Adaptive Parameter Estimation-Based Drug Delivery System for Blood Pressure Regulation

  • Bharat Singh
  • Shabana Urooj
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 701)


Controlled drug delivery system (DDS) is an electromechanical device that enables the injection of a therapeutic drug intravenously in the human body and improves its effectiveness and care by controlling the rate and time of drug release. Controlled operation of mean arterial blood pressure (MABP) and cardiac output (CO) is highly desired in clinical operation. Different methods have been proposed for controlling MABP; all methods have certain disadvantages according to patient model. In this paper, we have proposed blood pressure control using integral reinforcement learning-based fuzzy inference system (IRLFI) based on parameter estimation technique. To further increase the safety of the proposed method, a supervisory algorithm is implemented, which maintains the infusion rate within safety limit. MATLAB simulation depends the model of MABP, elucidate the ability of the suggested methodology in designing DDS and control postsurgical MABP.


Drug delivery system Fuzzy inference system Mean arterial blood pressure Sodium nitroprusside Maximum a posteriori estimators 

1 Introduction

The nonstop advancement of innovative medication delivery structures is driven by the essential expand remedial action while limiting negative symptoms [1]. Controlling MABP of a postsurgical patient is a key variable to describe the system’s operating point. MABP can be controlled by sodium nitroprusside (SNP), which is a drug that reduces the tension in the arteries walls; thus, blood pressure reduces [2].

A closed-loop proportional–integral–derivative controller (PID) control has been widely applied in many real applications. Also, some other well-known methods are Ziegler–Nichols method, IMC method, and loop-shaping method [3]. Since single controller is not capable of achieving the desired control objective of clinical person and this problem has been acknowledged by some authors, this prompts a variety of adaptive control systems being proposed in the course of the most recent three decades; researches are based on self-tuning regulators, multiple-model adaptive control (MMAC), model-reference adaptive control and model-predictive control, as well as fuzzy control and rule-based nonlinear control [4, 5]. Among these methods, fuzzy base PID controller can be used in drug delivery system for guaranteed stability and accuracy. It has been proposed by many researchers that use of fuzzy logic controller (FLC) improves the PID controller response as far as taking care of progress in a working point for nonlinear systems and update the controller parameters the problem associated with the PID-based tuning methods is that it avoids the essential property of fuzzy PID controller, which may result in poor response [6]. To overcome this problem a fractional order fuzzy proportional–integral–derivative (FOFPID) can be implemented with a digital filter [7]. FOFPID are also used in speed regulation of machines such as turbines. According to Takagi–Sugeno (TS), it makes controller more robust because of the presence of fractional order integral and derivative function and can easily model the nonlinear system [8].

Blood pressure as being an instable one, and hence in order to reduce the fluctuation, digital filters can be used. All the research so far is focused on predetermined response of patients. Somehow if we could determine the patient’s parameter-based current surroundings, health, and physiological condition. In this way, controller can optimize and can provide best control. For this purpose, in this work, we have used the parameter estimation technique to predict patient’s response. The oscillatory change in MABP through the administration of SNP can be decreased by using parameter estimation-based automatic drug delivery controller.

2 Problem Formulation

An intravenous drug delivery system intended here in such a way can only be more flexible and adaptive to any drug, since it is related to medical field and hence life of patients, a drug delivery system should be trustworthy with high accuracy and sensitivity. A patient’s response to mean arterial pressure for SNP infusion can be described by following dynamic model, originally proposed by slate [ 9].
$$MAP(t) = p_{o} + \Delta p(t) + p_{d} (t) + n(t)$$
Here, MAP is actually mean arterial pressure, initial blood pressure is p 0 , the change in blood pressure is Δp(t), and pd(t) is considered as rennin reflex action. When vasodilator drug is infused to body, then body reaction to particular drug is known as renin reflex, and n(t) is a stochastic noise. A continuous-time deterministic model describing the transfer function relating the output as change in MAP with respect to input as administration rate of SNP is given in (2) as [10]
$$\Delta p(t) = \frac{{ke^{ - ds} (1 - \alpha e^{ - ms} )}}{1 - \tau s}d(s)$$
where Δp(t) is the blood pressure change, drug sensitivity is k, d(s) is infusion rate of SNP, k is patient’s sensitivity, α is the recirculation constant of drug, d is the initial transport delay, m is considered as the recirculation time delay, and \(\tau\) is the system time constant. By Discretizing the continuous-time model shown in (2) with the sampling time of 15 s gives (3) the following discrete-time model [10]:
$$\Delta p(t) = \frac{{q^{{ - d_{1} }} (b_{o} + b_{1} q^{{ - d_{2} }} )}}{{1 - aq^{ - 1} }}I(t)$$
where q − 1 indicates a unit delay operator, are the parameters of the numerator, and b0, b1 are the delays achieved from the sampling the continuous-time model. A refined automated drug delivery system is necessary to overcome the limitations of the past research works such as latency and instability. A scale factor is introduced in order to equalize the quantity or to give more weightage or less weightage to a function by which it was multiplied with.

