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asymmetry on auto-oscillating replicas
Anne Bouvet, Isao Tokuda, Xavier Pelorson, Annemie van Hirtum
To cite this version:
replicas
Anne Bouvet,1 Isao Tokuda,2 Xavier Pelorson,1 and Annemie Van Hirtum1
1)LEGI, UMR CNRS 5519, Grenoble Alpes University,
France
2Dep. Mech. Eng., Ritsumeikan Univ., Nojihigashi, Kusatsu, Shiga 525-8577,
Dysphonia is often caused by level difference between left and right vocal folds, which
1
are positioned on different angles with respect to the transverse plane resulting in
2
angular asymmetry. Unilateral vocal fold paralysis may cause such angular
asymme-3
try. In this case the normal vocal fold is located on the transverse plane, whereas the
4
paralyzed vocal fold is rotated in the sagittal plane as its posterior edge is moved up
5
to the superior direction. The effect of such angular asymmetry (up to 25◦) between
6
the left and right vocal fold on the auto-oscillation is experimentally studied using
7
mechanical replicas. For all replicas, it is observed that, as full vocal folds contact is
8
lost, increase of angular asymmetry results in a decrease of the signal-to-noise ratio,
9
an increase of the total harmonic distortion rate, and an increase of the oscillation
10
threshold pressure. These general tendencies are in agreement with clinical findings
11
reported for vertical level difference during phonation. In analogy to the preceding
12
experimental study, in which vocal folds are spaced in parallel with a vertical
trade-13
off, a formula is proposed to describe the oscillation threshold as a function of angular
14
asymmetry.
I. INTRODUCTION
16
Vocal folds (VF) asymmetry is reported as one of the main causes of glottic insufficiency
17
resulting in dysphonia1. Glottic insufficiency is often caused by complete or partial VF
18
paralysis such as unilateral vocal fold paralysis (UVFP). UVFP has many causes and is
19
characterized by vocal fatigue and a breathy voice1.
20
21
Clinical examinations show that UVFP introduces left-right VF asymmetries to the
22
shape, tension and positioning. Videostroboscopic imaging (Fig.1) reveals that
asymmetri-23
cal positioning of the VFs in the sagittal plane prevents complete glottal closure1–3, which is
24
suggested to be the main physical cause of the observed breathy voice. The VF positioning
25
asymmetry results from both the normal and paralyzed VFs. In order to compensate the
re-26
duced movement of the paralyzed VF, the position of the normal VF in the inferior-superior
27
direction is changed. Consequently, a vertical level difference between the left and right
28
VF emerges, where the paralyzed VF is often inclined in the sagittal plane (right VF in
29
Fig. 1) and the normal VF remains in the transverse plane (left VF in Fig.1) as described
30
in literature3–5. Quantitative characterization of this spatial asymmetry from imaging on
31
human subjects is extremely tedious1–3. Nevertheless, the vertical level difference during
32
phonation is reported3 to yield several millimeters, i.e. 1.3 ± 1.5 mm or up to 3 mm for
33
UVFP subjects. The effect of positioning asymmetry on human vocalization can not be
34
studied independently from tension or shape asymmetry between left and right vocal folds.
35
In that sense, physical studies of auto-oscillation using mechanical replicas are of interest,
36
when aiming a systematic study of the influence of VF positioning asymmetry.
37
38
Recently4,5, the impact of level difference between parallelly located VFs, hereafter
re-39
ferred to as parallel level difference, was investigated. The preceding study, however, did
40
not take into account inclination of the paralyzed VF, which is physiologically more close to
41
reality. Therefore, in this work, the influence of inclination of one VF is studied, i.e. angular
42
level difference. The normal VF is kept in its transverse plane, whereas the paralyzed VF
43
is rotated in the sagittal plane so that its posterior edge is moved in the superior direction
44
imposing a left-right VF asymmetry angle (α). Our aim is to systematically study the
influ-45
ence of asymmetry angle α > 0◦ on the auto-oscillation of mechanical replicas. Quantified
46
objective oscillation features inform on the effect of α on the oscillation system (threshold
47
pressures, oscillation frequency) as well as on overall spectral properties (signal-to-noise
48
ratio, harmonic distortion rate) also used in voice quality studies13,14. There have been
49
numerous studies on left-right asymmetry focusing on other VF properties such as tension
50
and shape6–11. In the present study, symmetry of such properties is maintained so as to
concentrate only on the angular asymmetry between left and right VFs.
