Research on the Identification Method of Respiratory Characteristic Parameters during Mechanical Ventilation

Abstract

In order to enhance the accuracy of ventilator parameter setting, this paper analyzes two identification methods for respiratory characteristic parameters of non-invasive ventilators and invasive ventilators. For non-invasive ventilators, a respiratory characteristic parameter identification method based on a respiration model is established. In this method, the patient’s respiratory sample set is obtained through non-invasive measurements. Experimental results demonstrate that the mean relative error of pulmonary elastance identification was 14.25%, and the mean relative error of intrapulmonary pressure identification was 12.33% using the Romberg integral algorithm. For chronic patients using non-invasive ventilators, the fault-tolerant space for ventilator parameter setting is large; this method meets the requirement of auxiliary setting of non-invasive ventilator parameters. For invasive ventilators, a respiratory characteristic parameter identification method based on the AVOV–BP neural network is established. In this method, the patient’s respiratory sample set is obtained through real-time invasive measurements. Even with small sample datasets, experimental results show that the mean relative error of pulmonary elastance identification and intrapulmonary pressure identification were both 0.22%. For critically ill patients using invasive ventilators, the fault-tolerant space for ventilator parameter setting is small; this method meets the requirement of auxiliary setting of invasive ventilator parameters.

1. Introduction

In clinical practice, ventilators are widely used as an effective means of artificial replacement of autonomic ventilation. Improperly set ventilator parameters can lead to man–machine asynchronous problems, resulting in safety issues such as inadequate oxygen supply, lung damage, difficulty in weaning off the ventilator, and even life-threatening situations for patients. Medical staff typically set ventilator parameters such as ventilation volume and pressure based on the patient’s weight and their own clinical experience. However, the characteristics of the respiratory system vary from person to person, including differences in respiratory rhythm and lung elasticity based on factors such as age, gender, physical condition, and illness. Addressing these individual variations by obtaining each patient’s respiratory system characteristics is crucial for setting precise ventilator parameters and resolving man–machine asynchronous issues.

Respiratory compliance is a parameter used to describe pulmonary respiratory ability and is also known as pulmonary elastance. In numerical terms, respiratory compliance and pulmonary elastance are reciprocal to each other. Pulmonary elastance serves as an important basis for setting the airway pressure limit value of the ventilator. Experimental results from the literature [1] indicate that decreased pulmonary elastance leads to increased air flow amplitude and tidal volume, which potentially results in hyperventilation. Conversely, increased pulmonary elastance may result in hypoventilation [2]. Therefore, it is essential to set ventilator parameters according to the patient’s pulmonary elastance in order to maintain proper mechanical ventilation efficiency.

(1)

Identification of respiratory characteristic parameters of non-invasive ventilators

The structure of the human respiratory system is commonly described by respiration model; the respiratory parameters such as pulmonary elastance and airway resistance can be obtained by parameter identification of the respiration model. The mechanical structure of the human respiratory system is usually equivalent to the electrical form [2,3,4,5,6,7]. At present, there are three main kinds of research on respiration models during mechanical ventilation: respiration models based on pneumatic system theory, respiration models based on fluid mechanics theory, and respiration models based on differential equations.

Non-invasive ventilators are typically used for patients who have clear consciousness and unobstructed respiratory tracts. Patients generally receive respiratory support treatment through mouth and nose masks.

• Respiration model based on pneumatic system theory

In the literature [1,8,9,10], the mechanical ventilation system is equivalent to a pneumatic system. The established respiration model includes an airflow equation, a pressure equation, and a volume equation. Key parameters in the model include respiratory compliance, airway resistance, and equivalent effective area of throttle. Pulmonary elastance is defined as a constant that is easy to calculate but cannot describe its nonlinear characteristics.

In the literature [1], the air flow dynamic characteristics of the mechanical ventilation of a lung simulator are studied. In the literature [8], an aviation oxygen supply system based on a mechanical ventilation model is studied. In the literature [9], the dynamic characteristics of a mechanical ventilation system with spontaneous breathing are studied. In the above literature, the experimental apparatus comprises a ventilator, a flow sensor, a tube, an artificial simulating lung, a pressure sensor, a data acquisition card, and a computer. The simulation results of the respiration models align with experimental measurement results, verifying the accuracy of the respiration models. The influence of key parameters in the respiration model is studied in the papers.

In the literature [10], an online estimation method for respiratory parameters based on a pneumatic model is studied. The pressure and tidal volume measured by the experimental system are used as input and output data for the respiration model, followed by the use of the recursive least-square method to identify the parameters of the respiratory model.

• Respiration model based on fluid mechanics theory

In the literature [11,12,13], a time-switching mechanical ventilation piecewise model is established based on fluid mechanics theory. This model, hereinafter referred to as the TS respiration model, includes a flow velocity equation, a flow rate equation, a tidal volume equation, an intrapulmonary pressure equation, and an airway pressure equation. Pulmonary elastance is defined as a constant. By setting ventilator parameters, simulation results for airway pressure, tidal volume, and static pulmonary pressure at the end of the ventilator pipeline can be obtained. The simulation results of the TS respiration model are consistent with the experimental measurement results, verifying its accuracy.

• Respiration model based on differential equation

In the literature [14,15,16,17], the second-order linear ordinary differential equation models of the respiratory system are presented. This model is hereinafter referred to as the DE respiration model. In the literature, pulmonary elastance is defined as a polynomial [14,15], as a basis function form of the RBF network [16], and as an output function form of the GRNN network [17]. These varying definitions increase the complexity of the respiration model but accurately describe dynamic respiratory processes due to pulmonary elastance, being time-varying variables.

In the literature [14,15], the fuzzy reasoning method is used to identify the parameters of the respiratory model. In the literature [16], the recursive least-square method is used to identify the parameters of the respiratory model.

