A simple electrical lumped-element model simulates intra-cochlear sound pressures and cochlear impedance below 2 kHz Torsten Marquardta) UCL Ear Institute, University College London, 332 Gray’s Inn Road, London, WC1X8EE, United Kingdom
Johannes Hensel Physikalisch-Technische Bundesanstalt, WG 1.61, Bundesallee 100, D-38116 Braunschweig, Germany
(Received 19 March 2013; revised 16 September 2013; accepted 19 September 2013) Low-frequency sounds displace large parts of the basilar membrane (BM) and can have a modulating and possibly disturbing effect on hearing at other frequencies. A better understanding of the transfer of such sounds onto the BM is therefore desirable. Lumped-element models have previously been employed to determine the low-frequency acoustic properties of the cochlea. Although helpful in illustrating schematically the role of the helicotrema, BM compliance, and the round window on low-frequency hearing, these models, when applied quantitatively, have not been able to explain experimental data in detail. Building on these models, an extended electrical analog requires just 13 lumped elements to capture, in surprising detail, the physiologically determined frequencydependence of intra-cochlear pressure and cochlear impedance between 10 Hz and 2 kHz. The model’s verification is based on data from cat, guinea pig, and humans, who differ principally in their low-frequency cochlear acoustics. The modeling data suggest that damping within the helicotrema plays a less prominent role than previously assumed. A resonance feature, which is often observed experimentally near 150 Hz in these animals and near 50 Hz in humans, is presumably a phenomenon local to the apex and not the result of a standing wave between stapes and helicotrema. C 2013 Acoustical Society of America. [http://dx.doi.org/10.1121/1.4824154] V PACS number(s): 43.64.Bt, 43.64.Kc [CAS]
I. INTRODUCTION
A better understanding of the acoustic propagation of low-frequency sounds into the cochlea might help to explain the large range in susceptibility to environmental lowfrequency and infrasound observed among individuals (Schust, 2004). Based on the suppression of otoacoustic emissions, we previously developed a non-invasive technique to measure the forward middle-ear transfer function (fMETF) at low frequencies (Marquardt et al., 2007). We observed pronounced resonance features, centered at approximately 50 Hz in humans and 120 Hz in guinea pigs, and showed later that these impacted on judgments of subjective loudness (Marquardt and Jurado, 2011). A subsequent modeling study attributed this resonance to an oscillatory interaction between the stiffness of the apical basilar membrane (BM) and the inertia of the perilymph inside the helicotrema (Marquardt and Hensel, 2009). In the current study, we develop and apply a modified version of the model to simulate published measurements of intra-cochlear sound pressure and cochlear impedance in cat, guinea pig, and human ears. Although extensive distributed-element models of the cochlea have been applied previously in order to simulate these measurements (Puria and Allen, 1991), simple lumpedelement models can often describe the underlying acoustic principles in a more illustrative way. Our modeling results show that, at frequencies below the existence region of the cochlear traveling wave, the cochlea can be considered as a a)
Author to whom correspondence should be addressed. Electronic mail: [email protected]
3730
J. Acoust. Soc. Am. 134 (5), November 2013
Pages: 3730–3738
physical system no longer consisting of distributed elements but, rather, formed of lumped elements (Franke and Dancer, 1980). Previously proposed lumped-element models have, however, been unable to explain in detail the data recorded experimentally. Based on his extensive physiological and anatomical measurements, Dallos (1970) proposed a simple lumped-element model [Fig. 1(A)] to illustrate how, at very low frequencies, the helicotrema shunts the pressure difference between scala vestibuli (SV) and scala tympani (ST). This shunting not only prevents BM displacement caused by static pressure changes, but impacts generally on hearing sensitivity to low-frequency sounds. Dallos argued that the ratio between inertia (MH) and viscous friction (RH), both impeding the perilymph flow through the helicotrema— which differs between species—determines the slope of the fMETF below 80 Hz. Indeed, his experimental data show hearing sensitivity to drop at a faster rate (6 dB/octave steeper) toward low frequencies in cat and chinchilla than in guinea pig and kangaroo rat. The latter species have cochleae with more turns, narrower tapering, and a smaller helicotrema. All these geometrical features contribute to a larger RH. With only one reactive element (perilymph inertia), his model was not intended to explain the resonance feature that he observed in all four species. Lynch et al. (1982) extended Dallos’ circuit by the round window compliance (CRW) in order to describe the cochlear input impedance they derived from their measurements of intra-cochlear pressure and stapes velocity in cat [Fig. 1(B)]. Although their model helped determine the acoustic parameters underlying the cochlear impedance, it did not replicate the clearly visible resonance
0001-4966/2013/134(5)/3730/9/$30.00
C 2013 Acoustical Society of America V
FIG. 1. Lumped-element models of the apical cochlea explaining its principle acoustic properties below 2 kHz, as experimentally measured in the basal turn. Such models have been previously suggested by (A) Dallos (1970), (B) Lynch et al. (1982), and (C) Franke and Dancer (1982). (D) The model proposed in the current study.
from reaching the apical parts of the cochlear model. At lower frequencies, however, the acoustic energy passes through MP rather than CBM1, and thus reaches the cochlear apex, which is modeled by the apical BM compliance (CBM2) and damping (RBM2), in parallel with the perilymph mass (MH) and viscous friction (RH) inside the helicotrema. In order to describe the sound pressure in SV (pV) and ST (pT), the model must include a serial combination of elements representing the ST impedance basal from the measurement location toward the round window. These are the perilymph mass in ST between measurement location and the round window (MT), the round window compliance (CRW), and the damping (RT) of this local resonator. Where available, the model’s parameters were initialized with the experimentally derived values (see Table I). While keeping physiological plausibility in mind, the parameters were then manually adjusted to improve the model fit with the magnitude data, i.e., inconsistencies with the phase data were not taken into account during the fitting.
feature in their experimental data [Fig. 2(A)]. Whereas Dallos and Lynch et al., assumed that the cochlear impedance without the helicotrema remains resistive even at low frequencies, Franke and Dancer (1982) introduced a compliance element (CBM) in their model that represents the BM impedance at very low frequencies [Fig. 1(C)]. Because the perilymph flow through the guinea pig helicotrema was, at the time, thought to be resistive (Dallos, 1970), their circuit did not contain an inertia element. With the compliance element, these authors were the first to try to explain the observed irregularity around 100 Hz (Fig. 3), a plateau region that we consider resembles a damped version of the resonance reported by others. Their model explained acoustic principles, but was not applied in quantitative predictions. Lacking a mass element, it was of course only able to explain a plateau region, but not a non-monotonic resonance feature. Only combinations of both mass and compliance elements, as our model contains, are able to produce resonance phenomena. Like previous models, our model is an electric circuit analog, where voltage is analogous to sound pressure and current is analogous to acoustic volume velocity [Fig. 1(D)]. The middle ear is represented by a compliance (CME) and a mass element (MME). With the aim of keeping the circuit diagram simple, a middle ear transformer element has been omitted, but its gain (GME) is provided in Table I. A resistive element was considered unnecessary because the cochlea damps the middle ear resonance sufficiently. RBM1 represents the impedance of the resistive traveling wave along the basal part of the BM. CBM1 determines the frequency below which the impedance of this part of the BM becomes stiffness-dominated. MP combines the longitudinal perilymph inertia inside SV and ST, but excludes the helicotrema. It prevents high-frequency acoustic energy
The sound pressure at the tympanic membrane (pTM) causes movement of the ossicular chain and, consequently, a volume velocity at the stapes footplate (US). The pressure produced inside the basal SV (pV) depends on the cochlear input impedance (ZC ¼ pV / US). Figure 2(A) shows ZC(f) as it was measured in 29 cats (black dots show average data) and modeled (green solid line) by Lynch et al. (1982). It is not surprising that our higher-order model is capable of mimicking the oscillatory features in the ZC data in more detail (thick solid red line) than their simpler model. Impedances of the various lumped elements are illustrated by the
J. Acoust. Soc. Am., Vol. 134, No. 5, November 2013
T. Marquardt and J. Hensel: Lumped-element cochlear model
II. MODELING RESULTS
3731
FIG. 2. (A) Cochlear input impedance (ZC) derived by Lynch et al. (1982, Fig. 24; see also their footnote 5 for details) from physiological measurements in 29 cats (black dots). The solid green graph represents the input impedance of their model [shown in Fig. 1(B)], which aligns generally well with the data, but does not replicate the resonance features above 100 Hz. Bold red lines show ZC (solid), and the impedance across the BM (Z0 C, dashed) of our model [Fig. 1(D)], adjusted to fit the input impedance data by Lynch et al. (1982). The thin straight dotted lines show impedance functions of the model’s lumped elements. All blue lines show the respective impedances of the model (bold: ZC; dashed: Z0 C) when fitted to the single-cat pressure data shown in (B) (pressure fits also shown in blue). The middle ear compliance (CME) and mass (MME), shown in (A) as thin straight blue dotted lines, are only relevant for fitting these pressure functions. (B) Intra-cochlear pressures measured by Nedzelnitsky (1980, Fig. 14) in SV (pV, solid black) and ST (pT, dotted black) in one cat. The black dashed line indicates the median pressure difference across the BM (pVT) for six cats and, since normalized to the ear canal pressure (pTM), represents the cat’s fMETF. The blue lines show simulations of these pressure functions by our model when fitted to the single cat data (i.e., pV and pT only). The predicted impedance curves with this parameter set are shown in (A) in blue.