52
53
In the next section (SectionII), VF replicas are introduced, glottal replicas with left-right
54
angular asymmetry are described and mechanical resonances are characterized. Section III 55
reports experiments on fluid-structure interaction and their results. To elucidate the
oscilla-56
tion onset pressure, analogy of the angle asymmetry to parallel level difference is proposed
57
in Section IV. The final section is devoted to conclusions and discussions.
58
59
II. REPLICAS
60
A. Vocal folds replicas: M5, MRI and EPI
61
Three deformable silicone VF replicas are used, labelled M5, MRI and EPI, respectively.
62
A medio-frontal and top view of symmetrically mounted VF replicas (α = 0◦) and layer
63
thicknesses ld are illustrated in Fig. 2. The three replicas differ due to their multi-layer
64
composition as well as due to their geometry. In the next paragraphs, firstly the different
65
layers are described and next different casts are discussed.
66
67
Each silicone VF replica consists of an overlay of silicone molding layers obtained following
68
the procedure detailed in literature4,15–17. Concretely, two (M5), three (MRI) and four layers
69
(EPI) are superposed resulting in three different VF replicas labeled M5, MRI and EPI. The
70
layers aim to reproduce the properties (Young’s modulus E , thickness ld) of the different VF
constituents for an adult male18–21: the thyroid muscle (vocalis), the superficial layer of
72
the lamina propria (Reinke’s space), the underlying ligament of the lamina propria without
73
tension and the epithelium. Each layer is composed as a mixture of silicone thinner and
74
two parts of either A&B Ecoflex (03-00, Smooth-On, Inc, Easton, PA) or either two parts
75
A&B DragonSkin (FX Pro, Smooth-On, Inc, Easton, PA), labeled Ecoflex/Silicone (ES)
76
and Dragon Skin/Silicone (DS), respectively. The mixing ratio M (ES or DS Part A: Part
77
B: Silicone) differs between constituent layers as detailed in Table I. The properties (tensile
78
tests, electromechanical press INSTRON 3369) of each layer are detailed in Table I. Note
79
that the values of E are consistent with those reported in literature4,15–17.
80 81
(a)
(b)
FIG. 2. (color online) Symmetrically (α = 0◦) mounted vocal fold replicas M5, MRI and EPI: a)
TABLE I. Layer properties in human18–21 and in silicone vocal folds replicas (M5?, MRI†, EPI‡):
Young modulus E , ecoflex-to-silicone ratio M, layer thickness ld.
Male adult18–21 VF replica
Layer E [kPa] ld[mm] E [kPa] M [-] ld [mm]
Muscle 18-23 ± 2.4 6.0 10.4? 4.9† 21.9‡ ES 1:1:2? ES 1:1:4† ES 1:1:1‡ 6.4? 10.0† 6.4‡
Ligament 30-36 0.8 4.9‡ ES 1:1:4‡ 1.0‡
Superficial 35-49 7.1 0.6 4.9? 0.2†,‡ ES 1:1:4? ES 1:1:8† ‡ 1.5? 3.0†1.0‡
Epithelium 40-60 0.1 52†,‡ DS 1:1:1† ‡ 0.1†,‡
? M5: muscle and superficial
†MRI: muscle, superficial and epithelium
‡EPI: muscle, ligament, superficial and epithelium
ES Ecoflex/Silicone mixture
DS Dragonskin/Silicone mixture
The M5 VF replica is a two-layer model reproducing the muscle and superficial layer
82
of the lamina propria (see Table I) molded with the cast shaped in accordance with the
83
M5 geometry15,17,22. The MRI replica has a three-layer structure, obtained by adding a
84
thin surface layer representing the epithelium to the two-layer structure of the M5 replica.
85
This replica is molded using a realistic VF geometry cast obtained from magnetic resonance
86
imaging. The MRI geometry adopts a trapezoidal shape in the transverse plane as seen
from the top view in Fig. 2, whereas the M5 replica is rectangular. Therefore, the MRI
88
shape is more complex than the M5 shape and more faithful to the shape of the human
89
VF. Note that, compared to the MRI VF replicas reported in literature4,15,17, an epithelium
90
layer is added in this work. The EPI replica is molded with the M5 cast as compared in17.
