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Modelling and multivariable control in anaesthesia using neural-fuzzy paradigms
Artificial Intelligence in Medicine (2005) 35, 207—213
http://www.intl.elsevierhealth.com/journals/aiim
Modelling and multivariable control in anaesthesia using neural-fuzzy paradigms Part II. Closed-loop control of simultaneous administration of propofol and remifentanil Mahdi Mahfouf a,*, Catarina S. Nunes b, Derek A. Linkens a, John E. Peacock c a
Department of Automatic Control and Systems Engineering, The University of Sheffield, Mappin Street, Sheffield S1 3JD, UK b ´tica Aplicada, Faculdade de Cie ˆncias da Universidade do Porto, Departamento de Matema Rua do Campo Alegre 687, 4169-007 Porto, Portugal c Department of Anaesthesia, Royal Hallamshire Hospital, Glossop Road, Sheffield S10 2JF, UK Received 6 May 2004; received in revised form 24 November 2004; accepted 6 December 2004
KEYWORDS Patient model; Anaesthesia; Analgesia; Fuzzy logic; Feedback control; Multivariable
Summary Objective: Part II of this research study is concerned with the development of a closed-loop simulation linking the patient model as well as the fuzzy relational classifier already introduced in Part I with a control algorithm. The overall architecture is in fact a system advisor, which provides information to the anaesthetist about the adequate infusion-rates of propofol and remifentanil simultaneously. Methods and material: The developed fuzzy multivariable controller includes three rule-bases and takes into account the synergetic interactions between the above drugs and uses such knowledge to achieve rapidly the desired depth of anaesthesia (DOA) level. Results: The result of the study is a closed-loop control scheme, which adjusts efficiently the infusion-rates of two drugs in response to DOA changes. This controller can either be used in an advisory mode or closed-loop feedback mode in the operating theatre during surgery. Conclusion: It is hoped that this control scheme coupled with the patient model presented in Part I of this study will be used routinely in the operating theatre in the very near future. # 2005 Elsevier B.V. All rights reserved.
* Corresponding author. Tel.: +44 114 222 5607; fax: +44 114 222 5624. E-mail address: [email protected] (M. Mahfouf). 0933-3657/$ — see front matter # 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.artmed.2004.12.005
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1. Introduction The infusion-rate of the anaesthetic drug is titrated according to the patient’s requirements, so as to maintain a certain level of depth of anaesthesia (DOA). The patient’s clinical signs and/or brain signals are used by the anaesthetist to determine the adequate infusion-rate. In addition, the anaesthetist also establishes the required infusion-rate of the analgesic drug, based on the patient’s response to surgical stimuli. A closed-loop control system of DOA will help the anaesthetist adjust simultaneously the infusion-rates of the anaesthetic and analgesic drugs. The majority of the researches in the area are mainly concerned with the automatic control of the anaesthetic drug, whereas the analgesic is controlled manually by the anaesthetist. However, in this research the objective is a multivariable control structure for both drugs. The patient model presented in Part I of this research study [1], described adequately the effects and interactions of the two drugs in the presence of surgical stimuli. This model will be used to develop a control algorithm relating to the administration of both drugs. The study of the interactions between the anaesthetic propofol and the analgesic remifentanil helps to determine the ideal combination of infusion-rates. This study will also represent a practical guide for the anaesthetist, which would help him/her learn how to adjust the amount of drug infused and hence improve the patient’s comfort. First, the patient model will be tested using a series of open-loop simulations with different infusion profiles. Second, the closed-loop structure will be presented, showing the links between the patient model, the DOA classifier, i.e. the fuzzy relational classifier (FRC) presented in Part I of this study, and the controller. Third, a multivariable fuzzy controller, developed with the anaesthetist’s cooperation, is also described. This controller establishes the infusion-rates of propofol and remifentanil simultaneously based on the level of DOA, the concentrations of the drugs and the surgical stimuli. The performance of the controller is tested under different conditions.
Figure 1 files.
Propofol and remifentanil infusion-rate pro-
sampling-time of 30 s. The first 1500 s relate to the induction-phase, followed by the maintenancephase. It is worth noting that the recovery-phase is not simulated. The FRC for DOA developed in Part I is applied to the wavelet-extracted audio evoked potential (AEP) features, as determined by the maintenance-phase model. In the induction-phase, the FRC uses only the cardiovascular parameters, a change in systolic arterial pressure (DSAP) and a change in heart-rate (DHR), for the classification of DOA, since the AEP features are not modelled during this phase. Fig. 1 shows the infusion-rate profiles for propofol and remifentanil used in the first simulation. The propofol infusion-rate is very similar to the profile of patient Pat1, while the remifentanil infusion-rate follows a typical profile during a surgical procedure (i.e. high at induction and constant during maintenance). Fig. 2 shows the HR and SAP as simulated by the patient model, using the Infusion Profile 1. The wavelet-extracted AEP features as simulated by the patient model are used to classify the DOA level by the FRC during the maintenance-phase. The DOA level is shown in Fig. 3. The model performs
2. Open-loop simulation results using the patient model The patient model developed in Part I was tested in open-loop simulations with different infusion profiles for propofol and remifentanil. Although three different infusion profiles were studied only one profile will be presented and discussed here expressing lessons learned from the experiment. The simulations were performed for 7200 s (120 min) with a
Figure 2 SAP and HR as determined by the patient model following the infusion profile of Fig. 1.
