46 Design of the Saturated Interval Experiment for the Task of Recurrent Laryngeal Nerve Identification Mykola Dyvak, Iryna Oliynyk, Volodymyr Manzhula Department of Computer Science, Ternopil National Economic University, UKRAINE, Ternopil, 8 Chekhova str., email: mdy@tneu.edu.ua, ois@tneu.edu.ua, v.manzhula@tneu.edu.ua Abstract: the method of saturated design of experiment identify the type of tissue with the purpose of RLN with interval data and its application for reccurent identification. laryngeal nerve (RLN) location identification considered In articles [5-6] described the method for RLN in this paper. Built saturated plan of experiment makes it identification among tissues of a surgical wound. The scheme possible to reduce the duration of surgery by reducing the of this method on the fig 1 are shown. number of points of irritation woven surgical wounds to In respiratory tube 1 that inserted into larynx 2, the sound identify locations RLN and reduce the risk of damage. sensor 3 implemented and positioned above vocal cords 4. Keywords: neck surgery, recurrent laryngeal nerve, Probe 5 is connected to stimulation block. It the functions method of design if the saturated experiment, interval as a current generator controlled by the single-board analysis, interval model. computer 8. Surgical wound tissues are stimulated by the block 7 via probe. As a result, vocal cords 4 are stretched. I. INTRODUCTION Flow of air that passes through patient’s larynx, is RLN monitoring by using a special neuro monitors is very modulated by stretched vocal cords. The result is registered important procedure during the neck surgery. These monitors by voice sensor 3. Obtained signal is amplified and is work based on the principle of surgical wound tissues processed by single-board computer 8. stimulation and estimation of results of such stimulation [1- 4]. However, the monitoring procedure does not guarantee a reduction in the risk of RLN damage, but only establishes the fact of its damage (not damage). In this case, the procedures for RLN identification are actual. Іn the paper [5] the task of visualizing the RLN location based on evaluation the maximal amplitude of signal as response to its stimulation by alternating current was considered. In paper [6] the method of constructing the difference schemes as a model for RLN 1 is respiratory tube, 2 is larynx, 3 is sound sensor, 4 are vocal cords, 5 is probe, 6 is surgical wound, 7 is block for RLN stimulation, 8 is single-board location was considered. It should be noted that the computer, 9 is output part informative parameter in both methods used maximum amplitude of the signal as response to stimulation of tissues Fig. 1. Method of RLN identification among tissues in surgical wound in surgical wounds. The basis for localization the RLN damaging area assigned an interval model of distribution on The essence of processing the signal for a given simulation surgical wound surface the maximum amplitudes of point is to determine its maximum amplitude and then to information signals. However, both methods require creation construct it the interval model of distribution on surgical the uniform grid on surgical wound for tissues stimulation, wound surface the maximum amplitudes. The built model which substantially increases the time of surgical operation. makes it possible to determine the risk-sensitive area in Note that in both cases the amount of tissue stimulation of the which the RLN is localized. To construct this model, it is surgical wound is equal to the number of nodes in the grid. necessary to minimize the number of stimulations of the Such an approach greatly increases the time of the transaction tissues of the surgical wound. In the research [5-6] also found by means of the procedure for RLN identification. that the amplitude of the information signal depends on the Based on the case of build a mathematical model, distance from the stimulation point to the RLN. The near this described in [5], in this article proposed