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Computer Methods in Biomechanics and Biomedical

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Finite element studies of the mechanical behaviour of

the diaphragm in normal and pathological cases
a b b b c b d d
M. P.M. Pato , N. J.G. Santos , P. Areias , E. B. Pires , M. de Carvalho , S. Pinto
e
& D. S. Lopes
a
Instituto Superior de Engenharia de Lisboa, Rua Conselheiro Emdio Navarro, Lisbon,
Portugal
b
ICIST/Instituto Superior Tcnico, Technical University of Lisbon, Av. Rovisco Pais,
1049-001, Lisbon, Portugal
c
Universidade de vora, Largo dos Colegiais, 2, 7004-516, vora, Portugal
d
Neuromuscular Unit, Instituto de Medicina Molecular, University of Lisbon, Av. Professor
Egas Moniz, 1649-028, Lisbon, Portugal
e
IDMEC/Instituto Superior Tcnico, Technical University of Lisbon, Av. Rovisco Pais,
1049-001, Lisbon, Portugal

Available online: 15 Nov 2010

To cite this article: M. P.M. Pato, N. J.G. Santos, P. Areias, E. B. Pires, M. de Carvalho, S. Pinto & D. S. Lopes (2011): Finite
element studies of the mechanical behaviour of the diaphragm in normal and pathological cases, Computer Methods in
Biomechanics and Biomedical Engineering, 14:06, 505-513

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 Computer Methods in Biomechanics and Biomedical Engineering
Vol. 14, No. 6, June 2011, 505513

Finite element studies of the mechanical behaviour of the diaphragm in normal

and pathological cases
M.P.M. Patoa,b*, N.J.G. Santosb, P. Areiasb,c, E.B. Piresb, M. de Carvalhod, S. Pintod and D.S. Lopese
a
Instituto Superior de Engenharia de Lisboa, Rua Conselheiro Emdio Navarro, Lisbon, Portugal
b
ICIST/Instituto Superior Tecnico, Technical University of Lisbon, Av. Rovisco Pais, 1049-001 Lisbon, Portugal; cUniversidade de
Evora, Largo dos Colegiais, 2, 7004-516 Evora, Portugal; dNeuromuscular Unit, Instituto de Medicina Molecular, University of Lisbon,
Av. Professor Egas Moniz, 1649-028 Lisbon, Portugal; eIDMEC/Instituto Superior Tecnico, Technical University of Lisbon, Av. Rovisco
Pais, 1049-001 Lisbon, Portugal
Downloaded by [b-on: Biblioteca do conhecimento online UEvora] at 03:58 24 January 2012

(Received 2 January 2010; final version received 5 April 2010)

The diaphragm is a muscular membrane separating the abdominal and thoracic cavities, and its motion is directly linked to
respiration. In this study, using data from a 59-year-old female cadaver obtained from the Visible Human Project, the
diaphragm is reconstructed and, from the corresponding solid object, a shell finite element mesh is generated and used in
several analyses performed with the ABAQUS 6.7 software. These analyses consider the direction of the muscle fibres and
the incompressibility of the tissue. The constitutive model for the isotropic strain energy as well as the passive and active
strain energy stored in the fibres is adapted from Humphreys model for cardiac muscles. Furthermore, numerical results for
the diaphragmatic floor under pressure and active contraction in normal and pathological cases are presented.
Keywords: diaphragm; active behaviour; amyotrophic lateral sclerosis; right phrenic nerve lesion; shell finite elements

