CN116911212B - Respiratory system modeling method based on fractional calculus - Google Patents
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Abstract
The invention provides a respiratory system modeling method based on fractional calculus, which can acquire the pressure value, the flow velocity value and the accumulated gas quantity of a respiratory circuit in real time under any ventilation mode to quickly obtain a corresponding fractional mechanics model, wherein the fractional mechanics model comprises an optimal fractional order and various coefficient parameters. The method shows better fitting degree than the classical model in the measured data.
Description
Technical Field
The invention relates to a respiratory system model, in particular to a respiratory system modeling method based on fractional calculus.
Background
In the mechanical ventilation research of the respiratory system, lung tissues have certain power law characteristics and memory and path dependence properties and are not ideal viscoelastic substances, so that the characteristics are difficult to better describe and reflect by a classical lumped parameter model based on an integer order differential equation. Classical calculus is to solve integer differential and integral of a function, and if non-integer differential and integral is solved, the field of fractional differential integration is related. At present, no method for modeling respiratory system mechanics by using fractional calculus based on actual measurement data of any respiratory mode is found in the prior art, so that a modeling method based on fractional calculus is lacking in the field of respiratory system mechanics modeling at present.
Disclosure of Invention
The invention provides a respiratory system modeling method based on fractional calculus, which shows better fitting degree than a classical model in measured data.
In order to solve the technical problems, the technical scheme of the invention is as follows:
the invention discloses a respiratory system modeling method based on fractional calculus, wherein a fractional mechanics model is expressed as follows:
in the middle ofRepresenting V (t) to find alpha 1 Fractional order derivative, /)>Representing V (t) to find alpha 2 Fractional order derivative, /)>Representing V (t) to find alpha m The second order derivative, m is the model order, V is the respiration rate, V' is the 1 st derivative of V, i.e. the gas flow rate, E is the respiratory system elasticity, R is the airway resistance, P 0 To cross lung pressure, alpha 1 、α 2 、...、α m For fractional order, k 1 、k 2 、...、k m Coefficient parameters of the fractional order mechanical model;
by means ofObtaining fitting results of various coefficients, and repeatedly and iteratively searching the optimal order, wherein,
x is a data matrix of n rows m +3 columns,
is an n-dimensional barometric pressure vector, which is composed of values of breathing circuits acquired at n time points.
Optionally, in the method for modeling respiratory system based on fractional calculus, a fractional mechanical model of a fractional derivative term is expressed as:
optionally, in the method for modeling respiratory system based on fractional calculus, the fractional mechanical model of the two fractional derivative terms is expressed as:
optionally, in the method for modeling respiratory system based on fractional calculus, in the process of iteratively searching for the optimal order, the number of parameters to be optimized is a series of orders { α } 1 ,α 2 ,…,α m And each designated set of order numbers is based on all coefficient parameters of the obtained model to further obtain a residual square sum or root mean square error, and each parameter is continuously updated along with the residual square sum or root mean square error change trend so as to minimize the residual square sum or root mean square error.
The beneficial effects of the invention are as follows:
compared with a general integer calculus model, the method for modeling the respiratory system based on the fractional calculus can embody the power law and the memory property of the respiratory system, is easier to obtain better fitting degree than a classical integer calculus model, and does not require a specific respiratory mode for data acquisition.
Drawings
In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the embodiments or the prior art will be briefly described below.
FIG. 1 is a simulation result of a fractional mechanical model of the method of the present invention for respiratory modeling based on fractional calculus.
Detailed Description
In order to make the objects, technical solutions and advantages of the embodiments of the present invention more clear, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention.
According to the respiratory system modeling method based on fractional calculus, the pressure value, the flow velocity value and the accumulated gas quantity of the respiratory circuit are collected in real time under any ventilation mode, and the fractional mechanics model corresponding to the pressure value, the flow velocity value and the accumulated gas quantity can be obtained quickly, wherein the fractional mechanics model comprises the optimal fractional order and each coefficient parameter.
The invention discloses a respiratory system modeling method based on fractional calculus, which comprises the following steps:
s1, sorting measured data into corresponding item matrix form data according to a fractional order mechanical model form to serve as input
Fractional order mechanical models have the general form:
in the middle ofRepresenting V (t) to find alpha 1 The second fractional order derivatives, e.g. 0.23, 2.8, etc, +.>Representing V (t) to find alpha 2 Fractional order derivative, /)>Representing V (t) to find alpha m The second order derivative, m, is the model order; v is the respiration rate (singly)Bit L), V' is the 1 st derivative of V, i.e., gas flow rate, E is respiratory system elasticity, R is airway resistance, P 0 To cross lung pressure, alpha 1 、α 2 、...、α m For fractional order, k 1 、k 2 、...、k m Is a coefficient parameter of the fractional order mechanical model.