to use an method of point to the RLN, the greater the value of the amplitude of the design experiment with a minimum number of experimental information signal. points (stimulation points) in order to reduce the accuracy of Before the surgery on the neck, the surgeon determines the localization and at the same time ensure the highest possible area of the surgical intervention. Assume that in our case, this accuracy of RLN localization. Such plans are called area is square, given by coordinates: saturated I G -optimal. [ x1− ≤ x1 ≤ x1+ ; x2− ≤ x2 ≤ x2+ ] (1) II. TASK STATEMENT where ( x1− , x2− ) - coordinates of lower limit rectangle surgery; The stimulation of surgical wound tissues during the neck ( x1+ , x2+ ) - coordinates of upper limit rectangle surgery. surgery based on electrophysiological method allows to Based on the results of works [5,6], we will assume that the values of the maximum amplitude of the information ACIT 2018, June 1-3, 2018, Ceske Budejovice, Czech Republic 47    signal - the response to the irritation of the tissues of the [ y ( x )] = [ y − ( x ) ; y + ( x )], (9) surgical wound will be obtained in an interval form with a where constant error ∆ i = ∆, ∀i = 1,..., I for all stimulation points:      y − ( x ) min =  ϕT (x) ⋅ b , b∈Ω ( ) (10)   y1− ; y1+     and     x11 , x12     − +     y + ( x ) max =  ϕT (x) ⋅ b - b∈Ω ( ) (11) =X =  ; Y    yi ; yi   , (2)  x21 , х22    lower and upper bounds of functional corridor, respectively.   The error of prediction of the interval model of the  − +  distribution of the maximum amplitude of the information   yN ; yN     signal will be evaluated in the following form [7-8]:       where ( x1i , x2i ) - coordinates of stimulation points; [ yi− ; yi+ ] - = ∆ y ( x ) max  b∈Ω ( ϕ T ( x ) ⋅ b − min  ) ϕT (x) ⋅ b (12) b∈Ω ( ) interval value of maximum amplitude of information signal for i-th stimulation point/ As you can see, the minimum number of stimulation points Lets consider the mathematical model for RLN should be equal to the number of unknown m model identification in kind of algebraic equation described in [5]: parameters. Such experiments are called "saturated", the   matrix X of the set (2) is a matrix of the plan, and the matrix yo = β1 ⋅ ϕ1 ( x ) + ... + β m ⋅ ϕm ( x ) , (3)  F is an information matrix [7-8]. where β = ( β1 ,..., β m )T is the vector of unknown Thus, the purpose of this work is to find such a square     matrix of a plan or an informational square matrix F m of a parameters; ϕ T ( x ) = (ϕ1 ( x ),..., ϕ m ( x )) T is the vector of saturated experiment that, at a known interval error  known basic functions; x = ( x1 , x 2 ) is vector of stimulation ∆i = ∆, ∀i = 1,...m , would provide the best of the predictive point coordinates; yo is predicted value of maximal properties of the interval model of the distribution of the maximum amplitude of the information signal. This condition amplitude of information signal in point with coordinates (x1, will ensure the highest precision localization RLN using this x2). If for any i-th stimulation point the true unknown value of model.    information signal amplitude is= yoi ϕ T ( xi ) ⋅ β belongs to III. METHOD OF DESIGN I G -OPTIMAL SATURATED the interval [ yi− , yi+ ] , that is, we have the following EXPERIMENT condition: In the case of using a "saturated" block to estimate the yi− ≤ yoi ≤ yi+ , i=1,...,N (4) parameters of interval model, its minimum prediction error in  the input variable area x ∈ χ is reached at one of the points We get conditions:    of a given set of input variables x j (j=1,...,m) [7-11]:  y1− ≤ b1ϕ1 ( x1 ) +  + bm ϕ m ( x1 ) ≤ y1+ ;    ∆ min =  min { y ˆ + ( x j ) − yˆ − ( x j )} =  (5) x , j =1,.., mj  −   (13)  y N ≤ b1ϕ1 ( xN ) +  + bm ϕ m ( xN ) ≤ y N . = min{2∆ j } + j =1,.., m    x min arg min{ yˆ + ( x j ) − yˆ − ( x j )} System (5) is interval system of linear algebraic equations (ISLAE). The solution of this system will be obtained in the = (14) xi , j =1,.., m following form:    Procedures for calculating the maximum prediction error Y − ≤ F ⋅b ≤ Y + , (6) by interval models are much more complicated, even in the where case of estimating its parameters based on the "saturated"    ϕ1 ( x1 )ϕ m ( x1 )  block of ISLAE [7-11].   