Non-invasive ventilators are unable to measure the patient’s intrathoracic respiratory pressure, resulting in a crucial absence of respiratory data for identifying respiration model parameters. To address this issue, given that the literature [11,12,13] has confirmed consistency between the simulation data of the TS respiration model and measured data, this paper proposes to use the TS respiration model to simulate patients’ respiratory processes. The respiratory parameters of the respiration simulator are set, and airway pressure, flow rate, tidal volume, and pulmonary pressure data from the TS respiration model are sampled as input data for the DE respiration model. Pulmonary elastance and intrapulmonary pressure are then calculated using numerical integration algorithms and least-squares algorithms.

(2)

Identification of respiratory characteristic parameters of invasive ventilator

Invasive ventilators are commonly used for patients with impaired consciousness or obstructed airways who typically receive respiratory support treatment through tracheal intubation and tracheotomy. These invasive ventilators can provide real-time measurement of patients’ respiratory data, including intrathoracic respiratory pressure.

The neural network is a core technology in machine learning, which is widely applied in various fields such as pattern recognition, intelligent robots, prediction and estimation, biology, medicine, and economics.

In the literature [17], a respiratory sample set is composed of measured respiratory data, a PSO algorithm is used to optimize the smoothing factor of a GRNN network, and the PSO–GRNN network is used to predict pulmonary static pressure.

In the literature [18], the respiratory sample set is composed of oxygenation, ventilation, and acid-base balance respiration data. Based on fuzzy rules and neural networks, a ventilator parameter setting strategy is proposed using expert knowledge.

In the literature [19], a machine learning model is developed for estimating mechanical ventilation parameters for lung health. The model utilizes an inverse mapping of artificial neural networks with the Graded Particle Swarm Optimizer.

In the literature [20], an improved whale optimization algorithm is proposed and applied to optimize the parameters of the LSTM network. The SFE–LSTM model is established for predicting ventilator pressure. Experiments conducted on Kaggle’s open ventilator dataset verify the effectiveness of the model.

In this study, the neural network method was utilized to identify respiratory characteristic parameters of invasive ventilators. Firstly, a neural network model was established and then trained using measured respiratory samples. Subsequently, the model was tested with additional measured respiratory samples. Ultimately, the study achieved the identification of pulmonary elastance and the prediction of intrapulmonary pressure.

2. Materials and Methods

2.1. Identification of Respiratory Characteristic Parameters of Non-Invasive Ventilator

2.1.1. Construction of a Respiratory Sample Set Based on the TS Respiration Model

Mechanical ventilation is a type of fluid movement process that involves changes in the velocity of air flow ($v(t)$), flow rate ($F(t)$), tidal volume ($V(t)$), intrapulmonary pressure ($P_A(t)$), airway pressure ($P_{aw}(t)$), and their interrelationships. The physical process of increasing end-inspiratory breath-holding in intermittent positive pressure ventilation mode can be categorized into five distinct stages: inspiratory stage $(t_0,t_1)$, early end-inspiratory platform stage $(t_1,t_2)$, end-inspiratory equilibrium stage $(t_2,t_3)$, expiratory stage $(t_3,t_4)$, and end-expiratory stage $(t_4,t_5)$. According to the theory of fluid mechanics [11,12,13], $v(t)$, $F(t)$, $V(t)$, $P_A(t)$, and $P_{aw}(t)$ in the TS respiration model are as follows:

$$v(t)=\left\{\begin{array}{llll}\sqrt{K_1[P_S-P_A(t)]}& & & t\in (t_0,t_1)\\ \sqrt{K_3[P_{aw}(t)-P_A(t)]}& & & t\in (t_1,t_2)\\ 0& & & t\in (t_2,t_3)\\ \sqrt{K_4{P}_A(t)}& & & t\in (t_3,t_4)\\ 0& & & t\in (t_4,t_5)\end{array}\right.,$$

(1)

$$F(t)=-6\times {10}^{4}Av(t),$$

(2)

$$V(t)=\left\{\begin{array}{cc}0\hfill & \hfill t=t_0\\ [V(t-1)+{\displaystyle \frac{1000}{60}}Q(t)dt-L\cdot V(t-1)](1-{\displaystyle \frac{C_m}{C+C_m}})\hfill & \hfill t_0<t\le t_1\\ V(t-1)+{\displaystyle \frac{1000}{60}}Q(t)dt\hfill & \hfill t>t_1,V(t-1)+{\displaystyle \frac{1000}{60}}Q(t)>0\\ 0\hfill & \hfill t>t_1,V(t-1)+{\displaystyle \frac{1000}{60}}Q(t)>0\end{array}\right.,$$

(3)

$$P_A(t)=\left\{\begin{array}{ll}{\displaystyle \frac{V(t)}{C}}+PEEP& {\displaystyle \frac{V(t)}{C}}+PEEP>PEEP\\ PEEP& {\displaystyle \frac{V(t)}{C}}+PEEP\le PEEP\end{array}\right.,$$

(4)

$$P_{aw}(t)=\left\{\begin{array}{ll}0& t=t_0\\ P_A(t)+{\displaystyle \frac{K_2}{2}}\rho \cdot v^2(t)& t_0<t\le t_1\\ {\displaystyle \frac{P_{aw}(t-1)\cdot C_m-1000Q(t)/60}{C_m}}& t_1<t\le t_3,P_{aw}(t-1)\cdot C_m-1000Q(t)/60>P_A(t)\\ P_A(t)& t_1<t\le t_3,P_{aw}(t-1)\cdot C_m-1000Q(t)/60\le P_A(t)\\ P_A(t)-{\displaystyle \frac{K_5}{2}}\rho \cdot v^2(t)& t>t_3,P_A(t)-{\displaystyle \frac{K_5}{2}}\rho \cdot v^2(t)>PEEP\\ PEEP& t>t_3,P_A(t)-{\displaystyle \frac{K_5}{2}}\rho \cdot v^2(t)\le PEEP\end{array}\right.,$$

(5)

where L is the length of tube. $K_1$, the inspiratory ventilation resistance coefficient, can be calculated as follows:

$$K_1={\displaystyle \frac{16V_{T\max }^2{O}_p}{T_I^2{d}^4{\pi }^2{P}_s}},$$

(6)

where $V_{T_{\max }}$ is the maximum tidal volume, $O_p$ is the tidal volume adjustment opening, $T_I$ is the inspiratory time, and d is the inner diameter of the endotracheal tube.

