rad) = arg((jω)∓n)= ∓npi2(7)resulting in a straight line with a slope of ∓20n passingthrough 0 dB for ω = 1 for the magnitude (dB vs. log-frequency), respectively a horizontal line, thus independentwith frequency, with value ∓npi2for the phase (rad vs. log-frequency). The respective sketches are given in figure 8.Principles of Compartmental Fractional Derivative ModelsIn this section a two compartmental fractional derivativemodel is discussed. The basic idea behind this model can beused to model diffusion processes in the human body by amulti-compartmental model.The model is formulated so that the mass balance is!#"Fig. 8. Sketch representation of the FO integral and derivator operators infrequency domain, by means of the Bode plots (magnitude above and phasebellow).preserved. In figure 9, we see a conceptual schematic of amodel with two compartments. Assume that qi(t) = vici,for i = 1, 2 denote the amount of a drug in a specificcompartment. Here ci is the concentration of a drug and vi isthe volume of the i-th compartment and Kij is the fractionalrate of transfer from compartment i to compartment j.The first compartment represents the place where thedrug is applied i.e. muscle, subcutaneous tissue or digestivetract. The second compartment represents the plasma or anyother region in the body where the kinetics of the drug areuniform. Traditionally, the two compartments are describedby a system of differential equations of integer order.dq1(t)dt= K21q2(t)−K12q1(t)−K01q1(t) (8)dq2(t)dt= K12q1(t)−K21q2(t)−K02q2(t) (9)Recently, the fractional-order models seem to better suit thedynamics of biological systems than other integer models. Asimple model of a two-compartmental system is then givenby the following equations:τn1−11 0Dn1t q1(t) = −K12q1(t), (10)τn2−12 0Dn2t q2(t) = K12q1(t)−K02q2(t), (11)where we assumed K01 = 0, K21 = 0 and with the initialconditions q1(0) = dose, and q2(0) = 0. In these equationsτ1 and τ2 are time constants which represent the speed ofdiffusion, while n1 and n2 represent a non-integer betweenK12q1 K21q2 Muscle (1) Plasma  (2) K01 K02 Fig. 9. Conceptual schematic of a two-compartment model.0 and 1 and characterize the type of diffusion (sub-, super-,etc).IV. GRAND CHALLENGESSeveral grand challenges are encountered in the develop-ment of the proposed analgesic biosensor. We discuss themajor challenges in the remainder of this section.To be able to control the level of analgesia in the un-conscious patient, we apply a MPC strategy. This strategyneeds a reliable model of the process that it needs to control.In traditional, non-human, systems, this model identificationis done by sending excitation signals into the system andanalyzing the corresponding output signals of the open-loopsystem. However, as we are dealing with patients, it is notpossible to apply here the same strategy. Output signalsto analyze are available only after the nurse administersa certain amount of analgesic drug. As the nurse onlyadministers this drug after examination of the patient, thisis no longer an open-loop system and system identificationcan be compromised by this.Another difficulty in the model identification is the factthat every person reacts differently to a certain amount orcombination of drugs. Therefore, every model differs forevery patient i.e. inter-patient variability.Moreover, the conditions inside the body of every patientare changing as a result of accumulated drug effect i.e. intra-patient variability. The parameters of the patient model needto be updated regularly.The pain sensor is supposed to measure a pain signal inan unconscious patient. However, there is no reliable way tovalidate the pain sensor once it is developed as the patientis the only one who can feel the pain but he is not able toindicate it anymore because he is unconscious.Even if you can objectively proof that the pain sensorpicks up a pain signal in one patient, it is not certain that thesensor will have the same result in a different patient as thepain threshold for one person can be completely different foreach person. This is the result of the inter- and intra-patientvariabilities that pose an extra challenge on the developmentof the pain sensor.Another major challenge in this research direction is thefact that a combination of drugs is administered to thepatient. Therefore, it is difficult to completely separate andvalidate the effect of the analgesic drug.V. CONCLUSIONSThis paper proposes an early dawn pain sensor to measureanalgesia level in unconscious patients. The proposed sensorcan be used in combination with a model-based predictivecontrol strategy to control the level of anesthesia in anunconscious patient. To model the diffusion process in thehuman body a fractional-order impedance model is applied.The coinciding challenges in this research direction includeidentification of the patient model, validation of the painsensor and validation of the effect of the analgesic drug.!