Modelling and multivariable control in anaesthesia using neural-fuzzy paradigms
Figure 3 DOA as classified by the FRC using the waveletextracted AEP features.
adequately, describing the effects of the stimulus level on SAP and HR. The effect concentrations are within the ranges of the maintenance-phase and this is reflected in the model’s response. The DOA level as determined by the FRC cannot be compared with the anaesthetist’s classification, since this is a simulated infusion profile. However, the classification is not unreasonable as it reflects the AEP features from the model.
3. Closed-loop structure The closed-loop simulation system links the patient model, the FRC of DOA and the control algorithm. Fig. 4 shows the block diagram comprising the different components of the closed-loop system during the maintenance-phase of anaesthesia. The FRC uses the parameters from the patient model to classify DOA. Finally, a control structure maintains an adequate DOA level by adjusting the infusionrates of propofol and remifentanil, which are the inputs of the patient model. The fuzzy controller algorithm is discussed in the next section.
3.1. Multivariable fuzzy control In general, propofol is used for maintenance of anaesthesia in combination with an opioid, hence, the anaesthetist is confronted with the dilemma of whether to vary propofol or the opioid. In reference [2], Zhang and co-workers reported on a closed-loop system for total intravenous anaesthesia by simultaneously administering propofol and fentanyl.
Figure 4 system.
Block diagram of the closed-loop simulation
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They studied the interaction between propofol and fentanyl for loss of response to surgical stimuli using an unweighted least squares non-linear regression analysis with human data. A look-up table of optimal and awakening combinations of concentrations was built, and used to determine the fentanyl set point according to the propofol set point. To our knowledge, this is one of the only studies in simultaneous control of anaesthetic and analgesic drugs. Multivariable fuzzy controllers are not easy to design. There are several studies on the design of multivariable controllers, however, very few target biomedical systems. In reference [3], King et al. described an integrated development system, which permits the controller designer to test hypothesis, examine the effect of changes in the controller parameters and perform a complete off-line simulation of a proposed multivariable fuzzy controller. This system was applied to industrial processes and tested off-line with mathematical models. In reference [4] Linkens and Nyongesa developed a hierarchical multivariable fuzzy controller for learning with genetic algorithms (GAs). This controller was applied to a simulation case study in anaesthesia, using mathematical models describing the action of atracurium and isoflurane (i.e. a muscle relaxant and an inhalational anaesthetic). This structure decomposes a complex multivariable fuzzy controller into several simple fuzzy controllers. The anaesthesiologist is confronted with the dilemma of whether to vary propofol or the opioid. Anaesthesiologists’ knowledge and experience is incorporated into the fuzzy control system as a set of linguistic rules. In addition, the interaction between remifentanil and propofol provides information that can be used to determine an appropriate combination of the two drugs. The controller is activated only at the start of the maintenancephase. The infusion-rates of both drugs are determined according to the DOA level and the surgical stimulus. The controller comprises three different blocks, corresponding to three possible values of DOA, i.e. the DOA level is at the target level (ok), or the DOA is lighter than that desired, or the DOA is deeper than that desired. The controller acts differently according to these three stages. The linguistic scheme of the controller is described as follows: If DOA is ok then no change; If DOA is light and if: - stimulus present then increase remifentanil (via remifentanil rule-base 1); - no stimulus then increase propofol (via single input/single output–—SISO fuzzy proportional integral- PI controller);
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If DOA is deep and if: - no stimulus and remifentanil high then decrease remifentanil; - no stimulus and remifentanil normal and propofol high then decrease propofol; - no stimulus and remifentanil normal and propofol normal then decrease remifentanil; - stimulus and propofol High then decrease propofol (via SISO fuzzy PI controller) and increase remifentanil (via remifentanil rulebase 2); - stimulus and propofol normal then decrease remifentanil. The above multivariable controller is based on linguistic rules that interact with three decision tables, two of which are rule-bases for a change in remifentanil infusion-rate (rule-base 1 and rulebase 2) and the other represents a SISO fuzzy PI controller for a change in propofol infusion-rate. These rule-bases are described in the next sections. It is important to note that a minimum effect concentration of propofol and remifentanil is required at all times, in order to ensure unconsciousness and prevent arousal. The minimum values were established as 2250 and 3.5 ng/ml for propofol and remifentanil, respectively. If the concentrations decrease to these minimum values, then the infusion-rate of propofol will be increased by 0.2 mg/s and remifentanil will be increased by 0.025 mg/(kg min).
