An Analysis of the Error Associated to Single and Double Exponential Approximations of Theoretical Poroelastic Models

Abstract

The temporal evolutions of displacements, strains, stresses, and fluid pressure in a poroelastic material under sustained compression are theoretically expressed as sums of an infinite number of exponentials. However, estimation of an infinite number of time constants is impractical in experimental settings. In the past, empirical models containing a finite number of exponentials have been proposed and used to approximate the theoretical poroelastic models. At the present time, however, the degree of error associated with such approximations is unclear. In this paper, we present an analysis of the error encountered when approximating a poroelastic model containing an infinite number of exponentials with a single or double exponential model. As a testing platform, the presented error analysis is applied to the estimation of effective Poisson’s ratio (EPR) and fluid pressure in a uniform cylindrical poroelastic sample and a cylindrical poroelastic sample containing an inclusion under stress relaxation. Our results show that, when the infinite number of exponentials in the theoretical models are approximated with finite number of exponentials, significant error is invoked only in the first few time samples of the EPR and fluid pressure, while the error is negligible for the remaining time samples. We also show that, when estimating clinically relevant mechanical parameters such as Poisson’s ratio or the product of aggregate modulus and interstitial permeability, such approximation invokes small error (<3%) in comparison to the general model with infinite number of exponentials. Therefore, such approximation may be acceptable in techniques aiming at reconstructing mechanical parameters from poroelastographic data.

Keywords

mechanical model poroelasticity error analysis approximation infinite exponentials

Introduction

Several biological tissues have been modeled as poroelastic materials^1-3 using biphasic theories.^4-7 The first work on the theoretical development of the stress relaxation analysis of a two-dimensional (2D) poroelastic material using the biphasic theory was the one by Mow et al.,⁶ where expressions of the displacements, strains, and fluid pressure were determined. This study was then extended by Armstrong et al.⁸ to determine analytic expressions for a uniform cylindrical sample compressed with frictionless plates. The first work on the modeling of a poroelastic inclusion inside another poroelastic material was done by Rice et al.,⁹ where the authors used Eshelby’s formulation¹⁰ to derive expressions of the displacements, strains, and fluid pressure inside a poroelastic inclusion embedded in a poroelastic material even if no closed-form expressions in space and time domain were provided. Our group has recently reported the closed-form solutions of the effective Poisson’s ratio (EPR) and fluid pressure inside and outside a poroelastic inclusion embedded inside another poroelastic material under stress relaxation and creep compression.^11-14

When a poroelastic material is subjected to sustained compression, fluid exudation causes the stress to rise above the equilibrium value during the compression phase,^8,15 while in the relaxation phase, no fluid exudation occurs, and the internal fluid redistributes inside the sample. All aforementioned theoretical works demonstrate that the displacements, strains, and fluid pressure in a poroelastic material under sustained compression are temporally varying and that their theoretical temporal expressions contain an infinite number of exponentials.^8,11-15 The presence of an infinite number of exponentials in the analytic expressions of the displacements and strains creates hardship in the analysis and parameter estimation and is unpractical in clinical imaging settings. Parameter estimation is important for extracting diagnostically important tissue mechanical parameters such as Poisson’s ratio, product of aggregate modulus, and interstitial permeability, which have been found to be useful in diagnosis, prognosis, and treatment of diseases like cancer. However, it has been reported that estimation of mechanical parameters becomes computationally difficult and prone to errors and nonconvergence issues as the number of exponentials in the models increases.^16,17 Real-time implementation of data processing and visualization become impossible for a large number of exponentials. It is therefore clear that, in practical scenarios, poroelastic models containing an infinite number of exponentials need to be approximated with models containing a finite number of exponentials. The extent to which such approximations introduce inaccuracies in the estimation of the mechanical parameters of interest is currently unclear.