straight thin dotted lines. Table I lists the parameters of our model, as well as some previously derived physiological values. Below 30 Hz, the round window compliance (CRW) solely underlies ZC. In conjunction with the inertia of the perilymph flow through both cochlear scalae (MP) and the helicotrema (MH), this compliance contributes to a resonance at 35 Hz that is mainly damped by the viscous friction in scalae and helicotrema (RP and RH). Above this resonance, ZC follows the combined reactance of MP and MH. At 80 Hz, the compliance of the apical BM (CBM2) starts to impact on ZC, contributing to a “double-resonance” phenomenon, featuring an anti-resonance and a resonance. The center frequency of this feature is largely controlled by CBM2, whereas the positions of the þ6-dB/octave slopes on either side are defined by [MP þ MH] and MP, respectively. The anti-resonance at 100 Hz is caused by a parallel combination of CBM2 and MH. Both RBM2 and RH control the damping of this anti-resonance. The resonance near 150 Hz is created by a series combination of CBM2 and MP, and RBM2 contributes primarily to its damping. Above the “double resonance,” ZC rises with jxMP (where x is the angular frequency) until 1/(jxCBM1) becomes compliant enough such that the resistive impedance of the cochlear traveling wave (RBM1) 3732
J. Acoust. Soc. Am., Vol. 134, No. 5, November 2013
determines ZC. At this transition, MP and CBM1 create a highly damped anti-resonance, which is also evident in the experimental data. Above 2 kHz, MT causes a slight increase in ZC, likewise observed in the data. The bold dashed lines show data (black) and model results (red) of the so-called cochlear impedance “across the BM” (Z0 C ¼ pV-T/US). Such a definition of cochlear impedance is commonly applied in studies where the displacement of the BM is considered, or the fMETF is defined as pV-T/pTM (e.g., in Dallos, 1970; Marquardt et al., 2007). Toward 0 Hz, Z0 C asymptotically approaches [RP þ RH]. Note that ZC is merely the sum of Z0 C and ZT ¼ 1/(jxCRW) þ jxMT þ RT ¼ pT/US, which is the impedance seen from the ST measurement location toward the round window. The thick blue lines in Fig. 2(A) show the impedance curves of the model when fitted to the intra-cochlear pressure data measured in a single cat by Nedzelnitsky (1980) which are reproduced here in Fig. 2(B). The increased BM volume compliance (CBM1) and perilymph mass (MP) necessary to fit these data (see Table I) might indicate a more basal measurement location or simply a longer cochlea in this cat (both giving a larger perilymph volume and a larger BM area toward the apex). T. Marquardt and J. Hensel: Lumped-element cochlear model
J. Acoust. Soc. Am., Vol. 134, No. 5, November 2013
TABLE I. Model parameters and their physiologically derived values, where available. Cat
Guinea pig Human
Figure 2(A) (red) 5
1
T. Marquardt and J. Hensel: Lumped-element cochlear model
CME/(m N ) MME/(N s2 m5) CBM1/(m5 N1) RBM1/(N s m5) RP/(N s m5) MP/(N s2 m5) CBM2/(m5 N1) RBM2/(N s m5) MH/(N s2 m5) RH/(N s m5) MT/(N s2 m5) RT/(N s m5) CRW/(m5 N1) GME in dB a
— — 4 1015 11 1010 0.8 1010 5.3 107 38 1015 2.7 1010 6.9 107 1.6 1010 0.2 107 0.3 1010 110 1015 —
Lynch et al. (1982)
Figure 2(B) [and Fig. 2(A) blue]
Nedzelnitsky (1980)
15
12 1010
22107 a 2.8 1010 a
100 1015
1.2 10 1 107 5.9 1015 11 1010 1 1010 7.5 107 27 1015 1.3 1010 6.3 107 1.5 1010 0.2 107 0.3 1010 110 1015 27
Figure 3(A) 15
0.25 107 0.14 1010 160 1015 27
1 10 0 32 1015 4.5 1010 1.7 1010 5.6 107 80 1015 0.4 1010 3.5 107 1.1 1010 0.05 107 0.04 1010 250 1015 30.5c
Figure 3(B) intact 15
1 10 0 32 1015 4.5 1010 0.4 1010 b 3.8 107 55 1015 0.8 1010 3.5 107 1.1 1010 — — — —
Figure 3(B) sealed 15
1 10 0 32 1015 4.5 1010 0.1 1010 b 2.2 107 45 1015 0.6 1010 3.5 107 20 1010 — — — —
Figure 3(B) otoacoustic emissions
Franke and Dancer (1982)
Franke et al. (1985)
15
1 10 0 20 1015 2.5 1010 1.7 1010 3 107 60 1015 0.4 1010 3.5 107 1.1 1010 — — — —
4.5 1010
100 1015 a
10 a
9 10
150 1015
90 1015 a 7 107 a 5.2 1010 a
140 1015
Figure 5 7 1015 0.32 107 150 1015 2 1010 0.1 1010 3.0 107 630 1015 0.22 1010 1.7 107 0.02 1010 0.18 107 0.25 1010 90 1015 19.5
Physiologically derived values given here combine RP þ RH, MP þ MH, or CBM1 þ CBM2 of our model. The slope and the phase of the fMETF < 80 Hz of this particular guinea pig indicates a slightly more mass dominated perilymph flow, forcing the fitted RP to be uncharacteristically low. However, the possibility of a low-frequency roll-off in this CM measurement cannot be excluded. c In agreement with data shown in Fig. 4 of Decory et al. (1990). b
3733
When taking into consideration the 6 dB/octave increase in slope due to the middle ear compliance (CME), the sound pressure data and model function shown in Fig. 2(B) for the first turn of SV (pV) resemble the ZC function shown in Fig. 2(A). The pressure measured in the first turn of ST (pT), on the other hand, is determined by ZT. Below 300 Hz, pT is controlled by CRW. Above 300 Hz, the perilymph mass in ST between measurement location and the round window (MT) results in a 12 dB/octave increase in pT (see also, Fig. 16 of Nedzelnitsky, 1980). At 800 Hz, both intra-cochlear pressure functions describe the transition from stiffness to mass dominance of the middle ear impedance. As the frequency approaches zero, pV and pT are determined by CRW, and are completely equalized by the helicotrema shunt. It is worth noting, however, that, in contrast to the experimental data, the value of pV in the model drops below pT before it starts to rapidly rise with increasing frequency. Here, where inertia of the perilymph dominates Z0 C, the phases of pV-T and pT slightly oppose each other [i.e., they differ by considerably more than 90 degrees; see Fig. 2(B)]. Because pV is the sum of pT and pV-T, pV must therefore drop below pT at frequencies where pV-T is smaller than pT and Z0 C has a considerable inertia component. One might therefore conclude that the calibrations of the two pressure probes used to measure pV and pT possibly were slightly unmatched. It is worthwhile considering the pressure difference pV-T [re-plotted as a black dashed line in Fig. 2(B); from Nedzelnitsky, 1980, Fig. 15], because it drives the stiffness-controlled BM displacement at the basal location of these measurements and it also underlies our definitions of Z0 C and fMETF. Since a build-up of pT is effectively limited by the very compliant round window, the pressure difference across the BM above 80 Hz is clearly dominated by pV. But, whereas pV stabilizes below 40 Hz, the pressure difference continues to fall because of the shunting action of the helicotrema. It is the phase difference between the nearly magnitude-equalized pV and pT that initially maintains the differential pressure which, nonetheless, diminishes toward lower frequencies. Note that these pressure-difference data show a median for six cats. Here, and also in the average impedance function for 29 cats [Fig. 2(A)], the resonance feature is less pronounced than in the data from a single cat. Although the resonance in this particular cat might be exceptional, it is also possible that the feature is somewhat smoothed by averaging data from cats with individually differing resonance frequencies. In the model, however, the smoother appearance of the median data is easily mimicked by increasing RBM2. Note here that except for the damping, the model, when fitted to the single cat data (pV and pT only), simulates also the normalized differential pressure (pV-T/pTM) fairly well (blue dashed line). For this reason, these data have not been explicitly fitted. The corresponding Z0 C function for these six cats [Fig. 2(A), black dashed line], derived by Lynch et al. (1982, Fig. 21), was therefore also not explicitly fitted. Nonetheless, it is similar to the model’s prediction of Z0 C in the critical region below 100 Hz. Let us consider now the intra-cochlea pressure measurements in guinea pigs. The helicotrema and scalae of this 3734
J. Acoust. Soc. Am., Vol. 134, No. 5, November 2013
