=Paper=
{{Paper
|id=Vol-3302/paper10
|storemode=property
|title=Interval Non-linear Model of Information Signal Characteristics Distribution for Detection of Recurrent Laryngeal Nerve during Thyroid Surgery
|pdfUrl=https://ceur-ws.org/Vol-3302/short5.pdf
|volume=Vol-3302
|authors=Mykola Dyvak,Volodymyr Manzhula,Yulia Trufanova
|dblpUrl=https://dblp.org/rec/conf/iddm/DyvakMT22
}}
==Interval Non-linear Model of Information Signal Characteristics Distribution for Detection of Recurrent Laryngeal Nerve during Thyroid Surgery==
https://ceur-ws.org/Vol-3302/short5.pdf
Interval Non-linear Model of Information Signal Characteristics
Distribution for Detection of Recurrent Laryngeal Nerve during
Thyroid Surgery
Mykola Dyvaka, Volodymyr Manzhulaa, Yulia Trufanovaa
a
West Ukrainian National University, 11 Lvivska Str., Ternopil, 46009, Ukraine
Abstract
The work provides an analysis of known methods and technical means of identifying and
visualizing the recurrent laryngeal nerve during thyroid surgery. There's proposed a method
of building a non-linear model for detecting the location of the laryngeal nerve in the area of
thyroid surgery. It's based on the characteristics of the information signal from the
preliminary stimulation of the tissues of the surgical wound with an alternating current of a
fixed frequency and the subsequent construction of the distribution function of the response
to stimulation. The proposed method simplifies the procedures for identifying of the interval
model, in particular, due to the analytical representation of the objective function of the
optimization problem of their identification, and, accordingly, reduces the time spent on
building the non-linear models based on interval data. Based on real experimental data
obtained during thyroid surgery an interval non-linear model was built, which enables
detection and visualization of the location of the laryngeal nerve in the thyroid surgery area
and, accordingly, reducing the risk damage of its.
Keywords 1
Thyroid surgery, recurrent laryngeal nerve, information signal amplitude, interval data,
interval non-linear model, model identification, optimization problem
1. Introduction
The main problem when conducting thyroid surgery is the identification of the recurrent laryngeal
nerve, the damage of which leads to the patient losing his voice, as well as to other negative
consequences related to the functioning of the human respiratory system [1, 2]. As a rule, the means
used during such surgery make it impossible to visualize the process of identifying the laryngeal
nerve, also it are based on the dangerous procedure of introducing the patient to the third stage of
anesthesia, where there is a high risk of transition to a state of clinical death [3, 4].
The process of visualizing the laryngeal nerve is extremely complex and includes the procedure for
its detection [5, 6, 7]. The analysis of known technical means of detecting the recurrent laryngeal
nerve during thyroid surgery made it possible to establish the general principle of their surgery, which
is based on stimulation with a constant electric current in the area of surgery and evaluating the results
of this s on the vocal cords If the area of stimulation includes the recurrent laryngeal nerve, the vocal
cords are shortened, but if the stimulation is done on the muscle tissue, the reaction to the stimulation
will be insignificant. The basis of the method of identification of the laryngeal nerve from other
tissues of the area thyroid surgery than proposed by the authors [8, 9, 10] is to ensure the accuracy of
detection and visualization of the location of the laryngeal nerve in the surgical wound. The task is
solved by the fact that the tissue stimulation in the area thyroid surgery is carried out by an alternating
current of a fixed frequency which provides a low conductivity of the electrical signal through the
IDDM-2022: 5th International Conference on Informatics & Data-Driven Medicine, November 18–20, 2022, Lyon, France
EMAIL: mdy@wunu.edu.ua (A. 1); v.manzhula@wunu.edu.ua (A. 2); yu.trufanova@wunu.edu.ua (A. 3)
ORCID: 0000-0002-9049-4993 (A. 1); 0000-0001-5222-8443 (A. 2);
2022 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
muscle tissues and a high conductivity of the electrical signal through the laryngeal nerve and muscles
that control the tension of the vocal cords, followed by registration of the contraction of the vocal
cords at a given frequency by a sound sensor installed in the breathing tube placed in the patient's
larynx, followed by its conversion into an electrical signal, and the output information signal, which
characterizes the proximity to the laryngeal nerve, is determined by the change in the amplitude of the
electric current of a given frequency [8].