1. Introduction diaphragm are driven independently and form a two-sided

The diaphragm is the major muscle of respiration and muscle sheet. The tension within the diaphragmatic muscle
separates the thoracic and abdominal cavities. It is a thin fibres during contraction generates a caudal force on the
and flat modified half-dome of musculofibrous tissue that central tendon that descends in order to expand the thoracic
originates from the lower six ribs bilaterally, the posterior cavity along its craniocaudal axis. In addition, the costal
xiphoid process as well as the external and internal arcuate diaphragm fibres apply a force on the lower six ribs, which
ligaments. A number of different structures cross the lifts and rotates them outward (de Troyer et al. 1997).
diaphragm, but three distinct apertures allow the passage The body relies on the diaphragm for normal respiratory
of the aorta, oesophagus and vena cava. The aortic function. Contraction of the diaphragm has the following
aperture is the lowest and most posterior of the openings, functions: (1) decreasing intrapleural pressure, (2) expand-
lying at the level of the 12th thoracic vertebra. ing the rib cage through its region of apposition by
The oesophageal aperture is surrounded by diaphragmatic generating positive intra-abdominal pressure and
muscle and lies at the level of the tenth thoracic vertebra. (3) expanding the rib cage using the abdomen as a fulcrum.
The vena cava aperture is the highest of the three openings A respiratory dysfunction is observed when a decrease
and lies at the level of the disc space between the eighth in the diaphragmatic function occurs. The body possesses
and ninth thoracic vertebrae (Harrison 2005). inherent mechanisms of compensation for decreased
The respiratory muscles are skeletal muscles. diaphragmatic function, but none of the processes can
The inspiratory muscles include the diaphragm, external successfully prevent respiratory compromise if the
intercostal, parasternal, sternomastoid and scalene diaphragm excursion is diminished.
muscles. The expiratory muscles include the internal The diagnosis of the decreased function of diaphragm
intercostal, rectus abdominis, external and internal oblique typically consists of observation of both neurological and
and transverse abdominis muscles. During inspiration, the anatomical processes. Neurologic problems of the
diaphragm contracts, while expiration is generally a diaphragm occur when a traumatic injury or disease
passive process (Ratnovsky et al. 2008). process decreases or terminates the impulse of respiratory
The muscle fibres of the diaphragm radiate from the stimuli originating from the brainstem. Neurologic and
central tendon to either the three lumbar vertebral bodies other disorders decrease the integrity of the musculature of
(i.e. crural diaphragm) or to the inner surfaces of the lower the diaphragm, thus decreasing its excursion. These
six ribs (i.e. costal diaphragm). The costal fibres of the problems ultimately result in the inability of the

*Corresponding author. Email: mpato@civil.ist.utl.pt

ISSN 1025-5842 print/ISSN 1476-8259 online
q 2011 Taylor & Francis
DOI: 10.1080/10255842.2010.483683
http://www.informaworld.com
 506 M.P.M. Pato et al.

diaphragm to provide adequate negative intra-thoracic (1)

pressure, thereby decreasing the amount of oxygen
provided to the alveoli.
For instance, the amyotrophic lateral sclerosis (ALS) is
a neurological disease that causes respiratory failure and
death. This is a rapid progressive condition, which leads (3)
to severe weakness and disability, associated with a poor
quality of life. When the diaphragm fails, the patients (2)
manifest respiratory symptoms and eventually lose the
ability to breath without ventilatory support. The mean
survival after the onset of the disease is 3 5 years.
However, about 10% of the ALS patients survive for
(5)
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10 years or more (de Mamede 2002).

