In the field of general anesthesia, target-controlled infusion of anesthesia (TCIA) is a well-recognized technique, found among the most widely recognized closed-loop drug delivery methods [
Pharmacokinetics (PK) is a branch of pharmacology that is related to the study of how a living organism affects the infused drug, while pharmacodynamics (PD) is the branch of pharmacology that is concerned with the study of how the infused drug affects a living organism. Considering all the important parameters, precise pharmacokinetic (PK) and pharmacodynamic (PD) models are developed, on the basis of which TCIA is introduced. The efficacy of TCIA depends on the precision of the PK model, the mutual relation between the effect site concentration and measurement, and the device used for drug monitoring and control. To achieve radical control over the drug delivery, continuous propofol measurement is performed throughout total intravenous anesthesia (TIVA).
In [
Setting the drug concentration degree at the effect site to the desired targeted degree is precisely the main goal of TCIA. The drug is infused and measured in the blood that is the first compartment. In such case, by using the pharmacodynamic models, measurement of drug concentration in the blood provides an estimation of effect site drug concentration. Once the drug concentration is quantified in the blood, its control and regulation to a desired targeted level can also be accomplished.
In a closed-loop drug delivery system, different control techniques are employed in order to control the drug infusion rate, which leads to the control of drug concentration in the blood and therefore at the effect site. In [
The phase lead and phase lag are two of the most commonly utilized control architectures for designing control systems with the root locus or bode compensation approach used. In [
This paper focuses on the comparative analysis of five different control techniques and their design process, to automate the delivery of an infusion system, based on propofol measurement in a closed-loop feedback system. The comparative analysis is conducted with respect to different time domain specifications like gain, percentage overshoot, settling time, and rise time. The design process of phase-lead, lag, lead-lag, and cascaded lead controllers is performed by applying the principles of the root locus technique [
This paper is organized as follows: Section
The following equations, as shown by Myers et al. in [
After the introduction of a bolus injection or propofol infusion, a first-order kinetics model depicts the drug elimination dynamics in blood circulation, mathematically described as
The model represents an exponential decay in drug concentration over time
When the drug is injected in the blood at an infusion rate
In our simulation study, a dilution chamber of constant volume is used to mimic the human blood. The exponential decay of drug concentration in this chamber models the actual PK decay of the drug in the blood. The drug elimination rate
A particular amount of drug injection increases the concentration level inside the chamber. This new concentration of the chamber is defined as initial concentration, that is,
The dynamics of concentration in the dilution chamber can equally be symbolized in the derivative form as
The bolus dose, when merged with continuous infusion results in the time-dependent concentration, changes in the dilution chamber as
The system model in terms of its differential equation is represented by (
Figure A controller to achieve the target concentration of propofol A constant volume dilution chamber, representing the first compartment, which is our plant A dilution pump to regulate the continuous dilution rate of the background electrolyte A volume control pump to maintain a constant volume in the chamber.
Block diagram of a feedback-controlled drug infusion system.
To analyze the performance of the feedback-controlled drug infusion system, a dilution chamber with constant volume
The open-loop analysis can be described by the following two steps. The first step describes the exponential decay of the dilution chamber.
In the second step, the same volume is maintained in the dilution chamber, that is, 10 mL; however, the concentration of propofol in the chamber is set to zero, that is,
The design parameters of the open-loop analysis are shown in Table
Simulation parameters.
Variable | Description | Value |
---|---|---|
|
Infusion rate | Controller calculated infusion rate |
|
Concentration of the dilution chamber | Time dependent |
Target | Target concentration of the dilution chamber | 0.038 mMol |
|
Volume of the dilution chamber | 10 mL |
|
Speed of dilution pump/dilution flow rate | 0.04 mL/s |
|
Elimination rate |
0.004 |
Step response of the uncontrolled dilution chamber.
This section presents the closed-loop design analysis of the drug infusion system. In order to achieve the target-controlled drug infusion in the first compartment or the dilution chamber, five different control techniques are designed and implemented in a closed-loop feedback system. The fundamentals of the techniques employed in the meticulous design process of all five dynamic controllers are referred to [
The proportional-integral-derivative (PID) controller is one of the most commonly used and universally accepted control algorithm used in control industry. The PID controller is popular for its attributes partly to their wide-range operation of robust performance and partly to their simplified operation. The term PID describes its three main components, a proportional control term (
Effects of
Response | Rise time | Overshoot | Settling time | Steady state error |
---|---|---|---|---|
|
Decrease | Increase | No effect | Decrease |
|
Decrease | Increase | Increase | Eliminate |
|
No effect | Decrease | Decrease | No effect |
By keeping the same parameters, as used in the open-loop analysis, a PID controller is designed to yield the targeted concentration. In the study, conducted by Myers et al. in [
PI parameters.