$K_2$, the airway resistance coefficient during the inspiratory period, can be calculated as follows:

$$K_2=4-200d,$$

(7)

$K_3$, the resistance coefficient of the anatomical airway and endotracheal tube during the plateau period, can be calculated as follows:

$$K_3={\displaystyle \frac{2}{\rho K_2}},$$

(8)

$K_4$, the whole resistance coefficient during the expiratory period, can be calculated as follows:

$$K_4={\displaystyle \frac{1}{K_5}},$$

(9)

$$K_5=5-300d,$$

(10)

$K_5$, the resistance coefficient of the anatomical airway and endotracheal tube during the expiratory period, can be calculated as follows:

$L\cdot V(t)$ in Equation (3) can be calculated as follows:

$$L\cdot V(t)=6.2\times {10}^5({\displaystyle \frac{L}{2}}{)}^2\pi \sqrt{{\displaystyle \frac{2p_{aw}(t-1)dt}{\rho }}},$$

(11)

Based on the literature [11,12,13], the parameter values for Equations (1)–(5) are specified in

Table 1. Parameter values for the TS respiration model.

Symbol	Quantity	Value
$P_S$	gas source driving pressure	$2500 {\mathrm{P}}_{\mathrm{a}}$
C	respiratory compliance	$0.7 {\mathrm{mL}/\mathrm{P}}_{\mathrm{a}}$
$C_m$	ventilator compliance	$0.01 {\mathrm{mL}/\mathrm{P}}_{\mathrm{a}}$
$\rho$	gas density	$1.208 {\mathrm{kg}/\mathrm{m}}^3$
$T_I$	inspiratory time	1 s
$T_o$	expiratory time	2 s
$O_p$	adjustable opening of tidal volume	90%
PEEP	positive end-expiratory pressure	0 V
d	inner diameter of tracheal tube	7 mm
A	sectional area of tracheal tube	$\pi {\left(d/2\right)}^2$
L	pipe length (air leakage simulation)	0
EIP	percentage of end-inspiration platform time	40%
$V_{T_{\max }}$	maximum tidal volume	0.001

.

Figure 1 presents the waveform curves of $v(t)$, $F(t)$, $V(t)$, $P_A(t)$, and $P_{aw}(t)$ in four respiratory cycles.

2.1.2. The DE Respiration Model

The DE respiration model for patients undergoing mechanical ventilation without spontaneous respiration is as follows [14,15,16,17]:

$$P_{aw}(t)+a{\stackrel{·}{P}}_{aw}(t)=k_{1}V(t)+k_2{V}^2(t)+\dots +k_n{V}^n(t)+r_1\dot{V}(t)+r_2\left|F(t)\right|F(t)+b\dot{F}(t)+P_{eea}+e(t),$$

(12)

where the physical meanings of $F(t)$, $V(t)$, and $P_{aw}(t)$ are consistent with the TS respiration model. $P_{eea}$ is the alveolar pressure at the end of respiration. $\mathit{e}(t)$ represents error values, including equation error and measurement noise. $k_1,k_2,\dots ,k_n$, a, b, $r_1$, and $r_2$ are constant coefficients [18,19,20].

Pulmonary elastance is defined as a polynomial as follows:

$$f_E(V)=k_1+k_2\cdot V(t)+k_3\cdot V^2(t)+\dots +k_n\cdot V^{n-1}(t),$$

(13)

$P_A(t)$ is intrapulmonary pressure; it can be described as follows:

$$P_A(t)=f_E(V)V(t),$$

(14)

2.1.3. Matrix Representation of the DE Respiration Model

The matrix form of Equation (12) can be represented as follows:

$${\mathit{P}}_{aw}(t)={\mathit{\phi }}_p^{\prime }(t){\mathit{\theta }}_p+\mathit{e}(t),$$

(15)

$${\phi }_p(t)=\left[\begin{array}{l}-{\dot{P}}_{aw}(t)\\ V(t)\\ \dots \\ V^n(t)\\ \dot{V}(t)\\ \left|F(t)\right|F(t)\\ \dot{F}(t)\\ 1.0\end{array}\right],$$

(16)

$${\theta }_p=\left[\begin{array}{l}a\\ k_1\\ k_2\\ \dots \\ k_n\\ r_1\\ r_2\\ b\\ P_{eea}\end{array}\right],$$

(17)

where ${\phi }_p(t)$ is the given parameter matrix, and ${\theta }_p$ is the parameter matrix to be obtained.

2.1.4. Numerical Integration of the DE Respiration Model

The integral function is f(x), and compound numerical integration divides the integral interval [a,b] into n small intervals: $[x_{k-1},x_k]$, $(k=1,2,\dots ,n;x_0=a,x_n=b)$. Calculating the approximate value of the integral on each small interval ($I_k={\displaystyle {\int }_{x_{k-1}}^{x_k}f(x)dx}$), the approximate value of the integral on the interval [a, b] is ${\displaystyle {\int }_a^{b}f(x)dx}\approx {\displaystyle \sum _{k=1}^n{I}_k}$. This paper will compare four types of compound numerical integration formulas, including the trapezoidal formula, Simpson formula, Cotes formula, and Romberg formula. The definitions of the trapezoidal formula, Simpson formula, and Coates formula are shown in Equations (18)–(20).

$${\displaystyle {\int }_a^{b}f(x)dx}=T_n\approx {\displaystyle \frac{h}{2}}[f(a)+2{\displaystyle \sum _{k=1}^{n-1}f(x_k)+}f(b)],$$

(18)

$${\displaystyle {\int }_a^{b}f(x)dx}=S_n\approx {\displaystyle \frac{h}{6}}[f(a)+4{\displaystyle \sum _{k=0}^{n-1}f(x_{k+\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.})+}2{\displaystyle \sum _{k=1}^{n-1}f(x_k)+}f(b)],$$

(19)

$${\displaystyle {\int }_a^{b}f(x)dx}=C_n\approx {\displaystyle \frac{h}{90}}[7f(a)+32{\displaystyle \sum _{k=0}^{n-1}f(x_{k+\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.})+}12{\displaystyle \sum _{k=0}^{n-1}f(x_{k+\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.})+32{\displaystyle \sum _{k=0}^{n-1}f(x_{k+\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.})+}}14{\displaystyle \sum _{k=1}^{n-1}f(x_k)+}7f(b)],$$

(20)

where $h={\displaystyle \frac{b-a}{n}}$, $x_{k+{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}=x_k+{\displaystyle \frac{h}{2}}$, $x_{k+{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}}=x_k+{\displaystyle \frac{h}{4}}$, and $x_{k+{\displaystyle \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}}=x_k+{\displaystyle \frac{3h}{4}}$.