Ultrasound elastography is an imaging modality that can be used to assess and image the mechanical properties of biological tissues.^18,19 Poroelastography^20,21 is an elastographic technique used to image the time-dependent mechanical response of poroelastic tissues. In addition to the ultrasound field, poroelastography methods have also been developed for magnetic resonance imaging (MRI) applications.²² In poroelastography, approximation of models with an infinite series of exponentials by models with a finite number of exponentials (typically one or two) has several benefits. First, in poroelastography, we need to estimate the field variables (fluid pressure, EPR, etc.) for a large number of pixels (at least 128 × 128). Computation of the field variables with large number of exponentials can be computationally very intensive. Use of single or double exponential models can substantially decrease the computational burden. Second, in estimation of material properties by using curve fitting techniques, use of high number of exponentials is prone to errors and nonconvergence specially in noisy conditions.¹⁶ To our knowledge, however, no studies that focus on an investigation of the error invoked when using single or double exponential model in poroelastic estimation and imaging can be found in the literature.

In this paper, we analyze the errors that occur while estimating the EPR and fluid pressure using single and double exponential models in both uniform and inclusion-inserted poroelastic media under stress relaxation. We also analyze the error in the reconstruction of two mechanical parameters, that is, the Poisson’s ratio and the product of aggregate modulus and interstitial permeability in the material using single and double exponential models using both finite element and ultrasound poroelastographic simulation data.

Approximation of Analytical Solutions

The general poroelastic samples used for the analysis reported in this paper are shown in Figure 1. The radial direction is along r and the circumferential direction is along the angle θ. In a stress relaxation experiment, the cylindrical sample is compressed from the top with a frictionless compressor plate keeping the bottom fixed.

Figure 1.

(a) A schematic of a uniform cylindrical sample of a poroelastic material of radius a and (b) a schematic of a cylindrical sample of a poroelastic material of radius b with a cylindrical inclusion of radius a. The axial and radial directions are along the z-axis and r-axis, respectively. The circumferential direction is in the direction of the increasing angle.

Analytic models of the poroelastic samples shown in Figure 1 subjected to stress relaxation compression have been reported by Berry et al.¹⁵ and Islam et al.¹¹ The important equations, initial and boundary conditions, pertinent to the models are described below.

The first basic equation for cylindrical symmetric poroelasticity is the continuity equation of the pore fluid, which can be written as²³

ζ \frac{d ε}{d t} + S \frac{d p}{d t} = \frac{Ω}{γ_{f}} (\frac{d^{2} p}{d r^{2}} + \frac{1}{r} \frac{d p}{d r}),

(1)

where ε is the volume strain, p is the fluid pressure, $ζ$ is Biot’s coefficient, S is the storage coefficient, $Ω$ is the hydraulic conductivity, and $γ_{f}$ is the volumetric weight of the pore fluid. Equation (1) can also be written as

ζ \frac{d ε}{d t} + S \frac{d p}{d t} = k (\frac{d^{2} p}{d r^{2}} + \frac{1}{r} \frac{d p}{d r}),

(2)

where k is the interstitial permeability.

The second basic equation is the equation of radial equilibrium, which can be expressed as

\frac{d σ_{r r}}{d r} + \frac{σ_{r r} - σ_{θ θ}}{r} = 0,

(3)

where $σ_{r r}$ and $σ_{θ θ}$ are the total stresses in radial and tangential directions. The total stresses can be separated into the elastic stresses and the fluid pressure by

σ_{r r} = σ_{e, r r} + ζ p, σ_{θ θ} = σ_{e, θ θ} + ζ p .

(4)

The equation of radial equilibrium can be written now as

\frac{d σ_{e, r r}}{d r} + \frac{σ_{e, r r} - σ_{e, θ θ}}{r} + ζ \frac{d p}{d r} = 0 .

(5)

The elastic stress-strain relations can be written as

σ_{e, r r} = - (K - \frac{2}{3} G) ϵ - 2 G \frac{d u}{d r},

(6)

σ_{e, θ θ} = - (K - \frac{2}{3} G) ε - 2 G \frac{u}{r},

(7)

where u is the radial displacement and $K$ and $G$ are the bulk modulus and shear modulus of the solid matrix, respectively.

