Voice and speech have very important roles in human social life and professional performance. The negative impact of laryngeal nerve injury on voice is well known in thyroid surgery, but unfortunately, the correlation between them is little studied. The literature shows that altered voice is a common problem after thyroid surgery. The voice changes were reported in 25% to almost 90% of patients within the first few weeks after thyroidectomy (Henry et al., 2010). Other studies represent similar numbers (30–87%) (de Pedro Netto et al., 2006; Musholt et al., 2006; Stojadinovic et al., 2002; Page et al., 2007; Sinagra et al., 2004; Elsheikh et al., 2016). Voice changes can be classified as neural and non-neural related. The true incidence of recurrent laryngeal nerve injury following thyroid surgery is probably underrated, as it strongly depends on postoperative laryngeal examination. According to a systematic review (Jeannon et al., 2009), which included 27 articles and 25,000 patients, the average of temporary incidence of recurrent laryngeal nerve after thyroid operation was 9.8% and the incidence of permanent injury of the same nerve was 2.3%. The rate varied from 26% to 2.3%. The data of 3,605 patients from 5 high-volume centres in France (Lifante et al., 2017) shows similar results: immediate injury rate was 9.3% (range 3.8–21.8%), permanent rate was 3.1% (0–9.1%). The Scandinavian multicentre audit of 3,660 patients reports postoperative unilateral paresis of the recurrent laryngeal nerve in 3.9% of cases (Bergenfelz et al., 2008). It is very important to realize that vocal cord paralysis may occur without any voice changes. Voice could be normal in case of vocal cord paralysis in up to 28% of cases (Mihai and Randolph, 2009) or even in more than 50% (Ortega et al., 2009). Majority of endocrine and general surgeons agree that pre- and postoperative laryngoscopy should be mandatory in all patients undergoing thyroid surgery, as it is the most trustworthy method in determining vocal cord paralysis. Despite reliability of this method, it could be uncomfortable and unpleasant for the patient, adds extra costs, needs special instruments and trained personal, causes logistic problems (Ortega et al., 2009). Probably computerized acoustic voice analysis could be used as a screening method to select patients for laryngoscopic examination.

The idea to apply acoustical analysis of speech for voice disorder detection and evaluation is not new. Similar ideas were proposed 50–60 years ago (Lieberman, 1963; Koike, 1967), and has been studied since now. Various acoustic parameters were proposed and employed for this purpose. These include but are not limited to perturbations of fundamental tone (Kasuya et al., 1983), various noise estimation techniques (Yumoto et al., 1982; Kasuya et al., 1986; Fukazawa et al., 1988), cepstral features (Dejonckere and Wieneke, 1994; Hillenbrand et al., 1994), nonlinear operators and techniques (Cairns et al., 1994; Giovanni et al., 1999), MFCC features (Dibazar et al., 2002), fractal dimensions (Baljekar and Patil, 2012; Ali et al., 2016). During the last decade the task of acoustic analysis-based detection and evaluation of pathological voices was studied intensively. Vast majority of studies focus on combining various features without any physiological reasoning. Extensive and summarized reviews on acoustic analysis of pathological voice can be found in Arroyave et al. (2012), Vaičiukynas et al. (2015), Panek et al. (2015).

The speech signal is generated in two stages. Firstly, the so-called source signal is induced. The air flow generated by the lung causes the vibration of the vocal folds. This vibration is called phonation process, and its intensity is described by the fundamental frequency value. In the next step, the glottal flow is modulated by the voice tract. The result of this modulation is the speech signal, transmitting information on both the vocal fold and the voice tract resonant properties. Disorder of vocal folds (paralysis among them) affects the speech inevitably. The effect depends on dysfunction degree of the folds and can vary from inaudible changes up to severe changes of voice, for example, it becomes breathy, harsh, and weak.

Acoustical analysis of the speech signal is considered as an objective evaluation of the vocal tract functionality rather than perceptual analysis of the speech. Acoustic parameters represent generative and articulatory properties of the voice and thus could be applied for pathology detection and evaluation. Different acoustic parameters describe different stages of the speech signal production, thus should be chosen reasonably. To estimate the functionality (or immobility) of vocal folds, we have to analyse the glottal flow.

The most common technique to estimate the glottal flow is to employ source-filter production model. This model describes the speech signal as the convolution of the source signal (glottal flow) and a filter (vocal tract). Both source signal and vocal tract can be modelled using various joint estimation models or separately, ignoring or considering close phase of the glottal cycle (Walker and Murphy, 2007; Alku, 2011).

If we consider the glottal flow and the vocal tract as independent, the glottal flow can be extracted by inverse filtering of the speech signal (Alku, 2011). The inverse filter eliminates the effect of the vocal tract thus giving the estimate of the glottal flow. The process of inverse filtering can be simplified using linear modelling of the vocal tract.