In order to reduce the amount of computation, a recursive algorithm is employed. The equal division number of integral intervals is taken as the value of $2^0,2^1,2^2,\dots ,2^n$. The recursive formula of the trapezoidal algorithm is as follows:

$$\left\{\begin{array}{l}{T}_{2^0}={\displaystyle \frac{b-a}{2}}[f(a)+f(b)]\\ T_{2^k}={\displaystyle \frac{1}{2}}{T}_{2^{k-1}}+{\displaystyle \frac{b-a}{2^k}}{\displaystyle \sum _{i=1}^{2^{k-1}}f[a+(2i-1)\frac{b-a}{2^k})}\end{array}\right.,k=1,2,\dots ,$$

(21)

The recursive formula of the Simpson algorithm is

$$S_n={\displaystyle \frac{4}{3}}{T}_{2n}-{\displaystyle \frac{1}{3}}{T}_n,$$

(22)

The recursive formula of the Coates algorithm is

$$C_n={\displaystyle \frac{16}{15}}{S}_{2n}-{\displaystyle \frac{1}{15}}{S}_n,$$

(23)

The recursive formula of the Romberg algorithm is

$$R_n={\displaystyle \frac{64}{63}}{C}_{2n}-{\displaystyle \frac{1}{63}}{C}_n,$$

(24)

The DE respiration model is a continuous time function model. Firstly, each respiratory cycle is sampled at equal intervals (T) to obtain a numerical discrete respiratory sample set, $F(n)$, $V(n)$, and $P_{aw}(n)$, where $n=0,1,2,\dots ,N-1$. Subsequently, numerical integration is performed with the dichotomy degree of the integral interval being 4, that is, the integral length is 16.

The recursive formula for numerical integration can be uniformly written as follows [14,15,16,17]:

$$\left\{\begin{array}{l}{y}_{(0)}^{(0)}(n)=8[y(n)+y(n+16)]\\ y_{(0)}^{(l)}(n)={\displaystyle \frac{1}{2}}{y}_{(0)}^{(l-1)}+{\displaystyle \frac{16}{2^l}}{\displaystyle \sum _{i=1}^{2^{l-1}}y[n+(2i-1)\frac{16}{2^l}]}\\ y_{(m)}^{(k)}(n)={\displaystyle \frac{4^m{y}_{(m-1)}^{(k+1)}-y_{(m-1)}^{(k)}}{4^m-1}}\end{array}\right.,$$

(25)

where $n=0,1,\dots ,N-17$ and $l=1,2,3$. When $m=0$, $k=0,1,2,3,4$, and y(n) is a trapezoidal sequence. When $m=1$, $k=0,1,2,3$, and y(n) is a Simpson sequence. When $m=2$, $k=0,1,2$, and y(n) is a Coates sequences. When $m=3$, $k=0,1$, and y(n) is a Romberg sequences.

2.1.5. Parameter Identification of the DE Respiration Model Based on Least-Square Method

${\varphi }_p(n)$ is the numerical integral operation result of ${\phi }_p(t)$, and $\mathit{e}(n)$ is the numerical integration result of $\mathit{e}(t)$. The numerical integration of Equation (15) is as follows [14,15,16,17]:

$${\mathit{p}}_{aw}(n)={\varphi }_p^T(n){\theta }_p+\mathit{e}(n),$$

(26)

$p$ is the transposed matrix of ${\mathit{p}}_{aw}(n)$, and $p={[p_{aw}(0),p_{aw}(1),\dots ,p_{aw}(N-17)]}^T$. $\Phi$ is the transposed matrix of ${\mathit{\varphi }}_p(n)$, and $\Phi =[{\varphi }_p(0),{\varphi }_p(1),\dots ,{\varphi }_p(N-17){]}^T$. $e$ is the transposed matrix of $\mathit{e}(n)$, and $e={[e(0),e(1),\dots ,e(N-17)]}^T$. Equation (26) can be rewritten as follows:

$$p=\Phi {\theta }_p+e,$$

(27)

${\widehat{\theta }}_p$ is the least-square estimate of ${\theta }_p$; it can be calculated as follows:

$${\widehat{\theta }}_p=({\Phi }^T\Phi {)}^{-1}{\Phi }^{T}p,$$

(28)

where ${({\widehat{\theta }}_p)}^T=[\widehat{a},{\widehat{k}}_1,{\widehat{k}}_2,\dots ,{\widehat{k}}_n,{\widehat{r}}_1,{\widehat{r}}_2,\widehat{b},{\widehat{P}}_{eea}]$.

2.1.6. Simulation Experiment and Results

$F(t)$, $V(t)$, and $P_{aw}(t)$ are all periodic functions. The sample data for one period of these functions was collected with a sampling interval of 0.05 s. As the respiratory cycle was set to be 3 s, there were a total of 60 sampling points (N = 60). These sampling data $F(nT)$,$V(nT)$, and $P_{aw}(nT)$ were used as input data for Equation (26), and the parameter matrix ${\widehat{\theta }}_p$ of the respiratory equation was then identified according to Equation (28).

