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Journal of Vestibular Research 15 (2005) 203–214 203IOS Press

Biomechanics of horizontal canal benignparoxysmal positional vertigo

Suhrud M. Rajguru, Marytheresa A. Ifediba and Richard D. Rabbitt∗Department of Bioengineering, University of Utah, Salt Lake City, UT - 84112, USA

Received 6 August 2004

Accepted 20 May 2005

Abstract. Horizontal canal (HC) benign paroxysmal positional vertigo (HC-BPPV) is a vestibular disorder characterized bybouts of horizontal ocular nystagmus induced during reorientation of the head relative to gravity. The present report addressesthe application of a morphologically descriptive 3-canal biomechanical model of the human membranous labyrinth to studygravity-dependent semicircular canal responses during this condition. The model estimates dynamic cupular and endolymphdisplacements elicited during HC-BPPV provocative diagnostic maneuvers and canalith repositioning procedures (CRPs). Theactivation latencies in response to an HC-BPPV provocative diagnostic test were predicted to vary depending upon the initiallocation of the canalith debris (e.g. within the HC lumen vs. in the ampulla). Results may explain why the onset latency of ocularnystagmus evoked by the Dix-Hallpike provocative maneuver for posterior canal BPPV are typically longer than the latenciesevoked by analogous tests for HC-BPPV. The model was further applied to assess the efficacy of a 360-rotation CRP for thetreatment of canalithiasis HC-BPPV.

Keywords: HC-BPPV, BPPV, BPV, Semicircular canal, vestibular disorder, vestibular mechanics, otoconia, vertigo, dizziness

1. Introduction

Benign Paroxysmal Positional Vertigo (BPPV), acommon vestibular disorder, is commonly attributed tothe presence of basophilic particles in the semicircularcanals. These particles, most likely displaced otoconia(calcium carbonate crystals) from the utricle, render thesemicircular canal system sensitive to gravity. BPPV isoften classified according to the onset characteristics ofits primary symptoms: nausea, dizziness, vertigo andocular nystagmus. These symptoms, manifest duringprovocative tests for the disorder, fall into two maincategories: tonic and phasic. Tonic responses includemaintained ocular nystagmus that is initiated immedi-ately upon reorientation of the head relative to gravity.This condition, typically associated with the patholog-

∗Address for correspondence: Richard D. Rabbitt, Ph.D., Depart-ment of Bioengineering, 20 South 2030 East, Rm. 506 Salt LakeCity, UT - 84112, USA. Tel: +1 801 581 6968; Fax: +1 801 5818966; E-mail: [email protected].

ical presence of a heavy particulate mass adhered toa cupula, is termed cupulolithiasis [27]. In contrast,phasic symptoms occur with a delayed onset and sub-side over time. These responses, known as canalithi-asis, are typically attributed to the movement of free-floating particles within the lumen of the membranouslabyrinth [8,12,24]. Although the origins of either typeof BPPV are not known, clinical evidence indicatesthat they may arise in a majority of cases as the resultof head trauma [1,22]. Otoconia metabolism [17,21,29] and inner ear disorders (i.e. vestibular neuronitis,Meniere’s disease) have also been implicated in theincidence of the disorder.

BPPV develops most often in the posterior canal(PC-BPPV), as this canal is most susceptible anatomi-cally to the entrance of free-floating utricular particlesduring routine head movements. PC-BPPV is identi-fied clinically by transient vertical-torsional nystagmusevoked with the Dix-Hallpike maneuver, a diagnostictest during which the head is rapidly pitched back in theplane of the posterior canal to provoke symptoms. The

ISSN 0957-4271/05/$17.00 © 2005 – IOS Press and the authors. All rights reserved

204 S.M. Rajguru et al. / Biomechanics of horizontal canal benign paroxysmal positional vertigo

Rn

AC

CC

HC

PC

Ω . .