91
A four-layer structure is obtained by inserting an extremely soft ligament layer between the
92
vocalis muscle and the superficial layer. Epithelium layer is added to the superficial layer.
93
Each VF replica is mounted on a backing layer (DS, ratio M 1:1:1, thickness 4 mm) in
94
order to attach it to a rigid support.
95
96
B. Glottal replicas with imposed asymmetry angle α
97
TABLE II. Dimensional parameters (L, E, e, a) and critical asymmetry angles α(I,II),(II,III) for
all replicas (M5, MRI and EPI).
Replica L [mm] E [mm] e [mm] a [mm] αI,II [◦] αII,III [◦]
M5 17.0 10.2 2.99 4.5 8.1 29.2
MRI 18.0 10.0 3.00 4.5 7.8 28.8
EPI 17.0 10.2 1.09 4.5 3.0 14.0
Each VF is placed in a rigid holder which is adapted to the M5, MRI or EPI VF replica
98
geometry. Dimensions are summarized in Table II and depicted in Fig. 3: largest thickness
99
E and smallest thickness e along the inferior-superior direction and length L along the
posterior-anterior direction. The right VF (normal) is fixed in the transverse plane, whereas
101
the left VF (paralyzed) is rotated so that its posterior edge is lifted up in the superior
direc-102
tion resulting in left-right VF asymmetry angle α shown in Fig3. For all glottal VF replicas,
103
the rotation axis along the left-right direction is fixed in the rigid holder at a = 4.5 mm
104
from the anterior edge (Table II).
105
106
(a) general superior view, MRI
(b) sagittal side view (c) medio-frontal view
FIG. 3. (color online) Imposed asymmetry angle α, glottal gap parameters A, lA, hA and
geo-metrical parameters E, e, L and a: a) glottal gap (red triangle) for MRI replica with α = 20◦
(a) G(α)
(b) A(α)
(c) hA(α)
FIG. 4. Geometrical characterization of M5, MRI and EPI replicas as a function of asymmetry angle α: a) degree of contact G (Eq. (1)) and critical angles α(I,II),(II,III) (vertical lines), b) glottal gap
area A (Eq. (2)) and reported maximum normal pre-phonatory glottal area23 (5 mm2, horizontal
line) and c) glottal gap height hA (Eq. (4)) and maximum vertical level difference reported during
Experiments are performed for 13 different values of left-right asymmetry angle α ranging from α = 0◦ up to α = 24.6◦ affecting the glottal gap at rest. For symmetric angular
configuration (α = 0◦), the glottal gap area in the transverse plane is negligible as the vocal
folds are in full contact (Fig. 2). When α is increased, air leakage occurs through a glottal gap in the medio-sagittal plane since the vocal folds are no longer in full contact as depicted in Fig. 3 (red dashed triangle). The triangular gap with area A, base lA and height hA
appears near the posterior VF edge and extends towards the anterior edge as α increases. In this sense, lA indicates the glottal gap length in the posterior-anterior direction between the
normal right VF and the paralyzed left VF. Note that lA > L can occur due to the position
of the rotation axis at a distance a > 0 from the anterior VF edge as shown in Fig. 3(b). The degree of VF’s contact as a function of α is then expressed as G ∈ [0 1]:
G(α) = 0 if lA≥ L, 1 − lA(α) L otherwise. (1)
Three contact regimes (I, II and III) are identified as α increases: I) G = 1 when lA = 0,
i.e. full VF’s contact, II) 1 > G > 0 when 0 < lA < L, i.e. partial VF’s contact and III)
G = 0 when lA ≥ L, i.e. no VF contact. Note that full contact (regime I) implies no air
leakage and hence A = 0, whereas, for A > 0, either partial contact (regime II) or no contact (regime III) occurs depending on α. Two critical angles αI,II and αII,III are then defined:
αI,II indicating the largest angle, for which full contact occurs (shift from regime I to II) and
αII,III indicating the largest angle, for which partial contact occurs (shift from regime II to
lA(α), and triangle height hA(α) follow from trigonometric reasoning using geometrical VF
replica parameters L, E, e and a:
A(α) = 0 if tan(α) L + a + E/2−esin(α) < E2, 1 2·tan(α) · L + a + sin(α)1 · E 2 − e · tan(α) − E 2 2 +12 · tan(α) · (E − e) · e − 2 · tan α2 · (L + a) otherwise, (2) and lA(α) = 0 if tan(α) L + a + E/2 − e sin(α) < E 2 or A = 0, 2 v u u u t A(α) sin(2α) otherwise, (3)
so that hA depicted in Fig.3(b) varies as:
hA(α) = 0 if lA= 0, 2A(α) lA(α) otherwise, (4)
and thus ratio (hA/lA)(α) = 2A(α)/lA2(α).
