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Changes resulting from the publishing process, such as peer review, editing, corrections,

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Ultrasound, Volume 21, Issue 2 , Pages 81-91, (June 2013) doi:10.1016/j.jmu.2013.04.004

.

TITLE PAGE

Original Article

Vibration Analysis of Circular Membrane Model of Alveolar Wall in Examining

Ultrasound Induced Lung Haemorrhage

(Modelling of Ultrasound Induced Lung Haemorrhage)

D. John Jabaraj1 and Mohamad Suhaimi Jaafar2

1 Corresponding Author: D. John Jabaraj

MSI-UniKL Kedah, 09000 Kulim, Kedah & Dept. of Medical Physics, Faculty of Physical

Sciences, Universiti Sains Malaysia, 11800 USM, Pulau Pinang, Malaysia.

(Mail add: 449, Tmn Perdana, Jln Toh Kee Kah, 71000 P.Dickson, N. Sembilan, Malaysia.)

(e-mail: jojab77@yahoo.com) (Tel: +60174194717)

2Mohamad Suhaimi Jaafar

Dept. of Medical Physics, Universiti Sains Malaysia, 11800 USM, Penang, Malaysia. (e-mail:

msj@usm.my).

ABSTRACT AND KEYWORDS

Abstract

Background: The ultrasound induced lung haemorrhage is analyzed from the perspective of

alveolar resonance mechanism.

Method: The alveolar resonance mechanism is theoretically studied by modelling the alveolar

wall facet as circular membrane model vibrating at fundamental mode. Based on the vibration

analysis of this model, the equations of fundamental frequency and threshold pressure for the

occurrence of ultrasound induced lung haemorrhage are derived.

Results: The validity of the circular membrane model of the alveolar resonance mechanism is

demonstrated. This theoretical study predicts that the ultrasound induced lung haemorrhage

do not occur when the total lung capacity is < 20% (for mammals), and when the ultrasound

frequency is > 1.55 MHz at mechanical index of < 1.9 (for human only).

Conclusion: The alveolar resonance mechanism is a plausible mechanism of ultrasound

induced lung haemorrhage.

Keywords: Ultrasonic Therapy/adverse effects; Hemorrhage/pathology; Biomechanics; Lung

Injury; Ultrasonography/adverse effects;

MAIN TEXT

1. Introduction

1.1 The Ultrasound Induced Lung Haemorrhage

The ultrasound (US) induced lung damage has been observed in animal experimentations at

the exposure levels of diagnostic US imaging. The first reported US induced lung

haemorrhage occurred in adult mouse at exposure values of; pressure threshold, P = 1 MPa,

frequency, f = 2 MHz, pulse duration, PD = 10 µs, pulse repetition frequency, PRF = 100 Hz

and exposure duration, ED = 3 min [1]. Since then, the US induced lung haemorrhage has

been observed in other animals such as rats, rabbits, monkeys and pigs at similar exposure

conditions [2-5].

The US induced lung haemorrhage is characterized by localized extravasations of red blood

cells from the pulmonary capillaries into the alveolar spaces. ECMUS [6] states that the

rupture of capillaries and the fracture of alveolar epithelium and endothelium are the reasons

for this haemorrhage. The mechanical mechanism (non-thermal effects) is believed to cause

the US induced lung haemorrhage [6]. A mechanical mechanism known as the inertial

cavitations, has been widely researched in the study of US induced lung haemorrhage. Inertial

cavitations is the bursting of the micro-bubbles (in lungs) through resonance induced by the

US waves. This results in high energy emanations which damage surrounding tissues.

Nevertheless, several studies have subsequently ruled out the inertial cavitations as being the

mechanism of US induced lung haemorrhage [7, 8].

1.2 Alveolar Resonance Mechanism

There is another hypothetical mechanical mechanism known as the alveolar resonance which

may explain the US induced lung haemorrhage. The alveolar resonance mechanism suggests

that the exposure of US to lungs, results in oscillatory response of the alveolar structures

within to the varying compression and tensional US waves [9]. The alveolar structure

resonates and deforms rhythmically, producing local stresses and strains to its own and

surrounding tissues [10]. This is detrimental as the alveolar capillaries and membranes can be

damaged and bleeding occurs [6]. Nevertheless, ter Haar [11] states that the vibration and

resonance of the alveolus (~50 µm) is different than inertial cavitations of microbubbles (~1

µm) in respect to diagnostic US frequency range.