species have smaller cross-sectional areas than those of the cat. Their cochleae also have more turns. The increased viscous damping of the perilymph flow through scalae and helicotrema caused by these anatomical features is reflected in the experimental data of Franke and Dancer (1982). Below 80 Hz, the differential pressure (pV-T/pTM) has a slope of less than 12 dB/octave and a phase lead closer to 90 degrees [Fig. 3(B)]. Also, pV does not fall below pT [Fig. 3(A)] as is the case in the cat (at least in the model). Besides other surgical manipulations, Franke and Dancer (1982) studied the effect of sealing the helicotrema by injecting silicone gel into a hole drilled into the fourth turn [Fig. 3(B)]. In this experiment, they measured the cochlear microphonic potential (CM) signal in order to derive (indirectly) the frequency dependence of the pressure difference across the BM. The data show that by preventing perilymph flow through the helicotrema, the fMETF becomes almost frequency independent below 100 Hz, since Z0 C is now largely dominated by the BM stiffness. In the model, this surgical intervention is equivalent to setting RH to infinity. An even better fit to the data was obtained by maintaining a slight conductance (RH ¼ 20 1010 N s m5), hinting at a small remaining leak during their measurements. The fact that the effective longitudinal perilymph mass (MP) and apical BM compliance (CBM2) had to be decreased considerably in order to simulate the effect of the gel injection indicates that the silicone might have extended quite far into the cochlear turns. Note also the relatively large compliance values for the round window and the BM compared with those of the cat (Table I). One cannot exclude that air introduced into the perilymph during the surgical preparations has led to an increase in compliance so that these measurements might not reflect physiological properties. Figure 3(B) shows, furthermore, the fMETF shape obtained by Marquardt et al. (2007) using distortion-product otoacoustic emissions (DPOAE) in the guinea-pig. Their DPOAE suppression technique will be only briefly described here. The level of low frequency (LF) test tones of various frequencies was adjusted to result in a constant suppression of the DPOAE (in this particular case, 7 dB). The method is based on the assumption that this constant suppression of the DPOAE indicates an equal BM displacement caused by all the different LF test tones. The LF BM displacement is proportional to the differential pressure across the BM because it is impeded by stiffness at the relatively basal DPOAE generation site. [Primary tones were f1 ¼ 2525 Hz at 70 dB sound pressure level (SPL) and f2 ¼ 3015 Hz at 59 dB SPL, which produced in this ear an unsuppressed 2f1 - f2 level of 14 dB SPL.] This way, the ratio between ear canal pressure and differential BM pressure as a function of frequency is captured in its shape (i.e., as iso-output function). The more apical measurement location used in these experiments may explain the lower RBM1, CBM1, and MP compared to the data obtained by Franke and colleagues. The difference in measurement location might also be the reason for the relatively little damping of the “double resonance” feature, which was observed in all four ears of the two animals that were studied. Nevertheless, the model fit to these data (red) shows that the increased damping of perilymph flow necessary to T. Marquardt and J. Hensel: Lumped-element cochlear model
FIG. 3. (A) Average pressure in SV and ST of two guinea pigs (black) measured by Franke and Dancer (1982) with a constant ear canal pressure of 78 dB SPL, and simulated by our model (blue). (B) Changes in the fMETF induced by sealing the helicotrema with silicone (one guinea pig; reproduced from Franke and Dancer, 1982, Fig. 2). The shape of the fMETF is here obtained by measuring the CM, which is proportional to the pressure difference across the basal BM. The blue lines show our model fits. Additional fMETF shape data and their model fit (red) are shown for one guinea pig (dB with arbitrary reference), which exhibits a more pronounced resonance (Marquardt et al., 2007). The latter experiments employed a non-invasive DPOAE suppression technique (described in main text, Sec. II). Note that the measured phase values of Franke and Dancer (1982) and Marquardt et al. (2007) were latency-compensated by 150 ls and 750 ls, respectively.
decrease the fMETF slope below 80 Hz to 6 dB/octave is not necessarily accompanied by a damping of the resonance feature when the resistance in the scalae (RP) instead of that in the helicotrema (RH) is increased. In this context, it is worth noting that the model’s RH is in close agreement to the theoretical estimation of RH ¼ 1010 N s m5 based on a tube model with radius of 0.084 mm and length 0.1 mm (see Dallos, 1970, p. 497). Because the impedance of the model, when fitted to the intra-cochlear pressure data of the cat [Fig. 2(B)], also provides for a good representation of the cat cochlear impedance [Fig. 2(A)], we confidently assume that the impedance function of the model fitted to the guinea pig pressure data (Fig. 3) provides for a realistic estimation of the input impedance of the guinea pig cochlea (Fig. 4). Because the DPOAE suppression technique reveals the differential pressure across the BM, only Z0 C of the model fitted to these data is shown (dashed), featuring a more pronounced resonance at a slightly higher frequency. In order to determine a comparable model parameter set for low frequencies also for the human cochlea, data from two studies were combined. Direct measurements of intra-cochlear pressures and impedances in human cadaveric temporal bones are only available above 100 Hz (Merchant et al., 1996; Aibara et al., 2001; Nakajima et al., 2008). We, therefore, vertically aligned by eye the relative shape of the human fMETF, which Marquardt et al. (2007)
FIG. 4. Cochlear input impedance, ZC (solid), and the impedance across the BM, Z0 C (dotted), of our model, adjusted to fit the intra-cochlear pressure measured in the guinea pig by Franke and Dancer (1982) as shown in Fig. 3(A). The dashed line shows Z0 C of our model when fitted to the guinea pig data in Fig. 3(B) that were obtained by Marquardt et al. (2007) with the DPOAE-based method and showed a more pronounced resonance feature.
J. Acoust. Soc. Am., Vol. 134, No. 5, November 2013
T. Marquardt and J. Hensel: Lumped-element cochlear model
non-invasively measured between 15 Hz and 480 Hz, with the absolute differential pressure function measured for frequencies above 100 Hz by Nakajima et al. (2008). We were
3735
then able to satisfactorily fit our model between 10 Hz and 1 kHz [Fig. 5(A)], although data and model prediction of the pressure in ST start to divert already above 500 Hz (see discussion below). As was observed in almost all subjects (Marquardt et al., 2007; Marquardt and Pedersen, 2010; Marquardt and Jurado, 2011), the “double resonance” feature is located in humans below 100 Hz. The corresponding model impedance functions for the human cochlea (ZC, Z0 C) and the impedance function seen from the ST measurement location toward the round window (ZT) are shown together with the corresponding direct measurements in Fig. 5(B). The fact that the resistive impedance of the traveling wave is relatively low compared to the compliance of the round window gives the human input impedance function (ZC) a shape that distinctly differs from that of cat and guinea pig. Here, the resonance feature is superimposed on the downward-sloping round window compliance. In order to confirm these predictions, we hope that future measurements would include frequencies also below 100 Hz. III. DISCUSSION
The success in fitting the experimental data with our simple model demonstrates that below 2 kHz most global
acoustic properties of the cochlea can be described in principle by a system of few lumped elements. In the simulation of pT and the related ZT, the limit of this approach is, however, reached already at much lower frequencies. This becomes most obvious in the human data, where pT above 400 Hz shows not the characteristics of mass-dominance: the phase lead is only 45 degrees instead of 90 degrees, and the magnitude slope has only 6 dB/octave instead of 12 dB/octave [Fig. 5(A)]. Similarly, the measured ZT and modeled ZT deviate above this frequency [Fig. 5(B)], with an additional effect on ZC, which asymptotes toward ZT at high frequencies. This impact of MT on ZC above 2 kHz does obviously not replicate reality. In the distributed-element system of the real cochlea, a growing amount of volume is displaced across the BM already basally from the pT measurement location with increasing stimulation frequency. In effect, MT decreases with frequency so that, in reality, it has only a marginal impact on ZC. Nakajima and colleagues used a distributed Foster-iterated network to successfully model the frequency-dependence of MT (Nakajima et al., 2008). Note that the application of lumped elements is less problematic for the modeling of the animal cochleae. As evident by their higher upper frequency limit of hearing, the BM of these species is less compliant at its basal end, so that the impact