When the pulmonary elastance of the DE respiration model is second order, $f_E(V)=k_1+k_{2}V(t)$. The results of the parameter matrix ${\widehat{\theta }}_p$ identified by four numerical integration algorithms are presented in

Table 2. The parameter identification results of four numerical integration algorithms.

Parameter	Trapezoidal Algorithm	Simpson Algorithm	Coates Algorithm	Romberg Algorithm
$\widehat{a}$	0.6219	0.6196	0.6499	0.6589
${\widehat{k}}_1$	0.8658	1.0714	1.0731	1.0706
${\widehat{k}}_2$	0.0014	8.6052 × 10−4	8.8286 × 10−4	8.9825 × 10−4
${\widehat{k}}_3$	−8.9085 × 10−7	−5.8165 × 10−7	−6.1577 × 10−7	−6.3135 × 10−7
${\widehat{r}}_1$	−0.6079	−0.2818	−0.2475	−0.2381
${\widehat{r}}_2$	0.3472	0.3347	0.3346	0.3346
$\widehat{b}$	10.8471	5.9058	5.2852	5.1619
${\widehat{P}}_{eea}$	−123.8772	−57.4888	−41.8671	−37.6651

.

Pulmonary elastance was calculated based on the identified values of ${\widehat{k}}_1$, ${\widehat{k}}_2$, and ${\widehat{k}}_3$, and the resulting $f_E-V$ curve is shown in Figure 2. The set value of pulmonary elastance in the TS respiration model was 0.7, that is, the pulmonary elastance value was ${\displaystyle \frac{1}{0.7}}=1.428 {\mathrm{P}}_{\mathrm{a}}/\mathrm{mL}$.

The mean absolute error and the mean relative error of pulmonary elastance identification in Figure 2 are presented in

Table 3. Identification errors of pulmonary elastance.

	Trapezoidal Algorithm	Simpson Algorithm	Coates Algorithm	Romberg Algorithm
MAE	0.2922 (Pa/mL)	0.2023 (Pa/mL)	0.2020 (Pa/mL)	0.2033 (Pa/mL)
MRE	14.18%	14.18%	14.16%	14.25%

.

The intrapulmonary pressure was calculated based on the $f_E-V$ curve in Figure 2 and Equation (14). The curve of intrapulmonary pressure fitted for one cycle is shown in Figure 3.

For the four fitting curves of intrapulmonary pressure in Figure 3, the mean absolute error and the mean relative error were calculated. The results of the calculation are presented in

Table 4. Identification errors of intrapulmonary pressure.

	Trapezoidal Algorithm	Simpson Algorithm	Coates Algorithm	Romberg Algorithm
MAE	149.6749	87.2144	72.9494	69.0957
MRE	25.67%	15.18%	12.93%	12.33%

.

The relationship curves between tidal volume ($V(t)$) and fitted intrapulmonary pressure ($P_A(t)$) by four numerical integration algorithms are presented in Figure 4. The value of the TS respiration model curve corresponds to the “tidal volume curve” and the “intrapulmonary pressure curve” in Figure 1.

From the experimental results, it is evident that the final fitted $P_A(t)$ curve and $V-P_A$ curve closely align with the TS respiration model. Additionally, the errors of the Romberg numerical integration algorithm for the identification of pulmonary elastance and intrapulmonary pressure are minimal. Although the mean absolute error of the identification results of pulmonary elastance in Table 3 is small, it is evident from Figure 2 that there is significant fluctuation in the identification results of pulmonary elastance within one cycle, and the fitting results exhibit instability.

2.2. Identification of Respiratory Characteristic Parameters of Invasive Ventilator

The BP neural network is widely utilized in pattern recognition and model parameter identification due to its simple structure, as well as its strong approximation and generalization abilities.

However, the weight regulation method of the BP neural network itself tends to be slow and prone to falling into the local minimum. In this paper, the African Vulture algorithm (AVOV) was employed to optimize the weights and thresholds of BP neural networks [21]. The AVOV–BP neural network was established to enhance the convergence speed and identification accuracy of the network [22]. Experimental comparisons were made on the identification results of respiratory characteristic parameters using three types of neural networks: the RNN network, the LSTM network, and the AVOV–BP network.

2.2.1. The AVOV Optimization Algorithm

The principles and steps of the AVOV optimization algorithm are as follows:

(1)

Determine the group optimal:

After initialization, the vultures should be grouped based on their mass. Vultures corresponding to the optimal solution will be placed in the first set, and those corresponding to the suboptimal solution will be placed in the second set.

$$\begin{array}{l}\mathrm{R}(\mathrm{i})=\left\{\begin{array}{c}{\mathrm{Best}}_1, if p_i=L_1\\ {\mathrm{Best}}_2, if p_i=L_2\end{array}\right.\\ (p(i)=F/{\displaystyle \sum _{i=1}^n{F}_i})\end{array},$$

(29)

where Best₁ represents the best vultures and Best₂ represents the next best vultures. L₁ and L₂ are two random numbers in the range [0,1], with their sum being equal to 1. F_i represents the fitness values of the first and second groups of vultures, and n represents the total number of vultures in both groups;

(2)

Calculate the population hunger rate:

$$t=h({\sin }^w({\displaystyle \frac{\pi I_i}{2M}}))+\cos ({\displaystyle \frac{\pi I_i}{2M}}-1),$$

(30)

$$F=(2k_1+1)\times z\times (1-{\displaystyle \frac{I_i}{M}})+t,$$

(31)

where F indicates that the vulture is full, $I_i$ is the current iteration number, M is the maximum iteration number, z is a random number between −1 and 1, h is a random number between −2 and 2, and $k_1$ is a random number between 0 and 1. When z < 0, the vultures are in a state of starvation, and when z > 0, the vultures are in full state. When $\left|F\right|>1$, the AVOA algorithm is in the exploratory phase. When $\left|F\right|<1$, the AVOV algorithm is in the development phase, and vultures forage around the optimal solution;

(3)

Exploration phase:

$$P(i+1)=\left\{\begin{array}{l}R(i)-D(i)\times F, if P_1\ge k_{P_1}\hfill \\ R(i)-F+k_2\times ((ub-lb)\times k_3+lb),else\hfill \end{array}\right.,$$