There are other causes of diaphragm paralysis, namely,
(4)
those derived from phrenic nerve injury. The central
nervous system and the diaphragm are connected by Figure 1. Five sets of curves identifying the diaphragm:
the phrenic nerve. Each half of the diaphragm is innervated (1) sagittal; (2) and (3) axial apposition region; (4) and (5) axial
by the left and right phrenic nerves. When one of the view of pillars inserted in the vertebral column.
phrenic nerves is injured as a consequence of a surgical
intervention or an inflammatory process, the correspond- resolution of 24 bits, are available in the Visible Human
ing hemidiaphragm paralyses. Several studies have been Project. The 3D reconstruction was performed as follows:
performed to obtain the respiratory consequence encoun- (1) segmentation of axial and sagittal images, delimiting
tered in the unilateral paralysis of the diaphragm. In one of the outside border with cubic splines obtaining a contour
these studies (Laroche et al. 1998), 11 patients experien- network (Figure 1) and (2) interpolation of neighbouring
cing unilateral diaphragm paralysis were monitored. curves with non-uniform rational basis spline (NURBS)
Increase in breathlessness and decrease in exercise surfaces. These steps were performed with the
tolerance were reported, but the patients could breathe Rhinocerosw software (http://www.rhino3d.com). From
by themselves and the need of auxiliary ventilation was the segmented images, five sets of curves were created: the
not referred.
sagittal curves represent almost the totality of the anterior,
In this work, a previously developed computational
posterior and superior parts (set 1 in Figure 1); the two
model (dAulignac et al. 2005; Martins et al. 2006, 2007)
lateral sets of curves (sets 2 and 3 in Figure 1) form the
for the passive and active contraction of skeletal muscles
more vertical part, called apposition region, while the
is considered with the objective of studying the behaviour
other two axial sets of segments represent (sets 4 and 5 in
of the diaphragm during the respiratory cycle in normal
Figure 1) the pillars of the diaphragm. The interpolated
and pathological cases. In the next section, a geometrical
surfaces were discretised in triangular meshes. Mesh
model of the diaphragm is presented. In Section 3, the
imperfections, e.g. holes and incoherent C0 and C1
constitutive equations are described. In Section 4, the
continuity unification between sets, were corrected with
computational model is applied to the reconstructed
the Blender software (http://www.blender.org/) (Figure 2)
surface of the diaphragm and the corresponding
numerical results are presented. Finally, in Section 5,
some conclusions are drawn and future developments are
listed.

2. Geometrical model
Owing to its anatomical complexity and dynamic mobility,
the construction of a geometrical model of the diaphragm
is a difficult task when relying on standard medical
imaging. Here, data from the Visible Female Project
(Ackerman 1998; Spitzer et al. 1996) are used to
reconstruct the outer surface of the diaphragm. The
images are real-colour cryosections containing a para-
mount of anatomical information. A total of 905 axial
images of the thoracic and abdominal body segments, with Figure 2. Triangular surface of the diaphragm obtained with
0.33 0.33 0.33 mm3 voxel dimensions and a colour Blender.
 Computer Methods in Biomechanics and Biomedical Engineering 507

PC
muscle fibres:
Z
X s sincomp smatrix sfibre : 1
Y

The first contribution, in the case of perfect incompres-

sibility, has the form:

sincomp 2pI; 2

M where p is the hydrostatic pressure and I the second-order

Z
X identity tensor.
Y
The matrix contribution is assumed to be hyperelastic
and isotropic. This term has the same exponential form
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T adopted by Humphrey and Yin (1987), but with material

Z parameters obtained in some compression experiments
X
Y with skeletal muscles available in the literature (Grieve and
Armstrong 1988), which can be given by:
Figure 3. 3D solid object of the diaphragm obtained from 

ABAQUS: PC, phrenic centre; M, muscle; T, tendon. smatrix 2bc exp b I C1 2 3 devB
; 3