Z

X

z

x

y

Earth-fixed system

Head-fixed system

1

2

3 snU

Y

τ

A

PQ

Fd

Fg

ds

Q

P + P dss

A + A dss

Fig. 1. Membranous labyrinth geometry. The centerlines of the mem-branous labyrinth geometry of the right ear used in the present modelare illustrated with respect to the head-fixed (xyz) and earth-fixed(XYZ) co-ordinate systems. HC – horizontal (lateral) canal, PC –posterior canal, AC – anterior canal, CC – common crus, utricularsection. The trajectories of Rn define the centerline coordinates sn

and the vector Ω is the angular acceleration of the head, both resolvedin the head-fixed coordinate system. An expanded view of a shortsection of the canal (length ds) with a free-floating particle is shown.P represents pressure along the centerlines of the canal, τ is the shearstresses at the wall, Q is the endolymph volume displacement andA is the local cross-sectional area of the canal. Fg is the force dueto gravito-inertial acceleration and Fd is the interaction drag forcebetween the particle and the endolymph.

horizontal canal variant of BPPV (HC-BPPV), is char-acterized by acute vestibular symptoms induced duringactivities that reorient the HC relative to gravity, such asrolling over in bed [2,18–20,23]. The typical provoca-tive test for HC-BPPV involves turning the head to ei-ther side while the patient lies supine. Bursts of hori-zontal nystagmus, typically with a shorter onset latencyand duration than those of PC-BPPV, beat toward theaffected ear, indicating the presence of HC canalithia-sis [2,16]. In some cases, tonic horizontal nystagmus isevoked when the head is oriented with the HC cupulaaligned with the horizontal plane (perpendicular to thegravity vector). These responses are consistent withHC cupulolithiasis. In the case of canalithiasis, non-surgical canalith repositioning procedures (CRP, a se-ries of timed head reorientations designed to relocatethe particles under the action of gravity) may be appliedto move the particles from the HC lumen to the utric-

ular vestibule where they no longer generate anoma-lous gravity-sensitive angular motion sensations [5,9,16]. The most common HC CRP, the 270 head rota-tion [15,16], has been shown to relieve the symptomsof canalithiasis HC-BPPV in approximately 73% ofpatients [20].

The biomechanics underlying BPPV is well es-tablished and therefore vestibular symptoms may bepredicted, at least approximately, using mathemati-cal modeling. Most studies have confined their focusto PC canalithiasis and have employed mathematicalmodels to predict pathological semicircular canal re-sponses [13,26,28]. The present work takes a simi-lar approach to investigate the HC version of BPPV.The model, described extensively in a previous pa-per [26], estimates the time-dependent position of par-ticles within the membranous labyrinth and the cupu-lae volume displacements during the HC-BPPV diag-nostic maneuver and CRPs. Present model responsesare compared to clinically observed ocular nystagmusto evaluate the canalithiasis mechanism of HC-BPPV.The model was further applied to examine responsesfor cupulolithiasis during diagnostic testing.

2. Methods

2.1. Model membranous labyrinth geometry

The main input to the model was the 3D geometry ofthe human membranous labyrinth, extracted from tem-poral bone histological sections through manual seg-mentation and adjusted for tissue shrinkage [14,26].The reconstructed labyrinth consisted of five main seg-ments: horizontal canal (HC), posterior canal (PC),anterior canal (AC), common crus (CC) and utricularvestibule (U). Curved centerlines (Sn) were defined torun along each labyrinth component. Tangent, normal,and binormal vectors were defined at points along thecenterline. These formed local coordinate systems thatserved as a basis for data fitting of elliptical tubes tothe reconstruction [30]. Data fitting consisted of se-lecting a thin 500 µm slice of data points perpendicu-lar to the tangent vector at each center point, project-ing the points into the normal-binormal plane and leastsquares fitting an ellipse to the selection. The final re-sult was a parameterized surface reconstruction of themembranous labyrinth, shown in Fig. 1. The membra-nous labyrinth was oriented in a human head (FlashFire Designs, Staunton, VA) according to stereotacticbony canal planes determined from 3D multiplanar CTmeasurements by Della Santina et al. [7].

S.M. Rajguru et al. / Biomechanics of horizontal canal benign paroxysmal positional vertigo 205

Fig. 2. Diagnostic maneuver for HC-BPPV. The orientations of the membranous labyrinth at five instants in time (I-V) for the diagnostic maneuver(re: gound-fixed system from the left-side Y-axis, and back X-axis). Rotation matrices corresponding to the maneuver are given in Table 1.Arrows and symbols (⊗) indicate the instantaneous particle location at five different instants in time.

