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A mathematical model of mechanical ventilator describes its behavior during artificial ventilation. This paper purposes to create and simulate Mathematical Model (MM) of Pressure Controlled Ventilator (PCV) signal. This MM represents the respiratory activities and an important controlled parameter during mechanical ventilation—Positive End Expiration Pressure (PEEP). The MM is expressed and modelled using periodic functions with inequalities to control the beginning of inspiration and expiration durations. The created MM of PCV signal is combined with an existing multi compartmental model of respiratory system that is modified and developed in the internal parameters—compliances (C) to test created MM. The created MM and model of respiratory system are constructed and simulated using Simulink package in MATLAB platform. The obtained simulator of mechnical ventilation system could potentially represent the pressure signal of PVC as a complete respiratory cycle and continuance waveform. This simulator is also able to reflect a respiratory mechanic by changing some input variables such as inspiration pressure (IP), PEEP and C, which are monitored in volume, flow, pressure and PV loop waveforms. The obtained simulator has provided a simple environment for testing and monitoring PCV signal and other parameters (volume, flow and dynamic compliance) during artificial ventilation. Furthermore, the simulator may be used for studying in the laboratory and training ventilator’s operators.

Mathematical model (MM) of the respiratory system plays an actual important role in developing and correcting ventilators work that supports breathing during patient’s treatment. The pressure controlled ventilation (PCV) is considered as one of mechanical ventilators, used for supporting many respiratory failures. The MM of mechanical ventilations has been proposed in several medical and scientific studies [

Previous studies [

Recently, modelling and representation of the dynamic characteristics of PCV signal have considered mechanical ventilation system as a pure pneumatic system [

This paper demonstrates a modelling and simulation of PCV signal, which includes the respiratory activities and an important controlled parameter during ventilator support―Positive End Expiration Pressure (PEEP) [

The main novelty of this work is the new method of MM and its simulation. Moreover, it is combined with developed lung simulator to obtain mechanical ventilation system simulator. This new simulator can monitor the modeled pressure signal of PCV and output modulated signal (flow and volume) as continuous waveforms. Also, it can present the pressure wave of respiratory cycle and PV loop, which are viewed as important indicators of the patient’s response to mechanical ventilation [

The rest of this paper consists of two sections and a conclusion. The first section describes the detailed process of MM, followed by an MM simulation process. The second section presents the results through illustration curves e.g. volume and flow, followed by the interpretation of the obtained results. The conclusion provides the significant achievements and suggestions for future work.

The mathematical model for mechanical ventilation system in this work includes the pressure support signal from pressure controlled ventilation (PCV) device, which is applied to a modified compartment model of respiratory system.

The pressure signal generated by PCV in reality takes the same breathing behavior as the respiratory activities. The suggested MM for the PCV pressure signal should also represent the breathing behavior and activities.

_{in}), and rise time of pressure (t).

The pressure signal of PCV depends on setting parameters in real PCV device that formulates the typical waveform shown in

was used to express MM of PCV pressure signal as seen in the Equations (1), (2) and (3).

where

P(t)―the pressure signal of PCV, PEEP―positive end-expiratory pressure, P_{aw}―the pressure in respiratory airway;

t―time equal to (T_{in}-t), and T_{ex}―expiration time.

As shown from these equations, the ventilation pressure or supported pressure of PCV was represented as time based function P(t) to reflect the respiratory activities―inspiration and expiration during ventilation. These activities are generated from a change in the pressure amount in lungs during inspiration and expiration pro- cesses. The inspiratory pressure (IP) and expiration pressure (EP) were represented in mathematical model by P_{aw} and PEEP. Thus, the IP was represented using Equation (1) and (2), during inspiration time (T_{in}), whereas EP was represented using Equation (3) assuming that EP = PEEP.