107
108
For each replica (M5, MRI and EPI), G(α) (Eq. (1)), A(α) (Eq. (2)) and hA(α) (Eq. (4))
109
are plotted in Fig 4. The curves correspond to analytic expressions, while the symbols
110
indicate asymmetry angles (α < 25◦), at which experiments were carried out. Critical
111
angles αI,II and αII,III are summarized in TableII and plotted (vertical annotated lines) in
112
Fig. 4(a). Values of G(α), A(α) and α(I,II),(II,III) for M5 and MRI are close to each other,
113
since their geometrical parameters (L, E, e and a in Table II) are similar. For EPI, on
the other hand, e yields about one-third of the value for M5 and MRI so that glottal gap
115
A(α) increases more rapidly and VF’s contact degree G(α) reduces more rapidly than those
116
of M5 and MRI. As a consequence, in the range of our study α < 25◦, all three contact
117
regimes occur for EPI as αEP I
II,III < 25
◦ (Table II), whereas regime III (no contact, G = 0)
118
does not occur for M5 and MRI, since αM RI,M 5II,III > 25◦ holds (Table II). Considering that
119
initial pre-phonatory glottal gap area for healthy human speakers23 yields less than 5 mm2
120
(dashed horizontal line in Fig. 4(b)), it follows that the glottal gap area for α = 24.6◦
121
is increased by a factor of 8 (for M5) or more (for EPI and MRI). Height hA (Fig. 4(c))
122
quantifies the maximum distance between both VFs along the inferior-superior direction
123
and therefore provides a rough (since e is neglected) approximation for the vertical level
124
difference measured during disordered human phonation3, i.e. 1.3 ± 1.5 or up to about 3 mm
125
(dashed horizontal line in Fig. 4(c)). For α = 24.6◦, the height of hA ≈ 6 mm is twice as
126
much as the value reported on humans. Thus, the range of asymmetry angle α studied in
127
our experiments comprises the range observed for human phonation. Note that hA≤ 3 mm
128
corresponds to α ≤ 13◦ for EPI and to α ≤ 17◦ for M5 and MRI. The ratio (hA/lA)(α)
129
increases from 0 up to 0.4 for all replicas, indicating that height hA of the leakage triangle
130
remains less than half of its base lA for all α.
131
132
C. Mechanical characterization of replicas
133
Mechanical resonance properties of glottal replicas are examined for symmetric angular
134
configuration (α = 0◦). A shaker (Modalshop K2007E) equipped with a mobile bar (length
230 mm and diameter 3.7 mm) is positioned so that the free bar end is in contact with
136
the VF replica. Sinusoidal signals from 80 Hz to 350 Hz with a frequency step of 1Hz
137
and duration 1 s are fed to the shaker in order to excite the structure. An accelerometer
138
(PCB 482A21) is attached to the mobile bar to measure the applied excitation x(t). A laser
139
(wavelength 635 nm) and photo-diode (BPW34) are aligned along the inferior and superior
140
side of the replica, respectively. The response y(t) of the replica to the excitation is then
141
gathered from the photo-diode (BPW34) as the amount of laser light modulated by the
142
replica varies during excitation.
143
144
Frequency transfer functions H(f ) = X(f )/Y (f ), representing the ratio of the Fourier
145
transforms of response X(f ) = F (x(t)) to excitation Y (f ) = F (y(t)), are estimated in order
146
to characterize the mechanical resonances. Frequency fM associated with the maximum
147
amplitude |H(f )| and -3 dB frequency bandwidth ∆fM (|H(f )| > |H(fM)| − 3 dB) are
ex-148
tracted. The resonance quality factor QM yields QM = fM/∆fM. The estimated amplitude
149
|H(f )| and the detected resonances for EPI are illustrated in Fig.5. Mechanical resonance
150
properties and standard deviations for all replicas (M5, MRI and EPI) are summarized in
151
Table III. For all replicas, two mechanical resonances are identified.