In practical applications, the respiratory system power law and memory properties can be simply represented by a few fractional order items, for example:
or (b)
For a model with m fractional derivative terms (adding 3 non-fractional derivative terms and m+3 total terms), the invention acquires the pressure value, the flow velocity value and the accumulated gas value of the breathing circuit in real time, and according to the fractional derivative number m of the preset model, the corresponding form and the designated group of fractional orders { alpha } 1 ，α 2 ，…，α m And the obtained fractional differential values are arranged into corresponding matrix form data to be used as algorithm input. If the value of the acquired breathing circuit is n time points, the input data is n-dimensional air pressure vectorAnd n rows m+3 columns of data matrix X:
for example, V as above m The row (second row of right numbers) is to calculate alpha for the air volume V (t) m The numerical value of the fractional derivative is the same as the other corresponding fractional derivative terms.
Note that: if the flow rate value and the accumulated gas value are limited by the sampling condition, only one of the flow rate value and the accumulated gas value, the other missing value can be obtained by a proper value differentiation and value integration method.
S2, calculating model coefficients of designated orders
For fractional models with m fractional derivatives, when a set of initialization fractional orders { alpha } is specified 1 ，α 2 ，…，α m And after the step, calculating according to the following matrix to obtain each coefficient fitting result.
Wherein each coefficient vector a is as follows:
the calculation method proposed by the above formula is derived from extreme points with the bias of 0 when each coefficient takes the minimum value based on the sum of squares SSR (sum of squared residuals) of the residual errors. By the step, various parameters of the appointed fractional order model are obtained, and the fractional order can be further optimized.
S3, repeatedly and iteratively searching the optimal order based on the method
After each coefficient of the specified fractional order model is obtained, the actual measurement data can be substituted into model simulation to calculate the corresponding model residual error square sum (sum of squared residuals, SSR) and root mean square error (the root mean square error, RMSE). In order to optimize the most suitable series of orders based on the data, the calculation efficiency is considered, and a particle swarm algorithm is preferably selected from various machine learning algorithms. In the genetic algorithm, simulated annealing and other similar algorithms, although the particle swarm algorithm may fall into a locally optimal solution, the particle swarm algorithm converges faster, and the timeliness is more emphasized in practical application.
In the iterative optimization process, the number of parameters to be optimized is a series of orders { alpha } 1 ,α 2 ,…,α m Every designated group of order numbers, all coefficient parameters of the model can be obtained based on the step S2 so as to obtain SSR/RMSE, various parameters are updated continuously along with the SSR or RMSE change trend, and the model optimization target is to make the SSR or RMSE the mostAnd the model with better initial state can be obtained after iterative calculation.
Note that: after the model coefficient of the designated order is calculated in the step S2, in the step S3, other similar machine learning algorithms can be selected to further optimize the fractional order, which machine learning algorithm is not the core of the method, and the whole framework of the same-class method after the replacement and change still belongs to the category of the method.
The calculation amount of the method is higher than that of a typical single-chamber model in respiratory mechanics modeling, and the calculation amount of tens of seconds is correspondingly more, so that a better fitting effect can be obtained in measured data.
In the following, classical single-chamber model p=ev+rv' +p in respiratory mechanics modeling 0 And second order model p=ev+rv' +iv "+p 0 These two types of traditional models are used as a reference, 2 examples are supplemented to show calculation results, and the method of the invention is used for carrying out parameter estimation on 2 models of the following formulas (1) and (2):
the parameter estimation is based on animal experimental measured data (obtained by measuring the Delge Savina 300 ventilator) and the pressure P (in mbar), the flow rate V' (in L/s) and the respiration rate V (in L) data are recorded as one line every 10 milliseconds in time sequence.