3.2. Remifentanil rule-base 1 This determines the increment in the remifentanil infusion-rate when the DOA level is light in the presence of a stimulus. The rule-base is shown in Table 1, and uses the perceived stimulus and the change in DOA error as inputs. The change in remifentanil infusion-rate is normalised between [0,1], but the maximum value of the variable was established as 0.03 mg/kg/min. When DOA is light, higher changes in remifentanil infusion-rate are required in order to respond to the stimulus and increase the level of DOA. Table 1 Rule-base 1 describing the change in remifentanil infusion-rate using the stimulus level and the change-in-error of DOA Stimulus Change in DOA error Negative Zero
Positive small Positive big
Low Small Small Medium Medium Small Medium Medium High Medium Big Big
Big Big Big
Table 2 Rule-base 2 describing the change in remifentanil infusion-rate using the stimulus level and the change-in-error of DOA Stimulus
Low Medium High
Change in DOA error Negative big
Negative small
Zero
Positive
Small Small Small
Small Small Medium
Small Medium Medium
Small Medium Big
3.3. Remifentanil rule-base 2 This is used when the DOA is deep, but there is a stimulus present and an increment in the remifentanil infusion-rate is necessary. Rule-base 2 is shown in Table 2. The perceived stimulus level and the change-in-DOA error are used to determine the change in remifentanil infusion-rate, similarly to rule-base 1. In rule-base 2, the smaller increment in the infusion-rate is predominant, since the DOA is deep and the remifentanil is increased for its analgesic properties. In rule-base 1, the DOA is light, hence, remifentanil is increased as a response to the stimulus and for its synergistic properties in order to increase the DOA level.
3.4. SISO fuzzy PI controller for propofol The fuzzy PI controller was designed to control the change in the infusion-rate of propofol, in order to achieve and maintain a steady level of DOA and to minimise the amount of drug infused. The fuzzy PI controller uses the error (target DOA minus measured DOA) and the change-of-error as inputs. The controller’s rule-base is presented in Table 3. The variables are normalised between [1,1]. The maximum level of change in the infusion-rate was set to 4000 mg/h considering the maximum
Table 3 Rule-base of the fuzzy proportional integral controller for DOA; the inputs are the error and the change-of-error, and the output is the change-in-propofol infusion-rate; the membership functions are labelled: negative big (NB), negative small (NS), zero (ZE), positive small (PS) and positive big (PB) DOA error NB NS ZE PS PB
Change of DOA error NB
NS
ZE
PS
PB
NB NB NB NS ZE
NB NS NS ZE PS
NB NS ZE PS PB
NS ZE PS PS PB
ZE PS PB PB PB
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conditions pre-set by the anaesthesiologist. The input scaling factors of the controller were optimised using a Genetic Algorithm (GA) [5] with a performance index given by: N1 N1 X X P index ¼ l1 kjeðkÞj þ l2 uðkÞ (1) k¼0
k¼0
where e(k) is the error, u(k) the propofol infusionrate and N is the number of simulation samples. The weighting parameters l1 and l2 were chosen to place more emphasis on the error or on the infusion-rate, so that an ideal balance between them can be reached. The GA was implemented using MATLAB1 for a population of 40 strings each of length 20, with a probability of crossover of 0.95 and a probability of mutation of 0.06. The optimisation was run with parameters l1 = 0.4 and l2 = 0.6, which were found to be representative of the specifications for the control system, giving a bigger weight to the infusion profile. The GA optimisation led to the value of 0.375 for the error scaling factor and 7.77 for the change of error scaling factor.
Figure 6 Propofol and remifentanil infusion-rates as determined by the multivariable controller in Simulation 1.
Figure 7
SAP and HR for Simulation 1.
4. Simulation results The multivariable fuzzy controller was used in the closed-loop simulations with the patient model. Note that the controller only starts acting at 1500 s (i.e. after the induction-phase).
4.1. Simulation 1
Figure 8 Propofol and remifentanil infusion-rates as determined by the multivariable controller during the maintenance-phase; DOA target change to ok/deep at 3000 s.
In the first simulation shown here, a typical infusion profile of both drugs was used for the inductionphase. The DOA level for this simulation is shown in Fig. 5. The ok DOA level was achieved at 1740 s. Fig. 6 shows the infusion-rates of propofol and remifentanil for this simulation. The ok DOA level was achieved rapidly, and the controller maintained efficiently a stable DOA level by keeping both infusion-rates constant. In fact, the controller only changed the propofol infusion-rate in
this simulation. Fig. 7 shows the variations in the cardiovascular variables during this simulation. Next, the same conditions as in Simulation 1 were considered, but with a set-point change to ok/deep level at 3000 s. Fig. 8 shows the infusion-rates of propofol and remifentanil as determined by the multivariable controller during the maintenancephase. The DOA level is shown in Fig. 9.
Figure 5 DOA level using the multivariable controller in Simulation 1.