There are no mathematical calculations or theoretical models that support or disprove the

hypothetical alveolar resonance mechanism in producing the US induced lung haemorrhage

[6]. A thorough survey of literature gave no indication that the alveolar resonance mechanism

hypothesis has been experimentally tested or studied in any way.

1.3 The Alveolus

In three-dimension, the typical alveolus is seen as 14-faceted polyhedral with one open-end

for alveolar duct [12]. The facets of alveolar wall are shaped as irregular pentagons. The

structural framework of the alveolar wall contains elastins, which are mechanically connected

to collagens in order to provide elasticity. Within the alveolar wall, there is blood gas barrier

(BGB) made of the alveolar epithelium and capillary endothelium. The BGB is relatively very

thin (about 200 nm) and due to its inability to expand and shear, is susceptible to rupture.

The elastic tension of the alveolar wall depends on the strain caused by normal tidal

breathing. During recruitment and de-recruitment of the alveolus, the Total Lung Capacity

(TLC) varies from 0% to 20%. However the linear strain is about zero, because the alveolar

walls are not stretched but are unfolded. The normal tidal breathing is from 20% TLC

onwards (after recruitment) and the alveolus is stretched slightly, resulting in linear strain of 0

– 0.05 [13].

Belete [14] states that cell wounding in parenchyma tissues occurs significantly when linear

strain is 0.08. Vlahakis & Hubmayr [15] also summarized that an excess linear strain

percentage of 2% to 3% (above the linear strain of tidal breath) forms fractures in the plasma

membrane of the alveolar epithelial cells. Thus, this total linear strain of 0.08 that will damage

elements within the alveolar wall and probably cause haemorrhage, will be called here as the

yield linear strain.

1.4 Objectives of Study

This study is focused on the sparsely researched alveolar resonance mechanism and its

possible role in US induced lung haemorrhage. The objectives of this study are;

i. To develop a vibration model of the alveolar structure in order to theoretically

analyze its resonant frequency and the threshold pressure needed to cause the US

induced lung haemorrhage.

ii. To validate the developed vibration model of the alveolar structure for studying

the US induced lung haemorrhage and to further provide predictions and

precautionary recommendation.

2. Model Theory Establishment

2.1 The Modelling of the Alveolar Wall Structure

The defining structure of the alveolus is its wall. Several assumptions are made concerning

the alveolar wall and its behaviour for the purpose of modelling.

i) The alveolar wall is assumed to be isotropic and linearly elastic.

The alveolar wall contains elastin fibres and plasma membrane with lipid bi-layers. If the

deformation of the alveolar wall is reversible and small, then it obeys the Hooke’s Law [16].

Therefore, the linear theory of elasticity can be applied unto the alveolar wall. The alveolar

wall can be assumed as isotropic too as it is homogenous [17].

ii) The linear theory of vibration is assumed to be applicable to the alveolar wall.

The alveolar wall is a continuous material governed by partial differential equations (PDE),

which depend on time and spatial coordinates. If an alveolar wall facet vibrates transversely

with small amplitudes (compared to its dimension) and the elasticity of the alveolar wall

provides linear restoring force while damping is also linear, then the vibration is considered as

linear [18].

iii) The alveolar wall is assumed as clamped membranes.

The alveolar wall is modelled as flat membranes instead as plates because its stiffness is

determined to be too small, approaching zero (Appendix A). The alveoli are interconnected

alike the honeycomb structure, hence the alveolar walls are shared among alveoli [19]. This

provides radial support to the alveolar wall facets at its borders.

iv) The geometry of the alveolar wall facet is assumed as circular.

The facets of the alveolar wall are in reality shaped as flat irregular pentagon [12].

Nevertheless, the application of the theory of vibration on membrane has a certain restriction.

A suitable coordinate system is needed to spesify the boundary condition of the membrane.

This is to ensure the resulting PDE arising from the application of vibration theory on the

membrane will not be partially or approximately solvable [20]. The simple polar coordinate

system adequately describes the boundaries of the circular shape and thus is chosen.

