# Extended Poroelasticity: An Analytical Solution and Its Application to p-Wave Propagation in Cervical Tissues

Authors: Roberto Palma; Antonio M. Callejas
Journal: Mathematical Problems in Engineering
 (2018)

---

## Abstract

This work presents a semianalytical solution based on Laplace transform to study the behaviour of poroelastic materials in the context of the Extended Nonequilibrium Thermodynamics. In this framework, the fluid phase incorporates a relaxation time and, consequently, a frequency-dependence appears. This rheological behaviour could explain the frequency-dependence experimentally observed in biological tissues, which has traditionally attributed to the solid phase of tissues. In particular, the analytical solution is applied to two cases, heaviside and sinusoidal inputs, of a semi-infinite domain, which is filled with a material such as the human cervix. From the results, it is observed that the frequency-dependence of the fluid phase could be relevant to high relaxation times while for null relaxation times the classical poroelastic theory is recovered. Finally, the present analytical solution could be used to validate future computational codes and experimental settings.


---

## Body

## 1. Introduction

Nowadays, understanding mechanical behaviour of the human cervix is a challenge for theoretical, computational, and experimental communities since it could be used to develop functional anticipation diagnostic tools, which will be applied to reduce the main cause of infant mortality: preterm birth, according to [1]. Notice that the understanding of cervical tissues is considered one of the most pressing problems in obstetrics [2].According to [3], the human cervix is composed of a distribution of cells embedded in an extracellular matrix of fibrillar collagen, which can be considered as the solid phase and represent 20-30% of the tissue and water with dissolved salts (fluid phase). The latter phase is the responsible for the cervix incompressibility [4].There exist many models to study dynamic responses of tissues in specialised literature. However, most of them are contradictory since they are based on phenomenological equations, which lack robustness. In particular, measured values in cervix tissues differ several orders of magnitude for medium and high frequencies [5]. In this connection, there are several viscoelastic approaches to properly characterise the human cervix: a linear viscoelastic model was proposed in [6], an in-homogenous model in [7], and several approaches by the Rus’ group; see [8–10].Despite the fact that the cervix is composed of solid and fluid, all the previous works are only focused on the solid contribution. On the contrary, the Extended Nonequilibrium Thermodynamics (ENET) [11] incorporates a viscous-like term to the fluid phase, which could explain the frequency-dependence of the cervix. For instance, R. Palma et al. have developed analytical and numerical solutions to study thesecond sound in thermoelasticity (see [12, 13]), which can be considered as a viscothermal effect. Also, ENET is applied in [14] to study the Debye relaxation: a viscoelectric effect.On this ground, this paper presents a poroelastic formulation based on the ENET to study the frequency-dependence of cervix tissues due to the fluid phase, and it could be considered the main novelty of the present work. Then, the formulation is expressed in one-dimension in order to obtain a dynamical solution by a semianalytical approach based on Laplace transform. Finally, two numerical examples are developed to highlight the importance of the fluid-relaxation in the response of the cervix.The present work assumes small strains and linear elasticity for the solid phase. For the fluid, constant properties are considered and the effects of temperature are neglected.