152
TABLE III. Mechanical resonance properties (resonance frequencies f1,2M, bandwidths ∆f1,2M, quality
factor QM1,2) and standard deviations of replicas for α = 0◦.
first resonance peak second resonance peak
Replica f1M [Hz] ∆f1M [Hz] QM1 [-] f2M [Hz] ∆f2M [Hz] QM2 [-]
M5 142 ± 3 16 ± 7 10 ± 5 264 ± 3 9 ± 3 29 ± 2
MRI 144 ± 2 13 ± 4 11 ± 3 262 ± 2 8 ± 2 32 ± 5
EPI 141 ± 2 13 ± 2 11 ± 2 264 ± 2 15 ± 6 21 ± 9
FIG. 5. (color online) Estimated amplitude |H(f )| for the EPI replica with α = 0◦ with resonance
frequencies (fM
1 , f2M) and bandwidths (∆f1M, ∆f2M).
III. FLUID-STRUCTURE INTERACTION EXPERIMENTS
154
A. Experimental approach
155
In order to reproduce a fluid-structure interaction leading to auto-oscillation, glottal
156
VF replicas are positioned vertically as shown in Fig. 3(a). A rigid uniform tracheal tube
157
(diameter 16 mm, length 460 mm, acoustic resonance frequency 185 Hz) is attached to the
inferior end of the glottal VF replica. Continuous steady airflow (density ρG = 1.2 kg·m−3,
159
dynamic viscosity µG = 1.8 × 10−5 Pa·s, temperature 24 ± 2◦C) is provided by an air
160
compressor (Hitachi SC820) connected to a pressure reservoir (volume 0.04 m3) to which
161
the tracheal tube is mounted4. The transition is smoothed by a cone between the reservoir
162
and the tracheal tube (section ratio 0.3). The pressure reservoir is filled with acoustic
163
foam in order to avoid parasite acoustic resonances. The compressor is equipped with a
164
pressure regulator (10202U, Fairchild, Winston-salem, NC). A pressure transducer (Kyowa
165
PDS-70GA, accuracy ±5 Pa) is positioned in a pressure tap in the tracheal wall, 185 mm
166
upstream of the glottal VF replica, in order to measure upstream pressure Pu with sampling
167
frequency fs = 10 kHz.
168
169
For each VF replica, 13 different asymmetry angles α < 25◦ are imposed as outlined in
sectionII Band a rigid uniform vocal tract tube of length 170 mm (diameter 16 mm, acoustic resonance frequency 500 Hz) can be attached airtight along the superior end of the VF replica. The effect of angular asymmetry α on the fluid-structure interaction is characterized by analyzing the measured upstream pressure Pu(t). Upstream pressures associated with
auto-oscillation onset (POn) and offset (POf f) are sought. Next, auto-oscillation features are
analyzed for 5 s of steady-state upstream pressure signal Pu(t), while the mean upstream
pressure Pu ≈ POnholds as illustrated in Fig.6. First harmonic (or fundamental) frequency
f0 is extracted. Second f1 and third f2 harmonic frequencies of f0 are detected when its
power (in dB) yields at least half the power of f0. The harmonic contents of Pu(t) is then
FIG. 6. (color online) Measured upstream pressure time signal of Pu(t) for the EPI replica with
α = 0◦. Oscillation onset POn and offset POf f pressures are indicated (square) as is the steady
state oscillatory signal portion (Pu ≈ POn) used for analysis (5 s). For clarity, a zoom of Pu(t) near
oscillation onset (left), during steady state oscillation (middle) and near oscillation offset (right) is provided.
ratio between the summed power of higher harmonic frequencies Pharm and the power of the
first harmonic frequency Pf0 :
THD = 10 log10 Pharm Pf0
. (5)
The power ratio Pf1/Pf0 between the second and first harmonics is quantified as well. Next,
the signal-to-noise ratio SNR (in dB) is obtained as the ratio of the summed power of all signal harmonics Psignal based on the first harmonic frequency f0 to the summed power of
the remaining noise Pnoise:
SNR = 10 log10 Psignal Pnoise
All experiments are repeated 3 times so that the average and standard deviation of the
170
extracted features are plotted. Experiments are performed with and without vocal tract
171
tube. Since their resulting features are similar, only experimental results with vocal tract
172
tube are shown. It follows that the results are not much affected by acoustical coupling
173
with the supraglottal vocal tract.