2.2 Resonant Frequencies of the Modelled Alveolar Wall

The two dimensional wave PDE for a membrane on x-y axis with deflection axis as z at time,

t, is [20],

])/[]/([/ 2222222 yzxzctz T ∂∂+∂∂=∂∂ (1)

The transverse wave velocity in membrane is denoted as cT. The equation of resonant

frequency of the circular membrane model is obtained when the wave equation is solved,

using the separation of variables method together with boundary condition specifications.

The resonant frequency at n-th mode of the clamped circular membrane model is [20];

¼)(½ -1 −= nrcf on T (2)

The frequency mode is denoted as n and ro is radius of membrane model. Meanwhile, the

transverse wave velocity on the membrane with tension (T) is;

½)/( σTcT = (3)

with area density (σ) being related to density (ρ) and thickness (d) by, dρσ = (4)

2.3 Transverse Wave Velocity

The elastic tension in the membrane model of alveolar wall during normal tidal breath is

dependent on the Young’s modulus, E, the areal strain of tidal breath, εb and the membrane

thickness, d (Appendix B).

dET bε= (5)

By inserting E.4 & E.5 into E.3, the transverse wave velocity of the membrane model is;

½

½

)]/()[(

)/(

ddE

Tc

b

T

ρε

σ

=

=

½)/( ρεbEcT = (6)

An indirect calculation was conducted to obtain the equation of Young’s modulus for the

membrane model. The longitudinal speed (cL) of sound waves is dependent on the bulk

modulus (K) and density (ρ) of the material of propagation.

½)/( ρKcL = (7)

2LcK ρ= (8)

The equation of Young’s modulus can then be determined by utilizing its relationship with the

bulk modulus. The Poisson ratio of material is denoted as u.

)21(3 uKE −= (9)

)21)((3 2 ucE L −= ρ (10)

A relationship between transverse wave velocity and the areal strains of tidal breath for the

circular membrane model of the alveolar wall facet was derived by applying E.10 into E.6;

½2

½

)/)]21)((3([

)/(

ρερ

ρε

b

b

uc

Ec

L

T

−=

=

½½)63( bLT cuc ε−= (11)

2.4 Forced Vibration of the Modelled Alveolar Wall Facet

At fundamental mode, the point in the centre of membrane and its deflection imitates the

motion of the single degree of freedom (SDOF) vibration. So according to Bachmann [21],

the vibration of a continuous system at fundamental frequency can be approximated as an

equivalent SDOF vibration. The equation of amplitude for the equivalent SDOF forced

vibration damped system is [22];

½2222 ])()/[()/(

wQwmFZ

oo

eq0

−+=

ωω (12)

with F0 as peak force and w as the angular frequency of the applied driving wave. The

fundamental angular frequency of the membrane model is denoted as ωo while its equivalent

mass is meq. The equivalent mass depends on the shape of the membrane vibration at

fundamental mode. The quality (Q) factor of the vibrating system is denoted as Q.

Here in this study, the applied driving waves represent the US waves incident on the alveolar

wall surface. If the US wavelength is larger than the dimension of alveolar wall surface area,

it can be considered that the force imparted to the surface is uniformly distributed. Thus, the

peak rarefaction pressure (Po) of US waves applied to a membrane surface area (A) is related

to the peak rarefaction force (Fo) by; AFP oo /= .

Thus E.12 morphs into;

½2222 ])()/[()/(

wQwPZ

oo

eq0

−+=

ωωσ (13)

The equivalent area density is the equivalent mass over area, Am eqeq /=σ . (14)

Maximum amplitude, Zmax is the resonant amplitude and it occurs when the frequency of

driving waves approaches the fundamental frequency of membrane model. Thus when w ~

ωo, the E.13 becomes;

eqomax

QPZ 0

σω 2•

= (15)

i) The Q-factor

The dominant Q-factors of high frequency micro-resonators existing in non-vacuum medium

are Q-factor of surrounding medium damping (Qmedium) and Q-factor of membrane’s support

loss (Qsupport) [23]. Hence the total Q-factor (Q) is;

supportmedium QQQ /1/1/1 += (16)

However, the Qmedium factor value is negligible when it is much larger than Qsupport (Appendix

C). Thus, the Q-factor total can be approximated to be equal to Qsupport. The Q-factor of the

vibrating circular membrane is (Appendix D),

)/(64.0 drQQ osupport == (17)

where d is the thickness, ro is the radius of membrane.