where I C1 is the first invariant of C, the right Cauchy

(Lopes et al. 2008). This triangular surface provided a Green strain tensor, B is the left Cauchy Green strain
discrete representation of the geometric locus of the entire tensor and b and c are the constants. The deviatoric
diaphragm floor. operator in the current configuration can be expressed by:
Further modelling of the muscular and tendinous parts
1
was performed with the ScanTo3D toolboxw (Figure 3), dev
2 trI;
thus building a 3D solid model (NURBS patches). This 3
object was finally imported to ABAQUSw 6.7 (see Santos with [ Lin. The operator dev[] is a projection linear
2009) to generate a finite element (FE) mesh. Manual transformation.
segmentation and 3D model reconstruction were always Much of the research on the contraction of skeletal
performed under the supervision of skilled anatomists. muscles has focused on their 1D behaviour along the
fibres. One of the most well known of the proposed models
is Hills muscle model (Hill 1938; Pandy et al. 1990;
3. Mechanical model Fung 1993), shown in Figure 4.
The diaphragm is mainly formed by two types of tissues: The stress contribution of the muscle fibres has the form:
tendon and muscle. Both the phrenic centre on the top and
sfibre devlf Tn ^ n
; 4
the tissue connected to the bones are composed of tendon.
The remaining part of the diaphragm is formed by muscle. where lf is the stretch ratio in the muscle fibres that have the
Accordingly, three main regions are created: the phrenic direction of the unit vector N in the undeformed
centre, the muscle and the tendon region in the inferior configuration. As incompressibility is assumed, lf is given by
border of the diaphragm, shortly referred to as the inferior
tendon (see Figure 3). These three regions are identified in p
lf N T CN:
the model by comparing it with the images provided in
Netter (http://www.netterimages.com/image/atlas.htm) The scalar T represents the nominal stress in the fibre and is
(see Santos 2009 for details). the sum of the stresses T PE and T SE in the parallel and series
The constitutive equation for 3D skeletal muscles
adopted in the present work incorporates both passive and
active behaviours. Following previous works (dAulignac
SE CE
et al. 2005; Martins et al. 2006, 2007), a modified form of
the incompressible transversely isotropic hyperelastic T T
model proposed by Humphrey and Yin (1987) for passive
PE
cardiac muscle has been considered.
The constitutive equation for the Cauchy stress is the Figure 4. Hills three-element muscle model: SE, series
sum of an incompressible term, an assumed embedding (elastic) element; CE, contractile element; PE, parallel (elastic)
matrix term and a stress contribution term from the element; T, nominal stress.
 508 M.P.M. Pato et al.

elements, respectively, variable a [ [amin, 1], with amin $ 0:

T T PE T SE : 5 T CE l CE ; l_ CE ; a T M CE CE CE _ CE
0 f L l f V l a; 11

The stress T CE in the contractile element satisfies: where

8
T CE T SE : >
2
l CE 2 l CE 0:25; 0:5 # l CE , 0:75
>
>
>
>
The unit vector, n, in the current direction of the muscle fibre < 2l CE2 2l CE 20:875; 0:75 # l CE , 1:25
is defined by f CE
L l
CE
8 2 ;
>
> l CE 23l CE 2:25; 1:25 # l CE , 1:5
>
>
>
: 0;
FN otherwise
n ; 6
lf 12
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where F is the deformation gradient. 8 CE

>
> l_ 101; l_ CE # 210s21
Because of the large deformations involved, the fibre >
< arctan20:5l_ CE
stretch lf is decomposed into a multiplicative split of the f CE _ CE 2 arctan5 1; 210s21 , l_ CE , 2s21
V l
contractile stretch followed by an elastic stretch: >
>
>
: l_ CE 221 4arctan5
p
1; l_ CE $ 2s21
lf l CE l SE ;
13
which corresponds to an additive Hencky strain
and 1 is a sufficiently small scalar introduced to avoid
decomposition. _ CE # 10 s21 .
f CE
V 0 when l
The stresses T PE and T SE are highly non-linear
The time-dependent activation process involves the
functions of their elongations and essentially vanish in
contractile element and is caused by neural excitation.
compression. They are strictly positive, i.e. the fibres can
At the macroscopic level, it is described by the first-order
only work in tension. The stress T PE is the product of the
ordinary differential equation (ODE) proposed by Pandy
maximum tensile stress produced by the muscle at resting
et al. (1990):
length, T M
0 , and a function of the stretch ratio:
1
T PE lf T M
0 f
PE
lf ; 7 a_ u; t 1 2 atut
trise
where 1
amin 2 at1 2 ut: 14
tfall
(  
2aA exp alf 2 12 lf 2 1; lf . 1
f PE lf In Equation (14), trise and tfall are the characteristic
0; otherwise time constants for the activation and deactivation of the
8 muscle and amin is the minimum value of activation.
The function u(t) ranges from 0 to 1, represents the
and a and A are constants. The stress T SE is given by an neural excitation and is a part of the models input
analogous decomposition: data.
The approach followed in the present study assumes
T SE lf ; l CE T M
0 f
SE
lf ; l CE ; 9 that the strain energy is stored isotropically in the material
as well as in the direction of the muscle fibres and is given
with by