In work [8] expressions are given to calculate the error F =  , (7) value at any point for the specified event:  ϕ ( x )ϕ ( x )   1 N m N  m   − + ∆ y ( x ) = 2 ⋅ ∑ α j ( x ) ⋅ ∆ j , x ∈ χ = Y {= yi , i 1,, N } ,= − Y {= yi , i 1,, N } + – vectors j =1 (15) composed of upper and lower bounds of intervals [ yi− , yi+ ] , T    α= ( x ) ϕ T ( x ) ⋅ Fm−1 (16)    respectively; F – known matrix of values for basic functions, where α j ( x ) – j-th component of vector α ( x ) , which in the set by expression (7). If system (6) has solutions (or one general case depends on the choice of a point on the solution), then the area of these solutions is denoted by Ω:      + − experiment area; ∆ j= 0,5 ⋅ ( y j − y j ) - interval errors in x j { Ω= b ∈ R m Y − ≤ F ⋅ b ≤ Y + . } (8) observation points. The set of all solutions Ω of ISLAE (6) makes it Based on the expressions (15), (16), we formulate a possible to determine the set of equivalent (in terms of the condition for choosing a "saturated" block to minimize the existing interval uncertainty) interval models of static maximum prediction error in the area of the values of the systems belonging to the functional corridor: input variables that determine the matrix X rows: ACIT 2018, June 1-3, 2018, Ceske Budejovice, Czech Republic 48  m   Fm of ISLAE is constructed, then it would set the point with the max   2 ⋅ ∑ α j ( x ) ⋅ ∆ j   → min, minimum value of the prediction error. It is advisable to x∈ X  j =1  . (17) replace one of the interval equations in the ISLAE(5) by T  T  α= ( x ) ϕ ( x ) ⋅ Fm −1 interval equation with the vector of values of the input  As you can see, the maximum value of prediction error in variables x max defined by the expression (18) in the current  the area of input variables x ∈ X depends on the chosen "saturated" block. By analogy with the procedure of "saturated" block. We will denote it: ∆ max ( Fm ) . successive I G -optimal design, we simulate the additional  Expression (17) provides minimization of the maximum measurement procedure for the vector-row x max with the error of prediction of the interval model among all possible maximum error of prediction of the interval model, obtaining "saturated" blocks selected from ISLAE (5). measurements with a minimum interval error according to the Let's make an analogy with the theory of design successive expression: I G -optimal interval experiment plans that minimize the maximum error of prediction of interval models [7-11]. In our case, the essence is: design some series of experiments with a max  x∈ X (     ) ∆ ⋅ ϕ T ( x ) ⋅ ( FmT ⋅ Fm ) −1 ⋅ ϕ ( x ) ⋅ m → Fm min . (19) small amount of observations (for example, a saturated However, in contrast to the I G -optimal sequential design experiment); get the corridor of interval models; analysis of procedure of an experiment, we choose a given point on a the predictive properties of these models and on this basis the  discrete set of vector-rows xi (i = 1, ...,N) of matrix X. design of the next one observation [7-11]. Considering the requirements of providing optimal Denote the lower and upper bound for the resulting interval − + prognostic properties of the interval model (minimizing the by [ yˆ min ; yˆ min ]. maximum of prediction error) in the area of input variables, it We will carry out the above procedure for each interval is advisable to use this approach to select the "saturated" equation of the "saturated" block. We get p new "saturated" block from ISLAE (5) in order to simplify the task (17). blocks (p=m). As a result, for each of the m "saturated" Note that in the procedure of I G -optimal design on the first blocks we obtain m values of maximum errors for the iteration, the "saturated" block is also chosen according to the corresponding interval models: I G -criterion, the expression for which is represented (17). In  m   our case, such iteration is meaningless due to high ∆ max= ( Fm ( p )) max 2 ⋅ ∑ α jp ( xi ) ⋅ ∆ j  , xi , i =1,..., N computational complexity. Therefore, in the first step of the  j =1  , (20) method of estimating the set of values of the parameters of T  T α p ( xi ) = ϕ ( xi ) ⋅ Fm ( p), p = −1 1,..., m the interval models of static systems, the "saturated" block of ISLAE will be chosen arbitrarily. where p is index, which means a number of “saturated” Let the structure of the mathematical model of the static block, Fm ( p ) is matrix of basic functions values for p-th   system be defined by the expression (3) with unknown block, α jp ( xi ) is i-th component of vector α for p-th parameters, given interval data (4) and ISLAE formed in the “saturated” block. form (5). To choose the optimal "saturated" block in this step, Choose from ISLAE (5) an arbitrarily "saturated" block instead of a complex computational procedure (17), it is and compute its area of solutions, construct a prediction sufficient to choose from the m "saturated" blocks the one corridor with interval models in the form (9), where ∆ y ( x ) are that provides the lowest value of the sequence (23), that is: defined by expressions (15), (16). arg min {∆ max ( Fm ( p )), p = Fm opt = 1,..., m} , (21) By analogy with the procedure of successive I G -optimal p =1,..., m  design, based on the expressions (15), (16), among (i=1,...,N) Applying procedure (18), we obtain x max - the point at rows of the matrix X values of the input variables for which  which the maximum error of the prediction by the interval ISLAE (5) is composed, choose the vector-row x max for model, the estimation of the set of values of parameters which we calculate the maximum prediction error, that is: which is calculated from the chosen "saturated" block in the   m    above-described method. Further iterations are continued =x max arg  max 2 ⋅ ∑ α j ( x= i ) ⋅ ∆ j , xi , i 1,..., N  , (18) xi =1,..., N  j =1  until such a "saturated" block is obtained, the replacement of T    interval equations which does not lead to a decrease in the α= ( xi ) ϕ T ( xi ) ⋅ Fm−1 maximum prediction error by interval models. The vector obtained from expression (18) is a vector of The algorithm of realization of proposed method described values of input variables, which defines a certain interval below [8]: equation in ISLAE (5). According to the procedure of Step 1. Choose the “saturated” block of ISLAE (5). successive I G -optimal design, it is necessary to carry out the  Step 2. Determination of the vector-row x max of the matrix following measurement with respect to this vector-row. X for solving the task (18). Based on expression (14) to determine the vector of values Step 3. The iteration of the phased replacement of each of of input variables, where the prediction error is minimal in the m interval equations of the "saturated" block on the  the experiment area, we can state that if the vector x max ISLAE (5) to the interval equation with a vector of values of coincides with the vector of values of input variables, for  input variables x max (forming a set of "saturated" blocks) and which one of the interval equations of the "saturated" block calculating maximum prediction errors (15) for interval ACIT 2018, June 1-3, 2018, Ceske Budejovice, Czech Republic 49 models. well as its application for the problem of locating RLN in the Step 4. Choose the optimal “saturated” block based on process of operation on the neck organs considered in this expression (21). article. This saturated experimental plan allows to reduce the Transitioning to step 2. duration of surgical surgery by reducing the number of Note that when transitioning to step 2, for a received imitation points of a wound surgical wound to detect a RLN. "saturated" block, the estimations of the set of parameters of If for the cases considered in other works construct a grid of the interval model and the maximum