##ACKNOWLEDGMENTC. M. Ionescu acknowledges the Flanders Research Center(FWO) for its financial support.REFERENCES[1] J. M. Bailey and W. M. Haddad, ”Drug dosing control in clinicalpharmacology”, IEEE Control Systems Magazine, vol. 25, no. 2, pp.35-51, Apr. 2005.[2] W. M. Haddad, T. Hayakawa and J. M. Bailey, ”Nonlinear adaptivecontrol for intensive care unit sedation and operating room hypnosis”,American Control Conference, vol. 2, pp. 1808-1813, 2003.[3] D. O’Hara, D. Bogen and A. Noordergraaf, ”The use of computersfor controlling the delivery of anesthesia”, Anesthesiology, vol. 77, pp.563-581, 1992.[4] S. G. Greenhow, D. A. Linkens and A. J. Asbury, ”Pilot study ofan expert system adviser for controlling general anesthesia”, BritishJournal of Anesthesia, vol. 71, pp. 359-365, 1993.[5] M. Struys, H. Vereecke, A. Moerman, E. W. Jensen, D. Verhaeghen,N. De Neve, F. Dumortier and E. Mortier, ”Ability of the bispectralindex, autoregressive modelling with exogenous input-derived auditoryevoked potentials, and predicted propofol concentrations to measurepatient responsiveness during anesthesia with propofol and remifen-tanil”, Anesthesiology, vol. 99, pp. 802-814, 2003.[6] R. B. Northrop, Endogenous and Exogenous Regulation and Controlof Physiological Systems. Boca Raton, FL:CRC, 2000.[7] R. De Keyser, ”Model based predictive control”, UNESCO Encyclo-pedia Life Support System, vol. 6.43.16.1, pp. 802-812, 2003.[8] J. Kress, A. Pohlman and J. Hall, ”Sedation and analgesia in theintensive care unit”, American Journal of Respiratory and CriticalCare Medicine, vol. 166, pp. 1024-1028, 2002.[9] O. Steen-Knudsen, Biological Membranes. Theory of transport, po-tentials and electric impulses. Cambridge: Cambridge UniversityPress, 2002.[10] J. M. Berg, ed., Biochemistry, 6th ed. New York City, NY: W.H.Freeman and Company, 2002.[11] R. De Keyser and A. Van Cauwenberghe, ”A self-tuning multisteppredictor application”, Automatica, vol. 17, pp. 167-174, 1981.[12] T. M. Hemmerling, E. Salhab, G. Aoun , S.Charabati and P. Mathieu, ”The Analgoscore: a novel score to monitor intraoperative pain andits use for Remifentanil closed loop application”, IEEE ProceedingsCat., ref nr. 1-4244-0991-8/07, pp. 1494-1499, 2007.[13] C. N. Sessler, M. Gosnell, M. J. Grap, G. T. Brophy, P. V. O’Neal,E. Tesoro and R.K. Elswick, ”A new Agitation-Sedation Scale forcritically ill patients: development and testing of validity and inter-rater reliability”, American Journal of Respiratory and Critical CareMedicine, vol. 161, pp. A506, 2000.[14] P. S. Glass, M. Bloom, L. Kearse, C. Rosow, P. Sebel and P. Man-berg, ”Bispectral analysis measures sedation and memory effects ofpropofol, midazolam, isoflurane, and alfentanil in healthy volunteers”,Anesthesiology, vol. 86, pp. 836-847, 1997.[15] G. A. Losa, D. Merlini, T. F. Nonnenmacher and E. R. Weibel, Fractalsin Biology and Medicine, vol IV. Berlin: Birkhaser, 2005.[16] A. Oustaloup, La derivation non entiere. Hermes, Paris, 1995. (inFrench)[17] C. M. Ionescu, J. T. Machado and R. De Keyser, ”Modeling ofthe Lung Impedance using a Fractional Order Ladder Network withConstant Phase Elements”, IEEE Transactions on Biomedical Circuitsand Systems, vol. 5, no. 1, pp. 83-89, 2011.[18] B. Lundstrom, M. Higgs, W. Spain and A. Fairhall, ”Fractional dif-ferentiation by neocortical pyramidal neurons”, Nature Neuroscience,vol. 11, pp. 1135-1342, 2008.[19] C. M. Ionescu, ”Phase constancy in a ladder model of neural dynam-ics”, IEEE Transactions on Systems, Man and Cybernetics, part A:Systems and Humans, vol. 42, no. 6, 2012.[20] A. Dokoumetzidis, R. Magin and P. Macheras, ”A commentary onfractionalization of multi-compartmental models”, Journal of Pharma-cokinetics & Pharmacodynamics, vol. 37, no. 2, pp. 203-207, April2010.!#!

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Fractional order impedance models as rising tools for quantification of unconscious analgesia

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Ghent University Academic Bibliography