 1  
 
f SE lf ; l CE exp 100 lf 2 l CE 2 1 ; U UC; l CE
10 10

U matrix I C1 U PE lf U SE l SE ; l CE
lf $ l CE : |{z} |{z}
UI Uf
Both expressions (8) and (10) were obtained from the work
carried out by Pandy et al. (1990). where each of these energy terms originates the stress
The stress T CE is given by the product of (1) a function contributions (3) and (4), respectively.
of the contractile stretch, lCE, with maximum value at the The Cauchy stress tensor can also be written as:
muscle rest length, (2) a function of the strain rate of the

contractile stretch, lCE, which, for contracting rates, U T

s 2pI 2 dev F F : 15
corresponds to Hills hyperbolic law and (3) an activation C
 Computer Methods in Biomechanics and Biomedical Engineering 509

As the diaphragm is very thin (3 5 mm), the above- The simulations performed are based on the constitutive
mentioned 3D model is modified to agree with the thin Equation (17).
shell theory (dAulignac et al. 2005; Martins et al. 2006, Owing to the strict monotonicity and invertibility in R
CE
2007). of f CE
V , the time rate of change of the internal variable l
In the case of a shell without shear deformation, the is governed by the first-order ODE:
deformation gradient F is given by

 21

2 3 l_ CE lf ; l CE ; a; l_ f ; u f CE
V lf ; l CE ; a; l_ f ; u : 18
F 11 F 12 0 " #
6 7 Fp 0
F6 F
4 21
F 22 0 7
5 : To solve the two ODEs (14) and (18), the backward-Euler
0 F 33
0 0 F 33 scheme is used. The tangent modulus consists of the
derivative of sp with respect to the strain rate.
As we assume perfect incompressibility, J det F 1,
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hence:
4. Application to the diaphragmatic floor
21 4.1 Input data
F 33 det F p :
The diaphragm was modelled using triangular S3 shell
Therefore, the Cauchy Green strain tensors are given by elements and the muscle behaviour was incorporated in
2 3 2 3 ABAQUS by implementing a UMAT subroutine.
F p FTp 0 FTp F p 0 The tendinous part of the diaphragm was considered
6 |{z} 7 6 |{z} 7
6 7 linear, elastic and isotropic with values for Youngs modulus
B6
4 p
B 7;
5 C 6 Cp 7:
4 5 and Poissons coefficient of 33 MPa and 0.33, respectively
0 F 233 0 F 233 (Behr et al. 2006). The material parameters used for the
muscle were chosen from Humphrey and Yin (1987):
The first invariant can be calculated as: b 23.46; c 3.79517355 1024 MPa; A 8.73206
1024; a 12.43 and T M 0 6:5586872 10
21
MPa. The
IC1 trC p C 33 values for the constants in Equation (14) were trise
20 1023 s; tfall 20 10 3 s and amin 0.01. The value
The stretch ratio in the reference configuration can now be for the parameter 1 in definition (13) was 1 1 10 4.
defined as A constant excitation with value umax was applied
q during inspiration for a period of time of 2 s. After this
lf NTp C p N p period of time, the neural excitation was set to zero during
the remaining 3 s of the complete respiratory cycle.
The pressure profile in the surface of the diaphragm
and the muscle fibre direction in the deformed
was difficult to obtain. Instead, the pressure between the
configuration is given by
inside and outside of the diaphragmatic surfaces, called the
FpNp transdiaphragmatic pressure (Pdi), was used. The Pdi
np ; during normal and quiet breathing had a value of about
lf
5 cm H2O (490.333 MPa; Tobin et al. 2002). The pressure
where the subscript p in Np and np concerns the in-plane profile along the surface was not uniform. For simplifica-
(1,2) directions. tion, only two regions (apposition and diaphragmatic
To obtain p, the conditions si3 0, i [ {1,2,3} must regions) were considered (Figure 5), and a constant
be introduced, where the direction 3 of the orthonormal pressure was applied in their outer surfaces. During
reference frame (1,2,3) follows the normal to the middle inspiration, pressures of 4.9033 10 4 MPa and
surface of the shell. The plane stress condition s33 0 2 4.9033 10 4 MPa were applied to the diaphragmatic
imposes a particular form for the pressure. From (15), we and apposition zones, respectively. During expiration,
obtain: opposite pressures were applied to the same regions.
In the present study, all nodes at the inferior borders of
1 0  the model were considered fixed in displacement. The
p 2U I trB p
2 2C33 lf U 0f n p ^n p ; 16
3 vena cava and oesophageal apertures as well as the aortic
opening were kept fixed along the sagittal and coronal
with U 0I U I =I C1 and U 0f U f =lf . axes, but the axial displacement of their borders was kept
Therefore, the Cauchy stress in the plane is given by free. The boundary was free to rotate with the exception of
border regions, A and L, of the inferior border (see
sp 2U 0I B p 2 B33 I p lf U 0f n p ^ n p : 17 Figure 6). In Region A (anterior), the nodes could not
 510 M.P.M. Pato et al.