ii) The equivalent area density

The equivalent mass for circular membrane vibrating at fundamental frequency is [24];

dAm eq ρ613.0= (18)

The density is denoted as ρ while d is thickness and A is surface area. Thus by using E.14 the

equivalent area density for the vibrating circular membrane at fundamental frequency is,

AdAAm eqeq /613.0/ ρσ ==

deq ρσ 613.0= (19)

2.5 Relation between Resonant Amplitude and Areal Strain

The alveolar wall facet is modelled as two-dimensional thin flat elastic membrane. Thus the

type of strain analyzed here will be the areal strain, which is measured as;

001 /)( AAA −=ε (20)

The initial area is denoted as A0 while A1 is the deformed area.

The membranes’ shape while at rest is a flat circular disc. Thus, the initial area is,

20 orA π= (21)

where ro is the radius.

The circular membranes’ shape while vibrating at fundamental frequency mode is a circular

based dome. If the height, Z is smaller than the dimension of the membrane, then the shape

can be simplified into a spherical cap as an approximation (Fig. 1). Thus the deformed area is;

)( 221 ZrA o += π (22)

The areal strain at resonance (εr) for deformed circular membrane at resonant amplitude (Zmax)

in the fundamental vibration mode is calculated by applying E.21 & E.22 into E.20.

)()())((

/)(

2

222

001

o

omaxor

rrZr

AAA

πππε

ε

−+=

−=

Resonant amplitude at fundamental mode, ½romax rZ ε= (23)

2.6 The Threshold Pressure Equation

The yield areal strain in the alveolar wall that is needed to cause alveolar damage is 0.1664

(Appendix E). This yield areal strain is the total of the areal strain during tidal breath (εb) and

the areal strain of resonant amplitude (εr).

Thus, br εε −= 1664.0 (24)

The minimum peak pressure of driving waves which will result in enough resonant amplitude

to achieve the yield areal strain and damage the circular membrane model of alveolar wall

facet, will be called here as the threshold pressure. This represents the threshold pressure of

US waves that experimentally produce detectable lesions of US induced lung haemorrhage in

animals.

Meanwhile the alveolar wall (soft tissue) density is 1026 – 1068 kg / m3 [25]. This range can

be denoted in interval notation as 1026, 1068 kg / m3, to be used in calculations.

The threshold pressure equation for producing damage in the circular membrane model of the

alveolar wall facet is derived by substituting E.17, 19, 23 & 24 and alveolar density (ρ) range

into E.15;

½22

½12

max12

2

][40384.201 ,38796.058

][)/64.0)(613.0()2(

r1

roo1

eqo

eqomax

df

rdrdf

ZQP

QPZ

0

0

ε

ερπ

σω

σω

=

=

=

=

−

−

•

½2 )]1664.0[()(05838796 b1df.P0 ε−= (25)

Only the lower limit is retained as the threshold pressure equation should be concerning the

minimum pressure possible in inducing damage. The threshold pressure formulas for specific

mammals are obtained by inserting alveolar dimension values (Table 1) into E.25.

2.7 Validation Method

The theoretical modelling of this study needs to be validated. The validation comprises of

three steps. First, a comparison test between experimental and theoretical values is conducted.

Theoretical data are simulated from the derived equations obtained through modelling. These

are compared with past experimental data by using the chi-square goodness of fit method

[26]. The chi-square for p-1 degrees of freedom is;

∑=

−=

p

i i

ii2

SSO

1

2)(χ (26)

where p is number of sets of data, O is the observed empirical data and S is the simulated

theoretical data [27].

The calculated chi-square value is then compared with the critical chi-square value at

significance of 0.05. If it is lower, then this implies that the differences of values are

attributed to chance fluctuations. Therefore the null hypothesis is retained and not rejected.

The experimental data is consistent with the theory (hypothesis) and the theory is acceptable

with 95% probability or confidence. Secondly, experiments that can test and prove the theory

incorrect should be constructed. This is to provide a method of disproving the theory. Finally,

the theory must give predictions of the real world phenomena that can be confirmed in future.

Confirmed predictions will provide support concerning the validity of the theory.

2.8 Alveolar Dimensions

The alveolar dimensions for various adult mammals including human are given in Table 1.