Figure 9 DOA level using the multivariable controller during the maintenance-phase; DOA target change to ok/ deep at 3000 s.
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Figure 10 SAP during the maintenance-phase; DOA target change to ok/deep at 3000 s.
The ok/deep DOA level was reached at 3060 s, i.e. only 60 s after the set-point change. The multivariable controller reacted to the set-point change by increasing the remifentanil infusion-rate, due to the stimulus level present in the system. This increase in the remifentanil infusion-rate increased the level of analgesia and also potentiated the effect of propofol. Fig. 10 shows the SAP during the maintenance-phase. The decrease in SAP following an increase in the remifentanil infusion-rate shows how a deeper level of depression was achieved, leading to the ok/deep DOA level. Therefore, the multivariable controller is taking advantage of the synergism between propofol and remifentanil for efficient control of DOA.
M. Mahfouf et al.
Figure 13 Propofol and remifentanil infusion-rates as determined by the multivariable controller in Simulation 2 during the maintenance-phase (zoom of Fig. 12).
contrast to the previous simulation, the multivariable controller changed both infusion-rates in response to the ok/deep DOA level at approximately 1700 s. Fig. 13 shows the infusion-rate of both drugs during the maintenance-phase. The multivariable controller decreased the propofol infusionrate first and then gradually decreased the remifentanil infusion-rate. The controller was able to determine an adequate combination of the two drugs, thus achieving and maintaining the ok DOA level.
4.2. Simulation 2 Simulation 2 considered a different remifentanil infusion profile during the induction-phase, while the propofol infusion profile was the same as the one used in Simulation 1 thus producing different initial conditions for the controller. Fig. 11 shows the DOA level in Simulation 2, while the propofol and remifentanil infusion-rates are shown in Fig. 12. In
Figure 11 DOA level using the multivariable controller in Simulation 2.
Figure 12 Propofol and remifentanil infusion-rates as determined by the multivariable controller in Simulation 2.
5. Discussions and conclusions The objective of a control system for DOA is to determine the best infusion-rates of the anaesthesiologist and analgesic drugs, helping the anaesthetist to decide which drug should be changed in response to different events. The developed patient model was used to construct and test a multivariable controller for simultaneous administration of remifentanil and propofol during the maintenancephase. Anaesthesiologists’ experience was incorporated into the control structure using linguistic rules. According to the different profiles for the DOA level and for the surgical stimuli, the multivariable controller defines the required change in the infusion-rates of the two drugs. The controller was able to adjust the remifentanil infusion-rate according to the stimulus intensity, and takes advantage of the synergistic interaction to change appropriately the propofol infusion-rate. Propofol is titrated to lower infusion-rates, decreasing the amount of drug infused, and speeding up recovery. In addition, the controller ensures adequate analgesia by titrating the remifentanil according to the stimulus. The multivariable fuzzy controller was tested under different simulations, and responded efficiently to different induction profiles and set-point changes.
Modelling and multivariable control in anaesthesia using neural-fuzzy paradigms
Fuzzy logic techniques have been shown to be efficient in incorporating human knowledge for better solutions in biomedicine. The complexity of DOA and the unavailability of large data sets make this an ideal application for fuzzy logic based concepts. The FRC, the Takagi—Sugeno—Kang pharmacodynamic models and the multivariable fuzzy controller were combined successfully in an integrated structure to provide a closed-loop simulation platform for future anaesthesia monitoring and control in the operating theatre during surgery. It is hoped that such clinical trials will take place in the near future.
Acknowledgements The authors would like to thank the anonymous reviewers for their comments, which helped to improve the quality of this paper. In addition, the second author wishes to acknowledge the Portuguese Foundation for Science and Technology for their financial support during the course of this
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project: Fundac¸a ˜o para a Cie ˆncia e a Tecnologia, Ministe ´rio da Cie ˆncia e da Tecnologia, Portugal.
References [1] Nunes CS, Mahfouf M, Linkens DA, Peacock JE. Modelling and multivariable control in anaesthesia using neural-fuzzy paradigms: Part I–—classification of depth of aneasthesia and development of a patient model. Artif Intell Med 2005 [in this issue]. [2] Zhang XS, Roy RJ, Huang JW. Closed-loop system for total intravenous anesthesia by simultaneously administering two anesthetic drugs. In: Chang HK, Zhang YT, editors. Proceedings of the 20th annual international conference of the ieee engineering in medicine and biology society, vol. 20, biomedical engineering towards the year 2000 and beyond piscataway. 1998. p. 3052—5. [3] King RE, Magoulas GD, Stathaki AA. Multivariable fuzzy controller design. Control Eng Pract 1994;2:431—7. [4] Linkens DA, Nyongesa HO. A hierarchical multivariable fuzzy controller for learning with genetic algorithms. Int J Control 1996;63:865—83. [5] Goldberg DE. Genetic algorithms in search, optimization and machine learning. Reading: Addison Wesley, 1989.