Equations (2) to (7) are the equations that are used to develop analytical expressions of fluid pressure and EPR for an applied axial strain of $ε_{z z} (t) = - ε_{0} H (t)$ in uniform and inclusion-inserted media in Berry et al.¹⁵ and Islam et al.¹¹ Here, H(t) is the Heaviside step function. The assumptions of material properties, boundary conditions, and initial conditions associated to the problem are described below.

The analysis is based on the assumption of intrinsic incompressibility for both the solid phase and the fluid phase. The incompressibility of solid and fluid phase is incorporated in the analytical model by using $ζ = 1,$ where $ζ = 1 - K / K_{s}$ and $S = (ϕ) 1 / K_{f} + (ζ - ϕ) 1 / k_{s} = 0$ , where $K_{s}$ and $K_{f}$ are the bulk modulus of the solid grains and fluid phase and φ is the porosity.²³ The fluid of interest is of unit volumetric weight, that is, $γ_{f} = 1 .$

The boundary conditions for a poroelastic sample in an elastography experiment are (a) on the bottom plane of the sample, there is no axial displacement; (b) on the right edge of the sample, there is zero fluid pressure¹⁵; and (c) a constant uniaxial stress is present at the top boundary of the sample. Perfect bonding as well as continuous stress, fluid pressure, and displacement between inclusion and background should also be considered in case of inclusion-inserted media.^24,25

The initial conditions in the problem are assumed as follows: (a) the volumetric strain is 0 and (b) the fluid pressure is equal to the product of shear modulus of the sample and applied strain at $t {= 0}^{+} .$

Below, we summarize the resulting equations of these models, which are relevant for the error analysis reported in this paper. As previously mentioned, our analysis will refer to the EPR and fluid pressure. As the expressions of the EPR and fluid pressure in uniform medium and inside inclusion of inclusion-inserted medium are similar, we write the expressions of EPR and fluid pressure in only inclusion-inserted medium below.

The expression for the EPR inside the inclusion in the inclusion-inserted medium can be written as¹¹

\begin{matrix} Ψ_{i} (r, t) = - \frac{ε_{r r, i}}{ε_{z z}} (r, t) \\ = ν_{i} + \sum_{n = 1}^{\infty} \frac{J_{0} (α_{n} a) - \frac{1 - 2 ν_{i}}{1 - ν_{i}} {J_{0} (α_{n} \frac{r}{a}) - \frac{J_{1} (α_{n} \frac{r}{a})}{α_{n} \frac{r}{a}} + \frac{J_{1} (α_{n})}{α_{n}}}}{\frac{1}{1 - ν_{i}} J_{0} (α_{n} a) - α_{n} J_{1} (α_{n} a)} \exp (- \frac{α_{n}^{2} H_{A i} k_{i} t}{a^{2}}), \end{matrix}

(8)

where $ε_{r r, i}$ is the radial strain inside the inclusion and $ν_{i},$ $H_{A i},$ and $k_{i}$ denote Poisson’s ratio, aggregate modulus, and interstitial permeability of the inclusion material. In this case, $α_{n}$ are the roots of the characteristics equation, $J_{1} (x) - (1 - ν_{i} / 1 - 2 ν_{i}) x J_{0} (x) = 0$ . Here $J_{0}$ and $J_{1}$ are the Bessel functions of the first kind of order 0 and 1. The fluid pressure inside the inclusion can be expressed as¹¹

p_{i} (r, t) = - μ_{i} ε_{0} \sum_{n = 1}^{\infty} \frac{J_{0} (α_{n} r)}{\frac{1}{2 (1 - ν_{i})} J_{0} (α_{n} a) - \frac{α_{n}}{2} J_{1} (α_{n} a)} \exp (- \frac{α_{n}^{2} H_{A i} k_{i} t}{a^{2}}),