FIG. 5. (A) Mean intra-cochlear pressures measured by Nakajima et al. (2008) in SV (pV, solid black) and ST (pT, dotted black) in six human cadaveric temporal bones. Their calculated differential pressure across the BM (pVT, dashed black without markers) was extended toward lower frequencies by the shape of the differential pressure function of a typical human ear (dashed black with round markers) as was non-invasively obtained by Marquardt et al. (2007) using a non-invasive DPOAE suppression technique. These relative shape data were vertically aligned with the absolute differential pressure function. The red lines show our model fits. (B) Cochlear input impedance (ZC, solid black), the impedance across the BM (Z0 C, dashed black), and the impedance seen from the ST measurement location toward the round window (ZT, dotted black) derived for the human cochlea by Nakajima et al. (2008). The red lines show the impedance of our model when fitted to the intra-cochlear pressure data in (A). Note that the measured pressure phase values of Nakajima et al. (2008) and the DPOAE suppression phases of Marquardt et al. (2007) were latency-compensated by 83 ls and 500 ls, respectively. 3736
J. Acoust. Soc. Am., Vol. 134, No. 5, November 2013
T. Marquardt and J. Hensel: Lumped-element cochlear model
of MT, at least in the guinea pig, extents to at least 2 kHz, and our lumped-element model describes these data well [Fig. 3(A)]. For the single cat data by Nedzelnitsky (1980), replicated here in Fig. 2(B), it is difficult to judge how the pT phase is impacted by the prominent notch that is evident in both pressure functions between 1 kHz and 2 kHz in this particular animal. Here, the pT phase does not even depart much from zero [Fig. 2(B)] in an area where a large mass-induced phase lead is expected. It is possible that the poles and zeros underlying this nearby notch prevent the phase to reach larger values. Note that the magnitude slope of ZT, derived from ten cats by Nedzelnitsky (1980, Fig. 16) actually agrees with an inertia dominance. But again, the ZT phase lead reaches just 45 instead of 90 degrees. Such discrepancies between magnitude and phase data were often observed during our attempts to fit the data. We believe, however, that they do not necessarily reveal experimental measurement errors, but could also be explained by the fact that in reality the real cochlea is a distributedelement system with a much larger degree of freedom than our constrained lumped-element model. With its strictly defined relationship between magnitude slope and phase, in almost all cases we could not find parameters that satisfactorily fit both magnitude and phase data. We therefore fitted our model to the magnitude data only. Nevertheless, the resulting model phase functions are in good qualitative agreement with the physiological phase data—an indication that the model is close to physical reality. Dallos (1970) assumed that, for low frequencies, Z0 C is dominated by the impedance of the helicotrema (MH and RH). However, in our model, we found that RH had a very strong impact on the damping of the “double resonance” while trying to adjust the slope of the fMETF below 80 Hz with this parameter. The presence of the “double resonance” even in the guinea pig data supports the argument made by Lynch et al. (1982) that mainly viscosity in the perilymphatic scalae (RP) and not in the helicotrema (RH) damps the perilymph flow at low frequencies and influences the fMETF slope in this frequency region. Although RP also has a small effect, RH and also RBM2 dominate the damping of the “double resonance.” Whereas RH influences the anti-resonance and also the slope of the fMETF below the resonance frequency, RBM2 influences both anti-resonance and resonance, without impacting on the fMETF slope. Since RH and RBM2 are located within the apical resonance-circuit of CBM2 and MH, the phenomenon of the anti-resonance appears to be created locally, rather than being the result of a standing wave along the length of the cochlea, as has been previously suggested by Puria and Allen (1991). In this context, Puria and Allen (1991) mentioned a necessary impedance mismatch at the cochlea-stapes boundary causing wave reflections. Although the resonance in our model could be seen as a kind of k/4 resonance, the fact that only a minor increase in damping was observed during the substantial increase of RP necessary to reproduce the guinea pig data speaks against a standing wave. It is important to mention that CRW cannot be a contributor to the “double-resonance” phenomenon because it is,
like in all previous models, not part of the sub-circuit that determines the pressure difference, pVT (Fig. 1). Still, the pressure difference data [Fig. 2(B)], and those indirectly reproducing its shape [e.g., the DPOAE and CM data in Fig. 3(B), or the CM data of Dallos, 1970] do exhibit this resonance feature. In summary, our new model goes beyond previous models by combining anatomically motivated mass and compliance elements, and consequently is able to reproduce resonances that are frequently observed in the experimental data as well as in distributed-element simulations (e.g., Puria and Allen, 1991). As the entries in Table I show, the fitting to the available experimental data allowed the derivation of several additional, or more detailed, physiological values that have not been measured explicitly. Some of the physiologically derived values required substantial adjustment to ensure a sensible interaction of the lumped acoustical elements and a good fit with the experimental data. Subsequently, the model enabled us to predict currently unavailable impedance functions of the guinea pig cochlea between 10 Hz and 2 kHz, and to extend measured intracochlear pressure and impedance function of the human cochlea down to infrasonic frequencies.
J. Acoust. Soc. Am., Vol. 134, No. 5, November 2013
T. Marquardt and J. Hensel: Lumped-element cochlear model
ACKNOWLEDGMENTS
J.H. is supported by the EMRP project HLT01 (EARS), and T.M. is supported by the related EMRP Researcher Excellence Grant HLT01-Reg1. The EMRP is jointly funded by the EMRP participating countries within EURAMET and the European Union. The suggestion to model human data has considerably enhanced the manuscript.
Aibara, R., Welsh, J. T., Puria, S., and Goode, R. L. (2001). “Human middle-ear sound transfer function and cochlear input impedance,” Hear. Res. 152, 100–109. Dallos, P. (1970). “Low-frequency auditory characteristics: Species dependence,” J. Acoust. Soc. Am. 48, 489–499. Decory, L., Franke, R. B., and Dancer, A. L. (1990). “Measurement of the middle ear transfer function in cat, chinchilla and guinea pig,” in The Mechanics and Biophysics of Hearing, edited by P. Dallos, C. D. Geisler, J. W. Matthews, M. A. Ruggero, and C. R. Steele (Springer, Berlin), pp. 270–277. Franke, R., and Dancer, A. (1980). “Cochlear microphonic potential and intracochlear sound pressure measurements at low frequencies in guinea pig,” in Proceedings of the Conference on Low Frequency Noise and Hearing, edited by H. Møller and P. Rubak (Aalborg, Denmark), pp. 337–339. Franke, R., and Dancer, A. (1982). “Cochlear mechanisms at low frequencies in the guinea pig,” Arch. Oto-Rhino-Laryngol. 234, 213–218. Franke, R., Dancer, A., Buck, K., Evrard, G., and Lenoir, M. (1985). “Hydromechanical cochlear phenomena at low frequencies in guinea pig,” Acustica 59, 30–41. Lynch, T. J., Nedzelnitsky, V., and Peake, W. T. (1982). “Input impedance of the cochlea in cat,” J. Acoust. Soc. Am. 72, 108–130. Marquardt, T., and Hensel, J. (2009). “A lumped-element model of the apical cochlea at low frequencies,” in Concepts and Challenges into the Biophysics of Hearing, edited by N. P. Cooper and D. T. Kemp (World Scientific, Singapore), pp. 337–339. Marquardt, T., Hensel, J., Mrowinski, D., and Scholz, G. (2007). “Low-frequency characteristics of human and guinea pig cochleae,” J. Acoust. Soc. Am. 121, 3628–3638. Marquardt, T., and Jurado, C. A. (2011). “The effect of the helicotrema on low-frequency cochlear mechanics and hearing,” in What Fire is in My 3737
Ears?: Progress in Auditory Biomechanics, edited by C. A. Shera and E. S. Olsen (AIP, Melville, New York), pp. 324–329. Marquardt, T., and Pedersen, C. S. (2010). “The influence of the helicotrema on low-frequency hearing,” in The Neurophysiological Bases of Auditory Perception, edited by E. A. Lopez-Poveda, A. R. Palmer, and R. Meddis (Springer, New York), pp. 25–36. Merchant, S. N., Ravicz, M. E., and Rosowki, J. J. (1996). “Acoustic input impedance of the stapes and cochlea in human temporal bones,” Hear. Res. 97, 30–45.
3738
J. Acoust. Soc. Am., Vol. 134, No. 5, November 2013
Nakajima, H. H., Dong, W., Olson, E. S., Merchant, S. N., Ravicz, M. E., and Rosowski, J. J. (2008). “Differential intracochlear sound pressure measurements in normal human temporal bones,” J. Assoc. Res. Otolaryngol. 10(1), 23–36. Nedzelnitsky, V. (1980). “Sound pressures in the basal turn of the cat cochlea,” J. Acoust. Soc. Am. 68, 1676–1689. Puria, S., and Allen, J. B. (1991). “A parametric study of cochlear input impedance,” J. Acoust. Soc. Am. 89, 287–309. Schust, M. (2004). “Effects of low-frequency noise up to 100 Hz,” Noise Health 6(23), 73–85.