## 2. Three-Dimensional Governing Equations

Consider an arbitrary domainΩ and boundary Γ for which the governing equations are composed of equilibrium and constitutive equations and of the boundary conditions.With regard to the equilibrium equation and since the domain contains solid and fluid constituents, two equations must be enforced: linear momentum and mass conservation. Mathematically, they are expressed in local form as follows:(1)
ρ
u
¨
=
∇
·
σ
⊤
+
f
,
c
p
u
ϵ
˙
+
c
p
p
p
˙
=
-
∇
·
d
+
D
,where ρ, u
¨, and f denote mass density (including solid and fluid), acceleration, and body forces, respectively; σ
=
σ
⊤ is the Cauchy stress tensor; d and D denote the rate of change in fluid mass through the boundary and the production of fluid from an external source, respectively. Finally, ϵ, p, c
p
u, and c
p
p denote volumetric strain, fluid pressure, and two constants closely related to the porosity and fluid bulk modulus, respectively.Obviously, two constitutive equations, solid and fluid phases, are required to model poroelasticity; these equations read(2)
σ
=
C
:
ϵ
-
c
u
p
p
I
,
τ
d
˙
+
d
=
-
K
·
∇
p
.At this point, it is necessary to define all terms in (2): (i)
C denotes the elastic fourth order tensor, which is composed of matrix C
m
t and of fibre C
f
b (both solid phases) by the rule of mixture:(3)
C
=
ξ
C
m
t
+
1
-
ξ
C
f
b
,
w
i
t
h
:
C
m
t
=
2
μ
m
t
1
+
λ
m
t
I
⊗
I
,
C
f
b
=
2
μ
f
b
1
+
λ
f
b
I
⊗
I
,
whereξ
=
0.88 is the percentage of matrix, 1 denotes the fourth-order identity tensor, and μ, λ are the Lamé parameters for matrix and fibre constituents, respectively.
(ii)
ϵ
=
1
/
2
(
∇
⊗
u
+
u
⊗
∇
)
=
∇
s
u is the small strain second order tensor and ∇
s denotes the symmetric part of the gradient of displacements.
(iii)
K
=
1
/
ν
f
κ denotes solid permeability and it is closely related to the fluid viscosity ν
f.
(iv)
τ is the relaxation times, which is introduced by the assumption of a mixed entropy; see [15]. This empirical parameter is responsible for viscosity in the fluid phase and, consequently, for the frequency-dependence. Notice that the classical poroelasticity theory is recovered by imposing τ
=
0.Finally, the Dirichlet and Neumann boundary conditions for the extended poroelastic problem read(4)
u
=
u
¯
,
σ
⊤
·
n
=
t
,
p
=
p
¯
,
d
·
n
=
d
c
,where u
¯ and p
¯ denote prescribed displacements and pressure, respectively, and t and d
c prescribed traction and fluid flux on the boundary with outward normal n.


## 3. Analytical Solution

This section presents a one-dimensional, semianalytical, and dynamical solution for a half-space filled with a poroelastic material. For this purpose, the three-dimension extended poroelastic equations reported in Section2 are rewritten along the x-axis in order to apply the state space technique; see [16]. The three-dimensional Euclidean coordinates become(5)
x
≡
x
,
0,0
,
t
⇒
u
x
,
t
≡
u
x
,
t
,
p
=
p
x
,
t
,and the components of strain tensor are reduced to(6)
ϵ
⇒
ϵ
1
=
∂
u
x
,
t
∂
x
=
ϵ
,
ϵ
2
=
ϵ
3
=
ϵ
4
=
ϵ
5
=
ϵ
6
=
0
,and, finally, the constitutive equation of (2) (upper) becomes(7)
σ
1
=
C
11
ϵ
1
-
c
u
p
p
→
σ
=
C
ϵ
-
c
u
p
p
,
σ
2
=
σ
3
=
σ
4
=
σ
5
=
σ
6
=
0
.In absence of body forcesf
=
0 and mass sources D
=
0, the equilibrium equations of (1) are reduced to(8)
∂
σ
∂
x
=
ρ
u
¨
x
⇒
∂
2
σ
∂
x
2
=
ρ
∂
u
¨
x
∂
x
=
ρ
ϵ
¨
,
K
∂
2
p
∂
x
2
=
∂
∂
t
+
τ
∂
2
∂
t
2
c
p
u
ϵ
+
c
p
p
p
,Now, equations (6), (7), and (8) are rewritten in the Laplace domain by applying the transformation f
¯
(
s
)
=
∫
0
∞
‍
f
(
t
)
e
-
s
t
d
t:(9)
σ
¯
=
C
ϵ
¯
-
c
u
p
p
¯
,
∂
2
σ
¯
∂
x
2
=
s
2
ρ
ϵ
¯
,
K
∂
2
p
¯
∂
x
2
=
s
+
τ
s
2
c
p
u
ϵ
¯
+
c
p
p
p
¯
.These equations can be expressed in compact form by introducing the coefficients L
1, L
2, M
1, and M
2:(10)
L
1
=
c
p
u
c
u
p
+
C
c
p
p
K
C
s
+
τ
s
2
,
L
2
=
c
p
u
K
C
s
+
τ
s
2
,
M
1
=
c
u
p
ρ
s
2
C
,
M
2
=
s
2
ρ
C
,to give(11)
∂
2
p
¯
∂
x
2
=
L
1
p
¯
+
L
2
σ
¯
,
∂
2
σ
¯
∂
x
2
=
M
1
p
¯
+
M
2
σ
¯
,and the closed solution of this system of two couple equations can be expressed as follows:(12)
p
¯
x
,
s
σ
¯
x
,
s
=
exp
⁡
L
1
L
2
M
1
M
2
x
p
¯
0
,
s
=
p
¯
0
σ
¯
0
,
s
=
σ
¯
0
.The solution of this system is obtained by applying the Cayley-Hamilton theorem [16] to give(13)
p
¯
x
,
s
σ
¯
x
,
s
=
L
11
L
12
L
21
L
22
p
¯
0
σ
¯
0
,where p
¯
0 and σ
¯
0 are the boundary conditions and the coefficient and are explicitly given by(14)
L
11
=
e
-
η
2
x
η
1
-
L
1
-
e
-
η
1
x
η
2
-
L
1
η
1
-
η
2
,
L
22
=
e
-
η
1
x
η
2
-
M
2
-
e
-
η
2
x
η
1
-
M
2
η
2
-
η
1
,
L
12
=
L
2
e
-
η
1
x
-
e
-
η
2
x
η
1
-
η
2
,
L
21
=
M
1
e
-
η
1
x
-
e
-
η
2
x
η
1
-
η
2
,where η
1 and η
2 are the solutions of the following characteristic equations:(15)
η
1
+
η
2
=
L
1
+
M
2
,
η
1
η
2
=
L
1
M
2
-
L
2
M
1
.Finally, the semianalytical solution is attained by imposing boundary conditions and by inverting the Laplace transform using Riemann-sum approximations, as in [12].