2.2. Biomechanics

Following the approach described previously [25],the membranous labyrinth was modeled as a rigid struc-ture fixed firmly to the temporal bone and filled withviscous endolymph undergoing low Reynolds, numberPoiseuille flow. The forces acting during BPPV on theendolymph are illustrated in Fig. 1 as a free-body dia-gram of a short canal segment. These forces arise dueto the viscous interaction of endolymph and the free-floating particles, shear stresses acting at the duct walland a pressure drop along the canal centerline. We as-sumed ideal Stokes’ drag on the particles and neglectedthe interaction between the particles and the membra-nous duct wall. Specific model equations are providedin the Appendix (from [26]). Model equations were

solved in the time domain using a fifth-order Runga-Kutta-Fuehlberg method (Wavemetrics, Igor Pro).

2.3. Maneuvers

We applied the model to predict the location of par-ticle debris within the labyrinth and cupulae volumedisplacements during the HC-BPPV diagnostic maneu-ver and CRPs. The following standard definitions wereused to describe head rotations: pitch, rotation aboutan axis out the ear (head-fixed y-axis); roll, rotationabout an axis out the eye (head-fixed x-axis); and yaw,rotation about an axis pointing out the top of the head(head-fixed z-axis).

For the purpose of discussion, we defined a standardHC-BPPV diagnostic test consisting of four head rota-


Table 1Diagnostic maneuver for HC-BPPV

Finish (s) Move (s) Delay (s) R R11 R12 R13 R21 R22 R23 R31 R32 R33

tj ∆tmovej ∆tdelay

j

t0 = 0 0 1 R0 1 0 0 0 1 0 0 0 1t1 = 4 2 1 R1 0 0 −1 0 1 0 1 0 0t2 = 48 2 42 R2 0 −1 0 0 0 −1 1 0 0t3 = 50 2 35 R3 0 1 0 0 0 1 1 0 0t4 = 88 2 1 R4 0 0 −1 0 1 0 1 0 0t5 = 100 2 10 R5 1 0 0 0 1 0 0 0 1

tions illustrated in Fig. 2. The upper body was quicklypitched backwards until the patient was in the supineposition. After a brief delay, the head was rotated inyaw 90 toward the affected ear, and then 180 yaw inthe opposite direction. We modeled these movementsby defining the series of consecutive discrete rotationsteps with the instantaneous rotation matrix at each timedefined by R in Table 1 (R11 – R33 define the compo-nents of each 3 × 3 rotation matrix). The delay time∆tdelay

j is the time over which head is held stationaryin space (until the nystagmus subsides) and movementtime ∆tmove

j is the time over which a head rotation isexecuted. Each rotation was linearly interpolated withrespect to the previous one and smoothed in time togenerate continuous movements following Rajguru etal. [26].

The model was also applied to study a 270 rota-tion and a 360 rotation CRP [15,16,20] designed tomove the particle(s) from the HC lumen to the utricularvestibule (Table 2). The model 270 CRP movementconsisted of three consecutive 90 yaw head rotationsexecuted while the body was in the supine position,with each rotation toward the unaffected ear. The headwas kept stationary in each position for 30–60 secondsallowing time for the particle(s) to fall under the ac-tion of gravity. A 360 CRP was also investigated(Fig. 3 [5]). For this, the head was first rotated in yaw90 towards the affected ear while the subject was insupine position. The head was then turned 45 yawtowards the unaffected ear. These initial steps are key,as they facilitate the movement of the debris away fromthe ampulla. Subsequent steps are the same as those inthe 270 CRP. The instantaneous rotation matrices (R)for the 360 CRP are provided in Table 2.

In addition to estimating particle position during di-agnostic and repositioning procedures, we applied themodel to study the influence of particle size, numberand initial location on cupular responses. Simulationswere carried out for both the HC and PC variants ofBPPV and their results were used to estimate and com-pare ocular nystagmus magnitude and temporal char-acteristics (Table 3).

3. Results

3.1. Diagnostic maneuver – canalithiasis

The location of particle debris, indicated by an ar-row and denoted by the symbol ⊗, is shown in Fig. 2during each head position of the HC-BPPV provoca-tive maneuver. For this and subsequent simulations,a 6-particle aggregate (collective radius 12 µm, totalmass = 0.117 µg) was initially positioned in the HClumen (Fig. 2I). Figure 4 details the cupulae volumedisplacements evoked in response to the provocativemaneuver for (a) a no-particle control and (b) canalithi-asis conditions. The volume displacements predictedby the model for the HC are shown as solid curves.Figure 4(c), (d) show the difference between volumedisplacements in the canalithiasis and control condi-tions – a difference that corresponds to the patholog-ical cupula displacement due to the presence of theparticles. This pathological mechanical response es-timated by the model would be expected to directlylead to pathological afferent input to the brainstem andclinically observed vestibular symptoms.