http://www.intl.elsevierhealth.com/journals/aiim
Modelling and multivariable control in anaesthesia using neural-fuzzy paradigms Part II. Closed-loop control of simultaneous administration of propofol and remifentanil Mahdi Mahfouf a,*, Catarina S. Nunes b, Derek A. Linkens a, John E. Peacock c a
Department of Automatic Control and Systems Engineering, The University of Sheffield, Mappin Street, Sheffield S1 3JD, UK b ´tica Aplicada, Faculdade de Cie ˆncias da Universidade do Porto, Departamento de Matema Rua do Campo Alegre 687, 4169-007 Porto, Portugal c Department of Anaesthesia, Royal Hallamshire Hospital, Glossop Road, Sheffield S10 2JF, UK Received 6 May 2004; received in revised form 24 November 2004; accepted 6 December 2004
KEYWORDS Patient model; Anaesthesia; Analgesia; Fuzzy logic; Feedback control; Multivariable
Summary Objective: Part II of this research study is concerned with the development of a closed-loop simulation linking the patient model as well as the fuzzy relational classifier already introduced in Part I with a control algorithm. The overall architecture is in fact a system advisor, which provides information to the anaesthetist about the adequate infusion-rates of propofol and remifentanil simultaneously. Methods and material: The developed fuzzy multivariable controller includes three rule-bases and takes into account the synergetic interactions between the above drugs and uses such knowledge to achieve rapidly the desired depth of anaesthesia (DOA) level. Results: The result of the study is a closed-loop control scheme, which adjusts efficiently the infusion-rates of two drugs in response to DOA changes. This controller can either be used in an advisory mode or closed-loop feedback mode in the operating theatre during surgery. Conclusion: It is hoped that this control scheme coupled with the patient model presented in Part I of this study will be used routinely in the operating theatre in the very near future. # 2005 Elsevier B.V. All rights reserved.
* Corresponding author. Tel.: +44 114 222 5607; fax: +44 114 222 5624. E-mail address: [email protected] (M. Mahfouf). 0933-3657/$ — see front matter # 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.artmed.2004.12.005
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1. Introduction The infusion-rate of the anaesthetic drug is titrated according to the patient’s requirements, so as to maintain a certain level of depth of anaesthesia (DOA). The patient’s clinical signs and/or brain signals are used by the anaesthetist to determine the adequate infusion-rate. In addition, the anaesthetist also establishes the required infusion-rate of the analgesic drug, based on the patient’s response to surgical stimuli. A closed-loop control system of DOA will help the anaesthetist adjust simultaneously the infusion-rates of the anaesthetic and analgesic drugs. The majority of the researches in the area are mainly concerned with the automatic control of the anaesthetic drug, whereas the analgesic is controlled manually by the anaesthetist. However, in this research the objective is a multivariable control structure for both drugs. The patient model presented in Part I of this research study [1], described adequately the effects and interactions of the two drugs in the presence of surgical stimuli. This model will be used to develop a control algorithm relating to the administration of both drugs. The study of the interactions between the anaesthetic propofol and the analgesic remifentanil helps to determine the ideal combination of infusion-rates. This study will also represent a practical guide for the anaesthetist, which would help him/her learn how to adjust the amount of drug infused and hence improve the patient’s comfort. First, the patient model will be tested using a series of open-loop simulations with different infusion profiles. Second, the closed-loop structure will be presented, showing the links between the patient model, the DOA classifier, i.e. the fuzzy relational classifier (FRC) presented in Part I of this study, and the controller. Third, a multivariable fuzzy controller, developed with the anaesthetist’s cooperation, is also described. This controller establishes the infusion-rates of propofol and remifentanil simultaneously based on the level of DOA, the concentrations of the drugs and the surgical stimuli. The performance of the controller is tested under different conditions.
Figure 1 files.
Propofol and remifentanil infusion-rate pro-
sampling-time of 30 s. The first 1500 s relate to the induction-phase, followed by the maintenancephase. It is worth noting that the recovery-phase is not simulated. The FRC for DOA developed in Part I is applied to the wavelet-extracted audio evoked potential (AEP) features, as determined by the maintenance-phase model. In the induction-phase, the FRC uses only the cardiovascular parameters, a change in systolic arterial pressure (DSAP) and a change in heart-rate (DHR), for the classification of DOA, since the AEP features are not modelled during this phase. Fig. 1 shows the infusion-rate profiles for propofol and remifentanil used in the first simulation. The propofol infusion-rate is very similar to the profile of patient Pat1, while the remifentanil infusion-rate follows a typical profile during a surgical procedure (i.e. high at induction and constant during maintenance). Fig. 2 shows the HR and SAP as simulated by the patient model, using the Infusion Profile 1. The wavelet-extracted AEP features as simulated by the patient model are used to classify the DOA level by the FRC during the maintenance-phase. The DOA level is shown in Fig. 3. The model performs
2. Open-loop simulation results using the patient model The patient model developed in Part I was tested in open-loop simulations with different infusion profiles for propofol and remifentanil. Although three different infusion profiles were studied only one profile will be presented and discussed here expressing lessons learned from the experiment. The simulations were performed for 7200 s (120 min) with a
Figure 2 SAP and HR as determined by the patient model following the infusion profile of Fig. 1.