## 4. Results

This section presents two analytical solutions, called cases, in order to highlight the main features of the present formulation. For this purpose, the material properties are obtained from the literature (see Table1), and they are real measured variables of the human cervix. In particular, Lamé parameters for fibre and matrix phases are obtained from [8, 9], bulk modulus of fluid from [17], solid permeability from [18], and fluid viscosity from [19]. Finally, the coefficient c
u
p
=
0.75 obeys the composition of the cervix, namely, 80-70% of fluid phase; see [20].Table 1
Material properties of the human cervix.
Magnitude
Value
Units
ρ
1000
[kg/m
3]
ρ
f
1000
[kg/m
3]
c
u
p
0.75
[–]
ν
f
1.2
×
10
-
3
[Pa⋅s]
μ
f
b
6.45
×
10
7
[Pa]
λ
f
b
1.47
×
10
8
[Pa]
μ
m
t
7.47
×
10
3
[Pa]
λ
m
t
1.7
×
10
9
[Pa]
B
f
2.2
×
10
9
[Pa]
κ
7.2
×
10
-
14
[m
2]


### 4.1. Case I

For case I, the boundary conditions are assumed to be a heaviside unit step functionH
(
t
):(16)
p
¯
0
=
p
0
s
,
σ
¯
0
=
0
.Introducing (16) in (12) and taking into account (14), the solutions for p
¯ and σ
¯ in the Laplace domain read(17)
p
¯
=
p
0
η
1
-
L
1
e
-
η
2
x
-
η
2
-
L
1
e
-
η
1
x
s
η
1
-
η
2
,
σ
¯
=
p
0
M
1
e
-
η
1
x
-
e
-
η
2
x
s
η
1
-
η
2
.Furthermore, the mechanical displacement and the flux can be obtained taking into account(18)
d
¯
=
-
K
∂
p
¯
∂
x
,
∂
2
σ
¯
∂
x
2
=
s
3
ρ
u
¯
⇒
u
¯
=
1
s
3
ρ
∂
2
σ
¯
∂
x
2
,to give(19)
d
¯
=
K
p
0
η
1
-
L
1
η
2
e
-
η
2
x
-
η
2
-
L
1
η
1
e
-
η
1
x
s
η
1
-
η
2
,
u
¯
=
p
0
M
1
η
1
e
-
η
1
x
-
η
2
e
-
η
2
x
ρ
s
4
η
1
-
η
2
.Consider a one-dimensional and semi-infinite domain, which is filled with a material as that of cervix. Att
=
0, a fluid pressure of heaviside type is applied and, then, the evolution of all variables of (17) and (18) is numerically studied. In particular, Figure 1 shows the time evolution of p (left column), u (middle column), and σ (right column) for three different relaxation times: τ
=0.5, 0,1, 0 [s] (each one in a row). In addition, each figure contains three different curves at three time instants t
=0.3, 0.5, 1 [s].Figure 1
Case I. Fluid pressure (left column), mechanical displacement (middle), and mechanical stress (right) versus distance for three relaxation times:τ
=
0.5 (top row), τ
=
0.1 (middle) and τ
=
0 [s] (bottom). Each figure shows three different curves at three time instants: t
=0.3, 0.5, 1 [s].Forτ
≠
0 [s] (extended poroelasticity) a hyperbolic behaviour is attained in the evolution of p and, consequently, in u. It is observed in the wave front, which is moving at each time instant. On the contrary, for the case τ
=
0 a parabolic behaviour is observed, absence of wave front, and the classical poroelastic theory is recovered. In this sense, the extended poroelasticity removes the paradox of infinite velocity, which lacks physical meaning: in nature, there are no infinity velocities.Since the problem is coupled, the stresses forτ
≠
0 [s] show a viscous-like behaviour that could explain the frequency-dependence experimentally observed in cervical tissues. On the contrary, for τ
=
0 [s], a linear behaviour is observed and, therefore, it is concluded that the classical poroelasticity can not be used to model the human cervix.