PI parameters |
|
|
---|---|---|
1 | 3,000,000 | 55 |
2 | 6,000,000 | 55 |
3 | 750,000 | 55 |
4 | 4,500,000 | 5.5 |
5 | 4,500,000 | 55 |
6 | 4,500,000 | 550 |
7 | 4,500,000 | 5500 |
8 | 4,500,000 | 55,000 |
9 | 4,500,000 | 550,000 |
The open-loop dynamics plus the system modeling represented in (
Block diagram of a PID controller connected with the dilution chamber in a closed-loop feedback system.
By adding equal numbers of poles and zeros, a phase-lag controller provides an appreciable amount of relative stability to a system, yielding slow response time. In a phase-lag controller, the pole of the controller is placed closer to the origin as compared to the zero of the controller. The generalized transfer function of a phase-lag controller is given as
To address the steady state error, a phase-lag controller is designed, using the root locus method. The target is to achieve a compensated steady state error value of
In the first step, the
Using the final value theorem [
For step input,
Now, by solving for the ratio of the zero to the pole, we get the following final equation:
In the second step, the location of the zero and pole of the controller is determined using an algebraic approach. In order for a point
Graphical illustration of root locus existence shaping and the concept of pole zero placement using a pseudo system
Root locus existence using two poles of a system
Pole zero placement using system
Pole zero placement using system
Mathematically,
Now, to delineate the pole zero placement (design process) of the
Now, considering the
The appropriate placement and zero-angle contribution ( The The
Root locus of the system incorporated with the designed lag controller.
The design details of the phase-lag controller are illustrated in Figure
Illustration of the design details of the phase-lag controller.
The controller, when merged with the dilution chamber, resulted in the following closed-loop transfer function
By adding equal numbers of poles and zeros, a phase-lead controller provides an appreciable improvement in the transient response of a system, increasing the open-loop gain in some cases. Increasing gain leads to more susceptibility to noise. In case of a lead controller, the zero of the controller is placed closer to the origin as compared to the pole of the controller. The generalized transfer function of a phase-lead controller is given as
To achieve a faster response time, a phase-lead controller is designed, using the root locus method. In a root locus plot, the location where the asymptotes cross the real line is called the center of gravity or the centroid, defined in (
Graphical illustration of effects of a phase-lead controller on root locus using a pseudo system
Root locus of the pseudo system
Root locus of the pseudo system
Now, to achieve the desired output, the actual phase-lead controller is designed using the following steps:
The zero of the controller is kept near the imaginary axis, as compared to the pole of the controller. The pole of the controller is kept far from both the zero of the controller and the imaginary axis. The distance between the pole and the zero of the controller is kept large. The added pole has a larger negative value than the added zero, as zero resides near the origin.
The design details of the phase-lead controller are illustrated in Figure
Illustration of the design details of the phase-lead controller.
The designed phase-lead controller and the dilution chamber in a closed-loop system, resulted in the following transfer function:
Both phase-lead and lag controllers have their own advantages and disadvantages as discussed above. Practical systems often demand certain rigorous specifications, where a combination of both lead and lag controllers can be practicable. The generalized transfer function of a phase-lead-lag controller is
To attain both faster response and relative stability, a phase-lead-lag controller is designed using the same design principles as have been discussed in the phase-lead and phase-lag sections. The following two steps describe the whole design process.
In the first step, a phase-lead controller is designed using the following procedure:
The pole of the controller is placed farther into the left half plane away from the imaginary axis. The zero of the controller is placed near the imaginary axis. The pole and zero of the controller are placed at such points, so that the distance between them is large enough and the centroid is shifted further to the left half plane.
In the second step, a phase-lag controller is designed with a different approach due to the succeeding reason. In a traditional lag controller design, the main target is to eliminate steady state error without varying the transients and root locus shaping. Instead of implementing the conventional approach, here, the main target is to achieve Unlike the traditional pole zero placement of a lag controller, zero of the controller is placed away from both the pole of the controller and the imaginary axis. Unlike in (
A small amount of root locus shaping due to
Root locus of the system incorporated with the designed lead lag controller.
The design details of the phase-lead-lag controller are illustrated in Figure
Illustration of the design details of the phase-lead lag controller.