3. Results

3.1 Determination of Transverse Wave Velocity

The Poisson’s ratio for the alveolar wall (soft tissues) ranges from 0.35 – 0.45 [37]. This

range can be denoted in interval notation as 0.35, 0.45 to be used in calculations. The

longitudinal speed of sound wave (cL) in alveolar wall (soft tissues) ranges from 1490, 1610

m/s at 37oC body temperature [25]. Whenever calculations involve ranges, the interval

arithmetic should be used (Appendix F). The substituting of alveolar wall’s characteristic

interval values into E.11 results in;

½½

½½

1610 1490,)0.45 0.35,63(

)63(

b

bLT cuc

ε

ε

−=

−=

½1527.38 816.11, bTc ε= (27)

The areal strain of tidal breath can range from 0 – 0.1025 (Appendix F). Thus, the transverse

wave velocity is determined to range from 0 m/s at 0.00 areal strain to 489.00 m/s at 0.1025

areal strain during normal tidal breath.

3.2 The Fundamental Frequency Equation

The values of alveolar radius (ro) depend on the species of mammals. The fundamental

frequency mode for the circular membrane model is 1=n . Thus the fundamental frequency

equation for circular membrane model is obtained by applying the mode value and E.27 into

E.2.

¼)1()1527.38 ½(816.11,

¼)(½

1-½

-1

−=

−=

ob1

on

rf

nrcf T

ε

-1½572.77 306.04, ob1 rf ε= Hz (28)

The fundamental frequency formulas for specific mammals are obtained by inserting alveolar

dimension values from Table 1 into E.28 (Table 2). The other higher modes that can also

cause resonances are ignored. This is because the vibration analysis at fundamental frequency

and its effects provide the benchmark for structural integrity and failure [38].

3.3 The Threshold Pressure for Various Mammals and Human

The threshold pressure formulas for specific mammals and human are obtained by inserting

the alveolar dimension values from Table 1 into E.25 (Table 2). The threshold pressure vs.

fundamental frequency graphs of the circular membrane model of alveolar resonance are

depicted for the adult mouse and rat (Fig. 2 and 3).

3.4 Validation of the Membrane Model of Alveolar Resonance Mechanism

The experimental data of past US induced lung haemorrhage studies are compiled by Church

& O’Brien [39]. Only the lowest threshold pressure value at each frequency from each

experimental study was chosen. There are however only sufficient data for the adult mouse

and rat (Table 3) for comparison, and these were plotted in the threshold pressure vs.

frequency graphs of the circular membrane model of alveolar resonance (Fig. 2 and 3). The

equation of chi-square (E.26) is used for comparison testing.

3.5 The Calculated Threshold Pressures for Experimental Testing

The threshold pressures at 1 MHz fundamental frequency for the seven species of adult

mammals and human are calculated in Table 4. These calculated theoretical values are to be

used for future experimental invalidation testing of the theoretical modelling of the alveolar

resonance. An example of invalidation testing is; experiments are conducted below the

predicted threshold pressure values to obtain US induced lung haemorrhage in animals.

3.6 The MI Index

The mechanical index, MI = 1.9 is plotted in the threshold pressure vs. fundamental frequency

graph for the circular membrane model of the human alveolar wall, in order to analyze the MI

relation with US induced lung haemorrhage for human (Fig. 4).

4. Discussion

The thickness and size of the alveolar wall varies in species. According to this study, the

threshold pressure for US induced lung haemorrhage depends primarily on the alveolar wall

thickness while the resonant frequency of alveolar structure depends primarily on the length

dimension of the alveolar wall (Table 2). ECMUS [6] supports these findings by stating that

US induced lung haemorrhage depends on the species of animal.

This study shows that the varying tidal breath changes the resonant frequency of the alveolar

wall, which then affects the US induced lung haemorrhage. This agrees with the experimental

result of O’Brien [40], where the breathing pattern in mice is seen to alter the results of US

induced lung haemorrhage. This study also demonstrates that the resonance frequencies of the

alveolar wall are within the range of diagnostic US. Thus the alveolus structure is not too

large to oscillate in regard of the diagnostic US frequency range as stated by ter Haar [11].