To visualize the location of the laryngeal nerve in the surgical wound, information signal
processing tools are used [10]. A signal processing software module includes filtering the signal at the
excitation frequency, determining the maximum amplitude of the filtered signal for each observation,
and recording the received data in interval form, taking into account errors of various nature.
Moreover, the measurement of the interval value of the information signal amplitude is carried out
according to the coordinates on the thyroid surgery area, which are fixed on a sterile grid,
respectively, placed on the wound.
In papers [9, 10], the authors have proposed methods for constructing an interval models that
describe the maximum amplitude of the information signal depending on the coordinates on the
surgical wound, based on tolerance and guaranteed interval or ellipsoidal estimates of the parameters
of the algebraic expression. However, the computational complexity of implementing these methods
complicates online-visualization of the location of the recurrent laryngeal nerve during thyroid
surgery. Therefore, the actual task is to develop a method for identifying the model of the information
signal characteristics distribution with minimal computational costs for visualizing the location of the
laryngeal nerve in the thyroid surgery area.
2. Method for building an interval model of the distribution of information
signal characteristics for the recurrent laryngeal nerve’s detection in the
thyroid surgery area
The process of building mathematical models includes solving two problems: structural and
parametric identification [11]. At the same time, the task of identification the model structure is more
difficult and primary, since it’s necessary to first define the basic functions, generate the structure of
the model, and then calculate parameter estimates for selecting the optimal or "best". The most
effective methods of structural identification of interval models are based on self-adaptation and self-
organization procedures by analogy with the behavioral models of a bee colony. Complex
optimization problems are solved for this [12, 13, 14, 15].
The distribution of the information signal characteristics in the thyroid surgery area we'll be
described by interval mathematical models in the form of a nonlinear algebraic expression. Then we'll
search for the resulting information signal characteristic (maximum amplitude) in the non-linear
algebraic expression form of following kind:
(1)
where – is a set of unknown basis
functions (of a known class), and the basis functions relate to
unknown model parameters and the result of experimental measurements obtained in
interval form:
(2)
where – is number of measurements.
Let's set the conditions for the consistency of the model with experimental interval data as it's
customary in interval analysis:
(3)
where – means the true value of the information signal output characteristic for a fixed
model structure and for fixed input variables' values .
Parameter values of the model only remain unknown in this case. Taking into account
conditions (3) and replacing with expression (1) for fixed values of the input variables ,
we'll obtain the following system of interval non-linear algebraic equations (ISNAE):
(4)
Thus, the general form of the parametric identification problem of the interval model of the
information signal characteristics when it's distributed to the thyroid surgery area in the interval
system of non-linear algebraic equations form has been obtained. As is known [11], the solutions of
this system they’re obtained as a result of the implementation of an iterative procedure at each
iteration of its they calculate the function "quality" of estimation of mathematical model
parameters. Accordingly, there’s the task of structural identification of the interval model of the
distribution of information signal characteristics as a task of repeatedly solving the problems of
parametric identification of this model.
Let's assume that the solution of ISNAE (4) is obtained in the form of value intervals of model
parameter estimates . Let's substitute the obtained interval estimates into
expression (4) with the fixed values of the input variables (at the points of the experiment).
Because of these substitutions, we'll get estimates of the information signal output characteristic in the
interval form:
. (5)
Thus, the compatibility of ISNAE (4) means that the intervals of the values
of the predictive characteristics of the information signal at the points of
experimental measurements including to the intervals obtained
experimentally, that is, if the following conditions are satisfied:
(6)
Stating the fact that the structural identification problem of interval models of an information
signal characteristics is a problem of repeatedly solving parametric identification problems of this
model, and therefore from a computational point of view, it is NP complete. The complexity of the
problem related to the complexity of the objective function, which is given algorithmically, is discrete
and does not have an analytical representation, therefore it complicates the calculation [11].