The volume displacements of the PC and AC cupulaewere essentially unaffected by the presence of particlesin the HC (Fig. 4(c) and (d)). The pathological volumedisplacement of the HC cupula, however, was consid-erable. Note also that the latency-to-peak inhibitory re-sponse, computed as the time interval between provoca-tive position achievement and peak cupula volume dis-placement, was less than one second (arrow A, Fig. 4c).It is interesting that this latency increases if the particlesare initially located in the ampulla (arrow B, Fig. 4(d))rather than in the canal lumen (4(c)). The long latencywhen the particles are initially located in the ampulla ispredicted to be due to the time required for the particlesto reach the narrow canal lumen. Figure 4 also illus-trates that particles have the maximal effect on cupuladisplacements when moving within the slender lumenof the canal, and have a relatively small effect whenthey are free-floating within larger regions such as theampulla and/or utricular vestibule.


Table 2360-rotation CRP for HC-BPPV

Finish (s) Move (s) Delay (s) R R11 R12 R13 R21 R22 R23 R31 R32 R33

tj ∆tmovej ∆tdelay

j

t0 = 0 0 1 R0 1 0 0 0 1 0 0 0 1t1 = 4 1 2 R1 0 0 −1 0 1 0 1 0 0t2 = 25 3 18 R2 0 −1 0 0 0 −1 1 0 0t3 = 80 2 53 R3 0 0.707 −0.707 0 0.707 0.707 1 0 0t4 = 120 2 38 R4 0 0 1 0 −1 0 1 0 0t5 = 150 2 28 R5 0 −1 0 0 0 −1 1 0 0t6 = 153 2 1 R6 0 0 −1 0 1 0 1 0 0t7 = 180 2 25 R7 1 0 0 0 1 0 0 0 1

Table 3Comparison of model predictions of latency-to-peak response, duration and estimated magnitude of nystagmus for canalithiasiscondition due to various particle sizes for PC/HC variant of BPPV

Size Number of Initial Latency to Latency-to Duration of Non-physiological Estimated Nystagmus(µm) Particles Location Onset (s) -Peak (s) Response (s) (/s)

5 62 Ampulla 20/17 75/42 84.44/80 34.42/28.727.5 18 8/11 33/22 48.15/40.50 29.48/22.41

10 8 4/8 15/14 29.97/24.80 26.25/18.2112 5 3/6 11/11 21.79/17.70 23.98/16.2115 3 3/5 10/8 17.37/15.40 21.54/14.4518 1 3/5 6/6 8.69/4.70 9.77/6.8520 1 3/5 5/5 6.68/< 1 11.38/7.485 62 Canal Lumen 4/< 1 35/11 81.0/63.1 30.73/22.917.5 18 4/< 1 16/4 38.12/30.70 25.31/16.92

10 8 4/< 1 11/2 23.95/17.30 20.91/14.2012 5 4/< 1 10/1 17.29/10.80 18.48/13.8915 3 3/< 1 6/1 13.15/8.50 16.53/16.418 1 4/< 1 5/< 1 4.59/2.40 7.58/8.4620 1 4/< 1 4/< 1 1.00/1.90 9.10/10.69

3.2. Canalithiasis latency

The latency-to-peak response is a distinctive featureof canalithiasis BPPV and warrants further mention.Since it remains unclear the number or size of particlesinvolved in BPPV occurrences, we applied the model topredict latency changes as a function of particle dimen-sions and number. Latency was studied using particleswith radii of 5, 7.5, 10, 12, 18, and 20 µm [6]. Re-sponses during the provocative maneuver, with the par-ticle(s) initially located within the HC ampulla, werepredicted for multiple particles (Fig. 5a) with the totalmass held constant at ∼ 0.0879 µg [13] by varying thenumber of particles with changes in radii. The modelwas also used to predict the responses for a single par-ticle of different radii mentioned above (Fig. 5b). Thefirst vertical dashed line (denoted by 0 at the top of thefigure) indicates the time at which the 90 provocativehead position toward the affected side was achieved.Dashed lines labeled 1–7 indicate the incidences ofpeak HC cupula volume displacements predicted by themodel. In general, an aggregate of smaller particlesresults in longer latencies and larger HC cupula dis-

placements. The middle panel (b) shows the influenceof a single particle. In this case, single small particlesgive rise to longer latencies but a much smaller mag-nitude of response than a single large particle. This isbecause large particles fall faster and quickly come torest at the lowest position within the HC lumen. Thisrapid particle movement does not allow the cupula suf-ficient time to come to its steady-state position becausethe viscous fluid in the slender duct limits the mag-nitude of the response. This dependence on particlesize was expected since the particle terminal free-fallvelocity at low Reynolds’ number varies in proportionto the square of the radius. Consequently, the time atwhich peak cupula volume displacement occurs variesas ∼1/a2 as observed from the curve fit using a powerfunction (Fig. 5c).