Based on the example python+matlab code main.m (other. Py and. M are python and matlab subfunctions called by the main code. Slx are model files), three model parameter calculation results after reading actual measurement data are as follows in table 1:
TABLE 1 estimation results of model parameters
* Reserving 4-bit decimal, converting RMSE into milliliter unit, run_time representing running time (common PC running, CPU i7-1165G7@2.80GHz,16GB memory)
The result shows that the method is effective, the fitting effect of the fractional order model is better, and the method has lower RMSE value. The measured data (solid line) fit to the flow V (broken line) of equation (1), as shown in fig. 1, is obtained by taking 10 particle counts and iterating 10 rounds.
In addition, the calculation time consumption and the fitting degree are comprehensively considered, 10 rounds of 10-particle number iteration based on PSO in the formula (1) can be generally considered by default, stable optimization results can be basically obtained, and the calculation is faster and the results are better. With the increase of the number of fractional differential terms and the increase of the number of to-be-estimated steps, the machine learning algorithm needs more calculation amount to obtain a stable result, and for the PSO method to be represented by the formula (2), better results can be obtained by more particle numbers and iteration rounds or by narrowing the search range based on pre-calculation features.
In addition, one of a plurality of implementations of fractional calculus numerical computation is supplemented as a reference. Fractional calculus has different definition modes: grunwald-Letnikov definition, riemann-Liouville definition, caputo definition, erd lyi-Kober definition, etc., each with different application scenarios and numerical calculation methods. Taking the first class of G-L definitions as one possible calculation description example, the alpha derivative of a given function f (t) is:
where h is a small enough time step and Γ represents the Gamma function, i.eThe definition assumes that at t.ltoreq.t 0 The time function f (t) has a value of 0, which relates to the time value from t 0 All function values starting at the moment can be considered as having memory of fractional derivatives, and are applicable to alpha>Differentiation of 0 and alpha<Integration case of 0.
For the case of numerical calculations, when the step h is small enough there is:
and the coefficient ω in the above (4) j The recursive formula is convenient for numerical calculation:
since the Gamma function has the property Γ (z+1) =zΓ (z) according to the fractional integration method in differential integration, there is:
compared with a general integer calculus model, the method for modeling the respiratory system based on the fractional calculus can embody the power law and the memory property of the respiratory system, is easier to obtain better fitting degree than a classical integer calculus model, and does not require a specific respiratory mode for data acquisition.
The above examples are only specific embodiments of the present invention for illustrating the technical solution of the present invention, but not for limiting the scope of the present invention, and although the present invention has been described in detail with reference to the foregoing examples, it will be understood by those skilled in the art that the present invention is not limited thereto: any person skilled in the art may modify or improve the technical solution described in the foregoing embodiments, or perform equivalent replacement of some technical features thereof, while remaining within the technical scope of the disclosure; such modifications, changes or substitutions do not depart from the spirit and scope of the technical solutions of the embodiments of the present invention, and are intended to be included in the scope of the present invention. Therefore, the protection scope of the invention is subject to the protection scope of the claims.
Claims (4)
1. A method for modeling respiratory system based on fractional calculus is characterized in that,
the fractional order mechanical model is expressed as:
in the middle ofRepresenting V (t) to find alpha 1 Fractional order derivative, /)>Representing V (t) to find alpha 2 Fractional order derivative, /)>Representing V (t) to find alpha m The second order derivative, m is the model order, V is the respiration rate, V' is the 1 st derivative of V, i.e. the gas flow rate, E is the respiratory system elasticity, R is the airway resistance, P 0 To cross lung pressure, alpha 1 、α 2 、...、α m For fractional order, k 1 、k 2 、...、k m Coefficient parameters of the fractional order mechanical model;
by means ofObtaining fitting results of various coefficients, and repeatedly and iteratively searching the optimal order, wherein,
x is a data matrix of n rows m +3 columns,
is an n-dimensional barometric pressure vector, which is composed of values of breathing circuits acquired at n time points.
2. The method of modeling respiratory system based on fractional calculus of claim 1, wherein the fractional mechanical model of a fractional derivative term is expressed as:
3. the method for respiratory modeling based on fractional calculus of claim 1 wherein the fractional mechanical model of the two fractional derivative terms is expressed as:
4. the method for respiratory modeling based on fractional calculus of claim 1, wherein in the iterative search for the optimal order, the number of parameters to be optimized is a series of orders { α } 1 ，α 2 ，…，α m And each designated set of order numbers is based on all coefficient parameters of the obtained model to further obtain a residual square sum or root mean square error, and each parameter is continuously updated along with the residual square sum or root mean square error change trend so as to minimize the residual square sum or root mean square error.
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