At the same time, in the vast majority of problems of both structural and parametric identification
of mathematical models, the criterion of minimizing the mean square deviation is used. On the other
hand-side, it's mostly sufficient to find at least one model even with the interval formulation of the
problem in the sense of solving ISNAE (4). In this case, the interval model (5) will be able written in
the following form:
. (7)
The task of identifying the interval model of the information signal characteristic distribution in
the area of surgery we'll formulate in the optimization problem form:
(8)
,
where
(9)
where – is set of potential model’s
structural elements; – is the parameters vector components of the sth model;
– is set minimum and maximum value for each model's parameter; – is the parameters
number of the interval model.
There's what the smaller value that the "better" structure of the interval model. If
the equality is fulfilled
(10)
then the structure is guaranteed to allow building an adequate interval model of the information signal
characteristic distribution, since the existence of the ISNAE (4) solution means that the condition (6)
is satisfied, which in this case will have the following form:
, (11)
and it’s equivalent condition
(12)
since expression (12) is always a linear combination of limits of experimental values at measurement
points .
The advantage of using the objective function in the form (9) is that it's in an analytical form and
quadratic at least relative to the coefficients , .
Thus, the task of model's parametric identification for a fixed structure is the search for the
optimization problem solution:
(13)
To calculate the parameters of the interval model based on the optimization problem (13) and the
given structure should be used nonlinear optimization methods, such as the Gradient Descent method,
the Newton method or a combination of theirs [17, 18, 19]. The implementation of structural
identification consists in selecting the structure of the interval model by reducing or increasing
structural elements [20].
3. The interval non-linear model of the information signal characteristics
distribution for the detection of the recurrent laryngeal nerve during
thyroid surgery.
There were carried out the construction of an interval non-linear model of the characteristics of the
information signal distribution in the area thyroid surgery based on the developed method.
Experimental measurements on a sterile grid in the area of surgical intervention we’ve carried out
based on of two coordinates:
(14)
The data were obtained in interval form based on information signal processing taking into account
measurement errors and noises are presented in Table 1.
Detailed analysis of the data in the Table 1 showed that the structure of an adequate model of the
maximum amplitudes of the information signal distribution in the thyroid surgery area should be to
search with the inclusion of trigonometric basic functions. To reduce the number of such structural
elements we’ve added for the parameters in the power function form .
Accordingly, a set of potential structural elements we’ve formed in this form:
(15)
In the process of selecting and increasing the model structure with elements from the set F, we've
obtained a model structure based on the convolution of the following form:
(16)
Since the value of the objective function of the optimization problem (13) for this model’s
structure is close to zero (please, check Figure 1) and the condition (12) is satisfied, therefore, we
obtained the optimal solution based on it.
Table 1
Results of experimental measurements of information signal characteristics
Measurement Measurement coordinates, Interval value of the maximum signal
number amplitude
1 1 1 8,0974 11,5326
2 1 2 9,5576 13,6124
3 1 3 13,0391 18,5709
4 1 4 15,2955 21,7845
5 1 5 19,8619 28,2881
6 1 6 25,6492 36,5308
7 2 1 9,0626 12,9074
8 2 2 14,421 20,539
9 2 3 21,7099 30,9201
10 2 4 28,5656 40,6844
11 2 5 33,8498 48,2103
12 2 6 43,032 61,288
13 3 1 10,1516 14,4584
14 3 2 18,1335 25,8265
15 3 3 31,5892 44,9907
16 3 4 37,8716 53,9384
17 3 5 47,8912 68,2087
18 3 6 50,7004 72,2096
19 4 1 13,1134 18,6766
20 4 2 23,3475 33,2525
21 4 3 43,1516 61,4584
22 4 4 47,4375 67,5625
23 4 5 49,5001 70,4985
24 4 6 43,3125 61,6875
25 5 1 13,2021 18,8001
26 5 2 32,1750 45,825
27 5 3 43,3125 61,6875
28 5 4 47,0250 66,975
29 5 5 44,5501 63,4495
30 5 6 26,8125 38,1875
31 6 1 16,5001 23,4996
32 6 2 37,5375 53,4625
33 6 3 46,4063 66,0938
34 6 4 42,0750 59,925
35 6 5 24,3375 34,6625
36 6 6 16,9125 24,0875
Based on calculated parameter estimates the model of
the information signal amplitude distribution on the thyroid surgery area in this form was constructed:
. (17)
Figure 1: The objective function value in the process of calculating parameter
estimates and coefficients for the resulting model
There’re given the predictive values of the information signal amplitude that it's obtained
based on the constructed model and, accordingly, calculated coefficients in process solving
optimization problem (13) for parameter estimates of the resulting model in Table 2.