X The initial conditions used in (14) and (18) were

a(0) 0.01, lf(0) 1.0 and l CE(0) 0.982082.
Y Z
The initial directions of the muscle fibres were
determined according to a radial pattern (see Figure 7).
In fact, the direction of the majority of the fibres was radial,
X starting at the phrenic centre and ending at the inferior
Y Z borders of the diaphragm (for more details, see Santos 2009).

4.2 Numerical results

First FE simulations of the diaphragm were performed
A considering different values for the threshold, umax, in the
X
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D neural excitation u. The results obtained for the tidal

Y Z
volume (TV) as well as for the maximum axial
Figure 5. Apposition (A) and diaphragmatic (D) regions in the displacement (MAD) of the diaphragm are shown in
geometric model. Figure 8. The values given in the literature (Segars et al.
2001; Gaillard 2003) and the results obtained suggest that
rotate about the x-axis, and in Region L (lateral left and the best interval for umax, to characterise the excitation in a
right), the nodes could not rotate about the y-axis. These normal quiet breathing, is between 20 and 30%.
boundary conditions did not change with time during the The simulation of a normal quiet breathing was
respiratory cycle. performed during a period of 5 s and with umax 0.3.
The evolution of the volume and MAD of the top of the
diaphragm is shown in Figure 9. The volume of the
diaphragm under analysis was calculated as the difference
between two volumes: one obtained from its initial shape
and the other from its deformed configuration in a later
phase of respiration. The TV was approximately 564 cm3,
while the MAD was about 10.6 mm.
During inspiration, the diaphragm contracted, the domes
of the diaphragm descended and its apposition as well as its
anterior and posterior zones shrunk. The decrease in the
height of the domes was larger in the right dome, where
the MAD was obtained. The horizontal displacements of the
lateral sides of the diaphragm were two to three times smaller
than the axial displacement of the domes. In fact, the latter
was approximately two to three times greater than the
Figure 6. Boundary Conditions: A-anterior, ALR-anterior former. The posterior zone of the diaphragm demonstrated a
left/right, PLR-posterior left/right, L-lateral, OA-oesophageal smaller contraction when compared with the displacement of
aperture, AO-aortic opening and Pi-pillars. the apposition and anterior parts of the diaphragm. During
expiration, the diaphragm demonstrated a quasi-symmetric
Z behaviour relatively to the inspiration. The domes ascended
Y X
and the lateral sides moved to the outside due to the
relaxation of the muscle. Both the volume and MAD
evolutions presented four different regions during the
respiratory cycle. During inspiration, all the variations in
volume and displacement took place approximately at the
first second of the analysis. Likewise, during expiration, all
the variation occurred approximately during the first one and
a half seconds. During these periods of time, the evolution
showed a quasi-exponential behaviour. At the remaining
periods of time, the variation was inexistent.
Z Two pathological cases were analysed: a diaphragmatic
Y X
dysfunction due to an ALS disease and a complete right
phrenic nerve lesion (RPNL). The difference between these
Figure 7. Radial pattern of the direction of the fibres. two cases and the normal case was the inactivation of the
 Computer Methods in Biomechanics and Biomedical Engineering 511

