Is the time-dependent behaviour of the aortic valve intrinsically quasilinear?
Afshin Anssari-Benam1,*
1
School of Engineering,
University of Portsmouth,
Anglesea Road,
Portsmouth
PO1 3DJ
United Kingdom
*
Address for correspondence:
Afshin Anssari-Benam,
School of Engineering,
University of Portsmouth,
Anglesea Road,
Portsmouth
PO1 3DJ
United Kingdom
Tel:
+44 (0)23 9284 2187
E-mail: afshin.anssari-benam@port.ac.uk
Word count (Introduction to conclusion): 2919
Abstract
The widely popular quasi-linear viscoelasticity (QLV) theory has been employed
extensively in the literature for characterising the time-dependent behaviour of many
biological tissues, including the aortic valve (AV). However, in contrast to other
tissues, application of QLV to AV data has been met with varying success, with studies
reporting discrepancies in the values of the associated quantified parameters for data
collected from different time-scales in experiments. Furthermore, some studies
investigating the stress-relaxation phenomenon in valvular tissues have suggested
discrete relaxation spectra, as an alternative to the continuous spectrum proposed by
the QLV. These indications put forward a more fundamental question: Is the timedependent behaviour of the aortic valve intrinsically quasi-linear? In other words, can
the inherent characteristics of the tissue that govern its biomechanical behaviour
facilitate a quasi-linear time-dependent behaviour? This paper attempts to address
these questions by presenting a mathematical analysis to derive the expressions for the
stress-relaxation G(t) and creep J(t) functions for the AV tissue within the QLV theory.
The principal inherent characteristic of the tissue is incorporated into the QLV
formulation in the form of the well-established gradual fibre recruitment model, and
the corresponding expressions for G(t) and J(t) are derived. The outcomes indicate that
the resulting stress-relaxation and creep functions do not appear to voluntarily follow
the observed experimental trends reported in previous studies. These results highlight
that the time-dependent behaviour of the AV may not be quasi-linear, and more
suitable theoretical criteria and models may be required to explain the phenomenon
based on tissue’s microstructure, and for more accurate estimation of the associated
material parameters. In general, these results may further be applicable to other planar
soft tissues of the same class, i.e. with the same representation for fibre recruitment
mechanism and discrete time-dependent spectra.
Keywords: time-dependent behaviour, aortic valve, quasi-linear viscoelasticity (QLV),
stress relaxation, creep, microstructure.
2
Is the time-dependent behaviour of the aortic valve intrinsically quasilinear?
1. Introduction
In a broad classification, the biomechanical behaviour of soft tissues can be
considered in two distinct categories; quasi-static and time-dependent behaviour.
While quasi-static loading may replicate the main functional mode of many connective
tissues more effectively, characterising time-dependent behaviour is also important,
particularly in tissues that undergo long-term repetitive dynamic loading in vivo, such
as the aortic valve (AV). Structural durability of the AV is generally thought to be
linked to its ability to dampen (relax) the transient stresses created by the sudden
change in pressure gradient at systole during each cardiac cycle (Sacks 2001; Robinson
et al. 2009). Moreover, structural failure of substitute bioprosthetic valves has been
hypothesised to be a consequence of their reduced stress-relaxation characteristics
compared to the native tissue (Sacks 2001).
Time-dependent spectra cannot be measured directly through experimentation alone
(Baumgaertel and Winter 1992), and models are subsequently required to compliment
the estimation of time-dependent parameters. With recent studies underlining the
contribution of the AV microstructure to the bulk tissue quasi-static behaviour (Billiar
and Sacks 2000; Sacks 2003; Stella and Sacks 2007), it is appropriate to model the
time-dependent behaviour of the valve using theoretical criteria and models that are
intrinsic to the tissue microstructure, and to the underlying microstructural mechanisms
facilitating such characteristics. This approach would lead to more accurate and robust
characterisation of AV time-dependent behaviour, with structurally meaningful
material parameters. Such data can subsequently contribute towards gaining better
insights into the in vivo biomechanical function of the valve.
Perhaps the most celebrated microstructural based models applied in studies of tissue
biomechanics were pioneered by Lanir (1979, 1983), addressing structure-function
relationships in fibrous connective tissues under quasi-static loading. These models
were based on the gradual recruitment of collagen fibres as load is applied at the tissue
3
level, and have been validated with experimental observations. Following the same
approach, similar models have been proposed to characterise the quasi-static
