Human cardiovascular system helps in supplying sufficient amount of oxygen to all vital organs and tissues through the flow of blood. Due to current lifestyle and other circumstances, people are suffering from various diseases related to cardiovascular system of which surgery remains to be the only treatment. The most common heart surgeries performed in humans irrespective of age are coronary artery bypass grafting and

The initial work is carried out by observing the clinical data such as MAP, CO and dosage of NAR and NG during surgery. The modeling of cardiovascular system (CVS) is performed by considering four compartments such as left heart, systemic circulation, right heart and pulmonary circulation as shown in ^{1, 2, 3, 4}.

The differential equations were derived based on the following three equations-

Where

where

cardiovascular system which helps in short term regulation of MAP by altering the circulatory parameters such as Systemic Resistance (R_{sys}), Maximum Elastance (Els_{max}) and Unstressed Ventricular Volume (V_{us_ven}) ^{5, 6}. The change in circulatory parameters is provided based on the feedback action called as baroreflex action ^{7}. This feedback system based on circulatory parameters is modeled as-

Where

Where _{as}) and CO. Sensitivity analysis is performed to verify the tolerance of the baroreceptor to continuous regulation of MAP by modifying the circulatory parameters based on the MAP obtained from cardiovascular model. The circulatory parameters are modified continuously using the Equations (8) – (10). This analysis is carried out only with MAP because CO values increased with respective MAP value and MAP is the major parameter that has to be regulated during high and low pressure conditions. From this analysis, it is verified that baroreceptor is responsible for short term regulation of MAP and the dynamic action provided by baroreceptor. This has been presented in

Drug modeling equations for NAR and NG are derived based on the pharmacokinetics and pharmacodynamics of the drugs^{8}. The infusion of drug along with the blood flow in the left heart Q_{lv} is represented as-

where x represents the compartment being considered,

Where Conc_{dg}= M/V, M is the mass of drug in compartment and V is the volume of the compartment and _{1/2 }is the half-life of the drug in the compartment. This drug model is then combined with CVS-BR model^{9, 10, 11}. The open-loop response is obtained by varying the infusion rate based on the clinical analysis. The response depicted the effect of drugs on CVS-BR model. From the response, MAP and CO is observed and the transfer function model is developed as-

Since there are two manipulated variables and two control variables, it becomes a multi-input-multi-output (MIMO) system. To determine the maximum effect of drug on hemodynamic variables and to eliminate loop interactions, relative gain array (RGA) analysis was utilised and stability analysis performed using Neider-Linski index (NI) as shown in

RGA (Ʌ) | Loop Interaction | NI |

0.5233 | Cardiac Output (CO)/ Noradrenaline Mean Arterial Pressure (MAP)/ Nitroglycerine | 1.911 |

In this work, the conventional PI controller is developed based on optimization algorithm^{12}. BFOA is used to determine the optimal values of PI controller in regulating MAP and CO by controlling the infusion rate of NAR and NG ^{13, 14}. This algorithm is developed based on the foraging behaviour of

This process involves the movement of ^{15}. The bacterium alters between these two modes in its entire lifetime. Assuming that h^{th} bacterium is taking l^{th }chemotatic, m^{th} reproductive, v^{th} elimination and dispersal step which is represented as

where δ represents the vector of elements in random direction which lies between (-1 1).

Here a group of bacteria convene together to form a swarm, and they travel in stable spatio-temporal patterns. They move in nutrient gradient with concentric patterns of swarms. They appear like a semisolid matrix when placed with single nutrient chemo-effector. The bacteria move in groups with high bacterial density in search of the richest food location.

This reproduction stage is reached at every chemotactic step. At this stage, the healthier bacterium gets divided into two bacteria whereas the low prioritized healthy bacterium ultimately dies i.e., the bacterium yielding lowest value of objective function and this stage maintains the population size of the bacteria at constant value.

When some sudden changes occur in the environment where a group of bacteria live, it may kill them or disperse them into a new environment. Suppose if there is a rise in temperature in the location where bacteria live with high nutrient concentration, it may lead to death of bacteria. Sometimes they get relocated as it happens in human intestines. This process is simulated in the algorithm by randomly initializing the new replacements over the search space. The parameters involved in BFOA to determine the optimized value of PI controller is shown in

Parameters | Values |

Dimension of search space, d | 2 |

Total number of bacteria in the population, Tb | 10 |

number of chemotactic steps, Nch | 5 |

Swimming length, Nsl | 4 |

Number of reproduction steps, Nrep | 4 |

Number of elimination and dispersal events. Ned | 2 |

Probability of elimination and dispersal, Ped | 0.25 |

Number of steps involved in tumble movement, x(h) | 0.05 |

PI controller is the most commonly used control algorithm among the conventional controllers. It works with the combination of proportional and integral mode^{16, 17, 18}. The ideal form of PI controller is represented as

Where, C_{out} is the controller output

e(t) is the process error where e(t)=SV-PV

SV is the reference set point and PV the measured process output

K_{c} is the controller gain (tuning parameter) and τ_{i} is the integral time (tuning parameter), where K_{C}=K_{P} and K_{I}=1/τ_{i}.

BFOA is employed to search for optimal controller parameters to minimize the time domain objective function. A performance index can be defined by the Integral of Time Absolute Error (ITAE) of the drug infusion and hemodynamic variables (MAP and CO). Accordingly, the objective function J is set to be-

Based on this performance index J, optimization problem can be stated as: Minimize J subjected to-

Based on BFOA, the PI values are optimized and best solution is obtained. The closed loop response obtained using BFOA is then compared with conventional PI controller tuned using relay tuning method as shown in

The corresponding controlled infusion rate of NAR and NG are depicted in

The performance of the controller is evaluated using time domain specifications and their respective values, discussed in

Controllers | Hemodynamic Parameters | Rise Time (secs) | Peak Time (secs) | Settling Time (secs) | Steady state error (Ess) |

PI-ZN | Mean Arterial Pressure (MAP) (mmHg) | 66.9761 | 231.4122 | 330.8738 | 0.0754 |

Cardiac Output (CO) (ml) | 152.1225 | 314.4125 | 505.5532 | 3.5953 | |

PI-BFOA | Mean Arterial Pressure (MAP) (mmHg) | 4.3561 | 7.2387 | 17.4577 | 1.5342e-6 |

Cardiac Output (CO) (ml) | 4.8997 | 8.4 | 18.5881 | 0.0148 |

A combined four compartmental model of cardiovascular system with baroreceptor model and drug model is presented. In this work, the optimization technique is used to determine the optimal values of PI controllers in regulating Mean Arterial Pressure (MAP) and Cardiac Output (CO) by controlling the infusion rate of Noradrenaline (NAR) and Nitroglycerine (NG). This simulation study helped in analyzing automatic regulation of hemodynamic parameters which can provide better control allowing anaesthetists to focus on more critical issues which will result in reduction in amount of drugs infused and their side effects. This will lead to less time spent by patients during post surgical treatment and above all provides a safer platform during surgical procedure. This work also helps in analyzing the effects of drugs on physiological variables instead of conducting clinical trials on animals as the initial step. This work can be further extended by developing a switching based controller for the infusion of two drugs simultaneously. In order to carry out this work, interaction between the two drugs must be analyzed ad clinical trials can be carried out which helps in fine-tuning of the controller.