3.3. Canalithiasis CRP

The model was applied to estimate cupula and parti-cle displacements for 270 and 360 CRPs. The 270

CRP was successful in relocating debris to the utricle ifthe particles were initially located in the canal lumen,


I0.01 s

II4 s

III25 s

IV80 s

V120 s

VI150 s

VII180 s

S I D E - V I E W B A C K - V I E W

Fig. 3. 360 HC canalith repositioning procedure (CRP). Orientations of the head and membranous labyrinth for 7 instants of time defining the360 CRP (see Table 2 for rotation matrix values). Same notation as Fig. 2. The arrows and symbols (⊗) indicate the location of the particles.Note that this CRP was effective in moving the particles from the canal lumen to the utricular vestibule.

but failed when the particles were initially located inthe HC ampulla. Therefore we propose a 360 CRP,shown in Fig. 3, that was successful in simulations re-gardless of initial particle location. For this CRP, thepatient is first placed in the supine position (Fig. 3, II)and then subjected to four consecutive rotations. Thefirst 90 rotation toward the affected ear (Fig. 3, III) wasdesigned to move the particle(s) from the HC ampullainto the slender lumen of the canal. The remaining

steps (Fig. 3, IV, V, VI) define a roll-over sequence to-ward the unaffected ear designed to move the particlesalong the canal lumen and ultimately into the utricle.The first rotation toward the affected ear is unconven-tional, but simulations showed that this step is essen-tial when particles are initially located deep within theampulla. Predicted responses of all three canals duringthe 360 CRP are shown in Fig. 6. Differences be-tween the (a) control and (b) canalithiasis condition are


Time (s)

Particle(s) intially in the lumen.

Particle(s) intially in the ampulla.

II IIII

HCACPC

Cup

ulae

Vol

ume

Dis

plac

emen

t (nL

)

(b)

(a)

(c)

Control

BPPV

Difference (b-a)

3

2

1

0

-1

-2

3

21

0-1-2

100806040200

1

0.5

0

-0.5

-1

IV

-3

-3

(d)

DifferenceTime (s)

0

-0.2

-0.4

-0.6

-0.8

-1

V

A

B

Fig. 4. Canalithiasis responses to diagnostic maneuver. Cupulae volume displacements for the control (a) and the canalithiasis (b) conditions inresponse to the provocative maneuver (Fig. 2 and Table 1) when the particles were initially located in the HC lumen (response of HC – solid, PC– short dashed, AC – long dashed). I-V correspond to head positions illustrated in Fig. 2. Specific results were computed for 6 particles, eachof 12 µm radius. The difference between the control and canalithiasis conditions is a measure of the pathological response. This difference isshown in the third panel (c) for the case when the particles were initially located in the HC lumen, and shown in the bottom panel (d) for the casewhen the particles were initially located in the HC ampulla. Note the increased latency (arrow B vs. A) when the particles are initially located inthe ampulla vs. in the slender lumen of the canal.


(c)

(b)

Particle Radius (µm)

34Late

ncy

(s)

y = y0 + A xb

5

"Single Particle"

Time (s)

Cup

ulae

Vol

ume

Dis

p. (

nL)

2

6

Cup

ulae

Vol

ume

Dis

p. (

nL)

4 53

Time (s)

2

-1

0

1

806040200

6

"Multiple Particles, total mass = 0.0879 µg"

12 µm10 µm

15 µm 18 µm

7.5 µm

20 µm

5 µm

1.5

1

0.5

0

806040200

1

2

2.5

2

-0.5

0

1

123456

(a)

7

7

45

30

2015

181614121086

7

4 532 610 7

4035

25

20

Fig. 5. Canalithiasis response latency. HC cupula volume displace-ments estimated in response to the diagnostic maneuver (Fig. 2)for (a) multiple particles of various radii (constant total mass of0.0879 µg) and (b) single particle of various radii. The provocativeposition was achieved at the time denoted by (φ). Points (1–7) de-note the peak HC cupula volume displacement for the particle sizesindicated. Bottom panel (c) shows the latency to the peak cupulavolume displacement (latency times of points 1–7) as a function ofparticle size. The latency varied as the inverse of the radius squared(∼1/a2 ; curve fit parameters: b = −2, A = 961, y0 = 9.1).