mechanical behaviour of AV tissue (Sacks 2003). However, microstructural based
models and relationships have been considerably less developed and addressed for
describing the time-dependent behaviour of the AV. Indeed, with studies reporting a
‘no-creep’ behaviour for the tissue under equi-biaxial loading conditions (Stella et al.
2007), and exhibiting both the stress-relaxation and creep phenomena when loaded
uniaxially (Anssari-Benam et al. 2011a), the structural mechanisms and origins of the
AV time-dependent behaviour are yet to be fully understood, and have been a subject
of active research (Stella et al. 2007; Sacks et al. 2009).
Nonetheless, a commonly used criterion in characterising time-dependent behaviour
in soft tissues is the well-established Quasi-Linear Viscoelasticity (QLV) theory,
successfully applied to a wide range of tissues such as ligament (Woo et al. 1993),
tendon (Sarver et al. 2003), and cardiac muscle (Pinto and Patitucci 1980). Under the
assumption that the viscoelastic behaviour of the subject tissue is quasi-linear, QLV
provides a powerful mathematical tool for estimating the time-dependent parameters of
tissues, assuming a continuous relaxation spectrum (Rajagopal and Wineman 2008).
However, experimental procedures to ensure an accurate characterisation of the timedependent behaviour of biological tissues by the QLV theory are not straight forward
in practice. For example, for QLV to accurately characterise the experimental
relaxation data, an ideal instantaneous step displacement should be applied in the
experiments, to ensure a complete separation of the elastic and time-dependent
responses (Doehring et al. 2004). However, instantaneous steps are experimentally
difficult to implement, and generally the loading phase is limited to a finite-time ramp,
during the application of which the sample begins to relax. While this may lead to
small errors in estimating the relaxation parameters in unidirectional tissues such as
tendons, in which the population of fibres are already mainly aligned in the principal
loading direction, planar tissues may deviate more notably from the assumption of
complete separation of the elastic and time-dependent responses, since fibre realignment occurs during the application of displacement to the tissue.
4
For a planar tissue such as the AV, application of QLV has been met with varying
success, with studies reporting discrepancies in the time-dependent parameters
calculated for different experimental time-scales, as a consequence of the relatively
poor fit of the model to the experimental data (Sauren et al. 1983; Rousseau et al.
1983; Sauren and Rousseau 1983). In a recent study it was shown that the timedependent behaviour of the AV, similar to other valvular tissues (Liao et al. 2007),
may be better characterised using discrete spectra than the continuous spectrum offered
by the QLV model (Anssari-Benam et al. 2011a). Such data indicate that the AV tissue
may not fundamentally facilitate quasi-linear viscoelastic behaviour as an intrinsic
feature inherent to the valve’s time-dependent characteristics, and put forward the
important question: Is the time-dependent behaviour of the aortic valve intrinsically
quasi-linear? From a biomechanical perspective, the answer to this question may prove
valuable, as it may further assist a better understanding of the AV time-dependent
behaviour, and highlight the need for development of alternative approaches in order to
obtain more accurate estimations of the associated material parameters.
One analytical way to address this question may be to incorporate the inherent
characteristics of the tissue which govern its biomechanical behaviour into the QLV
formulation, account for the associated mathematics, and analytically derive the
resulting expressions for stress-relaxation and creep functions. The ensuing functions
can then be compared against the experimental data, to investigate and verify the
extent to which they match the observed modes in stress-relaxation and creep tests.
This, in effect, is the crux of the current study. The principal inherent characteristic of
the AV tissue, which has been shown to govern its biomechanical behaviour, is a
feature that is known in the literature as ‘gradual fibre recruitment’ (Billiar and Sacks
2000; Sacks 2003). In the following, the associated well-established gradual fibre
recruitment model is incorporated into the QLV, accounting for the elastic response
 e ( ) in the model. The corresponding stress-relaxation G (t ) and creep J (t )
functions are then mathematically derived, and are compared with the experimental
results previously reported in the literature.
5
2. QLV criterion
The hereditary integral constitutive equation for viscoelastic response, under the
assumption that the response to a multi-step strain history can be approximated by a
linear combination of the responses R[ (t ), t ] to single strain histories, is (Pipkin and
Rogers 1968; Rajagopal and Wineman):
t
 (t )   d  R[ ( s ), t  s ]
(1)