T. Marquardt and J. Hensel: Lumped-element cochlear model
Copyright of Journal of the Acoustical Society of America is the property of American Institute of Physics and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use.
Johannes Hensel Physikalisch-Technische Bundesanstalt, WG 1.61, Bundesallee 100, D-38116 Braunschweig, Germany
(Received 19 March 2013; revised 16 September 2013; accepted 19 September 2013) Low-frequency sounds displace large parts of the basilar membrane (BM) and can have a modulating and possibly disturbing effect on hearing at other frequencies. A better understanding of the transfer of such sounds onto the BM is therefore desirable. Lumped-element models have previously been employed to determine the low-frequency acoustic properties of the cochlea. Although helpful in illustrating schematically the role of the helicotrema, BM compliance, and the round window on low-frequency hearing, these models, when applied quantitatively, have not been able to explain experimental data in detail. Building on these models, an extended electrical analog requires just 13 lumped elements to capture, in surprising detail, the physiologically determined frequencydependence of intra-cochlear pressure and cochlear impedance between 10 Hz and 2 kHz. The model’s verification is based on data from cat, guinea pig, and humans, who differ principally in their low-frequency cochlear acoustics. The modeling data suggest that damping within the helicotrema plays a less prominent role than previously assumed. A resonance feature, which is often observed experimentally near 150 Hz in these animals and near 50 Hz in humans, is presumably a phenomenon local to the apex and not the result of a standing wave between stapes and helicotrema. C 2013 Acoustical Society of America. [http://dx.doi.org/10.1121/1.4824154] V PACS number(s): 43.64.Bt, 43.64.Kc [CAS]
I. INTRODUCTION
A better understanding of the acoustic propagation of low-frequency sounds into the cochlea might help to explain the large range in susceptibility to environmental lowfrequency and infrasound observed among individuals (Schust, 2004). Based on the suppression of otoacoustic emissions, we previously developed a non-invasive technique to measure the forward middle-ear transfer function (fMETF) at low frequencies (Marquardt et al., 2007). We observed pronounced resonance features, centered at approximately 50 Hz in humans and 120 Hz in guinea pigs, and showed later that these impacted on judgments of subjective loudness (Marquardt and Jurado, 2011). A subsequent modeling study attributed this resonance to an oscillatory interaction between the stiffness of the apical basilar membrane (BM) and the inertia of the perilymph inside the helicotrema (Marquardt and Hensel, 2009). In the current study, we develop and apply a modified version of the model to simulate published measurements of intra-cochlear sound pressure and cochlear impedance in cat, guinea pig, and human ears. Although extensive distributed-element models of the cochlea have been applied previously in order to simulate these measurements (Puria and Allen, 1991), simple lumpedelement models can often describe the underlying acoustic principles in a more illustrative way. Our modeling results show that, at frequencies below the existence region of the cochlear traveling wave, the cochlea can be considered as a a)
Author to whom correspondence should be addressed. Electronic mail: [email protected]
3730
J. Acoust. Soc. Am. 134 (5), November 2013
Pages: 3730–3738
physical system no longer consisting of distributed elements but, rather, formed of lumped elements (Franke and Dancer, 1980). Previously proposed lumped-element models have, however, been unable to explain in detail the data recorded experimentally. Based on his extensive physiological and anatomical measurements, Dallos (1970) proposed a simple lumped-element model [Fig. 1(A)] to illustrate how, at very low frequencies, the helicotrema shunts the pressure difference between scala vestibuli (SV) and scala tympani (ST). This shunting not only prevents BM displacement caused by static pressure changes, but impacts generally on hearing sensitivity to low-frequency sounds. Dallos argued that the ratio between inertia (MH) and viscous friction (RH), both impeding the perilymph flow through the helicotrema— which differs between species—determines the slope of the fMETF below 80 Hz. Indeed, his experimental data show hearing sensitivity to drop at a faster rate (6 dB/octave steeper) toward low frequencies in cat and chinchilla than in guinea pig and kangaroo rat. The latter species have cochleae with more turns, narrower tapering, and a smaller helicotrema. All these geometrical features contribute to a larger RH. With only one reactive element (perilymph inertia), his model was not intended to explain the resonance feature that he observed in all four species. Lynch et al. (1982) extended Dallos’ circuit by the round window compliance (CRW) in order to describe the cochlear input impedance they derived from their measurements of intra-cochlear pressure and stapes velocity in cat [Fig. 1(B)]. Although their model helped determine the acoustic parameters underlying the cochlear impedance, it did not replicate the clearly visible resonance
0001-4966/2013/134(5)/3730/9/$30.00
C 2013 Acoustical Society of America V
FIG. 1. Lumped-element models of the apical cochlea explaining its principle acoustic properties below 2 kHz, as experimentally measured in the basal turn. Such models have been previously suggested by (A) Dallos (1970), (B) Lynch et al. (1982), and (C) Franke and Dancer (1982). (D) The model proposed in the current study.
from reaching the apical parts of the cochlear model. At lower frequencies, however, the acoustic energy passes through MP rather than CBM1, and thus reaches the cochlear apex, which is modeled by the apical BM compliance (CBM2) and damping (RBM2), in parallel with the perilymph mass (MH) and viscous friction (RH) inside the helicotrema. In order to describe the sound pressure in SV (pV) and ST (pT), the model must include a serial combination of elements representing the ST impedance basal from the measurement location toward the round window. These are the perilymph mass in ST between measurement location and the round window (MT), the round window compliance (CRW), and the damping (RT) of this local resonator. Where available, the model’s parameters were initialized with the experimentally derived values (see Table I). While keeping physiological plausibility in mind, the parameters were then manually adjusted to improve the model fit with the magnitude data, i.e., inconsistencies with the phase data were not taken into account during the fitting.
feature in their experimental data [Fig. 2(A)]. Whereas Dallos and Lynch et al., assumed that the cochlear impedance without the helicotrema remains resistive even at low frequencies, Franke and Dancer (1982) introduced a compliance element (CBM) in their model that represents the BM impedance at very low frequencies [Fig. 1(C)]. Because the perilymph flow through the guinea pig helicotrema was, at the time, thought to be resistive (Dallos, 1970), their circuit did not contain an inertia element. With the compliance element, these authors were the first to try to explain the observed irregularity around 100 Hz (Fig. 3), a plateau region that we consider resembles a damped version of the resonance reported by others. Their model explained acoustic principles, but was not applied in quantitative predictions. Lacking a mass element, it was of course only able to explain a plateau region, but not a non-monotonic resonance feature. Only combinations of both mass and compliance elements, as our model contains, are able to produce resonance phenomena. Like previous models, our model is an electric circuit analog, where voltage is analogous to sound pressure and current is analogous to acoustic volume velocity [Fig. 1(D)]. The middle ear is represented by a compliance (CME) and a mass element (MME). With the aim of keeping the circuit diagram simple, a middle ear transformer element has been omitted, but its gain (GME) is provided in Table I. A resistive element was considered unnecessary because the cochlea damps the middle ear resonance sufficiently. RBM1 represents the impedance of the resistive traveling wave along the basal part of the BM. CBM1 determines the frequency below which the impedance of this part of the BM becomes stiffness-dominated. MP combines the longitudinal perilymph inertia inside SV and ST, but excludes the helicotrema. It prevents high-frequency acoustic energy
The sound pressure at the tympanic membrane (pTM) causes movement of the ossicular chain and, consequently, a volume velocity at the stapes footplate (US). The pressure produced inside the basal SV (pV) depends on the cochlear input impedance (ZC ¼ pV / US). Figure 2(A) shows ZC(f) as it was measured in 29 cats (black dots show average data) and modeled (green solid line) by Lynch et al. (1982). It is not surprising that our higher-order model is capable of mimicking the oscillatory features in the ZC data in more detail (thick solid red line) than their simpler model. Impedances of the various lumped elements are illustrated by the
J. Acoust. Soc. Am., Vol. 134, No. 5, November 2013
T. Marquardt and J. Hensel: Lumped-element cochlear model
II. MODELING RESULTS
3731
FIG. 2. (A) Cochlear input impedance (ZC) derived by Lynch et al. (1982, Fig. 24; see also their footnote 5 for details) from physiological measurements in 29 cats (black dots). The solid green graph represents the input impedance of their model [shown in Fig. 1(B)], which aligns generally well with the data, but does not replicate the resonance features above 100 Hz. Bold red lines show ZC (solid), and the impedance across the BM (Z0 C, dashed) of our model [Fig. 1(D)], adjusted to fit the input impedance data by Lynch et al. (1982). The thin straight dotted lines show impedance functions of the model’s lumped elements. All blue lines show the respective impedances of the model (bold: ZC; dashed: Z0 C) when fitted to the single-cat pressure data shown in (B) (pressure fits also shown in blue). The middle ear compliance (CME) and mass (MME), shown in (A) as thin straight blue dotted lines, are only relevant for fitting these pressure functions. (B) Intra-cochlear pressures measured by Nedzelnitsky (1980, Fig. 14) in SV (pV, solid black) and ST (pT, dotted black) in one cat. The black dashed line indicates the median pressure difference across the BM (pVT) for six cats and, since normalized to the ear canal pressure (pTM), represents the cat’s fMETF. The blue lines show simulations of these pressure functions by our model when fitted to the single cat data (i.e., pV and pT only). The predicted impedance curves with this parameter set are shown in (A) in blue.