The resonant frequency of the alveolar wall (as according to the circular membrane model)

depends indirectly to the areal strain of tidal breath. When areal strain of tidal breath is 0.00

(at < 20% TLC); the resonant frequency will be non-existent and the alveolar wall will not

resonate or be affected by US waves. Thus, this theoretical study predicts that it is possible to

avoid the US induced lung haemorrhage by keeping the TLC < 20%. Nonetheless, the 20%

TLC is actually the residual volume (RV) of the end of maximal expiration [41], and so

maintaining it during US exposure will not be an easy task.

This study also predicts that if diagnostic US imaging for humans is kept above 1.55 MHz,

the US induced lung haemorrhage will not occur when MI < 1.9 (Fig. 4). As the US

frequencies for diagnostic imaging of abdominal and thoracic regions are > 2.5 MHz while

for pneumothorax evaluation is > 2 MHz [42, 43], the risk of US induced lung haemorrhage

should be non-existent for human during medical US imaging. There is no evidence of US

induced lung haemorrhage occurring in human, mainly due to the lack of experimental studies

on humans. Only a single study reports that US induced lung haemorrhage do not occur in

humans at 3.5 MHz minimum frequency while maximum pressure is 2.4 MPa [44].

The comparison tests show that the circular membrane model of the alveolar resonance to be

acceptable with 95% probability. The model is found to be valid and it describes the US

induced lung haemorrhage adequately. Therefore, the alveolar resonance should be seriously

considered as a mechanism of the US induced lung haemorrhage based on the theoretical

validation of this circular membrane model.

Experiments that can test the invalidity of this theoretical study should be conducted in future.

If invalidation test results are negative, then it will strengthen the theory of alveolar resonance

as the mechanism of US induced lung haemorrhage and that the circular membrane model of

alveolar wall facet adequately describes it. If the invalidation test results are positive, then the

circular membrane model is to be deemed not suitable to study the alveolar resonance

mechanism or the alveolar resonance is an unlikely mechanism of the US induced lung

haemorrhage.

Several assumptions and approximations are made during modelling process; hence their

usefulness and limitations should be discussed. The accuracy of this study and its results are

limited by those assumptions and approximations. The assumption of the alveolar wall being

linearly elastic had simplified the derivations of the equations in this study. Nonlinear

elasticity treatment would have been highly accurate but very complex. Nevertheless, at small

deformations, the linear elasticity is applicable with certain reduced accuracy. The alveolar

wall is assumed isotropic; however it is too small for direct stress-strain measurement to be

verified [17].

The assumption of the alveolar wall facet undergoing linear vibration has also simplified the

derivations of the equations in this study. Throughout the study, the deflections can be seen

smaller than the alveolar dimension (radius) by analyzing the E.23. Thus it is acceptable to

apply linear vibration for simplicity. The alveolar wall is further assumed as membranes

instead as plates. As membranes, it is allowed to have deflection larger than its thickness

which will produce enough yield areal strain for alveolar wall damage. This do not occur in

thin plates as deflections must be smaller than its thickness.

The assumption of the alveolar facet having fixed boundary narrows down the scope of this

study by ignoring other vibrating states such as un-clamped or simply-supported. The facet of

the alveolar wall is also assumed as circular. The simplification of the geometrical shape

allows simplified derivation of equations throughout this study. The alveolar wall facets are

actually shaped as irregular pentagon, hence the calculations and results in this study should

differ slightly from reality.

APPENDICES

Appendix A: Stiffness calculation for human alveolar wall

The calculation utilizes this equation of membrane/plate stiffness;

)1(12/ 23 uEdDe −=

The stiffness of the membrane or plate depends on the Young’s modulus (E),

thickness (d) and Poisson ratio (u). The human alveolar wall minimum thickness is 10

x 10-6 m (Table 1).

The Young’s modulus value of the alveolar wall can be determined from its density, ρ

= 1026, 1068 kg / m3 [25]; its longitudinal wave speed, cL = 1490, 1610[25]; and

its Poisson’s ratio, u = 0.35, 0.45[37].

Pa

ucE L

9

2

2

10x2.49 0.68,

)0.45 0.35,21)(1610 ,10681490 1026,(3

)21)((3

=

−=

−= ρ

Thus the alveolar wall stiffness is;

Nm26.01x10 6.46,

)0.45) 0.35,(1(12)1010)(10x2.49 (0.68,

)1(12/

6-

2

369

23

=

−=

−=

−x

uEdDe

Appendix B: The elastic tension in membranes

Assume that an arbitrary piece of membrane which at zero external stress has an area,

A0, is stretched to a size A1 > A0.