error of the interval m2 stimulation points, then in the case of a saturated plan, the model constructed for this block will be known. number of points is equal to the number of unknown We implement a sequence of steps until the "saturated" coefficients of the model, which is tens of times smaller. block is received at the last step, any replacement of interval equations does not lead to a decrease in the maximum REFERENCES prediction error for the interval-based models constructed on [1] M.C.D. Poveda, G. Dionigi, A. Sitges-Serra, M. its basis. Barczynski, P. Angelos, H. Dralle, E. Phelan and G. IV. EXAMPLE OF APPLICATION THE PROPOSED Randolph. “Intraoperative Monitoring of the Recurrent Laryngeal Nerve during Thyroidectomy: A Standardized METHOD FOR RLN IDENTIFICATION Approach (Part 2)”. World Journal of Endocrine Surgery, The example of constructing a model of distribution on the vol. 4, no. 1, pp. 33-40, 2012. surface of a surgical wound the maximum signal amplitudes [2] V.K. Dhillon and R.P. Tufano. “The pros and cons to real- as reaction to stimulation of surgical wound tissues described time nerve monitoring during recurrent laryngeal nerve in article [5]. The structure of this mathematical model, dissection: an analysis of the data from a series of obtained from the work [5], has the following form: thyroidectomy patients”. Gland Surgery, vol. 6, no. 6, pp.  π y ( x ) = b0 + b1 ⋅ sin 2 ( x1 ⋅ x2 ⋅ ) + b2 ⋅ x2 + b3 ⋅ ( x22 ) (24) 608-610, 2017. 36 [3] H.Y. Kim, X. Liu, C.W. Wu, Y.J. Chai and G. Dionigi. A fragment of data obtained during the surgical operation “Future Directions of Neural Monitoring in Thyroid on thyroid gland is given in Table 1 in [5]. Surgery”. Journal of Endocrine Surgery, vol. 17, no. 3, We apply the method of selection the “saturated” block pp. 96-103, 2017. from ISLAE to find the most informative points. As a result, the coordinates of the points of the saturated [4] W.E. Davis, J.L. Rea, J. Templer,. “Recurrent laryngeal plan are found: [1.5;3], [3;6], [6;0.5], [6;6] (Fig. 2). nerve localization using a microlaryngeal electrode”. Otolaryngology – Head and Neck Surgery, vol, 87, no. 3, pp. 330-333, 1979. [5] M. Dyvak, O. Kozak, A. Pukas. “Interval model for identification of laryngeal nerves”. Przegląd Elektrotechniczny, vol. 86, no. 1, pp. 139-140, 2010. [6] N. Porplytsya, M. Dyvak. “Interval difference operator for the task of identification recurrent laryngeal nerve”, 16th International Conference On Computational Problems of Electrical Engineering (CPEE), pp. 156-158, 2015. [7] M. Dyvak, I. Oliynyk. “Estimation Method for a Set of Solutions to Interval System of Linear Algebraic Equations with Optimized “Saturated Block” Selection Procedure”. Computational Problems of Electrical Engineering, Lviv, 2017. – V. 7, No. 1. – P. 17-24. Fig. 2. The coordinates of the points found in the saturated plan for [8] Dyvak M., "Tasks of mathematical modeling the static the RLN identification task systems with interval data", Economic thought, Ternopil, 2011, 216 p. (in Ukrainian). V. ACKNOWLEDGMENT [9] Wu C. F. J., Hamada M. S., “Experiments: Planning, This research was supported by National Grant of Ministry Analysis and Optimization”, Wiley, 2009, 743 p. of Education and Science of Ukraine “Mathematical tools [10] Shary S.P., “Algebraic Approach to the Interval Linear and software for classification of tissues in surgical wound Static Identification, Tolerance, and Control Problems, or during surgery on the neck organs” (0117U000410). One More Application of Kaucher Arithmetic”, Reliable Computing 2(1) (1996), p. 3–33. VI. CONCLUSIONS [11]Walter E. and Pronzato L., “Identification of parametric model from experimental data”. London, Berlin, The method of design the saturated experiment with Heidelberg, New York, Paris, Tokyo: Springer, 1997. – interval data on the principles of formation of a saturated 413 p. block of the interval system of linear algebraic equations, as ACIT 2018, June 1-3, 2018, Ceske Budejovice, Czech Republic