3 Control Algorithms

Patient can have different sensitivities to particular drug, and response may vary with time, so closed-loop monitoring model has been presented to improve the patient’s response with respect to intravenously infused drug. Parameter estimation technique has been used to detect the patient’s response, based on response integral learning-based fuzzy system is implemented to regulate whole process. We provide an automated drug delivery system using parameter estimation with fuzzy inference system which is shown in Fig. 1.
Fig. 1

Automated drug delivery model using parameter estimation-based learning with fuzzy inference system

3.1 Fuzzification

Fuzzy logic controller is implemented here by considering two inputs: one is error and another one is rate of change of error required to be fuzzified in addition to a single output. The entire process requires a sufficient number of membership functions (MFs), thereby causing in a larger rule base for two inputs. These membership functions have their own functional area in the input space as a membership value. Fuzzy rules depend on membership function and fuzzy rules, each MF is assigned an SNP infusion rates in (mcg/kg/min). Initially at normal blood pressure (90 mm Hg), the administration rate is zero, because the difference between the MABP and set point is zero. While the MABP increases through its membership curve, the infusion rate also increases with respect to the fuzzy rules. Another important thing is the infusion time with MABP of 160 mm Hg an infusion rate of 5 mcg/kg/min may reduce the MABP to normal level (90 mm Hg). The infusion time is adopted from IRL, which also has its effect on the decision of infusion rate as long as it updates the membership function by its learned things from the real time process, there upon it increases the robustness of the system in any uncertain conditions.

The all useful condition can be described by combination of these variables without doubt. After the selecting input variables, the shape and numbers of fuzzy sets have to be decided to sense them. A robust fuzzy partition has been proposed to keep the rule legibility (4)
$$\forall S_{n} \, \in \,S_{n} ,\sum\limits_{j = 1}^{{N_{L} (n)}} {\mu L_{n}^{j} (S_{n} ) = 1}$$
where the input variable S n is defined as N L (n) the fuzzy sets number. This type of separation suggests that there will be no additional fuzzy sets “activated” for an input variable (Fig. 2).
Fig. 2

Fuzzy partition of variable. a Input membership function. b Output membership function

With the help of input variables N1 and number of labels help us determine the number of rules N of the database. They can be defined as fuzzy sets with a membership function of the following properties:
$$\mu o_{m}^{i} (Y_{m} ) = \left\{ {\begin{array}{*{20}l} {1.0,} \hfill & {Y_{m} = o_{m}^{i} } \hfill \\ {0.0,} \hfill & {otherwise} \hfill \\ \end{array} } \right.$$

The function Y m is approximated by O m . Here, O m is considered as conclusion vector that is related with rules. The actuated rule set represented by A is characterized by rule set that authenticates this condition. The maximum number of rules in A is equal to 2 N T according to fuzzy partition.

3.2 Estimation of Parameters by Maximum a Posteriori Estimators (MAPE)

Parameter estimation helps the system in updating the online variation, being an integral reinforcement learning method the control decisions also depend on the variations is the real-time ongoing process. A posterior distribution contains all the knowledge about the unknown quantity; therefore, we can use the posterior distribution to find point or interval estimates of. To obtain a parameter estimate is to choose the value of that maximizes the posterior probability density function (pdf).

The parameters of premises and consequents and structure identification will be identified by MAPE, which concerns partitioning the input space. Suppose that there are N observations and y = [y(x1)…y(x N )] T have been obtained with the design X = (x1,…, x N ) and assume π(.) indicate a prior posterior density function for one or a subset of it. Bayesian estimation depends on the construction of the posterior (pdf) (6) for θ as [11],
$$\pi_{X,y} (\theta ) = \frac{{\varphi_{X,\varPhi } (y)\pi (\theta )}}{{\varphi_{X}^{*} (y)}}$$

With \(\varphi_{X}^{*} (y)\) the p.d.f. of the marginal distribution of the observations y and \(\varphi_{X,\varPhi }\) the p.d.f. of their conditional distribution given θ. When the y k are independent, y k  = y(x k ) having the density \(\varphi_{xk} ,\overline{\theta } \left( y \right)\) with respect to the measure \(\mu_{xK}\) on \({\text{Y}}_{\text{xk}} \, \subset \,R\).

4 Result and Discussion

Each pair of membership function, i.e., input as well as output membership function, is related with the fuzzy set rules; the relation is in the form of logic operator may be either “or”/“and”. Our proposed IRLFI model is verified for adaptive control of the drug delivery system, the reference path shows a drop of MAP from 150 to 90 mmHg initially, and finally settled the level at 90 mmHg. The desired change of normal blood pressure is shown in Fig. 3.
Fig. 3

Time versus normal mean arterial blood pressure

The rate of change of SNP infusion depends on the variation in MABP level at a discrete-time period, the actual infusion rate of the SNP is shown in Fig. 4. At any time period, the infusion rate would be zero if the MABP level is within 90 mmHg and in case if the MABP level of a patient increases beyond the set point, the infusion rate also increases with respect to the increase in MABP level and the path followed by infusion rate for a certain time interval is given in graph below.
Fig. 4

Time versus SNP infusion rate

5 Conclusion

In an automated system which controls physiological variables, there are essentially three components sensors, a controller, and an actuator or an infusion pump. In this paper, the main focus was on control strategies using IRLFI system, by directly controlling the mechanical determinants of infusion pump. Our automated drug delivery system using IRLFI allows simultaneous control of MABP with stability and accuracy, and therefore, it can be used for the controlling of MABP and can also be applicable for any other infusion rate control system. The proposed method displays the ability of controller to improve the patient’s response over less time, and also it improved the overall response of the closed-loop control of drug delivery system even in the presence of disturbances. In future, we can use this adaptive parameter estimation method for healing of different chronic diseases.


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© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Bharatividyapeeth’s College of EngineeringNew DelhiIndia
  2. 2.School of EngineeringGautam Buddha UniversityGreater NoidaIndia

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