The energy change at lowest form is;

02

01 /)(½ AAAkU stretchstretch −= ,

where the modulus, kstretch is the proportionality constant between a quadratic

deviation of the area from its unstressed state and the respective energy. The

additional A0-1 is a physics convention [45].

The tension, under which the membrane is subjected to, is the derivative of energy

with respect to area:

ε

ε

Ed

k

AAAk

AUT

stretch

stretch

stretch

=

=

−=

∂∂=

001

1

/)(

/

The defined dimensionless areal strain is, 001 /)( AAA −=ε . The constant of

proportionality is, Edk stretch = , where E is Young’s modulus and d is membrane

thickness. The membrane tension equation follows the Hooke’s Law for membrane

stretching, where stress is proportional to strain and to the constant of proportionality

kstretch.

Appendix C: The Qmedium of human alveolar wall

The Q-factor of air medium damping is;

Ω= /σωomediumQ

with Ω as the acoustic impendence of air = 420 Pa. s. m [23]. The density of the

alveolar wall is 1026, 1068 kg / m3 [25] while the human alveolar wall minimum

thickness is 10 x 10-6 m (Table 1).

Thus at 1 MHz frequency, the Q-factor of air medium damping for human alveolar

wall is;

1663 1527,420/)10x10)(1068 1026,)(10x1(2/ 66 ==Ω= −πσωomediumQ

Appendix D: The Qsupport of circular vibrating membrane

The Q-factor of support loss for the circular membrane with r as the radius, d as the

thickness of the membrane and J as support loss coefficient is as according to Park

[23]

3)/( drJQsupport =

In a loss factor study of eccentric loading of circular plate, Gupta & Nigam [46] gives

a loss factor value of 50.91 x 105 when loading is near the circular edge (0.96r) at

fundamental mode for a circular plate which radius and thickness is 0.6 m and 0.003

m respectively. This loading about the edge is assumed to deter the edge from

moving, hence representing the clamped resonator state. Thus;

64.0

)003.0/6.0(10x91.50

)/(

35

3

=

=

=

J

J

drJQsupport

When the support is as thin as the resonator, the cubic term degenerates to linearity

[47]. This is because the support is not rigid and isolated anymore against in-plane

bending of the resonator [48]. Thus;

)/(64.0)/( drdrJQ support ==

Appendix E: The areal strain in alveolus

The deformation in a system produces strain. The strain is measured by the ratio of

dimensional change in the system. The linear strain that quantizes the longitudinal

changes is;

001 /)( LLL −=α

The areal strain that quantizes the surface distortions is;

001 /)( AAA −=ε

The initial length and area are L0 and A0 respectively while the length and area after

deformation are L1 and A1 respectively. The relation between areal strain and linear

strain can be obtained starting with the stretch ratio equation. The stretch ratio is [17];

½0101 )/()/( AALL ==λ

Thus the relationship between linear strain and areal strain can be derived as;

1)1(1

)1(

)/(

½

½

½01

−+=−

+=

=

ελ

ελ

λ AA

1)1( ½ −+= εα

During normal tidal breathing, the linear strain changes [13] between 0 < α < 0.05

and so the areal strain changes;

1025.00

05.0]1)1[(0

05.00

½

≤≤

≤−+≤

≤≤

ε

ε

α

The wounding and breaks in cells in the plasma membrane of the alveolar epithelial

cells occurs at about a linear strain of α = 0.08 [14, 15]. Thus;

1664.0

08.0]1)1[(

08.0

½

=

=−+

=

ε

ε

α

Appendix F: Introduction to interval arithmetic

In order to have complete confidence in numerical calculations, interval arithmetic is

used to analyze the range of possible values changes throughout the calculation. If x =

a, b with x ranging between a and b while y = c, d with y ranging between c and

d, then [49];

, dbcayx ++=+

, cbdayx −−=+−

),,,max(),,,,min( bdbcadacbdbcadacyx =+

)/,/,/,/max(),/,/,/,/min( dbcbdacadbcbdacayx =+

DECLARATIONS

Competing interests: None declared.