straight thin dotted lines. Table I lists the parameters of our model, as well as some previously derived physiological values. Below 30 Hz, the round window compliance (CRW) solely underlies ZC. In conjunction with the inertia of the perilymph flow through both cochlear scalae (MP) and the helicotrema (MH), this compliance contributes to a resonance at 35 Hz that is mainly damped by the viscous friction in scalae and helicotrema (RP and RH). Above this resonance, ZC follows the combined reactance of MP and MH. At 80 Hz, the compliance of the apical BM (CBM2) starts to impact on ZC, contributing to a “double-resonance” phenomenon, featuring an anti-resonance and a resonance. The center frequency of this feature is largely controlled by CBM2, whereas the positions of the þ6-dB/octave slopes on either side are defined by [MP þ MH] and MP, respectively. The anti-resonance at 100 Hz is caused by a parallel combination of CBM2 and MH. Both RBM2 and RH control the damping of this anti-resonance. The resonance near 150 Hz is created by a series combination of CBM2 and MP, and RBM2 contributes primarily to its damping. Above the “double resonance,” ZC rises with jxMP (where x is the angular frequency) until 1/(jxCBM1) becomes compliant enough such that the resistive impedance of the cochlear traveling wave (RBM1) 3732
J. Acoust. Soc. Am., Vol. 134, No. 5, November 2013
determines ZC. At this transition, MP and CBM1 create a highly damped anti-resonance, which is also evident in the experimental data. Above 2 kHz, MT causes a slight increase in ZC, likewise observed in the data. The bold dashed lines show data (black) and model results (red) of the so-called cochlear impedance “across the BM” (Z0 C ¼ pV-T/US). Such a definition of cochlear impedance is commonly applied in studies where the displacement of the BM is considered, or the fMETF is defined as pV-T/pTM (e.g., in Dallos, 1970; Marquardt et al., 2007). Toward 0 Hz, Z0 C asymptotically approaches [RP þ RH]. Note that ZC is merely the sum of Z0 C and ZT ¼ 1/(jxCRW) þ jxMT þ RT ¼ pT/US, which is the impedance seen from the ST measurement location toward the round window. The thick blue lines in Fig. 2(A) show the impedance curves of the model when fitted to the intra-cochlear pressure data measured in a single cat by Nedzelnitsky (1980) which are reproduced here in Fig. 2(B). The increased BM volume compliance (CBM1) and perilymph mass (MP) necessary to fit these data (see Table I) might indicate a more basal measurement location or simply a longer cochlea in this cat (both giving a larger perilymph volume and a larger BM area toward the apex). T. Marquardt and J. Hensel: Lumped-element cochlear model
J. Acoust. Soc. Am., Vol. 134, No. 5, November 2013
TABLE I. Model parameters and their physiologically derived values, where available. Cat
Guinea pig Human
Figure 2(A) (red) 5
1
T. Marquardt and J. Hensel: Lumped-element cochlear model
CME/(m N ) MME/(N s2 m5) CBM1/(m5 N1) RBM1/(N s m5) RP/(N s m5) MP/(N s2 m5) CBM2/(m5 N1) RBM2/(N s m5) MH/(N s2 m5) RH/(N s m5) MT/(N s2 m5) RT/(N s m5) CRW/(m5 N1) GME in dB a
— — 4 1015 11 1010 0.8 1010 5.3 107 38 1015 2.7 1010 6.9 107 1.6 1010 0.2 107 0.3 1010 110 1015 —
Lynch et al. (1982)
Figure 2(B) [and Fig. 2(A) blue]
Nedzelnitsky (1980)
15
12 1010
22107 a 2.8 1010 a
100 1015
1.2 10 1 107 5.9 1015 11 1010 1 1010 7.5 107 27 1015 1.3 1010 6.3 107 1.5 1010 0.2 107 0.3 1010 110 1015 27
Figure 3(A) 15
0.25 107 0.14 1010 160 1015 27
1 10 0 32 1015 4.5 1010 1.7 1010 5.6 107 80 1015 0.4 1010 3.5 107 1.1 1010 0.05 107 0.04 1010 250 1015 30.5c
Figure 3(B) intact 15
1 10 0 32 1015 4.5 1010 0.4 1010 b 3.8 107 55 1015 0.8 1010 3.5 107 1.1 1010 — — — —
Figure 3(B) sealed 15
1 10 0 32 1015 4.5 1010 0.1 1010 b 2.2 107 45 1015 0.6 1010 3.5 107 20 1010 — — — —
Figure 3(B) otoacoustic emissions
Franke and Dancer (1982)
Franke et al. (1985)
15
1 10 0 20 1015 2.5 1010 1.7 1010 3 107 60 1015 0.4 1010 3.5 107 1.1 1010 — — — —
4.5 1010
100 1015 a
10 a
9 10
150 1015
90 1015 a 7 107 a 5.2 1010 a
140 1015
Figure 5 7 1015 0.32 107 150 1015 2 1010 0.1 1010 3.0 107 630 1015 0.22 1010 1.7 107 0.02 1010 0.18 107 0.25 1010 90 1015 19.5
Physiologically derived values given here combine RP þ RH, MP þ MH, or CBM1 þ CBM2 of our model. The slope and the phase of the fMETF < 80 Hz of this particular guinea pig indicates a slightly more mass dominated perilymph flow, forcing the fitted RP to be uncharacteristically low. However, the possibility of a low-frequency roll-off in this CM measurement cannot be excluded. c In agreement with data shown in Fig. 4 of Decory et al. (1990). b
3733
When taking into consideration the 6 dB/octave increase in slope due to the middle ear compliance (CME), the sound pressure data and model function shown in Fig. 2(B) for the first turn of SV (pV) resemble the ZC function shown in Fig. 2(A). The pressure measured in the first turn of ST (pT), on the other hand, is determined by ZT. Below 300 Hz, pT is controlled by CRW. Above 300 Hz, the perilymph mass in ST between measurement location and the round window (MT) results in a 12 dB/octave increase in pT (see also, Fig. 16 of Nedzelnitsky, 1980). At 800 Hz, both intra-cochlear pressure functions describe the transition from stiffness to mass dominance of the middle ear impedance. As the frequency approaches zero, pV and pT are determined by CRW, and are completely equalized by the helicotrema shunt. It is worth noting, however, that, in contrast to the experimental data, the value of pV in the model drops below pT before it starts to rapidly rise with increasing frequency. Here, where inertia of the perilymph dominates Z0 C, the phases of pV-T and pT slightly oppose each other [i.e., they differ by considerably more than 90 degrees; see Fig. 2(B)]. Because pV is the sum of pT and pV-T, pV must therefore drop below pT at frequencies where pV-T is smaller than pT and Z0 C has a considerable inertia component. One might therefore conclude that the calibrations of the two pressure probes used to measure pV and pT possibly were slightly unmatched. It is worthwhile considering the pressure difference pV-T [re-plotted as a black dashed line in Fig. 2(B); from Nedzelnitsky, 1980, Fig. 15], because it drives the stiffness-controlled BM displacement at the basal location of these measurements and it also underlies our definitions of Z0 C and fMETF. Since a build-up of pT is effectively limited by the very compliant round window, the pressure difference across the BM above 80 Hz is clearly dominated by pV. But, whereas pV stabilizes below 40 Hz, the pressure difference continues to fall because of the shunting action of the helicotrema. It is the phase difference between the nearly magnitude-equalized pV and pT that initially maintains the differential pressure which, nonetheless, diminishes toward lower frequencies. Note that these pressure-difference data show a median for six cats. Here, and also in the average impedance function for 29 cats [Fig. 2(A)], the resonance feature is less pronounced than in the data from a single cat. Although the resonance in this particular cat might be exceptional, it is also possible that the feature is somewhat smoothed by averaging data from cats with individually differing resonance frequencies. In the model, however, the smoother appearance of the median data is easily mimicked by increasing RBM2. Note here that except for the damping, the model, when fitted to the single cat data (pV and pT only), simulates also the normalized differential pressure (pV-T/pTM) fairly well (blue dashed line). For this reason, these data have not been explicitly fitted. The corresponding Z0 C function for these six cats [Fig. 2(A), black dashed line], derived by Lynch et al. (1982, Fig. 21), was therefore also not explicitly fitted. Nonetheless, it is similar to the model’s prediction of Z0 C in the critical region below 100 Hz. Let us consider now the intra-cochlea pressure measurements in guinea pigs. The helicotrema and scalae of this 3734
J. Acoust. Soc. Am., Vol. 134, No. 5, November 2013