### 4.2. Case II

For case II, a sinusoidal functionp
0
sin
⁡
(
ω
t
) of frequency ω is prescribed:(20)
p
¯
0
=
p
0
ω
s
2
+
ω
2
,
σ
¯
0
=
0
.Following a procedure similar to that of case I, the solutions for p
¯ and σ
¯ in the Laplace domain are(21)
p
¯
=
p
0
ω
η
1
-
L
1
e
-
η
2
x
-
η
2
-
L
1
e
-
η
1
x
s
2
+
ω
2
η
1
-
η
2
,
σ
¯
=
p
0
M
1
ω
e
-
η
1
x
-
e
-
η
2
x
s
2
+
ω
2
η
1
-
η
2
,and the mechanical displacement and the flux are given by(22)
d
¯
=
K
p
0
ω
η
1
-
L
1
η
2
e
-
η
2
x
-
η
2
-
L
1
η
1
e
-
η
1
x
s
2
+
ω
2
η
1
-
η
2
,
u
¯
=
p
0
M
1
ω
η
1
e
-
η
1
x
-
η
2
e
-
η
2
x
ρ
s
3
s
2
+
ω
2
η
1
-
η
2
.Consider the same semi-infinite domain as that in case I. Now, a fluid pressure of sinusoidal type is applied att
=
0 and all variables of (21) and (22) are shown in Figure 2. Again, the same conclusion as those in case I can be observed for this sinusoidal input. Nevertheless, the curves are smoother due to the nature of the sinusoidal signal. Therefore, these solutions are more amenable for future computational validations since it is not necessary to use regularisation schemes.Figure 2
Case II. Fluid pressure (left column), mechanical displacement (middle), and mechanical stress (right) versus distance for three relaxation times:τ
=
0.5 (top row), τ
=
0.1 (middle), and τ
=
0 [s] (bottom). Each figure shows three different curves at three time instants: t
=0.3, 0.5, 1 [s].