The parameter values of the new MM are variables and have some limits in this research work initial as values for normal adult and this can be clarified as follows:

Inspiratory time is usually set for adult in average 0.7 to 1.0 second, but can be increased to reach the targeted tidal volume (V_{T}) or when the patient remains hypoxic in spite of a plateau pressure (Pplat) > 30 cm H_{2}O; and usually keeps the I/E ratio at 1:2 or 1:3 [

The total cycle time (TCT) equals inspiratory time plus expiratory time (T_{in} + T_{ex} = TCT), then the TCT for one breath could be written by Equation (4)

The respiratory rate (RR) or frequency is obtained by dividing the number of breaths per minute to TCT as shown in Equation (5). Practically, the normal RR is 12 - 18 or 10 - 20 breaths/min [

The rise time of pressure (t) is one of the setting variables in PCV shown in

The IP is based on the setting pressure value that considers the working pressure on the supplying system. Practically, it is determined by looking for a V_{T} of 5 - 6 ml/kg [_{2}O taking into account its peak or plateau pressure (Pplat)―PEEP used as a starting point and modified to reach desired V_{T} [_{2}O (in some cases 35 cm H_{2}O as maximum) to prevent lung from injury.

PEEP set 5 - 10 cm H_{2}O, initially, has a value depending on other variables such the degree of hypoxemia and the expiration tidal volume, which keeps the range between 4 - 6 mL/kg [_{2}O.

Knowing all parameters related to time (T_{in}, T_{ex}, RR, ratio I/E), and other setting variables will determine the beginning of each breathing cycle as well as the beginning of inspiration and expiration using shown Inequalities (6), (7) and (8).

where n is the number of breathing cycle and T equals to 1 sec.

The suggested MM for PCV was integrated as a pressure that is applied to a multi compartment model for lungs. This model has been suggested by Michael C. [

Where: all parameters refer to normal lungs:

The current flow (I) represents airflow, and the voltage source (V) demonstrates the applied pressure produced by the ventilator.

R_{C} = 1 cm H_{2}O/L/s, shows the airflow resistance of the central airways.

R_{P} = 0.5 cm H_{2}O/L/s, demonstrates the resistance of the peripheral airways.

C_{L} = 200 ml/cm H_{2}O, represents the capacity of the alveoli.

C_{W} = 200 ml/cm H_{2}O, indicates the chest wall capacity, which is in series with the alveoli

C_{S} = 5 ml/cm H_{2}O represents a shunt capacitance known as “dead space” of air, which does not participate in the exchange of oxygen and carbon dioxide between air and blood.

C_{T} demonstrates the total compliance of airways and has variation values that depend on C_{L} and C_{W}.

Therefore, the transfer function has expressed the airflow output as a function of pressure input I(s)/P(s) using the transform function in Equation (10). This equation is assumed to represent a healthy respiratory system with above stated parameters as well as the total capacity of lungs equal to

Equations (9) and (10) can be modified to represent other cases of lungs. Basically, the change in lung status is associated with two airway categories: the larger or central airways and the smaller or peripheral airways. The incoming air in these ways is shown by the connection of the lung (C_{L}) and chest-wall (C_{w}) compliances in series [_{T}). _{T} in denoted and dashed line as a variable and therefore C_{T} is calculated according to

Thus, to get C_{T}, we make change in C_{L} and C_{w}, which were modified to reflect the lung mechanic status and examine the behavior of the created MM using lung simulator. In PCV case, the compliance is called elastic compliance because the IP is kept during the entire set inspiratory time and the driving pressure set on the ventilator corresponds to the alveolar pressure [_{T} has been calculated experimentally for lung simulator at about 10 mL/cm H_{2}O in normal case [_{T}, initially 6 mL/kg ideal body weight [

PCV parameters, particularly, IP and PEEP, were modified according to lung mechanic taking into account that the maximal lung distending pressures is 25 - 35 cm H_{2}O plateau [_{2}O for some cases such patients with acute respiratory distress syndrome (ARDS) [

The proposed method is based on deriving displaying signals and waveforms developed simulator to monitor dynamic characteristics behavior. This research work was limited to show the pressure wave of complete respiratory cycle and its activities and waveforms of P, V, F. In addition the dynamic compliance was provided to demonstrate the dynamic effect of change in PV loop that refers to the ratio change of applied pressure to the change of applied volume as shown in Equation (12).

The F, V and P signals were simulated as continuous waveforms with time course similar to reality; however, their simulation are based on the modelling of pressure as input signal presented in Section 2.1.

The created MM of PCV signal was modelled and simulated as a generator for the pressure, which was combined with developed lung simulator to obtain simulator of the mechanical ventilation system.