174
175
B. Experimental results
176
TABLE IV. Minimum and maximum onset pressures relative [%] to the value for α = 0◦ and
associated α [◦]. minimum maximum Replica [%] [◦] [%] [◦] M5 -10.5 10.2 46.9 24.6 MRI -2.0 1.7 29.5 22.0 EPI -12.9 11.7 45.0 24.6
Oscillation onset (POn) and offset (POf f) pressures are plotted as a function of α in Fig.7.
177
Associated fundamental (f0) frequencies and detected second (f1) and third (f2) harmonics
178
are shown in Fig.8. The minimum and maximum onset pressures POn(α) relative to the value
179
for symmetric angular configuration (α = 0◦) are quantified in Table IV. The minimum of
180
the onset pressure (min(POn(α))) occurs for α > 0◦: within regime I (full contact, G = 1) for
(a) M5
(b) MRI
(c) EPI
FIG. 7. Mean (symbols) and standard deviation (error bars) of onset POnand offset POf fpressures
as a function of asymmetry angle α: a) M5, b) MRI and c) EPI. Critical angles (vertical lines) indicate contact regime changes between G = 1 (I), G ∈]0, 1[ (II) and G = 0 (III).
the MRI replica and within regime II (partial contact, G ∈]0, 1[) for the M5 and EPI replicas.
182
This suggests that for small values of α (under 10◦) has a beneficial effect. Nevertheless, the
(a) M5
(b) MRI
(c) EPI
FIG. 8. Mean (symbols) and standard deviation (error bars) of harmonic frequencies (fN) at the
phonation onset, as a function of asymmetry angle α: a) M5, b) MRI and c) EPI. Critical angles (vertical lines) indicate contact regime changes between G = 1 (I), G ∈]0, 1[ (II) and G = 0 (III). Horizontal dashed lines indicate mechanical resonance frequencies f1,2M.
amount of decrease is limited to less than 13% for all replicas and is very low for MRI replica
TABLE V. Overall decrease of fundamental frequency f0 relative [%] to the value for α = 0◦ in
the full contact (I), partial contact (II) and no contact (III) regime.
Replica I II III
M5 0.3 -10.3 not measured
MRI -2.4 -19.8 not measured
EPI -0.5 -22.8 -34.2
which is the most realistic geometry compare to human VF. In addition other quantities
185
(SNR, THD, etc.) have to confirm this tendency.
186
The maximum of the onset pressure (max(POn(α))) occurs near the largest asymmetry
187
angle 24.6◦ and the relative increase yields more than 29.5% for all replicas. For all replicas,
188
the onset pressure increases for α ≥ 15◦. For α < 15◦, the shape of the Pon(α) curves varies
189
between replicas, because of its dependence on the detailed geometry and layer composition
190
of the individual replica. It is observed that more realistic replicas (MRI and EPI) exhibit a
191
more monotonic tendency than the M5 replica. For all replicas, hysteresis between the onset
192
and offset pressures is observed as expected. The degree of hysteresis is only marginally
193
affected by α in contact regimes I and II (all replicas: M5, MRI and EPI), whereas it is seen
194
to be reinforced in contact regime III (only EPI replica).
195
196
For symmetric angular configuration (α = 0◦), the detected oscillation frequencies f0
197
and f1 are closely located to the first and second mechanical resonance frequencies given
FIG. 9. Mean (symbols) and standard deviation (error bars) of total harmonic distortion (THD in [dB]) as a function of asymmetry angle α for all replicas. Critical angles (vertical lines) indicate contact regime changes between G = 1 (I), G ∈]0, 1[ (II) and G = 0 (III).
in Table III. This suggests that the auto-oscillation resulting from the fluid-structure
in-199
teraction is aerodynamically driven. Thereafter, the fundamental frequency f0 decreases
200
monotonically with α. The overall decrease of f0 relative to its value for symmetric
config-201
uration (α = 0◦) is quantified in Table V for all three replicas. It follows that the decrease
202
is negligible (<3%) in contact regime I (full contact, G = 1) and becomes more pronounced
203
(> 10%) in regimes II (partial contact, G ∈]0, 1[) and III (no contact, G = 0). The second
204
harmonic f1 is detected for all replicas and all asymmetry angles α of regimes I and II. For
205
the EPI replica, the third harmonic f2 is detected for all α in regime III, indicating that the
206
harmonic power increases with α.