Linear modelling has played a very important role in speech analysis domain because of its mathematical tractability and applicability, spectral estimation properties. For speech analysis purposes, the linear all-pole filter was applied mostly. Various linear prediction techniques were employed for glottal flow extraction: constrained linear prediction with reduced distortion of filter frequency response (Alku and Magi, 2009), weighted linear prediction with temporal weighting of the residual (Airaksinen et al., 2014) and its stabilized modification (Kafentzis et al., 2011).

All voice pathology detection and inverse filtering studies can be summarized as follows:

In this paper, we present the AR model-based inverse filtering approach for estimation of glottal flow and detection of vocal fold paralysis. A variable order AR model was employed to model the vocal tract and the glottal flow.

The glottal flow can be obtained from the voiced speech segments. Considering the source-filter approach, the speech signal $s(t)$ can be expressed as the convolution of the glottal flow $g(t)$ and the vocal tract filter $h(t)$ Here the lip radiation effect (modelled as a first-order differentiating filter) is included in the vocal tract processing and is not considered separately. Traditionally, the vocal tract is modelled using an all-pole filter for speech analysis purposes.

If we obtain an estimate of the inverse vocal tract filter ${\hat{h}^{-1}}(t)$ and apply it to the analysed speech signal $s(t)$, we will eliminate the effect of the vocal tract thus obtaining the estimate of the glottal flow

In this study, we applied AR model for the modelling of the vocal tract. The choice was due to the following reasons: The AR model parameter estimation technique is presented in the next subsection.

Let us explore the speech signal as the process $\{{S_{t}}\}$ with zero mean and describe it using the AR model where N is the length of the signal ${S_{t}}$, $\{{V_{t}},\hspace{2.5pt}t=1,2,\dots \}$ is the process of mutually independent and normally distributed random variables.

Our task is to estimate the model order M, the parameters $\{{a_{1}},{a_{2}},\dots ,{a_{M}}\}$ and b of the AR model.

From (3) we can obtain If we denote we get the following expression of the AR model

The equation is solved using the recurrent evaluation approach (Kaukėnas, 1983). The Efroymson matrix is composed where R is the cross-correlation matrix of ${X_{i}}$ and ${X_{j}}$, $i,j=1,\dots ,M$; T is the cross-correlation vector of Y and ${X_{i}}$, $i=1,\dots ,M$; O denotes zero vectors and matrices, and I is a unit matrix.

Each new sequence ${X_{i}}$ is included during the recurrent modification of the Efroymson matrix $E(i,j)$ denotes the Efroymson matrix before including ${X_{i}}$, $E{(i,j)^{\prime }}$ is an updated version of the Efroymson matrix with included ${X_{i}}$.

Finally, the model parameter ${a_{i}}$ is estimated The estimate of ${b^{2}}$ is obtained as follows

The model order estimation task is solved by comparing ${M_{1}}$ and ${M_{2}}$ order models. Usually, the estimated variances of prediction error ${\hat{b}_{M1}^{2}}$ and ${\hat{b}_{M2}^{2}}$ are compared. The following estimator for the model order was formulated in Kaukėnas (1983) where ${F_{cr}}(1,N-i)$ is the quantile of Fisher distribution with 1 and $(N-i)$ degrees of freedom; ${\hat{b}_{0}^{2}}$ is the estimate of variance, i.e. ${\hat{b}_{0}^{2}}=\hat{D}$, $\hat{D}$ is the estimated variance of the process. ${M_{\max }}$ is the maximum model order value, it is based on empirical knowledge of the signal.

In this study, we have chosen ${M_{\max }}=20$ for the vocal tract model. The filter with order up to 20${^{th}}$ will model up to 10 resonant frequencies (formants), which is completely sufficient for description of speaker’s individual articulation (vocal tract) properties.

For the modelling of the glottal flow we have chosen ${M_{\max }}=200$. The decision is based on the results obtained in Tamulevičius and Kaukėnas (2017), where description of individual speakers qualities demanded AR model order up to 170.

The quality of the estimated glottal flow was assessed on the basis of the ratio of estimated squared prediction error and estimated signal variance ${\hat{b}^{2}}/\hat{D}$. The value of ${\hat{b}^{2}}/\hat{D}$ indicates the relative part of the unmodelled signal: the higher ratio value we obtain, the higher signal prediction error is. Therefore, we can expect a low ${\hat{b}^{2}}/\hat{D}$ value for normal glottal flow and high values for pathological voices (with paralysis).

In this study, we will express this ratio in percentage and call it the estimated error of glottal flow. We think that for healthy and normal voices this ratio will approach towards zero level, and for pathological voices, it will converge to 100% (in case of full paralysis or dysfunction of vocal folds).