species have smaller cross-sectional areas than those of the cat. Their cochleae also have more turns. The increased viscous damping of the perilymph flow through scalae and helicotrema caused by these anatomical features is reflected in the experimental data of Franke and Dancer (1982). Below 80 Hz, the differential pressure (pV-T/pTM) has a slope of less than 12 dB/octave and a phase lead closer to 90 degrees [Fig. 3(B)]. Also, pV does not fall below pT [Fig. 3(A)] as is the case in the cat (at least in the model). Besides other surgical manipulations, Franke and Dancer (1982) studied the effect of sealing the helicotrema by injecting silicone gel into a hole drilled into the fourth turn [Fig. 3(B)]. In this experiment, they measured the cochlear microphonic potential (CM) signal in order to derive (indirectly) the frequency dependence of the pressure difference across the BM. The data show that by preventing perilymph flow through the helicotrema, the fMETF becomes almost frequency independent below 100 Hz, since Z0 C is now largely dominated by the BM stiffness. In the model, this surgical intervention is equivalent to setting RH to infinity. An even better fit to the data was obtained by maintaining a slight conductance (RH ¼ 20 1010 N s m5), hinting at a small remaining leak during their measurements. The fact that the effective longitudinal perilymph mass (MP) and apical BM compliance (CBM2) had to be decreased considerably in order to simulate the effect of the gel injection indicates that the silicone might have extended quite far into the cochlear turns. Note also the relatively large compliance values for the round window and the BM compared with those of the cat (Table I). One cannot exclude that air introduced into the perilymph during the surgical preparations has led to an increase in compliance so that these measurements might not reflect physiological properties. Figure 3(B) shows, furthermore, the fMETF shape obtained by Marquardt et al. (2007) using distortion-product otoacoustic emissions (DPOAE) in the guinea-pig. Their DPOAE suppression technique will be only briefly described here. The level of low frequency (LF) test tones of various frequencies was adjusted to result in a constant suppression of the DPOAE (in this particular case, 7 dB). The method is based on the assumption that this constant suppression of the DPOAE indicates an equal BM displacement caused by all the different LF test tones. The LF BM displacement is proportional to the differential pressure across the BM because it is impeded by stiffness at the relatively basal DPOAE generation site. [Primary tones were f1 ¼ 2525 Hz at 70 dB sound pressure level (SPL) and f2 ¼ 3015 Hz at 59 dB SPL, which produced in this ear an unsuppressed 2f1 - f2 level of 14 dB SPL.] This way, the ratio between ear canal pressure and differential BM pressure as a function of frequency is captured in its shape (i.e., as iso-output function). The more apical measurement location used in these experiments may explain the lower RBM1, CBM1, and MP compared to the data obtained by Franke and colleagues. The difference in measurement location might also be the reason for the relatively little damping of the “double resonance” feature, which was observed in all four ears of the two animals that were studied. Nevertheless, the model fit to these data (red) shows that the increased damping of perilymph flow necessary to T. Marquardt and J. Hensel: Lumped-element cochlear model
FIG. 3. (A) Average pressure in SV and ST of two guinea pigs (black) measured by Franke and Dancer (1982) with a constant ear canal pressure of 78 dB SPL, and simulated by our model (blue). (B) Changes in the fMETF induced by sealing the helicotrema with silicone (one guinea pig; reproduced from Franke and Dancer, 1982, Fig. 2). The shape of the fMETF is here obtained by measuring the CM, which is proportional to the pressure difference across the basal BM. The blue lines show our model fits. Additional fMETF shape data and their model fit (red) are shown for one guinea pig (dB with arbitrary reference), which exhibits a more pronounced resonance (Marquardt et al., 2007). The latter experiments employed a non-invasive DPOAE suppression technique (described in main text, Sec. II). Note that the measured phase values of Franke and Dancer (1982) and Marquardt et al. (2007) were latency-compensated by 150 ls and 750 ls, respectively.
decrease the fMETF slope below 80 Hz to 6 dB/octave is not necessarily accompanied by a damping of the resonance feature when the resistance in the scalae (RP) instead of that in the helicotrema (RH) is increased. In this context, it is worth noting that the model’s RH is in close agreement to the theoretical estimation of RH ¼ 1010 N s m5 based on a tube model with radius of 0.084 mm and length 0.1 mm (see Dallos, 1970, p. 497). Because the impedance of the model, when fitted to the intra-cochlear pressure data of the cat [Fig. 2(B)], also provides for a good representation of the cat cochlear impedance [Fig. 2(A)], we confidently assume that the impedance function of the model fitted to the guinea pig pressure data (Fig. 3) provides for a realistic estimation of the input impedance of the guinea pig cochlea (Fig. 4). Because the DPOAE suppression technique reveals the differential pressure across the BM, only Z0 C of the model fitted to these data is shown (dashed), featuring a more pronounced resonance at a slightly higher frequency. In order to determine a comparable model parameter set for low frequencies also for the human cochlea, data from two studies were combined. Direct measurements of intra-cochlear pressures and impedances in human cadaveric temporal bones are only available above 100 Hz (Merchant et al., 1996; Aibara et al., 2001; Nakajima et al., 2008). We, therefore, vertically aligned by eye the relative shape of the human fMETF, which Marquardt et al. (2007)
FIG. 4. Cochlear input impedance, ZC (solid), and the impedance across the BM, Z0 C (dotted), of our model, adjusted to fit the intra-cochlear pressure measured in the guinea pig by Franke and Dancer (1982) as shown in Fig. 3(A). The dashed line shows Z0 C of our model when fitted to the guinea pig data in Fig. 3(B) that were obtained by Marquardt et al. (2007) with the DPOAE-based method and showed a more pronounced resonance feature.
J. Acoust. Soc. Am., Vol. 134, No. 5, November 2013
T. Marquardt and J. Hensel: Lumped-element cochlear model
non-invasively measured between 15 Hz and 480 Hz, with the absolute differential pressure function measured for frequencies above 100 Hz by Nakajima et al. (2008). We were
3735
then able to satisfactorily fit our model between 10 Hz and 1 kHz [Fig. 5(A)], although data and model prediction of the pressure in ST start to divert already above 500 Hz (see discussion below). As was observed in almost all subjects (Marquardt et al., 2007; Marquardt and Pedersen, 2010; Marquardt and Jurado, 2011), the “double resonance” feature is located in humans below 100 Hz. The corresponding model impedance functions for the human cochlea (ZC, Z0 C) and the impedance function seen from the ST measurement location toward the round window (ZT) are shown together with the corresponding direct measurements in Fig. 5(B). The fact that the resistive impedance of the traveling wave is relatively low compared to the compliance of the round window gives the human input impedance function (ZC) a shape that distinctly differs from that of cat and guinea pig. Here, the resonance feature is superimposed on the downward-sloping round window compliance. In order to confirm these predictions, we hope that future measurements would include frequencies also below 100 Hz. III. DISCUSSION
The success in fitting the experimental data with our simple model demonstrates that below 2 kHz most global
acoustic properties of the cochlea can be described in principle by a system of few lumped elements. In the simulation of pT and the related ZT, the limit of this approach is, however, reached already at much lower frequencies. This becomes most obvious in the human data, where pT above 400 Hz shows not the characteristics of mass-dominance: the phase lead is only 45 degrees instead of 90 degrees, and the magnitude slope has only 6 dB/octave instead of 12 dB/octave [Fig. 5(A)]. Similarly, the measured ZT and modeled ZT deviate above this frequency [Fig. 5(B)], with an additional effect on ZC, which asymptotes toward ZT at high frequencies. This impact of MT on ZC above 2 kHz does obviously not replicate reality. In the distributed-element system of the real cochlea, a growing amount of volume is displaced across the BM already basally from the pT measurement location with increasing stimulation frequency. In effect, MT decreases with frequency so that, in reality, it has only a marginal impact on ZC. Nakajima and colleagues used a distributed Foster-iterated network to successfully model the frequency-dependence of MT (Nakajima et al., 2008). Note that the application of lumped elements is less problematic for the modeling of the animal cochleae. As evident by their higher upper frequency limit of hearing, the BM of these species is less compliant at its basal end, so that the impact