(32)

$$D(i)=\left|X\times R(i)-P(i)\right|,$$

(33)

where $k_{P1}$ is a random number between 0 and 1, P₁ is the predetermined exploration parameter, P(i + 1) is the vulture-position vector in the next iteration, F is the satiety rate of vultures in the current iteration, X is a constant, R(i) is one of the best vultures, k₂ and k₃ are random numbers between 0 and 1, and lb and ub are the upper and lower bounds of optimization, respectively;

(4)

Development phase:

• Stage I

When $\left|F\right|$ is between 0.5 and 1, AVOA enters stage I of the development phase, implementing two different rotational flight and siege strategies. The selection of the strategies is based on P₂, as expressed in Equation (34):

$$\mathrm{P}(i+1)=\left\{\begin{array}{l}\left\{\begin{array}{l}D(i)\times (F+k_4)-d(t)\hfill \\ d(t)=R(i)-P(i)\hfill \end{array}, \mathrm{if} P_2\ge k_{P_2}\right.\hfill \\ \left\{\begin{array}{l}{S}_1=R(i)\times {\displaystyle \frac{k_5\times P(i)}{2\pi }}\times \cos (P(i))\hfill \\ S_2=R(i)\times {\displaystyle \frac{k_5\times P(i)}{2\pi }}\times \sin (P(i)),else\hfill \\ R(i)-(S_1+S_2)\hfill \end{array}\right.\hfill \end{array}\right.$$

(34)

where $k_{P_2},k_4,k_5$ are all random numbers within the range of [0, 1], F is the satiety rate of vultures in the current iteration, and R(i) is one of the best vultures;

• Stage II

When $\left|\mathrm{F}\right|<0.5$, AVOA enters stage II of the development phase. The movement of the two vultures gathers many types of vultures on the food source, leading to a siege and an aggressive struggle for food among them. Strategies should be selected according to P₃, as expressed in Equation (35):

$$P(i+1)=\left\{\begin{array}{l}\left\{\begin{array}{l}{\displaystyle \frac{A_1+A_2}{2}}\hfill \\ A_1=Bes{t}_1(i)-{\displaystyle \frac{Bes{t}_1(i)\times P(i)\times F}{Bes{t}_1(i)-P{(i)}^2}},if P_3\ge k_{P_3}\hfill \\ A_2=Bes{t}_2(i)-{\displaystyle \frac{Bes{t}_2(i)\times P(i)}{Bes{t}_2(i)-P{(i)}^2}}\times F\hfill \end{array}\right.\hfill \\ \left\{\begin{array}{l}R(i)-\left|d(t)\right|\times F\times levy(d)\hfill \\ d(t)=R(i)-\mathrm{P}(i)\hfill \end{array},else\right.\hfill \end{array}\right.,$$

(35)

where levy(d) represents the flight operations, and d(t) is the distance between the vultures and the best vultures in both groups.

2.2.2. Implementation of AVOA Algorithm for Optimizing BP Neural Networks

The selection of initial weights and biases in the training process of BP neural networks significantly impacts the model’s performance. This paper adopts a method based on AVOA to optimize the initial weights and biases of BP neural networks, thereby enhancing the training efficiency and prediction performance of the model. The following sections will elaborate on how AVOA optimizes BP neural networks in detail:

(1)

Initialization of network structure and data normalization:

Firstly, the structural parameters of the BP neural network are set, including the number of neurons in the input layer, hidden layer, and output layer. It was assumed that the number of neurons in the input layer, hidden layer, and output layer are ${\mathrm{n}}_{\mathrm{i}\mathrm{n}},{\mathrm{n}}_{\mathrm{h}\mathrm{i}\mathrm{d}\mathrm{d}\mathrm{e}\mathrm{n}}$, and ${\mathrm{n}}_{\mathrm{o}}\mathrm{u}\mathrm{t}\mathsf{\iota }$, respectively. Then, the training data is normalized to improve the stability of the training process;

(2)

Random initialization of weights and biases:

The AVOA algorithm performs a global search by simulating the foraging behavior of vultures, where each vulture’s position represents a candidate solution for the BP neural network, i.e., the weights and biases. For the BP neural network, the total number of parameters is set to ${\mathrm{n}}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$, including the weights from the input layer to the hidden layer, the weights from the hidden layer to the output layer, and the biases of each layer. The initial positions of the vultures are randomly generated within the search space:

$${\mathrm{position}}_i=lb+(ub-lb)\cdot rand(n_{total})$$

(36)

The lower bound ($I_b$) and upper bound ($u_b$) represent the parameter limits;

(3)

Definition and calculation of the fitness function:

The fitness function is used to evaluate the quality of each vulture’s position. In this paper, the mean squared error (MSE) is used as the fitness function. For each vulture’s position, the corresponding weights and biases are assigned to the BP neural network, and the MSE on the training set is calculated as follows:

$$MSE={\displaystyle \frac{1}{N}}{\displaystyle {\sum }_{\mathrm{i}=1}^N(y_i-{\widehat{y}}_i}{)}^2$$

(37)

where $N$ is the number of training samples, $y_i$ is the true output of the $i$-th sample, and ${\widehat{y}}_i$ is the neural network’s predicted output;

(4)

Updating vulture positions:

According to the update rules of the AVOA algorithm, the position update of vultures is divided into two scenarios: random flight and approaching the optimal solution.

Random flight simulates the behavior of vultures randomly moving during the search process:

$${\mathrm{position}}_i^{new}=positio{n}_i\cdot \exp (-r\cdot t)\cdot \cos (2\pi r)$$

(38)

where $r$ is a random number, and $t$ is the current iteration number.