## 4.1. Case I

For case I, the boundary conditions are assumed to be a heaviside unit step functionH
(
t
):(16)
p
¯
0
=
p
0
s
,
σ
¯
0
=
0
.Introducing (16) in (12) and taking into account (14), the solutions for p
¯ and σ
¯ in the Laplace domain read(17)
p
¯
=
p
0
η
1
-
L
1
e
-
η
2
x
-
η
2
-
L
1
e
-
η
1
x
s
η
1
-
η
2
,
σ
¯
=
p
0
M
1
e
-
η
1
x
-
e
-
η
2
x
s
η
1
-
η
2
.Furthermore, the mechanical displacement and the flux can be obtained taking into account(18)
d
¯
=
-
K
∂
p
¯
∂
x
,
∂
2
σ
¯
∂
x
2
=
s
3
ρ
u
¯
⇒
u
¯
=
1
s
3
ρ
∂
2
σ
¯
∂
x
2
,to give(19)
d
¯
=
K
p
0
η
1
-
L
1
η
2
e
-
η
2
x
-
η
2
-
L
1
η
1
e
-
η
1
x
s
η
1
-
η
2
,
u
¯
=
p
0
M
1
η
1
e
-
η
1
x
-
η
2
e
-
η
2
x
ρ
s
4
η
1
-
η
2
.Consider a one-dimensional and semi-infinite domain, which is filled with a material as that of cervix. Att
=
0, a fluid pressure of heaviside type is applied and, then, the evolution of all variables of (17) and (18) is numerically studied. In particular, Figure 1 shows the time evolution of p (left column), u (middle column), and σ (right column) for three different relaxation times: τ
=0.5, 0,1, 0 [s] (each one in a row). In addition, each figure contains three different curves at three time instants t
=0.3, 0.5, 1 [s].Figure 1
Case I. Fluid pressure (left column), mechanical displacement (middle), and mechanical stress (right) versus distance for three relaxation times:τ
=
0.5 (top row), τ
=
0.1 (middle) and τ
=
0 [s] (bottom). Each figure shows three different curves at three time instants: t
=0.3, 0.5, 1 [s].Forτ
≠
0 [s] (extended poroelasticity) a hyperbolic behaviour is attained in the evolution of p and, consequently, in u. It is observed in the wave front, which is moving at each time instant. On the contrary, for the case τ
=
0 a parabolic behaviour is observed, absence of wave front, and the classical poroelastic theory is recovered. In this sense, the extended poroelasticity removes the paradox of infinite velocity, which lacks physical meaning: in nature, there are no infinity velocities.Since the problem is coupled, the stresses forτ
≠
0 [s] show a viscous-like behaviour that could explain the frequency-dependence experimentally observed in cervical tissues. On the contrary, for τ
=
0 [s], a linear behaviour is observed and, therefore, it is concluded that the classical poroelasticity can not be used to model the human cervix.


## 4.2. Case II

For case II, a sinusoidal functionp
0
sin
⁡
(
ω
t
) of frequency ω is prescribed:(20)
p
¯
0
=
p
0
ω
s
2
+
ω
2
,
σ
¯
0
=
0
.Following a procedure similar to that of case I, the solutions for p
¯ and σ
¯ in the Laplace domain are(21)
p
¯
=
p
0
ω
η
1
-
L
1
e
-
η
2
x
-
η
2
-
L
1
e
-
η
1
x
s
2
+
ω
2
η
1
-
η
2
,
σ
¯
=
p
0
M
1
ω
e
-
η
1
x
-
e
-
η
2
x
s
2
+
ω
2
η
1
-
η
2
,and the mechanical displacement and the flux are given by(22)
d
¯
=
K
p
0
ω
η
1
-
L
1
η
2
e
-
η
2
x
-
η
2
-
L
1
η
1
e
-
η
1
x
s
2
+
ω
2
η
1
-
η
2
,
u
¯
=
p
0
M
1
ω
η
1
e
-
η
1
x
-
η
2
e
-
η
2
x
ρ
s
3
s
2
+
ω
2
η
1
-
η
2
.Consider the same semi-infinite domain as that in case I. Now, a fluid pressure of sinusoidal type is applied att
=
0 and all variables of (21) and (22) are shown in Figure 2. Again, the same conclusion as those in case I can be observed for this sinusoidal input. Nevertheless, the curves are smoother due to the nature of the sinusoidal signal. Therefore, these solutions are more amenable for future computational validations since it is not necessary to use regularisation schemes.Figure 2
Case II. Fluid pressure (left column), mechanical displacement (middle), and mechanical stress (right) versus distance for three relaxation times:τ
=
0.5 (top row), τ
=
0.1 (middle), and τ
=
0 [s] (bottom). Each figure shows three different curves at three time instants: t
=0.3, 0.5, 1 [s].


## 5. Conclusions

This work has presented a theoretical approach based on Nonequilibrium Thermodynamics to study the behaviour of poroelastic materials taking into account the frequency-dependence of the fluid phase. In this connection, the main novelty of the present work is the incorporation of relaxation times for the fluid phase to perform a material constitution applied to biological tissues. Then, the three-dimensional governing equations are reduced to one dimension in order to obtain a semianalytical and dynamical solution based on Laplace transform. In particular, the solution is applied to simulate a semi-infinite domain, which is filled with a material such as the human cervix, and it is observed that the frequency-dependence also could be due to the fluid phase.



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*Source: 5280141-2018-10-29.xml*