Modelling and multivariable control in anaesthesia using neural-fuzzy paradigms
Figure 3 DOA as classified by the FRC using the waveletextracted AEP features.
adequately, describing the effects of the stimulus level on SAP and HR. The effect concentrations are within the ranges of the maintenance-phase and this is reflected in the model’s response. The DOA level as determined by the FRC cannot be compared with the anaesthetist’s classification, since this is a simulated infusion profile. However, the classification is not unreasonable as it reflects the AEP features from the model.
3. Closed-loop structure The closed-loop simulation system links the patient model, the FRC of DOA and the control algorithm. Fig. 4 shows the block diagram comprising the different components of the closed-loop system during the maintenance-phase of anaesthesia. The FRC uses the parameters from the patient model to classify DOA. Finally, a control structure maintains an adequate DOA level by adjusting the infusionrates of propofol and remifentanil, which are the inputs of the patient model. The fuzzy controller algorithm is discussed in the next section.
3.1. Multivariable fuzzy control In general, propofol is used for maintenance of anaesthesia in combination with an opioid, hence, the anaesthetist is confronted with the dilemma of whether to vary propofol or the opioid. In reference [2], Zhang and co-workers reported on a closed-loop system for total intravenous anaesthesia by simultaneously administering propofol and fentanyl.
Figure 4 system.
Block diagram of the closed-loop simulation
209
They studied the interaction between propofol and fentanyl for loss of response to surgical stimuli using an unweighted least squares non-linear regression analysis with human data. A look-up table of optimal and awakening combinations of concentrations was built, and used to determine the fentanyl set point according to the propofol set point. To our knowledge, this is one of the only studies in simultaneous control of anaesthetic and analgesic drugs. Multivariable fuzzy controllers are not easy to design. There are several studies on the design of multivariable controllers, however, very few target biomedical systems. In reference [3], King et al. described an integrated development system, which permits the controller designer to test hypothesis, examine the effect of changes in the controller parameters and perform a complete off-line simulation of a proposed multivariable fuzzy controller. This system was applied to industrial processes and tested off-line with mathematical models. In reference [4] Linkens and Nyongesa developed a hierarchical multivariable fuzzy controller for learning with genetic algorithms (GAs). This controller was applied to a simulation case study in anaesthesia, using mathematical models describing the action of atracurium and isoflurane (i.e. a muscle relaxant and an inhalational anaesthetic). This structure decomposes a complex multivariable fuzzy controller into several simple fuzzy controllers. The anaesthesiologist is confronted with the dilemma of whether to vary propofol or the opioid. Anaesthesiologists’ knowledge and experience is incorporated into the fuzzy control system as a set of linguistic rules. In addition, the interaction between remifentanil and propofol provides information that can be used to determine an appropriate combination of the two drugs. The controller is activated only at the start of the maintenancephase. The infusion-rates of both drugs are determined according to the DOA level and the surgical stimulus. The controller comprises three different blocks, corresponding to three possible values of DOA, i.e. the DOA level is at the target level (ok), or the DOA is lighter than that desired, or the DOA is deeper than that desired. The controller acts differently according to these three stages. The linguistic scheme of the controller is described as follows: If DOA is ok then no change; If DOA is light and if: - stimulus present then increase remifentanil (via remifentanil rule-base 1); - no stimulus then increase propofol (via single input/single output–—SISO fuzzy proportional integral- PI controller);
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If DOA is deep and if: - no stimulus and remifentanil high then decrease remifentanil; - no stimulus and remifentanil normal and propofol high then decrease propofol; - no stimulus and remifentanil normal and propofol normal then decrease remifentanil; - stimulus and propofol High then decrease propofol (via SISO fuzzy PI controller) and increase remifentanil (via remifentanil rulebase 2); - stimulus and propofol normal then decrease remifentanil. The above multivariable controller is based on linguistic rules that interact with three decision tables, two of which are rule-bases for a change in remifentanil infusion-rate (rule-base 1 and rulebase 2) and the other represents a SISO fuzzy PI controller for a change in propofol infusion-rate. These rule-bases are described in the next sections. It is important to note that a minimum effect concentration of propofol and remifentanil is required at all times, in order to ensure unconsciousness and prevent arousal. The minimum values were established as 2250 and 3.5 ng/ml for propofol and remifentanil, respectively. If the concentrations decrease to these minimum values, then the infusion-rate of propofol will be increased by 0.2 mg/s and remifentanil will be increased by 0.025 mg/(kg min).