FIG. 5. (A) Mean intra-cochlear pressures measured by Nakajima et al. (2008) in SV (pV, solid black) and ST (pT, dotted black) in six human cadaveric temporal bones. Their calculated differential pressure across the BM (pVT, dashed black without markers) was extended toward lower frequencies by the shape of the differential pressure function of a typical human ear (dashed black with round markers) as was non-invasively obtained by Marquardt et al. (2007) using a non-invasive DPOAE suppression technique. These relative shape data were vertically aligned with the absolute differential pressure function. The red lines show our model fits. (B) Cochlear input impedance (ZC, solid black), the impedance across the BM (Z0 C, dashed black), and the impedance seen from the ST measurement location toward the round window (ZT, dotted black) derived for the human cochlea by Nakajima et al. (2008). The red lines show the impedance of our model when fitted to the intra-cochlear pressure data in (A). Note that the measured pressure phase values of Nakajima et al. (2008) and the DPOAE suppression phases of Marquardt et al. (2007) were latency-compensated by 83 ls and 500 ls, respectively. 3736
J. Acoust. Soc. Am., Vol. 134, No. 5, November 2013
T. Marquardt and J. Hensel: Lumped-element cochlear model
of MT, at least in the guinea pig, extents to at least 2 kHz, and our lumped-element model describes these data well [Fig. 3(A)]. For the single cat data by Nedzelnitsky (1980), replicated here in Fig. 2(B), it is difficult to judge how the pT phase is impacted by the prominent notch that is evident in both pressure functions between 1 kHz and 2 kHz in this particular animal. Here, the pT phase does not even depart much from zero [Fig. 2(B)] in an area where a large mass-induced phase lead is expected. It is possible that the poles and zeros underlying this nearby notch prevent the phase to reach larger values. Note that the magnitude slope of ZT, derived from ten cats by Nedzelnitsky (1980, Fig. 16) actually agrees with an inertia dominance. But again, the ZT phase lead reaches just 45 instead of 90 degrees. Such discrepancies between magnitude and phase data were often observed during our attempts to fit the data. We believe, however, that they do not necessarily reveal experimental measurement errors, but could also be explained by the fact that in reality the real cochlea is a distributedelement system with a much larger degree of freedom than our constrained lumped-element model. With its strictly defined relationship between magnitude slope and phase, in almost all cases we could not find parameters that satisfactorily fit both magnitude and phase data. We therefore fitted our model to the magnitude data only. Nevertheless, the resulting model phase functions are in good qualitative agreement with the physiological phase data—an indication that the model is close to physical reality. Dallos (1970) assumed that, for low frequencies, Z0 C is dominated by the impedance of the helicotrema (MH and RH). However, in our model, we found that RH had a very strong impact on the damping of the “double resonance” while trying to adjust the slope of the fMETF below 80 Hz with this parameter. The presence of the “double resonance” even in the guinea pig data supports the argument made by Lynch et al. (1982) that mainly viscosity in the perilymphatic scalae (RP) and not in the helicotrema (RH) damps the perilymph flow at low frequencies and influences the fMETF slope in this frequency region. Although RP also has a small effect, RH and also RBM2 dominate the damping of the “double resonance.” Whereas RH influences the anti-resonance and also the slope of the fMETF below the resonance frequency, RBM2 influences both anti-resonance and resonance, without impacting on the fMETF slope. Since RH and RBM2 are located within the apical resonance-circuit of CBM2 and MH, the phenomenon of the anti-resonance appears to be created locally, rather than being the result of a standing wave along the length of the cochlea, as has been previously suggested by Puria and Allen (1991). In this context, Puria and Allen (1991) mentioned a necessary impedance mismatch at the cochlea-stapes boundary causing wave reflections. Although the resonance in our model could be seen as a kind of k/4 resonance, the fact that only a minor increase in damping was observed during the substantial increase of RP necessary to reproduce the guinea pig data speaks against a standing wave. It is important to mention that CRW cannot be a contributor to the “double-resonance” phenomenon because it is,
like in all previous models, not part of the sub-circuit that determines the pressure difference, pVT (Fig. 1). Still, the pressure difference data [Fig. 2(B)], and those indirectly reproducing its shape [e.g., the DPOAE and CM data in Fig. 3(B), or the CM data of Dallos, 1970] do exhibit this resonance feature. In summary, our new model goes beyond previous models by combining anatomically motivated mass and compliance elements, and consequently is able to reproduce resonances that are frequently observed in the experimental data as well as in distributed-element simulations (e.g., Puria and Allen, 1991). As the entries in Table I show, the fitting to the available experimental data allowed the derivation of several additional, or more detailed, physiological values that have not been measured explicitly. Some of the physiologically derived values required substantial adjustment to ensure a sensible interaction of the lumped acoustical elements and a good fit with the experimental data. Subsequently, the model enabled us to predict currently unavailable impedance functions of the guinea pig cochlea between 10 Hz and 2 kHz, and to extend measured intracochlear pressure and impedance function of the human cochlea down to infrasonic frequencies.
J. Acoust. Soc. Am., Vol. 134, No. 5, November 2013
T. Marquardt and J. Hensel: Lumped-element cochlear model
ACKNOWLEDGMENTS
J.H. is supported by the EMRP project HLT01 (EARS), and T.M. is supported by the related EMRP Researcher Excellence Grant HLT01-Reg1. The EMRP is jointly funded by the EMRP participating countries within EURAMET and the European Union. The suggestion to model human data has considerably enhanced the manuscript.
Aibara, R., Welsh, J. T., Puria, S., and Goode, R. L. (2001). “Human middle-ear sound transfer function and cochlear input impedance,” Hear. Res. 152, 100–109. Dallos, P. (1970). “Low-frequency auditory characteristics: Species dependence,” J. Acoust. Soc. Am. 48, 489–499. Decory, L., Franke, R. B., and Dancer, A. L. (1990). “Measurement of the middle ear transfer function in cat, chinchilla and guinea pig,” in The Mechanics and Biophysics of Hearing, edited by P. Dallos, C. D. Geisler, J. W. Matthews, M. A. Ruggero, and C. R. Steele (Springer, Berlin), pp. 270–277. Franke, R., and Dancer, A. (1980). “Cochlear microphonic potential and intracochlear sound pressure measurements at low frequencies in guinea pig,” in Proceedings of the Conference on Low Frequency Noise and Hearing, edited by H. Møller and P. Rubak (Aalborg, Denmark), pp. 337–339. Franke, R., and Dancer, A. (1982). “Cochlear mechanisms at low frequencies in the guinea pig,” Arch. Oto-Rhino-Laryngol. 234, 213–218. Franke, R., Dancer, A., Buck, K., Evrard, G., and Lenoir, M. (1985). “Hydromechanical cochlear phenomena at low frequencies in guinea pig,” Acustica 59, 30–41. Lynch, T. J., Nedzelnitsky, V., and Peake, W. T. (1982). “Input impedance of the cochlea in cat,” J. Acoust. Soc. Am. 72, 108–130. Marquardt, T., and Hensel, J. (2009). “A lumped-element model of the apical cochlea at low frequencies,” in Concepts and Challenges into the Biophysics of Hearing, edited by N. P. Cooper and D. T. Kemp (World Scientific, Singapore), pp. 337–339. Marquardt, T., Hensel, J., Mrowinski, D., and Scholz, G. (2007). “Low-frequency characteristics of human and guinea pig cochleae,” J. Acoust. Soc. Am. 121, 3628–3638. Marquardt, T., and Jurado, C. A. (2011). “The effect of the helicotrema on low-frequency cochlear mechanics and hearing,” in What Fire is in My 3737
Ears?: Progress in Auditory Biomechanics, edited by C. A. Shera and E. S. Olsen (AIP, Melville, New York), pp. 324–329. Marquardt, T., and Pedersen, C. S. (2010). “The influence of the helicotrema on low-frequency hearing,” in The Neurophysiological Bases of Auditory Perception, edited by E. A. Lopez-Poveda, A. R. Palmer, and R. Meddis (Springer, New York), pp. 25–36. Merchant, S. N., Ravicz, M. E., and Rosowki, J. J. (1996). “Acoustic input impedance of the stapes and cochlea in human temporal bones,” Hear. Res. 97, 30–45.
3738
J. Acoust. Soc. Am., Vol. 134, No. 5, November 2013
Nakajima, H. H., Dong, W., Olson, E. S., Merchant, S. N., Ravicz, M. E., and Rosowski, J. J. (2008). “Differential intracochlear sound pressure measurements in normal human temporal bones,” J. Assoc. Res. Otolaryngol. 10(1), 23–36. Nedzelnitsky, V. (1980). “Sound pressures in the basal turn of the cat cochlea,” J. Acoust. Soc. Am. 68, 1676–1689. Puria, S., and Allen, J. B. (1991). “A parametric study of cochlear input impedance,” J. Acoust. Soc. Am. 89, 287–309. Schust, M. (2004). “Effects of low-frequency noise up to 100 Hz,” Noise Health 6(23), 73–85.
T. Marquardt and J. Hensel: Lumped-element cochlear model
Copyright of Journal of the Acoustical Society of America is the property of American Institute of Physics and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use.