(9)

where $μ_{i}$ is the shear modulus of the inclusion material. The solution for the fluid pressure outside the inclusion can be written as¹¹

\begin{matrix} p_{b} (r, t) = μ_{i} ε_{0} π \sum_{n = 1}^{\infty} \frac{J_{0} (β_{n} a) J_{0} (β_{n} b) U (β_{n} r)}{J_{0}^{2} (β_{n} a) - J_{0}^{2} (β_{n} b)} \\ \times \sum_{m = 1}^{\infty} \frac{J_{0} (α_{m})}{\frac{1}{2 (1 - ν_{i})} J_{0} (α_{m}) - \frac{1}{2} α_{m} J_{1} (α_{m})} \frac{e^{- (α_{m}^{2} / a^{2}) C_{i} t} - e^{- β_{n}^{2} C_{b} t}}{\frac{β_{n}^{2} C_{b} a^{2}}{α_{m}^{2} C_{i}} - 1} f (t) \\ - μ_{i} ε_{0} \frac{\log (r / b)}{\log (a / b)} \sum_{m = 1}^{\infty} \frac{J_{0} (α_{m})}{\frac{1}{2 (1 - ν_{i})} J_{0} (α_{m}) - \frac{1}{2} α_{m} J_{1} (α_{m})} e^{- (α_{m}^{2} / a^{2}) C_{i} t} f (t) \\ + μ_{b} ε_{0} π \sum_{n = 1}^{\infty} \frac{J_{0} (β_{n} a) U (β_{n} r) e^{- β_{n}^{2} C_{b} t}}{J_{0} (β_{n} a) + J_{0} (β_{n} b)}, \end{matrix}

(10)

where $C_{i} = H_{A i} k_{i},$ $C_{b} = H_{A b} k_{b},$ $β_{n}$ are roots of the characteristics equation, $J_{0} (x a) Y_{0} (x b) - Y_{0} (x a) J_{0} (x b) = 0,$ and $Y_{0}$ is the Bessel function of second kind of order 0. $H_{A b},$ $μ_{b}$ , and $k_{b}$ denote the aggregate modulus, shear modulus, and interstitial permeability of the background. Here $f (t)$ is a function defined as

f (t) = (\begin{matrix} 0, t \leq 0 \\ 1, t > 0 \end{matrix} .

(11)

The expression for the radial strain outside the inclusion can be written as¹¹

ε_{r r, b} (r, t) = \frac{1 - ν_{b}}{H_{A b}} p_{b} (r, t) + ν_{b} ε_{0} - \frac{a ν_{b}}{(ν_{b} - 1) r} (\frac{1 - ν_{b}}{H_{A b}} p_{b} (a, t) + ν_{b} ε_{0} - \frac{1}{a} u_{i} (a, t)) .

(12)

where $ν_{b}$ is the Poisson’s ratio of the background material and $u_{i} (a, t)$ is the radial displacement at the interface of the inclusion and background. The EPR can be found by dividing this equation by $- ε_{z z} (t)$ as

\begin{matrix} Ψ_{b} (r, t) = - \frac{ε_{r r, b}}{ε_{z z}} (r, t) \\ = \frac{1 - ν_{b}}{H_{A b} ϵ_{0}} p_{b} (r, t) + ν_{b} - \frac{a ν_{b}}{(ν_{b} - 1) r} (\frac{1 - ν_{b}}{H_{A b} ϵ_{0}} p_{b} (a, t) + ν_{b} - \frac{1}{a ϵ_{0}} u_{i} (a, t)) . \end{matrix}

(13)

When finite number of exponentials are used in Equation (8), expression of the EPR inside the inclusion becomes

Ψ_{i, N} (r, t) = ν_{i} + \sum_{n = 1}^{N} \frac{J_{0} (α_{n} a) - \frac{1 - 2 ν_{i}}{1 - ν_{i}} {J_{0} (α_{n} \frac{r}{a}) - \frac{J_{1} (α_{n} \frac{r}{a})}{α_{n} \frac{r}{a}} + \frac{J_{1} (α_{n})}{α_{n}}}}{\frac{1}{1 - ν_{i}} J_{0} (α_{n} a) - α_{n} J_{1} (α_{n} a)} \exp (- \frac{α_{n}^{2} H_{A i} k_{i} t}{a^{2}}),

(14)

where N indicates a finite number. For a single exponential approximation, N = 1 and for double exponential approximation, N = 2. The fluid pressure and EPR in the uniform sample, fluid pressure inside and outside, and EPR outside the inclusion in the inclusion-inserted sample can be approximated using single or double exponential in a similar manner.