3.2. Remifentanil rule-base 1 This determines the increment in the remifentanil infusion-rate when the DOA level is light in the presence of a stimulus. The rule-base is shown in Table 1, and uses the perceived stimulus and the change in DOA error as inputs. The change in remifentanil infusion-rate is normalised between [0,1], but the maximum value of the variable was established as 0.03 mg/kg/min. When DOA is light, higher changes in remifentanil infusion-rate are required in order to respond to the stimulus and increase the level of DOA. Table 1 Rule-base 1 describing the change in remifentanil infusion-rate using the stimulus level and the change-in-error of DOA Stimulus Change in DOA error Negative Zero
Positive small Positive big
Low Small Small Medium Medium Small Medium Medium High Medium Big Big
Big Big Big
Table 2 Rule-base 2 describing the change in remifentanil infusion-rate using the stimulus level and the change-in-error of DOA Stimulus
Low Medium High
Change in DOA error Negative big
Negative small
Zero
Positive
Small Small Small
Small Small Medium
Small Medium Medium
Small Medium Big
3.3. Remifentanil rule-base 2 This is used when the DOA is deep, but there is a stimulus present and an increment in the remifentanil infusion-rate is necessary. Rule-base 2 is shown in Table 2. The perceived stimulus level and the change-in-DOA error are used to determine the change in remifentanil infusion-rate, similarly to rule-base 1. In rule-base 2, the smaller increment in the infusion-rate is predominant, since the DOA is deep and the remifentanil is increased for its analgesic properties. In rule-base 1, the DOA is light, hence, remifentanil is increased as a response to the stimulus and for its synergistic properties in order to increase the DOA level.
3.4. SISO fuzzy PI controller for propofol The fuzzy PI controller was designed to control the change in the infusion-rate of propofol, in order to achieve and maintain a steady level of DOA and to minimise the amount of drug infused. The fuzzy PI controller uses the error (target DOA minus measured DOA) and the change-of-error as inputs. The controller’s rule-base is presented in Table 3. The variables are normalised between [1,1]. The maximum level of change in the infusion-rate was set to 4000 mg/h considering the maximum
Table 3 Rule-base of the fuzzy proportional integral controller for DOA; the inputs are the error and the change-of-error, and the output is the change-in-propofol infusion-rate; the membership functions are labelled: negative big (NB), negative small (NS), zero (ZE), positive small (PS) and positive big (PB) DOA error NB NS ZE PS PB
Change of DOA error NB
NS
ZE
PS
PB
NB NB NB NS ZE
NB NS NS ZE PS
NB NS ZE PS PB
NS ZE PS PS PB
ZE PS PB PB PB
Modelling and multivariable control in anaesthesia using neural-fuzzy paradigms
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conditions pre-set by the anaesthesiologist. The input scaling factors of the controller were optimised using a Genetic Algorithm (GA) [5] with a performance index given by: N1 N1 X X P index ¼ l1 kjeðkÞj þ l2 uðkÞ (1) k¼0
k¼0
where e(k) is the error, u(k) the propofol infusionrate and N is the number of simulation samples. The weighting parameters l1 and l2 were chosen to place more emphasis on the error or on the infusion-rate, so that an ideal balance between them can be reached. The GA was implemented using MATLAB1 for a population of 40 strings each of length 20, with a probability of crossover of 0.95 and a probability of mutation of 0.06. The optimisation was run with parameters l1 = 0.4 and l2 = 0.6, which were found to be representative of the specifications for the control system, giving a bigger weight to the infusion profile. The GA optimisation led to the value of 0.375 for the error scaling factor and 7.77 for the change of error scaling factor.
Figure 6 Propofol and remifentanil infusion-rates as determined by the multivariable controller in Simulation 1.
Figure 7
SAP and HR for Simulation 1.
4. Simulation results The multivariable fuzzy controller was used in the closed-loop simulations with the patient model. Note that the controller only starts acting at 1500 s (i.e. after the induction-phase).
4.1. Simulation 1
Figure 8 Propofol and remifentanil infusion-rates as determined by the multivariable controller during the maintenance-phase; DOA target change to ok/deep at 3000 s.
In the first simulation shown here, a typical infusion profile of both drugs was used for the inductionphase. The DOA level for this simulation is shown in Fig. 5. The ok DOA level was achieved at 1740 s. Fig. 6 shows the infusion-rates of propofol and remifentanil for this simulation. The ok DOA level was achieved rapidly, and the controller maintained efficiently a stable DOA level by keeping both infusion-rates constant. In fact, the controller only changed the propofol infusion-rate in
this simulation. Fig. 7 shows the variations in the cardiovascular variables during this simulation. Next, the same conditions as in Simulation 1 were considered, but with a set-point change to ok/deep level at 3000 s. Fig. 8 shows the infusion-rates of propofol and remifentanil as determined by the multivariable controller during the maintenancephase. The DOA level is shown in Fig. 9.
Figure 5 DOA level using the multivariable controller in Simulation 1.