shown in the bottom panel (c). This difference is thefraction of the response that is not present in the con-trol condition and therefore represents the pathologicalresponse. Excitatory HC stimulation estimated by themodel in the initial 90 head turn would be expected

HCACPC

Cup

ulae

Vol

ume

Dis

plac

emen

t (nL

)

-2

-1

0

1

Time (s)

-1.5

-1

-0.5

0

(b)

(a)

(c)

Control2

-2

-1

0

1

BPPV

Difference (b-a)

III IV V VI VIIII

2

150100500

Fig. 6. Cupulae volume displacements during the CRP. Cupulae vol-ume displacements estimated for the control (a) and the canalithia-sis BPPV (b) conditions in response to the 360 CRP (HC – solid,PC – short dashed, AC – long dashed). Instants in time labeledII-VII correspond to the head positions illustrated in Fig. 3. The dif-ference between the canalithiasis and control volume displacements(i.e. pathological response) is shown in (c). Specific results werecomputed for 6 particles, each of 12 µm radius.

to lead to the characteristic horizontal ocular nystag-mus observed clinically. Subsequent rotations result insimilar responses, though with shorter latencies. Theinfluence of the particles is relatively small once posi-tioned within utricular vestibule, since this region hasa large cross sectional area.

3.4. Diagnostic maneuver – cupulolithiasis

We also applied the model to investigate cupulolithi-asis by adhering 6 particles, of 12 µm each, to the


-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

Cup

ulae

Vol

ume

Dis

plac

emen

t (nL

)

604020-20Time (s)

HC ACPC

800

Fig. 7. Cupulolithiasis responses to diagnostic maneuver. The dif-ference between the cupulae volume displacement estimated in thecontrol condition vs. the cupulolithiasis condition is shown for theHC diagnostic maneuver illustrated in Fig. 2. As expected, the modelpredicts maintained mechanical stimulation of the HC cupula whenthe head is held in the provocative position.

HC cupula. Figure 7 shows cupulae volume displace-ments of the three canals in the control and cupulolithi-asis conditions in response to the diagnostic maneu-ver (Fig. 2). Maintained pathological activation ofthe HC was predicted once the provocative positionwas achieved. The peak displacement of the cupula(∼320 pL) was less than that predicted under the sameconditions during canalithiasis (∼1200 pL). This dif-ference arises in the model due to a force amplifyinglever-effect that occurs in canalithiasis as the particlesmove from the ampulla into a smaller cross-section ofthe canal lumen, an occurrence that is absent in cupu-lolithiasis [13,28].

In addition to predicting cupular volumetric displace-ments, the model was also used to estimate ocular nys-tagmus evoked by diagnostic maneuvers in control andBPPV canalithiasis conditions. For this, we computedthe transfer function between cupula volume displace-ment and slow-phase angular eye velocity for sinu-soidal angular head oscillation in the control conditionat 2 Hz. This yielded ∼53 pL cupula displacementfor each 1 deg/s of angular head velocity. Therefore,a pathological cupula displacement of 53 pL would beestimated to induce pathological nystagmus of ∼1/s.For comparison to clinical data, we used a thresholdof 9/s ocular nystagmus (i.e. 473 pL cupula volumedisplacement) to estimate nystagmus onset latency andduration. Results are summarized in Table 3 for variousparticle sizes (radii, µm), numbers, and initial positions(in the ampulla or lumen of the canal). The table com-pares results for HC-BPPV (present manuscript) to pre-vious model predictions for PC-BPPV [20]. Latenciesare reported relative to achievement of the provocative

position. Both onset latency (to reach 9/s threshold)and latency-to-peak response are shown. Duration ofthe non-physiological response indicates the total timeover which the estimated nystagmus was above the 9/sthreshold. Under most conditions, the model predictsa shorter onset latency for HC-BPPV vs. PC-BPPV.This occurs because the orientation and geometry ofthe membranous labyrinth makes it likely for particlesin the PC to initially reside within the PC ampulla,whereas particles in the HC are likely to initially residein the HC canal lumen.