where d  R[ , t ] denotes a differential with respect to the first argument of R[ , t ] .
With a strain history such that:
 (s)  0
for
s  (,0)
the stress in equation (1) can be expressed as (Pipkin and Rogers 1968; Rajagopal and
Wineman 2008):
R[ ( s), t  s] d ( s)

 ds
 ( s)
ds
0
t
 (t )  R[ (0), t ]  
(2)
The QLV model assumes that R[ , t ] can be expressed by (Fung 1993):
R[ , t ]   e ( )  G(t )
(3)
where G (t ) is a stress-relaxation function and  e ( ) is the immediate elastic response
(Fung 1993). It can thus be observed that QLV is a special case of the nonlinear
viscoelastic model in equation (1).
By substituting (3) into (2):
t
 (t )   ( (0))  G(t )   G(t  s)
e
0
d e ( ( s)) d ( s)

 ds
d ( s)
ds
(4)
or to simplify the notation for  (s ) :
d e ( ) d
 (t )   ( (0))  G(t )   G(t  s)

 ds
d
ds
0
t
e
6
(5)
The elastic response  e ( ) of the AV tissue has been formulated based on a
stochastic representation of the process of gradual recruitment of the fibres under
deformation by Sacks (2003), based on the general theoretical criterion developed by
Lanir (1979, 1983). In the following, this relationship is employed as the elastic
response  e ( ) of the AV tissue in equation (5), as a representation of the tissue
microstructure. It is then investigated that if the experimentally observed exponential
relaxation and creep modes reported previously in the literature can be mathematically
derived within the QLV criterion.
3. Application to stress-relaxation
For tissues consisting of distributed arrays of wavy collagen fibres, such as the AV,
fibres are generally modelled not to contribute to the load bearing capacity of the tissue
until they are fully straightened. Straightening occurs gradually across the population
of fibres in the tissue continuum, when load or displacement is applied at the tissue
level. Stochastic approaches have been widely popular for modelling this gradual
recruitment of fibres, with a statistical distribution function representing the stretch at
which each fibre becomes uncrimped. Based on these approaches, different distribution
functions including gamma (Sacks 2003; Bischoff 2006), beta (Raz and Lanir 2009)
and Weibull (Hurschler et al. 1997) have been used to model fibre recruitment. Here, a
gamma distribution function of the following form is utilised, consistent with other
related studies of AV biomechanics (Sacks 2003):
D( s ) 
 
1
 s 1 exp   s
 ( )
 




(6)
where  s is the straightening fibre strain. The physical interpretation of D( s ) is a
straightening strain density distribution function, representing the fraction of fibre fully
straightened at fibre’s current strain (Sacks 2003; Raz and Lanir 2009).
 D(
s
) d s is
the cumulative distribution function, which gives the density of the volume fraction of
the fibre population that is straight at  s (Sacks 2003).
7
A fibre can support load only when fully uncrimped, at which point it is considered
to be linearly elastic (Bischoff 2006; Raz and Lanir 2009). Thus, the fibre stress s fibre
is related to the fibre true strain,  t , by:
s fibre  K t
(7)
where K is the elastic modulus. By definition,  t represents the strain in a fibre with
respect to its uncrimped length. The relationship between  t ,  s and  is (Raz and
Lanir 2009):
t 
 s
1 2 s
(8)
Therefore, the overall Lagrangian elastic stress of the tissue is:

 e ( )   V  K 
0
 s
D( s )d s
1  2 s
(9)
where V is the fibre volume fraction (Aspden 1986). Substituting equation (6) into (9):

 s

1
 
 s 1 exp(  s ) d s
1  2 s  ( )