Approaching the optimal solution simulates the behavior of vultures gradually approaching the best foraging spot:

$$positio{n}_i^{new}=bes{t}_{position}+randn(1,n_{total})\cdot \left.\left|bes{t}_{position}\right.-positio{n}_i\right|$$

(39)

where best_position is the current optimal position;

(5)

Assigning optimal weights and biases:

After multiple iterations of optimization, the best position found by the AVOA algorithm represents the optimal initial weights and biases for the BP neural network. These weights and biases are assigned to the network, followed by BP algorithm training to further optimize the network parameters. The ultimately optimized BP neural network can significantly enhance training efficiency and prediction accuracy.

By introducing the AVOA algorithm to optimize the initial weights and biases of the BP neural network, the training efficiency and prediction performance of the network can be significantly improved. The random search and gradual approach to the optimal solution characteristics of the AVOA algorithm effectively avoid the common local optimum issues encountered during traditional BP neural network training, thereby achieving better global optimization results. Experimental results demonstrate that the AVOA-optimized BP neural network outperforms traditional methods in complex nonlinear prediction tasks, indicating its broad application prospects.

2.2.3. Simulation Experiment and Results

The 26 groups of measured respiratory data of one respiratory cycle in the literature [14] were used as the sample set, as shown in Figure 5, in which 20 groups of data were used as the training set, and six groups of data were used as the test set.The numbers of neurons in the input layer, hidden layer, and output layer of the AVOV–BP neural network were set to 3, 5, and 1, respectively. The inputs of the AVOV–BP neural network were tidal volume, flow rate, and airway pressure. The output was intrapulmonary pressure. In the AVOV optimization algorithm, the number of vultures was set to 20, and the maximum number of iterations was set to 30.

The numbers of neurons in the input layer, hidden layer, and output layer of the RNN and LSTM neural network were set to 3, 100, and 1, respectively. The inputs of the neural network were tidal volume, flow rate, and airway pressure, and the output was intrapulmonary pressure. Using the Adam algorithm, the weight and bias parameters of the network were dynamically adjusted through a combination of the gradient descent method and the momentum method.

The experimental programs were executed multiple times, and the superior experimental results were recorded. The prediction results and prediction errors of intrapulmonary pressure are presented in

Table 5. Prediction results of intrapulmonary pressure.

Measured Value	384.7000	722.1000	795.3000	500.8000	155.9000	15.6000
AVOV–BP neural network	384.9361	722.1952	795.3560	500.5971	156.1584	15.7622
RNN neural network	384.7000	717.522	795.3004	500.313	153.988	15.6002
LSTM neural network	262.05	596.929	721.73	527.439	303.296	153.782

and

Table 6. Prediction errors of intrapulmonary pressure.

	AVOV–BP Neural Network	RNN Neural Network	LSTM Neural Network
MAE	0.1685 (Pa)	1.1629 (Pa)	105.6013 (Pa)
MRE	0.22%	0.33%	174.02%

.

According to Equation (14), the pulmonary elastance values are equal to $P_A(t)/V(t)$. Therefore, by dividing the output of the neural network by the corresponding input tidal volume, the predicted values of pulmonary elastance corresponding to the samples were obtained.

The experimental programs were executed multiple times, and the superior experimental results were recorded. The prediction results and prediction errors of pulmonary elastance are presented in

Table 7. Prediction results of pulmonary elastance.

Measured Value	1.4285	1.4285	1.4286	1.4288	1.4277	1.4312
AVOV–BP neural network	1.4294	1.4287	1.4287	1.4282	1.4300	1.4461
RNN neural network	1.4262	1.4193	1.4304	1.4277	1.4177	1.4132
LSTM neural network	1.0054	1.2084	1.3422	1.5759	2.622	1.4312

and

Table 8. Prediction errors of pulmonary elastance.

	AVOV–BP Neural Network	RNN Neural Network	LSTM Neural Network
MAE	0.1685 (Pa)	1.1629 (Pa)	105.6013 (Pa)
MRE	0.22%	0.49%	24.17%

.

The experimental results indicate that for small samples, the AVOV–BP neural network demonstrated higher prediction accuracy for pulmonary elastance and intrapulmonary pressure compared to both the RNN and LSTM neural networks.

3. Discussion

The main factors that affect the accuracy of identifying respiratory characteristic parameters in non-invasive ventilators are:

(1)

Equations (7) and (10) provide approximate expressions for resistance coefficients $K_2$ and $K_5$, which have been determined through a combination of fluid mechanics principles and practical measurements. The measured value and fitting accuracy directly impact the values of $v(t)$, $F(t)$,$V(t)$, $P_A(t)$, $P_{aw}(t)$, and the results of identifying respiratory characteristic parameters and fitting error in $P_A(t)$;

(2)

The values set for parameters in Table 1 directly impact the values of $v(t)$, $F(t)$, $V(t)$, $P_A(t)$, $P_{aw}(t)$, and the results of identifying respiratory characteristic parameters and fitting error in $P_A(t)$;

(3)

The TS respiration model and the DE respiration model discussed in this paper are both periodic functions, with the data from each respiratory cycle being consistent. However, there are perturbations in human respiration that cannot be ignored, and the data from each respiratory cycle may not be entirely consistent. As a result, the parameters identified by this method, such as pulmonary elastance, intrapulmonary pressure, and airway resistance, may constrain when compared to the actual situation of patients.

In the literature [10], experimental apparatus measurement data are utilized as the input and output for the respiration model. The parameters of the respiratory model are then identified using the recursive least-square algorithm, with an error rate ranging between 1% and 5%. In the literature [17], measurement data is also used as the input and output for the DE respiration model. The parameters of the DE respiratory model are identified using numerical integration and the recursive least-square algorithm; the average error of intrapulmonary pressure estimation is about 0.004.

In this paper, numerical integration and the least-square method are utilized. However, the results presented in Table 3 and Table 4 indicate large errors in the identification of pulmonary elastance and the prediction of intrapulmonary pressure. These errors arise from the differing definitions of pulmonary elastance in the two models: while it is defined as a constant in the TS respiration model, it is defined as a polynomial in the DE respiration model. Consequently, this disparity directly contributes to substantial errors in the identification of pulmonary elastance and intrapulmonary pressure.