207
208
The increased power of harmonics can be confirmed by the total harmonic distortion rate
209
(THD) plotted in Fig. 9. For all replicas, the THD increases as the asymmetry angle α is
210
increased. The increase is most prominent in contact regime II (G ∈]0, 1[) as it yields about
211
10 dB for the EPI replica and about 25 dB for the M5 and MRI replicas. The imprint of
(a) Steady state, α = 0◦ (b) Steady state, α = 20◦
FIG. 10. Measured upstream pressure Pu(t) for EPI steady state oscillation: a) α = 0◦ and b)
α = 20◦.
FIG. 11. Harmonics power ratio Pf1/Pf0 as a function of asymmetry angle α for M5, MRI and
EPI. Critical angles (vertical lines) indicate contact regime changes between G = 1 (I), G ∈]0, 1[ (II) and G = 0 (III).
higher harmonics on the temporal waveform shape of steady state Pu(t) for EPI is illustrated
213
in Fig. 10for α = 0◦ (few harmonics, low THD) and α = 20◦ (rich harmonic contents, large
214
THD). The ratio between first and second harmonic powers Pf1/Pf0 is plotted in Fig.11. It 215
is seen that the ratio increases with α towards 1 for all replicas in regime II (G ∈]0, 1[),
in-216
dicating that both harmonics contribute almost equally. Fig. 12shows that, for all replicas,
217
increased THD is accompanied by a reduced signal-to-noise (SNR) ratio of about 15 dB.
For all three replicas, angular asymmetry within contact regime I (full contact) showed only
219
FIG. 12. Mean (symbols) and standard deviation (error bars) of signal-to-noise ratio (SNR in [dB]) as a function of asymmetry angle α for all replicas. Critical angles (vertical lines) indicate contact regime changes between G = 1 (I), G ∈]0, 1[ (II) and G = 0 (III).
220 221
a minor influence on onset pressures POn, oscillation frequencies f0, THD and SNR, which
222
are close to those observed for symmetric angular configuration (α = 0◦). In contact regime
223
II (up to α ≤ 15◦), the THD and SNR start to deteriorate as α increases. For α > 15◦, all
224
features (POn, f0, THD and SNR) are affected by the angular asymmetry.
225
226
Some of our observations may elucidate medical findings reported on human patients
227
having UVFP1. For instance, reduced SNR due to air leakage in contact regime II and III
228
through the glottal gap is consistent with a breathy voice description. Increased
oscilla-229
tion onset pressure POn in contact regimes II and III may cause vocal fatigue (vocal folds
230
constantly subjecting to greater forces) . The decrease in fundamental frequency f0, on
231
the other hand, does not follow the unnatural high-pitched voice described for UVFP. This
232
might be due to several reasons such as the increased contribution of the observed higher
harmonics to the perceived pitch, mechanical property changes in the paralyzed vocal fold,
234
which are not considered in the current work or yet compensatory strategies involving
supra-235
laryngeal structures of human speakers. These aspects need further investigation. In our
236
studied range of asymmetry angles, aphonia (or oscillation death) could not be reproduced.
237
Finally, it is noted from Fig. 4(c) that α-ranges, for which hA ≤ 3 mm, i.e. vertical level
238
difference observed during human phonation3, extend up to contact regime II for all three
239
replicas: α ≤ 13◦ for EPI and α ≤ 17◦ for both EPI and MRI. Therefore, partial contact
240
(regime II) is of particular importance for phonation with UVFP.
241
242
IV. PARALLEL LEVEL DIFFERENCE ANALOGY FROM POn
243
In4,5, a parallel level difference ∆E was imposed by varying the distance in the
inferior-244
superior direction between both VFs, while they remain parallel to the posterior-anterior
245
axis. It follows that the glottal leakage due to parallel level difference ∆E is rectangular.