0
 ( )  K  V 
e
(10)
Now, substituting (10) into (5), and assuming that both stress and strain at 0 are equal
to zero, i.e.  (0)  0 ,  (0)  0 , and strain rate is constant upon applying the ramp, i.e.
d
 C  const , one will arrive at:
ds
t
 (t )   G(t  s)
0
d e ( )
 C  ds
d
(11)
From equation (10):


D( s )d s

d e
1
1
 K V 
 K V 
 s 1 exp(  s ) d s

d
1  2 s
1  2 s  ( )

0
0
(12)
Thus by substituting (12) into (11):
t



1
1
 s 1 exp(  s )d s  ds


 0 1  2 s  ( )

 (t )  K  V  C  G (t  s) 
0
8
(13)
Under the assumption of linear elasticity of the fibre, one can consider that the fibre
straightening strain,  s , is independent of time. The argument in the parenthesis in
equation (13) will therefore become a constant, leading to:
 (t )
t


1
1
K V  C 
 s 1 exp(  s )d s

1  2 s  ( )

0
  G(t  s) ds
(14)
0
In a stress-relaxation test, the left side of equation (14) describes the stress history of
the tissue specimen during relaxation. It has been experimentally observed that the
relaxation modes of the AV tissue typically follow a Maxwellian exponential decay
trend (Stella et al. 2007; Anssari-Benam et al. 2011a), generally represented by Prony
series of the form

i
exp( t /  i ) , where i is the number of Maxwell elements,  is
i
the amplitude of relaxation, and  is the characteristic relaxation time (Anssari-Benam
et al. 2011a). Thus, the integral on the right hand side of equation (14) should
mathematically result in the same exponential decay terms. However, the only
mathematically apparent exponential-dependence in equation (14) stems from strain
variables  and  s , and not the time variable t. This demonstrates that G (t ) is not
automatically constrained by equation (14) to result in an exponential decay function
of t. Thus, not only it contradicts the quasi-linearity assumption by which the stressrelaxation is independent of the strain, but also introduces arbitration for the choice of
G(t). Therefore, the resulting stress-relaxation function G(t) by the QLV principle does
not mathematically correlate with the relaxation modes incurred in experiments, and
does not conclude a Maxwell type relaxation. This implies that the response of the
tissue R[ , t ] may not necessarily be quasi-linear, so that it may not be substituted by
the specific quasi-linear relationship  e ( )  G(t ) in equation (3). Two representative
stress-relaxation responses reported in Anssari-Benam et al. (2011a) are given in
Figure 1 to highlight examples of discrete Maxwell-type relaxation modes observed in
the experiments.
9
4. Application to creep
Corresponding to the stress-relaxation function G (t ) is a creep function J (t ) . Under
the assumption of quasi-linear time-dependent behaviour, G (t ) and J (t ) are related by
(Rajagopal and Wineman 2008):
t
1  J (0)G(t )   G( s)
0
dJ (t  s)
dG(t  s)
ds = G(0) J (t )   J ( s)
ds
d (t  s)
d
(
t

s
)
0
t
(15)
By definition, the stress-relaxation at t=0 is G (0)  1 . Based on this, one can note
from equation (15) that: 1  J (0)G (0)  J (0)  1 . As such, and in light of equation
(15), equation (5) will take the following form when describing creep (Rajagopal and
Wineman 2008):
d e
 ( (t ))   (0) J (t )   J (t  s)
ds
ds
0
t
(16)
Equation (16) can be rearranged by the chain rule as:
t
 ( (t ))   (0) J (t )   J (t  s)
0
d e d s d
ds
d s d ds
(17)
d s
d e
can be calculated from equations (10), and
will be a constant
d s
d
The term
according to equation (8), shown hereafter as C1 . Additionally, similar to the
relaxation case, strain rate is assumed constant upon applying the creep loading
protocol, i.e.
d
 C 2  const . Equation (17) can now be re-written as:
ds

t
 ( (t ))   J (t  s) K  V

0
 s
 
1
 s 1 exp(  s ) C1C 2 ds

1  2 s  ( )
 