14 800
MAD, mm

Maximum axial displacement, MAD (mm)

TV, cm3
12 700

600

Tidal volume, TV (cm3)

10
500
8
400
6
300
4
200

2 100
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0 0
0 0.2 0.4 0.6 0.8 1.0
Neuronal excitation, umax

Figure 8. MAD ( ) and TV ( ) at the end of the inspiration as a function of umax.

relevant fibres in the pathological diaphragms. In the normal muscle. However, these movements were smaller than those
case, all the fibres may be activated. In the ALS case, the in the normal diaphragm. In the RPNL simulation, during
inactivation demonstrated a random pattern with about half inspiration, only the left side of the diaphragm contracted,
of the fibres of the muscle with zero excitation. In the RPNL and hence, suffered a larger deflection than the right side. The
case, all the fibres in the right side of the muscle were paralysed right hemidiaphragm did not shrink as much as the
inactivated. Figure 9 also shows the variations of the volume left side. Furthermore, the domes descended less than the
and MAD of the diaphragm for these two pathological healthy case. During expiration, the movement of the
conditions. The evolutions of volume and displacement in diaphragm in both the pathological cases was symmetric
both the cases were similar and analogous to the relatively to that during inspiration, like that during normal
corresponding evolutions in the normal case. In the ALS breathing.
disease, during inspiration, both the domes of the diaphragm In the pathological cases, both TV and MAD decreased
descended axially and the lateral parts of the diaphragm significantly. In the ALS case, the TV was approximately
shrunk as a result of the contraction of the healthy fibres of the 306 cm3 and the maximum descent of the domes was about

12 600
11.5
11 (ALS) Disp., mm
10.5 (RPNL) Disp., mm
10 500
(Normal) Disp., mm
9.5
(ALS) Vol., mm3
Maximum axial displacement (mm)

9
8.5 (RPNL) Vol., mm3
8 (Normal) Vol., cm3 400
Volume variation (cm3)

7.5
7
6.5
6 300
5.5
5
4.5
4 200
3.5
3
2.5
2 100
1.5
1
0.5
0 0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Time (s)

Figure 9. Mad and volume variations in the normal and two pathological cases for umax 0.3.
 512 M.P.M. Pato et al.

U, U3
A B +7.781e01
6.179e01
2.014e+00
3.410e+00
4.806e+00
6.202e+00
7.598e+00
8.994e+00
1.039e+01
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Y Z
Step: Exp
X
Y Increment 0: Step Time = 0.000
Primary Var: U, U3
X Deformed Var: U Deformation Scale Factor: +5.000e+00

C D
U, U3
U, U3 +8.031e01
+3.976e01 1.148e02
3.369e01 8.260e01
1.071e+00 1.641e+00
1.806e+00 2.455e+00
2.541e+00 3.270e+00
3.275e+00 4.084e+00
4.010e+00 4.899e+00
4.744e+00 5.713e+00
5.479e+00

Z Step: Ins Z Step: Ins

Increment 37: Step Time = 1.000 Increment 32: Step Time = 1.000
Y Primary Var: U, U3 Y Primary Var: U, U3
Deformed Var: U Deformation Scale Factor: +5.000e+00 Deformed Var: U Deformation Scale Factor: +5.000e+00
X X

Figure 10. Vertical displacement of the diaphragm (U3): (A) before inspiration and at the end of inspiration, (B) in the normal case,
(C) in the ALS case and (D) in the RPNL case.