Figure 9 DOA level using the multivariable controller during the maintenance-phase; DOA target change to ok/ deep at 3000 s.
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Figure 10 SAP during the maintenance-phase; DOA target change to ok/deep at 3000 s.
The ok/deep DOA level was reached at 3060 s, i.e. only 60 s after the set-point change. The multivariable controller reacted to the set-point change by increasing the remifentanil infusion-rate, due to the stimulus level present in the system. This increase in the remifentanil infusion-rate increased the level of analgesia and also potentiated the effect of propofol. Fig. 10 shows the SAP during the maintenance-phase. The decrease in SAP following an increase in the remifentanil infusion-rate shows how a deeper level of depression was achieved, leading to the ok/deep DOA level. Therefore, the multivariable controller is taking advantage of the synergism between propofol and remifentanil for efficient control of DOA.
M. Mahfouf et al.
Figure 13 Propofol and remifentanil infusion-rates as determined by the multivariable controller in Simulation 2 during the maintenance-phase (zoom of Fig. 12).
contrast to the previous simulation, the multivariable controller changed both infusion-rates in response to the ok/deep DOA level at approximately 1700 s. Fig. 13 shows the infusion-rate of both drugs during the maintenance-phase. The multivariable controller decreased the propofol infusionrate first and then gradually decreased the remifentanil infusion-rate. The controller was able to determine an adequate combination of the two drugs, thus achieving and maintaining the ok DOA level.
4.2. Simulation 2 Simulation 2 considered a different remifentanil infusion profile during the induction-phase, while the propofol infusion profile was the same as the one used in Simulation 1 thus producing different initial conditions for the controller. Fig. 11 shows the DOA level in Simulation 2, while the propofol and remifentanil infusion-rates are shown in Fig. 12. In
Figure 11 DOA level using the multivariable controller in Simulation 2.
Figure 12 Propofol and remifentanil infusion-rates as determined by the multivariable controller in Simulation 2.
5. Discussions and conclusions The objective of a control system for DOA is to determine the best infusion-rates of the anaesthesiologist and analgesic drugs, helping the anaesthetist to decide which drug should be changed in response to different events. The developed patient model was used to construct and test a multivariable controller for simultaneous administration of remifentanil and propofol during the maintenancephase. Anaesthesiologists’ experience was incorporated into the control structure using linguistic rules. According to the different profiles for the DOA level and for the surgical stimuli, the multivariable controller defines the required change in the infusion-rates of the two drugs. The controller was able to adjust the remifentanil infusion-rate according to the stimulus intensity, and takes advantage of the synergistic interaction to change appropriately the propofol infusion-rate. Propofol is titrated to lower infusion-rates, decreasing the amount of drug infused, and speeding up recovery. In addition, the controller ensures adequate analgesia by titrating the remifentanil according to the stimulus. The multivariable fuzzy controller was tested under different simulations, and responded efficiently to different induction profiles and set-point changes.
Modelling and multivariable control in anaesthesia using neural-fuzzy paradigms
Fuzzy logic techniques have been shown to be efficient in incorporating human knowledge for better solutions in biomedicine. The complexity of DOA and the unavailability of large data sets make this an ideal application for fuzzy logic based concepts. The FRC, the Takagi—Sugeno—Kang pharmacodynamic models and the multivariable fuzzy controller were combined successfully in an integrated structure to provide a closed-loop simulation platform for future anaesthesia monitoring and control in the operating theatre during surgery. It is hoped that such clinical trials will take place in the near future.
Acknowledgements The authors would like to thank the anonymous reviewers for their comments, which helped to improve the quality of this paper. In addition, the second author wishes to acknowledge the Portuguese Foundation for Science and Technology for their financial support during the course of this
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project: Fundac¸a ˜o para a Cie ˆncia e a Tecnologia, Ministe ´rio da Cie ˆncia e da Tecnologia, Portugal.
References [1] Nunes CS, Mahfouf M, Linkens DA, Peacock JE. Modelling and multivariable control in anaesthesia using neural-fuzzy paradigms: Part I–—classification of depth of aneasthesia and development of a patient model. Artif Intell Med 2005 [in this issue]. [2] Zhang XS, Roy RJ, Huang JW. Closed-loop system for total intravenous anesthesia by simultaneously administering two anesthetic drugs. In: Chang HK, Zhang YT, editors. Proceedings of the 20th annual international conference of the ieee engineering in medicine and biology society, vol. 20, biomedical engineering towards the year 2000 and beyond piscataway. 1998. p. 3052—5. [3] King RE, Magoulas GD, Stathaki AA. Multivariable fuzzy controller design. Control Eng Pract 1994;2:431—7. [4] Linkens DA, Nyongesa HO. A hierarchical multivariable fuzzy controller for learning with genetic algorithms. Int J Control 1996;63:865—83. [5] Goldberg DE. Genetic algorithms in search, optimization and machine learning. Reading: Addison Wesley, 1989.