(18)
Substituting for  in the above equation using equation (8):
t

 ( (t ))   J (t  s) K  V t
0

 
1
 s 1 exp(  s ) C1C 2 ds
 
 ( )

10
(19)
Under the assumption of linear elasticity of a fibre (equation (7)), one can consider  t
to be independent of time. Additionally,  s is an intrinsic property of the subject
tissue, primarily determined by the waviness, initial crimp period and the cross-links of
the fibres. Therefore,  s is also independent of time. Thus, equation (19) can be
rearranged as:
 ( (t ))  A

1
 s 1 exp(  s )  J (t  s)ds

 0
 ( )
t
(20)
where A stands for the multiplication of all the constants in (19). The creep function
J (t ) can then be calculated by:
  ( )
t
 J (t  s)ds   ( (t )) 
0
 1
A s

exp(  s )

(21)
The implications of equation (21) are interesting. The right hand-side of the equation
is constant, and independent of time, noting that the creep stress  ( (t )) is kept
constant during a creep test. Therefore, equation (21) underlines that the creep function
derived from the relationship furnished by the QLV (equation (15)) calculates a creep
response which remains constant over time. This is inconsistent with the principles of
viscoelasticity and time-dependent behaviour of soft tissues, as well as in contrast to
the reported experimental data. For example, the results reported by Anssari-Benam et
al. (2011a) indicate clear time dependence in the creep response of AV specimens,
both in values and creep stages. The primary creep was reported to be an exponential
function of time, while the secondary creep was observed to be characterised by a
linear relationship of creep rate s and time t in the form:  C (t )  s t , where  C (t ) is
the creep strain in time t. Two representative creep responses reported in that study are
shown in Figure 2. This analysis implies that the relationship between G(t) and J(t) for
the AV tissue could not be characterised under the assumption of quasi-linear timedependent behaviour, stated by equation (15).
11
5. Concluding remarks
This paper presented a mathematical analysis investigating the assumption of quasilinear viscoelasticity in AV time-dependent behaviour. The principal inherent
characteristic of the tissue, namely the gradual fibre recruitment, which primarily
governs its biomechanical behaviour, was incorporated into the QLV model
formulation and the corresponding stress-relaxation and creep functions were derived.
These functions were then compared with the relaxation and creep modes observed in
experiments reported previously in the literature through the works of Stella et al.
(2007) and Anssari-Benam et al. (2011a). The aim of this analysis was not to describe
or model the underlying mechanisms involved in uniaixal and planar stress-relaxation
and creep behaviour of the valve. Rather, this paper attempted to investigate the quasilinearity of the AV time-dependent behaviour.
Time-dependent behaviour of soft tissues has proved to be very complex, and
formulating models based on tissue structure that would result in the observed
experimental behaviour remains a challenging task. In a recent study a rheological
model was developed to account for the rate-dependency of the mechanical behaviour
of the AV specimens under tensile deformation (Anssari-Benam et al. 2011b).
However, such models may not elaborate the underlying microstructural mechanisms
that contribute to the overall time-dependent behaviour of the bulk tissue, and therefore
may not be suitable for general application to different tissues and under different
loading conditions. Studies investigating the micromechanics of connective tissues
such as tendons have reported fibre sliding during relaxation experiments, suggesting
that it may be a possible mechanism to contribute to the overall stress-relaxation of the
tissue (Screen 2008; Gupta et al. 2010; Screen et al. 2012). However, analytical models
to directly correlate the observed phenomena to the tissue relaxation modes have less
been addressed. More thorough experimental observations are required to investigate
the possible microstructural origins of the time-dependent behaviour in the AV tissue,
and appropriate models need to be developed to accommodate the identified
mechanisms.