4. Discussion

The present biomechanical model was developed toprovide a quantitative basis for the diagnosis and treat-ment of HC-BPPV, and to provide an understandingof differences between HC-BPPV vs. PC-BPPV. Oc-ular nystagmus onset latency during provocative testsis a key difference between HC- vs. PC-BPPV ob-served clinically. The present simulations indicate thatthis difference is probably due to the initial locationof canalithiasis particles prior to administration of theprovocative diagnostic test. In PC-BPPV the particlesare likely to be initially located in the PC ampulla dueto the geometry and orientation of the labyrinth in thehead [26]. In contrast, there is no compelling reasonwhy particles in HC-BPPV would be initially located inthe HC ampulla and, indeed under clinical conditions,may be inadvertently displaced to the HC lumen duringthe diagnostic test. For particles initially located withinthe ampulla, there would be a delay time during whichthe particles move from the larger-cross sectional areaof the ampulla into the narrow canal lumen and hence,a longer latency relative to particles initially located inthe lumen. The relatively short onset latency (< 1–3 s)of HC-BPPV relative to PC-BPPV may simply reflectthe initial location of the particles in the lumen vs. theampulla. For the special case of HC-BPPV when theparticles are initially located in the ampulla, the modelindicates that the use of a 360CRP to treat canalithiasismay increase the likelihood of clinical success relativeto a 270 CRP.

The model was also applied to estimate the timecourse of pathological ocular nystagmus that would beexpected to occur under conditions of HC-BPPV. Forexample, the model predicts ∼ 1264 pL of cupula dis-placement for the canalithiasis case in response to theprovocative maneuver (using particle mass ∼ 0.117 µg,6 particles, radius 12 µm each) – a response that would


be expected to produce slow-phase eye movements witha velocity of approximately 22/s. Similarly, for apathological mass of ∼0.0877 µg (62 particles, radius5 µm each), the model predicts the peak HC cupuladisplacement of∼1520 pL and a slow-phase eye veloc-ity on the order of 26/s. These velocity estimates fallwithin the clinically observed range of 17/s–72/s [2]and therefore provide confidence in the modeling ap-proach. Changing the number and/or sizes of particleswould of course, expand the range.

Although the model is quite detailed in its attentionto the geometry of the membranous labyrinth, there areseveral limitations that should be noted. The modelassumes low-frequency head movements, does not ac-count for the interaction of the particle(s) with the mem-branous wall, and is strictly biomechanical in nature.It has been shown previously that neglecting the wallinteractions is reasonable when the particles are lo-cated near the center of the canal lumen, but it willoverestimate the force transmitted to the endolymphfor small particles sliding or rolling along the canalwall [28]. Bungay and Brenner [3] predict the theoret-ical pressure drop due to a sphere settling in a verticaltube as a function of particle lateral position (radius ofsphere (a)/distance from wall of tube (h)) and particle-to-tube radius ratio (a/R). Experimental data availablefor larger particles are consistent with the theoreticalmodel of [3] (a/R = 0.13–0.53 [11]), but are not avail-able for smaller particles. For very small particles orparticles in the ampulla (a/R ∼0.05) our model wouldoverestimate the pressure drop by approximately 90%when the particles are sliding along the wall (a/h =1). And for the same case if the particle is one diam-eter away from the wall (a/h = 0.5), the model wouldoverestimate the pressure drop by about 60%. How-ever, for most cases of BPPV i.e., for larger particles orparticles in smaller canal lumen (a/R ∼0.11–0.13), ourmodel would overestimate the pressure drop by only10% when a/h = 0.5 and by 70% for a/h = 1. Qual-itatively this means that small particles in the vicinityof the canal wall would generate cupula displacementswith longer latencies and smaller magnitudes than pre-dicted by the present simplified model. This wouldbe similar to adjusting the particle size to a smaller“effective” diameter in the model. Also, the pressure(or form) drag arising from the pressure difference thatresults from the diversion of the fluid flow around thesmall spheres can be assumed small (unless the particleis sliding along the canal wall). Although theoreticallycontact with the canal wall would bring the particles torest since infinite force is then required to produce mo-