In this subsection we will present the algorithm of the inverse filtering of the speech signal and estimation of the glottal flow quality.

For experimental analysis of the proposed method, records of two voice types were collected.

Starting in 2016, patients scheduled for thyroidectomy and included in the study (study launched in Vilnius University Faculty of Medicine Institute of Clinical Medicine Clinics of Gastroenterology, Nephrourology and Surgery in cooperation with the Institute of Data Science and Digital Technologies) were selected for voice recording and vocal folds movement evaluation before and after the operation. Vilnius regional Biomedical research committee permission No. 158200-15-819-331 has been given in 2015.12.08. The interval comparison of sequential voice recording was matched against change in vocal folds movement. The vocal folds function was assessed by a laryngoscopy in each case before and after thyroidectomy procedure.

A prospective trial was launched in March 2016 and finished in May 2017. 112 patients with known thyroid pathology were prospectively enrolled in this study. All 112 patients were operated on in Vilnius University Hospital Santaros Klinikos. The study protocol included voice recording and laryngeal exam in all patients preoperatively and postoperatively by a qualified ENT specialist. 6 cases of temporary vocal cord palsy were diagnosed on postoperative examination (5.4% injury rate per patient and 3% per nerve at risk). No cases of permanent or bilateral vocal cord palsy were recognized postoperatively.

All the patient voices were recorded using headset microphones in a clinician’s room environment. There were 4 recording sessions organized: one day before surgery, one day, 2 weeks, and 4 months after surgery.

The control group consisted of healthy people with no complaints or throat/mouth surgery procedures in last 3 months. The voices of 10 female and 10 male speakers were recorded using voice recorder with an external microphone in a silent room environment.

All the recorded persons were asked to pronounce vowel [a] in a sustained manner for 3–4 seconds. This vowel is characterized by a minimal lip restriction during radiation phase and a fully expressed phonation level. Besides, vowel [a] is common for most languages, what makes it universal for comparison purposes.

For analysis of pathological and healthy voices, we have selected two voices for inverse filtering procedure and estimation of glottal flow. The estimated signal of glottal flow and its spectral density function were analysed to estimate the qualities of the pathological and healthy voices.

Figure 1 presents the results obtained for the healthy female’s voice. The estimated order of the vocal tract filter was 11 (i.e. the vocal tract had 6 resonant frequency values). The estimated glottal flow can be evaluated as periodic and normal (Fig. 1(b)). Spectral density function (Fig. 1(c)) is also periodic, the harmonic components are vivid through the entire frequency range of the signal.

The results of pathological male voice analysis are given in Fig. 2. Here we can see the distorted waveform of the utterance (Fig. 2(a)). The vocal tract was modelled by 20-th order model which means ten resonant frequencies of the tract. The estimated glottal (Fig. 2(b)) flow is noisy with no sign of periodicity (what is characteristic for the vocalized vowel). The spectral density (Fig. 2(c)) of the flow is noise-like, here we can see only 4–5 harmonic components. This is the evidence of vocal fold immobility which can be the result of the vocal fold paralysis.

Similar results were obtained for all pathological voices: non-periodicity of the estimated glottal flow, noise-like spectral density function. The degree of non-periodicity was different for the individual voices. This difference may be with individual characteristics of the voices and require a more detailed study with larger datasets.

First of all, we evaluated the error level of glottal flow for healthy and pathological voices. The averaged results are given in Fig. 3.

We can see the clear difference between healthy and pathological voices. The patients’ voices (before thyroidectomy surgery) have at 50% higher error level than healthy ones. The thyroidectomy procedure with the output of the immobility of the vocal folds increased the error level by 15–50% (by 2–3 times in comparison with healthy voices). Therefore, the prediction error level of the glottal flow enables us to identify the case of vocal fold paralysis.

Nevertheless, the amount of analysed data is not sufficient to make statistically reasoned conclusions and to propose some global criteria for detection of vocal fold paralysis. The main reason is the scattering of the results because of individual properties of the persons’ voice. Every person is characterized by his own inherent qualities of glottal flow, so the output of the surgery (which is also very characteristic to person) should be estimated individually, taking into account these qualities. To illustrate this statement the data about the status of 3 patient’s vocal folds is given in Fig. 4.

Comments in Fig. 4:

It is obvious that glottal flow prediction error-based estimation of the vocal fold functionality should be performed individually. As we can see in Fig. 4, the preoperative and postoperative status of vocal folds were different for patients, and the recovery process is also individual. Therefore, this assessment can be implemented as monitoring the dynamics of vocal fold functionality for screening examination method to select patients for laryngoscopic procedure. Relative change of the glottal flow prediction error reflects changes in glottal flow. For application purposes, the change should be parametrized.