Funding: This work was supported in part by the USM PGRS Grant and the UniKL

Financial Assistance Scheme.

Ethical approval: None declared.

Guarantor: DJJ.

Contributorship: DJJ was involved in the study concept and design, literature

search, theoretical modeling and analysis, manuscript preparation and editing. MSJ

played the supervisory role in this study

Acknowledgements:. D. John Jabaraj give praises to his Lord and Redeemer, the One

and Only, Jesus Christ.

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Table 1: The alveolar dimensions for various adult mammals and humans.

Alveolar Property

Adult Mouse

Adult Rat

Adult Rabbit

Adult Dog

Adult Pig

Adult Cat

Adult Monkey

Adult Human

Ratio of mean alveolar

diameter[28] 1 ≈1 ≈2 ≈2 ≈2.3 ≈2.5 ≈2.5 ≈5

Alveolar radius range

(µm)[29] 20, 40 20, 40 40, 80 40, 80 46, 92 50, 100 50, 100 100, 200

Minimum alveolar wall

thickness (µm)[30-36]

3 4 4 4 4.5 4 5 10

Table 2: The formulas of fundamental frequency and threshold pressure of the circular membrane model of alveolar resonance for various adult mammals and human.

Mammalian species

Fundamental frequency (MHz)

Threshold pressure (MPa)

Adult Mouse ½28.639 7.651, bε ½212 )1664.0(10x349.0 b1f ε−−

Adult Rat ½28.639 7.651, bε ½212 )1664.0(10x621.0 b1f ε−−

Adult Rabbit ½14.319 3.826, bε ½212 )1664.0(10x621.0 b1f ε−−

Adult Dog ½14.319 3.826, bε ½212 )1664.0(10x621.0 b1f ε−−

Adult Pig ½12.452 3.327, bε ½212 )1664.0(10x786.0 b1f ε−−

Adult Cat ½11.455 3.060, bε ½212 )1664.0(10x621.0 b1f ε−−

Adult Monkey ½11.455 3.060, bε ½212 )1664.0(10x970.0 b1f ε−−

Adult Human ½5.728 1.530, bε ½212 )1664.0(10x880.3 b1f ε−−

Table 3: Selected experimental data of lowest threshold pressure of the US induced lung haemorrhage for adult mouse and rat at certain frequencies (from data compiled by Church & O’Brien [39]).

Threshold Pressure (MPa) Frequency (MHz) Adult Mouse 3.6 2.8

3.0 5.6 1.4 3.7 0.4 1.1 0.7 1.2 0.6 2.3 1.3 3.5 1.0 3.7 1.1 1.2 0.7 2.3 0.8 2.3

Adult Rat 2.8 2.3 5.6 2.8 2.8 3.5 5.6 3.4 4.0 2.0 2.8 2.0

Table 4: Calculated threshold pressure at 1 MHz frequency for various adult mammals and human according to the circular membrane model of alveolar resonance.

Mammalian species Threshold pressure (MPa) Adult Mouse 0.135

Adult Rat 0.240 Adult Rabbit 0.194 Adult Dog 0.194Adult Pig 0.217 Adult Cat 0.157

Adult Monkey 0.245 Adult Human 0.981

Figure Legends

Fig. 1: a) circular membrane under deformation at fundamental frequency mode. b) spherical cap as an approximation. Fig. 2: The threshold pressure vs. fundamental frequency curves of the circular membrane model of adult mouse alveolar wall together with the plotted past experimental data of ultrasound induced lung haemorrhage in the adult mouse. An experimental data point (2.8, 3.6) is clearly an outlier and hence is omitted in the chi-square calculation. Chi-square (χ2) of data vs. circular membrane model is calculated as 7.89, while the critical χ2 (0.05 sig.) is 16.919. Fig. 3: The threshold pressure vs. fundamental frequency curves of the circular membrane model of adult rat alveolar wall together with the plotted past experimental data of ultrasound induced lung haemorrhage in the adult rat. Chi-square (χ2) of data vs. circular membrane models is calculated as 7.09, while the critical χ2 (0.05 sig.) is 11.070. Fig. 4: The threshold pressure vs. fundamental frequency curve of the circular membrane model of adult human alveolar wall together with the plotted MI = 1.9 limit line.