It was shown here that the assumption of quasi-linear viscoelastic behaviour for AV
tissue does not mathematically correlate with the experimentally observed modes and
12
trends, and the resulting stress-relaxation and creep functions do not appear to
voluntarily follow the experimental data as reported in the works of Stella et al. (2007)
and Anssari-Benam et al. (2011a). These results suggest that the quasi-linear
viscoelasticity may not be an intrinsic behaviour of the AV, and thus the application of
QLV model to characterise its time-dependent behaviour may not be theoretically
supported. This may be a contributing factor to the limited success of fitting the QLV
model to the AV stress-relaxation data, reported in some studies (Sauren et al. 1983;
Rousseau et al. 1983; Sauren and Rousseau 1983).
Concerning the creep behaviour, previous studies have reported particular poor
outcomes when fitting creep data to the QLV model (Haslach 2005). This is also
reflected in equation (21), as the resulting creep function suggests a constant creep
response, discounting for the primary and the secondary creep behaviours observed in
experimental investigations (Anssari-Benam et al. 2011a).
The stress-relaxation and creep modes exhibited in experiments by other connective
tissues such as tendons have been shown to be interrelated under the assumption of
quasi-linear viscoelasticity, by incorporating the experimentally observed structural
mechanism of gradual fibre recruitment during the both phenomena, into QLV
(Thornton et al. 2001). However, the analysis presented here shows that when the fibre
recruitment model is incorporated for the case of AV, mathematical formulations based
on the quasi-linearity assumption result in two uncoupled functions with no apparent
interrelations (equations 14 and 21), further reflecting on the limitations of such
assumption for the AV tissue.
While the analysis presented here was centred on AV tissue, the mathematical
principles and definitions are generic to all soft tissues of the same class, in which
time-dependent behaviour does not follow a continuous spectrum. Based on this
analysis, it may further be concluded that it is highly unlikely that the time-dependent
behaviour of this class of tissues could be assumed to be quasi-linear.
13
Acknowledgment
The author wishes to thank Dr. Hazel Screen for her helpful comments and feedback.
This work was part of a research project funded by the UK Engineering & Physical
Sciences Research Council (EPSRC), a Discipline Bridging Initiative (DBI) grant from
the EPSRC and the Medical Research Council (MRC).
14
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17
Figure legends
Figure 1- Two representative stress-relaxation curves for AV specimens loaded in
circumferential direction. Experimental data points are shown with hollow circles ( ),
and Maxwell relaxation modes in form of Prony series by continuous line. The y axis
represents the reduced stress relaxation G(t). At lower circumferential strains
(  5%  failure) , single relaxation mode describes the experimental data, while it
switches to double relaxation modes at higher strain levels (e.g.   80%  failure) . Data
were collated from Anssari-Benam et al. (2011a).
Figure 2- Two representative creep curves for AV specimens loaded in circumferential
direction: (a) creep under 50% of failure load (50% Ffailure); (b) creep under 70% of
failure load (70% Ffailure). Experimental data points are shown with hollow circles ( ).
Continuous line represents the exponential and linear models used to characterize the
primary and secondary creep phases, respectively. Primary and secondary creep stages
are shown in the graphs. Data were collated from Anssari-Benam et al. (2011a).
18
Figure 1
Stress-relaxation in circumferential direction
1
0.9
  80%  failure
0.8
Double exponential decay modes
G(t)
0.7
Single exponential decay mode
0.6
  5%  failure
0.5
0
50
100
150
200
Elapsed time (s)
19
250
300
Figure 2
Creep in circumferential direction: F = 50% Ffailure
(a)
30
29.5
29
J(t)
28.5
Primary creep
Secondary creep
28
27.5
27
0
50
100
150
200
250
300
Elapsed time (s)
Creep in circumferential direction: F = 70% Ffailure
(b)
36
35.5
35
J(t)
34.5
34
Primary creep
Secondary creep
33.5
33
0
50
100
150
Elapsed time (s)
20
200
250
300