246
This is in contrast to the triangular leakage area associated with angular asymmetry α in
247
this study. Consequently, when imposing ∆E, only regime I (full contact) and regime III
248
(no contact) occur for the parallel level difference. When imposing asymmetry angle α, on
249
the other hand, partial contact (regime II) can also happen. Nevertheless, major tendencies
250
observed for increasing asymmetry angle α (Section III B), such as decrease in
fundamen-251
tal frequency f0 and overall increase in onset pressure POn, have been also reported when
252
parallel level difference ∆E was increased4,5. Therefore, a parallel level difference analogy
253
∆E(α) is assessed expressing angular level difference in terms of an equivalent parallel level
difference as ∆E(α) ≈ hlA(α) with hlA(α) = A(α)/lA(α), i.e. the height of the rectangle 255
obtained from glottal gap area A(α) and width lA(α).
256
257
Using this analogy, the modeling approach outlined in4, under the assumptions of small amplitude approximation of the vocal folds oscillation12, no acoustical coupling with sub-and supra-glottal systems sub-and mechanical left-right VF symmetry, results in a theoretical expression of the oscillation onset threshold pressure POn as a function of hlA(α):
POn(hlA(α)) = ˜ w ˜E ˜ E − hlA(α) 2, (7)
with model parameter set { ˜w, ˜E}. The parameter ˜w accounts for mechanical damping
258
per unit area, mucosal surface wave velocity and pre-phonatory glottal half-width. The
259
parameter ˜E corresponds to the largest VF thickness.
260
Fitted curves for POn(hlA(α)) using (7) and experimental data are shown in Fig.13. The 261
fit accuracy expressed by the coefficient of determination R2 yields 89% for the M5 replica
262
(hlA ≤ 2.9 mm), 89% for the EPI replica (hlA ≤ 3.1 mm) and 81% for the MRI replica 263
(hlA ≤ 3.7 mm). The vocal fold thickness ˜E is estimated as 12 mm for M5, 26 mm for MRI 264
and 15 mm for EPI. Compared to the real values of E ≈ 10 mm (see TableII), the estimates
265
are of the same order of magnitude as the experimental measurements.
266
hlA [mm] 0 0.5 1 1.5 2 2.5 3 3.5 4 PO n [P a ] 0 500 1000 1500 2000 M5 MRI EPI
FIG. 13. (color online) Measured (symbols) POn(hlA(α)) and predicted (full lines) using Eq. (7)
for all replicas.
V. CONCLUSION
269
An experimental study is presented on the effect of angular level difference on the
270
auto-oscillation for three mechanical VF replicas. Spectral features of upstream pressure
271
(harmonic frequencies, THD and SNR) as well as upstream threshold pressures (oscillation
272
onset and offset) are analyzed. The same tendencies are observed for all three replicas.
273
As the asymmetry angle α is increased so that VFs are no longer in full contact (contact
274
regimes II and III), dynamics of the VF replica are altered clearly: SNR decreases, THD
275
increases, oscillation threshold pressures increase, fundamental frequencies decrease and
276
higher harmonics emerge. Geometrical details of each VF replica determine the leakage area
277
and critical asymmetry angles associated with a shift in contact regime. Expressions are
278
given to quantify these features from geometrical VF parameters. Apart from the decrease
279
in fundamental frequency, observed tendencies are in good agreement with those reported in
280
clinical studies on vertical level difference. The same geometrical reasoning and expressions
281
might be applied to characterize the glottal gap during phonation of patients with UVFP.
Based on analogy to parallel level difference, a formula is proposed to describe the oscillation
283
onset pressure as a function of the asymmetry angle.
284
285
Up to date, few measurements have been reported on level difference for UVFP. Various
286
attempts have been recently made to measure three-dimensional in-vivo geometry of the
287
VFs, e.g., laser systems25,27 and stereo endoscopy24. Such advanced measurements may
288
eventually provide more precise information about the left-right angular asymmetry of
pa-289
tients with voice pathology. Geometrical expressions derived in this study (A(α) in Eq. (2),
290
lA(α) in Eq. (3), etc.) could be applied to characterize VF contact (G and regime) for
291
patients. This study on the oscillation onset and voice quality features could be of great use
292
as a reference for breathy voice, vocal fatigue, and other symptoms of voice disorders.
293
294
ACKNOWLEDGMENTS
295
This work was partly supported by ArtSpeech project (ANR-15-CE23-0024), JSPS
296
International Research Fellowship (Graduate School of Science, Ritsumeikan University,
297
SP18205) and JSPS Grant-in-Aid for Scientific Research (No. 19H01002). Authors thank
298
T. Otsuka and Dr. K. Ishimura (Ritsumeikan Univ., Japan) for experimental support.
299
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