Parameters | IP, cm H_{2}O | PEEP, cm H_{2}O | Rise rate (time) | Inspiratory time (T_{in}) | Respiratory rate in breaths/min |
---|---|---|---|---|---|

Normal case | 25 | 5 | 1:2 | 1 | 20 |

Another case | 27 | 8 | 1:2 | 1 | 20 |

The results of study shown that the results of new MM of PCV signal and its simulator are linked with respiratory system to represent the artificial mechanical ventilation system and illustrate the curves of input pressure wave over complete respiratory cycle, waveforms of P, V, F and PV loop.

The setting parameter values illustrated in _{in}, T_{ex}, as well as complete cycle time (3 seconds) and RR, whereas IP and PEEP are changed as seen in _{2}O to prevent lungs from injuries.

The created and simulated MM reflected the different values in waveforms and showed convergence of the pressure waves of complete respiratory cycle and its activities with waveforms in reference standards [

Following up the waveforms of P, V and F is intended to test the effectiveness of created MM combined with lung simulator by changing their input parameters. _{T} 10 ml/cm H_{2}O. In this figure, time duration equals approximately 21 sec. for 7 complete respiratory cycles illustrating that RR repeats 20 per 60 sec. Also, this figure represents the pressure waveform (the first waveform from above) of created MM, which has constant pressure at 30 cm H_{2}O and constant width of inspiration and expiration time during the entire course (T_{in} = 1 sec. and T_{ex} = 2 sec.).

The general view of flow waveform in

Also,

The obtained result of P, F, V waveforms showed convergence and corresponding in their shapes and characteristics with standard reference curves [

_{2}O, PEEP = 8 cm H_{2}O and C_{T} = 10 mL/cm H_{2}O. The shown duration equals approximately 21 sec. for 7 complete respiratory cycles.

The P, V and F waveforms in this figure remain unchanged in time course. Pplat increased compared to P waveform in

The obtained results show clearly the ability of obtained simulator to reflect and represent the changes in its input parameters.

The PV loop of PCV is studied with different C_{T} values to test the ability of proposed method by monitoring the output variables behavior. _{T} at 0.5 L. In addition,

_{2}O, PEEP = 5 cm H_{2}O, T_{in} = 1 sec., t = 0.1, and C_{T} given in _{T} is set with different values as a change in lung mechanic to monitor the response of created MM and simulator. _{T}. The value of _{L} and C_{W} in simulator as seen in

The obtained curves, seen in _{T} (look at cases No. 1, 2, 3 in

Parameters of PCV signal | Case number | C_{T}, ml/cm H_{2}O | _{2}O | C_{L}, ml/cm H_{2}O | C_{w}, ml/cm H_{2}O | V_{T}, ml |
---|---|---|---|---|---|---|

IP = 25 cm H_{2}O, | 1 | 15 | 0.07 | 0.2 | 0.1 | 320 |

PEEP = 5 cm H_{2}O, | 2 | 10 | 0.1 | 0.2 | 0.2 | 500 |

T_{in} = 1 sec., t = 0.1, | 3 | 7 | 0.14 | 0.2 | 0.5 | 700 |

method are reflected the changes in lung mechanic.

The created MM of PCV signal and developed lung simulator composed the mechanical ventilation system simulator. This simulator provides the significant results of representation pressure signal of PCV and displays the output dynamic characteristics of pressure (flow and volume waveforms) and monitors them as the continuous waveforms.

Furthermore, simulator has the ability to monitor the PV loop through which approved it is that the proposed method reflects the changes in the lung mechanic.

In this work, we developed and simulated the mathematical model of the PCV pressure signal and combined it with modified lung simulator to obtain the mechanical ventilation system simulator. The MM is able to represent setting parameters and its limit values, and therefore represents the real PCV device by simulator. This simulator is able to monitor the input and output signals as continuous waveforms to mimic the real artificial ventilation process. Thus, it leads to the use of the simulator as a tool for studying the behavior of PCV ventilators in training students in laboratories to be familiar with curves and variables.

These model and simulator demonstrated the ability to reflect the changes in lung mechanic. However, the simulator requires expansion of the modulated internal parameters of respiratory mechanic. Thus, this new model requires further investigation and validation.

Noman Q.Al-Naggar, (2015) Modelling and Simulation of Pressure Controlled Mechanical Ventilation System. Journal of Biomedical Science and Engineering,08,707-716. doi: 10.4236/jbise.2015.810068