Table 2
The results of the interval model constructing of the maximum information signal amplitude
distribution
Measurement Predictive Coefficient Measurement Predictive Coefficient
number, amplitude estimates, number, amplitude estimates,
value, value,
1 10,0373 0,4354 19 14,2191 0,41
2 12,8713 0,1975 20 29,1968 0,7892
3 16,3145 0,4089 21 47,0182 0,3897
4 20,3638 0,2261 22 59,7198 0,4161
5 24,9817 0,3916 23 61,7681 0,464
6 30,1031 0,5902 24 53,1629 0,3642
7 10,823 0,5447 25 16,7887 0,3171
8 16,2673 0,6984 26 37,2303 0,3632
9 23,9583 0,7539 27 55,8709 0,8519
10 33,2932 0,609 28 59,7198 0,7099
11 43,3751 0,3362 29 47,348 0,5194
12 53,1629 0,4447 30 30,1031 0,5332
13 12,218 0,5094 31 19,8618 0,353
14 21,9101 0,7275 32 44,9699 0,6086
15 35,2319 0,3041 33 59,1501 0,6397
16 49,0664 0,4059 34 49,0664 0,5615
17 59,9674 0,3207 35 28,0548 0,3171
18 65,2948 0,7971 36 20,0517 0,3632
Figure 2 shows the graphs of the experimental interval corridor of the information signal amplitude
and the predictive values that are obtained based on the model. The presented graphs demonstrate the
inclusion of predictive values in the experimental corridor at each measurement point that satisfy
condition (11) and indicates the adequacy of the constructed model.
So, constructed the interval non-linear model based on of real experimental data that were obtained
during thyroid surgery will be able used to detect the placement of the laryngeal nerve in the thyroid
surgery area and, accordingly, reduce the risk damage of its. Figure 3 shows the 2- and 3-dimensional
visualization of the maximum amplitude distribution over the surgical area, which demonstrates the
possible placement of the laryngeal nerve.
Figure 2: Interval values of experimental measurements and model-based predictive values of the
information signal amplitude
4. Conclusions
There were proposed based on the known method and technical means of the recurrent laryngeal
nerve detecting during thyroid surgery the method and the non-linear model for predicting the
information signal characteristics in order to detect the laryngeal nerve.
The method of building the non-linear models was created to detect the laryngeal nerve location in
the area of surgical intervention based on the characteristics of the information signal. Signal was
gotten from the previous excitation of the tissues of the surgical wound with an alternating current of
a fixed frequency and the subsequent construction of the distribution function of the response to
excitation. The proposed method simplifies and, accordingly, reduces the time spent on building a
non-linear model based on interval data, in particular, due to the analytical representation of the
objective function of the optimization problem of identification model.
Based on real experimental data of thyroid surgery the interval non-linear model was built it made
possible to identify the placement of the laryngeal nerve in the thyroid surgery area, accordingly,
reducing the risk damage of its.
a)
b)
Figure 3: Visualization of the information signal amplitude distribution based on the constructed
model: a) two-dimensional image, b) 3d image.
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