Therefore, it can be inferred that the accuracy of numerical integration and least-squares algorithm for respiration model parameter identification is high. However, due to the use of non-invasive ventilators and functional limitations, it is not possible to directly measure invasive respiratory data. The use of the TS respiration model to simulate respiratory data leads to a significant error in subsequent identification. In future studies, addressing how to unify the consistency of the TS respiration model and the DE respiration model on the definition of pulmonary elastance can solve this problem.

The accuracy of respiratory characteristic parameter identification in invasive ventilators is influenced by several key factors:

(1)

The accuracy of respiratory data samples directly impacts the system parameter identification and data fitting results;

(2)

Increasing the quantity and diversity of respiratory samples can enhance the model’s universality;

(3)

Advancements in neural network and intelligent optimization algorithms have an impact on the results of parameter identification and data fitting.

In the literature [17], the PSO–GRNN network is established, the mean error between the predicted value and the true value of intrapulmonary pressure in the test set is 52.4 Pa, and the mean absolute error between the predicted value and the true value of intrapulmonary pressure in the train set is 8.2 Pa. However, the AVOV–BP network established in this paper demonstrated higher prediction accuracy, with a mean absolute error of only 0.1685 Pa for intrapulmonary pressure prediction.

In the literature [20], in the SFE–LSTM network, the mean absolute error of ventilator pressure prediction is 0.162115 Pa. In the WOA–SFE–LSTM network, the mean absolute error of ventilator pressure prediction is 0.157829 Pa. In the MWOA–SFE–LSTM network, the mean absolute error of ventilator pressure prediction is 0.140571 Pa. The prediction accuracy of ventilator pressure in all three network models is significantly higher than that of the LSTM network in Section 2.2.3 and also higher than that of the AVOV–BP network in this paper. The primary objective of this study is to enhance calculation speed. Therefore, this paper focuses on the parameter identification and intrapulmonary pressure prediction of invasive ventilators using a small sample respiratory dataset. The sample set in this paper only consists of 26 sets of data, while the literature [20] contains a much larger number of datasets, totaling 6,036,000. As a result, the calculation speed of the method in this paper is faster than that in the literature [20]. Considering the speed of calculation and the accuracy of prediction, the AVOV–BP network established in this paper demonstrates superior performance.

The experimental results in Section 2.1.6 and Section 2.2.3 of this paper demonstrate that the accuracy of parameter identification for invasive ventilators is significantly higher than that of non-invasive ventilators. This can be attributed to the fact that the respiratory sample data for invasive ventilators are actual measured values, whereas for non-invasive ventilators, there is a larger fault-tolerant space for parameter setting due to chronic patient usage. In contrast, critically ill patients using invasive ventilators have a smaller fault-tolerant space for parameter setting. Therefore, the two respiratory parameter identification methods meet the specific requirements for auxiliary settings of both invasive and non-invasive ventilators, respectively, in clinical practice.

4. Conclusions

The existing literature only studies one type of respiration model. This paper combined the respiration model based on fluid mechanics theory and the respiration model based on differential equations in the respiratory characteristic parameter identification of non-invasive ventilators. Pulmonary elastance was defined differently under the two respiration models, and the experimental results provided a quantitative relationship of pulmonary elastance in both models, verifying their correlation and consistency. This method simulated respiratory data by setting the resistance coefficient value of the TS respiration model. However, it should be noted that the simulated respiratory data are identical for each respiratory cycle without considering differences in the same patient’s respiratory data during different cycles, leading to potential errors compared to actual patient respiratory situations. Nonetheless, this method does not require the real-time measurement of respiratory data and offers an advantage by avoiding the discomfort caused by the invasive measurement of intrapulmonary pressure in patients. It is particularly suitable for chronic patients using non-invasive ventilators and for settings where high accuracy of ventilator parameter setting is not essential, such as in homes, nursing institutions, and community hospitals.

In this paper, a method based on the AVOV–BP neural network was proposed for the identification of respiratory characteristic parameters of invasive ventilators. The method required real-time monitoring of the patient’s respiratory data. Specifically, if a respiratory cycle lasts 3 s, the respiratory characteristic parameters need to be identified every 3 s using the patient’s actual respiratory data. However, it was found that under Matlab2017b software, the method took about 4 s to complete an operation. It does not meet the requirement for the real-time identification of respiratory parameters. In order to minimize the need for frequent adjustment of ventilator parameters, a designated observation period, such as one hour, can be established in clinical practice. During this time, the patient’s respiratory parameter identification results should be recorded, and the ventilator parameter settings should be adjusted based on both these results and clinical observations.

In this paper, two methods of identifying respiratory characteristic parameters during mechanical ventilation were studied. These methods are suitable for different types of ventilators and patients. Therefore, the superiority of one method over the other cannot be determined solely based on the accuracy of parameter identification. In practical applications, the parameter setting of non-invasive ventilators is simple to operate, allowing even household users to manage after basic training. On the other hand, the parameter setting of invasive ventilators is complex and requires completion by highly skilled medical staff.

In this paper, two respiratory parameter identification algorithms were analyzed and improved, and some results were obtained. However, in clinical applications, there are numerous influencing factors involved in the setting of ventilator parameters, such as individual differences in patients, equipment factors, and medical staff factors. The stability and security of the proposed respiratory characteristic parameter identification algorithm need to be further verified for practical application on a large scale. There is still a distance to go before it can be widely applied.

In our follow-up work, our focus will be on addressing the following issues:

(1)

More real patient data will be collected from clinical practice to enrich respiratory samples of different ages, genders, and disease;

(2)

The study aims to investigate the direct relationship between respiratory characteristic parameters and ventilator parameter settings, as well as to explore the specific operating rules of ventilator settings, so as to provide valuable reference for medical staff and patients’ families;

(3)

For the respiratory characteristic parameter identification algorithm based on the neural network, how to reduce the calculation amount will be studied.

The methods for identifying respiratory characteristic parameters in this paper will serve as a valuable reference for doctors when setting ventilator parameters. Additionally, they offer theoretical support for doctors conducting pathological research, physiological research, and clinical diagnosis.