The controller when incorporated with the dilution chamber resulted in the following closed-loop transfer function:
Generally, a cascaded phase-lead controller is applied to attain a very fast response time. Applying two lead controllers in series, merged with the plant in a closed-loop feedback system, can significantly accelerate the system response time. In case of a drug infusion system, a faster response is required because a small delay or overshoot in drug delivery can lead to severe consequences. The transfer function of a traditional cascaded lead controller is
To achieve a much higher response time, a cascaded phase-lead controller is designed. The design principles are similar to what has been discussed in the phase-lead section. To achieve maximum desired output, consider the following:
Both poles of the two lead controllers are placed farther from the imaginary axis into the left half plane. Both zeros of the two lead controllers are placed near the imaginary axis. Maximum distance between the poles (
Mathematically,
The design details of the cascaded lead controller are illustrated in Figure
Illustration of the design details of the cascaded lead controller.
The designed cascaded phase-lead controller, when incorporated with the drug infusion system, resulted in the following closed-loop transfer function:
To understand the first-order exponential decay of the dilution chamber, open-loop characteristics of the system were assessed in two steps in Section
Exponential decay in concentration with propofol concentration = 0.01 mM and dilution flow rate = 0.04 mL/s.
In the second step, identification and validation of the open-loop gain and steady state analysis of the system is performed. Theoretically, the steady state gain is calculated from (
Propofol infusion, in parallel with the continuous dilution, is carried out for a certain duration. After some time, an equilibration state is achieved between the infusion and elimination rate of propofol. At such point,
The above infusion-dilution equilibrium led to the succeeding conception. For a particular value of dilution flow rate, a defined infusion rate results into a specific steady state concentration point. For a dilution flow rate of 0.04 mL/s, at five different infusion rates, the open-loop gain and the corresponding steady state concentration is evaluated and illustrated in Figure
Steady state concentration at five different infusion rates with a flow rate,
Five different control schemes were employed in order to achieve the target drug concentration in a closed-loop feedback system. The PI controller was tuned with nine different parameters, shown in Table
Comparative analysis of closed-loop results using PI, lead, lag, and lead-lag controllers.
Closed-loop results using the PI controller tuned with 9 different parameters in Table
Closed-loop result using the PI controller tuned with the fifth parameter in Table
Comparative analysis of closed-loop results using lag, lead, and lead-lag controllers
Figure
Figure
Comparative analysis of closed-loop results using cascaded lead controller and PI controller.
Closed-loop result using the cascaded lead controller
Comparative analysis of closed-loop results using PI controller and cascaded lead controller
Figure
Controller performance characteristics (a comparative analysis).
Controller | Gain | Rise time (sec) | Settling time (sec) | Overshoot (%) |
---|---|---|---|---|
Lag | 30 | 2.9120 | 5.2720 | 0 |
Lead | 500 | 0.0573 | 0.2492 | 0 |
Lead-lag | 500 | 0.0880 | 0.1980 | 0.7164 |
PI | 4.5 × 106 | 4.8822 × 10−6 | 8.6935 × 10−6 | 0 |
Cascaded lead | 500 | 4.0862 × 10−7 | 7.2779 × 10−7 | 0 |
To target and maintain the drug concentration level in the human body, represented by a dilution chamber, five different control schemes are designed and implemented with a simulation-based comparative analysis between each of them. The comparative simulation study elucidates the effectiveness of each controller in targeting and maintaining the propofol level.
The drug concentration and dynamics are represented using the constant volume dilution chamber, due to the succeeding two
In our
The comparative analysis of the closed-loop results along with the time domain characteristics illustrated in Table
Targeting and maintaining the drug concentration level are indispensable in human anatomy and hold central importance as an application of control technology in biomedical engineering. For a closed-loop drug infusion system, simulation of five different controllers has been demonstrated using MATLAB, which shows encouraging results for the delivery of propofol. For faster and stable response of the drug infusion system, comparative analysis for abovementioned controllers has been performed and analyzed. Following the simulation analysis and the results obtained, the cascaded lead controller and PI controller show the finest results with the cascaded lead controller showing relatively better results than all the other control techniques. The assertive results provide a key platform to implement this model in real time for automatic drug delivery. Such quintessential paradigm in the domain of biomedical control jargon would lead to faster response induction and reduction of clinical workload in the field of total intravenous anesthesia.
The authors of this manuscript declare that there is no conflict of interest regarding the publication of this paper.