5.55 mm (54.3 and 52.6% of the value of the normal case, 5. Conclusions
respectively). In the RPNL case, the TV was about 196 cm3 During normal breathing (as well as in the pathological
and the descent was approximately 5.74 mm (34.8 and cases), periods of time with no variation in volume and MAD
54.4% of the value of the normal case, respectively). occur. This suggests that during a normal quiet breathing, the
The MAD occurred in the right dome in the ALS case as in time duration of the respiratory cycle is probably smaller
the normal case, but its location changed to the left dome in (,3 s) because the diaphragmatic muscle during inspiration
the RPNL case due to the inactivation of the right and the expiratory muscles during expiration require a
hemidiaphragm. smaller period of time to contract. However, the periods of
Figure 10 illustrates the movement of the diaphragm at time when there is no variation may represent periods for gas
the end of inspiration in all the three cases. Animations of exchange and muscle relaxation.
this movement as well as the maximum principal stresses In both the pathological cases, the behaviour and
during the respiratory cycle can be found in http://www. performance of the diaphragm presented differences during
civil.ist.utl.pt/mpato. respiration, although in the RPNL case, the differences were
 Computer Methods in Biomechanics and Biomedical Engineering 513

more significant. In both the cases, the TV was smaller than Gaillard L. 2003. Modele fonctionnel du diaphragme pour l
in the normal breathing case as expected. However, the TV of acquisition et le diagnostic en imagerie medicale
the ALS case was approximately 110 cm3 greater than that in [masters thesis]. [La Tronche (France)]: Universite Joseph
Fourier.
the RPNL case. It is known that the effect of the inactivation Grieve A, Armstrong C. 1988. Compressive properties of soft
of the fibres in one of the hemidiaphragms is not significant in tissues. In: Biomechanics XI-A, International series on
the performance of quiet breathing. This suggests that in the Biomechanics. Amsterdam: Free University Press, Vol. 53,
RPNL case, there is an active compensation from the other p. 1 536.
inspiratory muscles or even from a reinforcement of healthy Harrison G. 2005. The anatomy and physiology of the
fibres, which does not occur in the ALS cases. As a result, a diaphragm. In: Upper gastrointestinal surgery. London
(UK): Springer. p. 45 58.
value for umax greater than 0.3 should be considered in this Hill A. 1938. The heat of shortening and the dynamic constants of
case. muscle. Proc Roy Soc London. B129:136 195.
The diaphragm has a movement relative to the body as a Humphrey J, Yin F. 1987. On constitutive relations and finite
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consequence of the movement of the other inspiratory deformations of passive cardiac tissue: I. A pseudostrain-
muscles and ribs. As a future development, analyses should energy function. ASME J Biomech Eng. 109:298 304.
consider the borders of the diaphragm to displace during Laroche C, Mier A, Moxham J, Green M. 1998. Diaphragm
strength in patients with recent hemidiaphragm paralysis.
respiration. The values of the displacement of the inferior
Thorax. 43:170 174.
borders of the diaphragm may be determined from the Lopes D, Martins J, Pires E. 2008. Three-dimensional
medical imaging taken during quiet breathing or obtained reconstruction of biomechanical structures for finite element
from the literature. analysis. In: st Workshop on Computational Engineering:
Fluid Dynamics, Portugal-UT Austin CFD, Instituto Superior
Tecnico.
Martins J, Pato M, Pires E. 2006. A finite element model of
Acknowledgements
skeletal muscles. Virtual Phys Prototyping. 1:159 170.
The authors gratefully acknowledge the Fundacao para a Ciencia Martins J, Pato M, Pires E, Natal Jorge R, Parente M,
e a Tecnologia for the support granted in the context of POCI Mascarenhas T. 2007. Finite element studies of the
2010 and for the PhD grant SFRH/BD/47750/2008. deformation of the pelvic floor. Ann NY Acad Sci. 1101:
316 334.
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