Table 4Physical parameters

Parameter Value Units

ρe endolymph density 1.0 g.cm−3

ρe endolymph viscosity 8.5 × 10−3 dyn.s−1.cm−1

ρc cupula density 1.0 g.cm−3

γc cupula shear stiffness 3.7 dyn.cm−2

h cupula thickness 3.08 × 10−2 cmΓ permeability coef. 2.0 × 105 dyn.s−1.cm−1

ρs particle density 2.7 g.cm−3

a nom. particle radius 1.2 × 10−3 cmNs nom. particle number 6

tion of the spheres, it is unlikely to be observed in prac-tice (as discussed by [3] and [10], owing to the break-down in assumptions of perfect smoothness of the rigidsurfaces and the integrity and constancy of the physicalproperties of fluid phase). Therefore, for simplicity,we neglected wall interaction in the present model withthe understanding that predicted responses are approx-imate. A neural model was not included, so results donot predict the afferent responses or eye movementsdirectly. Eye movements were estimated from cupuladisplacements by assuming a perfect vestibulo-ocularreflex with a gain near unity. Although the model canbe applied to any individual human geometry, predic-tions shown here are based on one example subject.However, even with these simplifications both the la-tency and the magnitude of the predicted pathologi-cal responses fall within the range observed clinically.Differences between HC-BPPV and PC-BPPV are alsoconsistent with clinical observations, further support-ing canalithiasis and cupulolithiasis as common originsof BPPV.

Acknowledgments

This work was supported by the National Institutesof Health NIDCD R01-DC006685.

Appendix. Model Equations

The endolymph volume displacement, Qn, for eachnth segment of the membranous labyrinth was modeledusing:

mnd2Qn

dt2+ cn

dQn

dt+ knQn = Pn(ln)

(1)−Pn(0) + fn.

where mn, cn and kn were computed using [4]


mn =

ln∫0

ρ

A(s)ds, (2)

cn =

ln∫0

µλ

A(s)2ds, (3)

and

kn =

ln∫0

γλ

A(s)2ds. (4)

A(s) is the cross-sectional area as a function of the cen-terline coordinate s, ρ is the mass density, γ is the shearstiffness and µ is the viscosity (values of the physicalparameters are provided in Table 4). The coefficientλ = 8π, which is appropriate for low frequency headmovements used in the present CRP movements andprovocative tests. The cupulae were modeled as poroe-lastic, also using Eq. (1) but noting that the pressuredifference is the drag between the fluid and solid phasesrather than the fluid pressure [25]

∆P = Γ(

dQe

dt− dQc

dt

). (5)

Qe and Qc are the volume displacements of the cupulaand endolymph respectively and Γ is the inverse Darcycoefficient. The forcing term is the sum of inertial andparticle components:

fni =

ln∫0

ρ(Ω × R(s)) · ds + fns (6)

where Ω × R(s) is the local tangential acceleration of

the duct due to angular acceleration of the head ( Ω)(resolved into the head-fixed frame) [25]. The secondterm accounts for the force on the endolymph from theparticle [20]

fns =AsNs

A∗e

[43a(ρs − ρe)(g − Ω × R(s)) · n

(7)

+6aµe

(ξ − Qe

A∗e − As

)],

a is the radius of the particle(s), g is the gravitationalacceleration, n is the unit normal directed tangent to theduct centerline and Ns is the number of particles. Thefrontal area of a single particle is As = πa2 and thearea of the canal lumen at the position of the particle isA∗

e = Ae|s=ξ . This equation is nonlinear due to the factthat local cross-sectional area varies as the particle(s)

moves within the labyrinth. The coordinate ξ(t) de-fines the centerline “s” position of the particle(s). Theacceleration of the particle(s) tangent to the centerlineis

msd2ξ

dt2= FnsA

∗e . (8)

where ms is the mass of the particle. For cupulolithi-asis, the particle velocity matches the endolymph ve-locity such that ξ − Q/(A∗

e − As) = 0 and the secondterm in Eq. (7) is zero.

Equations for each segment were combined to obtaina coupled set of ordinary differential equations of theform [25]

Cd Q

dt+ K Q = F (9)

where C is the damping matrix, K is the stiffness ma-trix and F is the forcing vector. We have neglectedthe mass matrix M due to the low frequencies of in-terest in the present simulations. The dependent vari-ables are endolymph ( Qe) and cupulae ( Qc) volumedisplacements

Q =[

Qe

Qc

](10)

The damping and stiffness matrices are

C =[Ce + Γ −Γ−Γ Cc + Γ

](11)

and

K =[

0Kc

](12)

The above equations Eqs (1–4), (6–12) account forall three canals, common crus and utricle. The com-plete damping and stiffness matrices are provided inRajguru et al. [20] The forcing vector is:

F =[

F e

F c

](13)

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