Simulations

Finite Element Simulations

To compare the results of the EPR and fluid pressure for single and double exponential approximation with the results from finite element analysis, a commercial finite element modeling (FEM) software, namely, ABAQUS, Abaqus Inc, Providence, Rhode Island, was used. For our analysis, we have used the “coupled pore fluid diffusion and stress analysis” module of ABAQUS. Details of this simulation module along with necessary equations can be found in the documentation provided by ABAQUS²⁶ (6.7.1). In brief, a poroelastic material saturated with a fluid is modeled in ABAQUS as a biphasic material (consisting of a solid phase and a fluid phase) and an effective stress principle is adopted to describe its behavior. The material is modeled by attaching the finite element mesh to the solid phase, and fluid can flow through the mesh. The mechanical part of the model is based on the effective stress principle, which can be described as, “the total stress acting at a point, $σ$ , is assumed to be made up of an average pressure stress in the wetting fluid, $p,$ called the ‘wetting fluid pressure’ and an ‘elastic stress’ $σ_{e}$ on the solid matrix.” The effective stress principle can be expressed mathematically as²⁶ (2.8.1)

σ_{e} = σ + p I,

(15)

where I denotes the identity matrix.

The model also uses a continuity equation for the mass of wetting fluid in a unit volume of the poroelastic material,

\int_{V} [δ p {(ρ_{w} n_{w})}_{t + Δ t} - \frac{1}{J_{t + Δ t}} {(J ρ_{w} n_{w})}_{t}] d V + Δ t \int_{V} δ p {[\frac{d}{d x} \cdot (ρ_{w} n_{w} v_{w})]}_{t + Δ t} d V = 0,

(16)

where $ρ_{w}$ is the density of the fluid, $n_{w} = d V_{w} / d V,$ $V_{w}$ is the volume of the wetting fluid, $v_{w}$ is the average velocity of the wetting liquid relative to the solid phase and $J$ is the ratio of the material’s volume in the current configuration to its volume in the reference configuration $J = d V / d V^{0} \sim 1 + div (u),$ where $u$ is the solid phase displacement vector²⁶ (2.8.1). The constitutive behavior for the pore fluid flow in the above-mentioned biphasic model is governed by Darcy’s law stated as²⁶(6.7.1)

n_{w} v_{w} = - \frac{Ω}{γ_{f}} \nabla p .

(17)

The uniform sample and the inclusion-inserted sample were modeled as a linearly elastic, isotropic, incompressible, permeable solid phase saturated with an incompressible fluid. The samples were compressed from top and the bottom side was kept static. The interstitial permeability of the sample was assumed independent of the strain and void ratio. A single 2D plane of the 3D (three-dimensional) cylinder was analyzed in ABAQUS.¹⁵ The mesh element type used for this analysis was CAX4RP with 400 elements in a single row. A boundary condition of zero fluid pressure on the right-hand side of both samples was imposed to ensure flow of the fluid only in the right direction.¹⁵ The analysis is done in two steps. An instantaneous axial strain of 4% is applied in the first step, which remains constant in the second step. The first step is of 1 s and the second step continues up to 901 s. The specific weight of the fluid was taken as 1 Nm^–3. The incompressibility condition of solid and fluid phase is ensured in ABAQUS by fixing the bulk modulus of fluid and solid phase to a large value (10¹⁵ Pa).

Four uniform and five inclusion-inserted samples were chosen with different sizes and mechanical parameters. The radii of the uniform samples are given in Table 1 and the radii of inclusion-inserted samples and their inclusions are given in Table 2. The heights of the samples were always taken as equal to their radii. Young’s modulus of the background (simulating normal tissue) in the inclusion-inserted samples was chosen as 32.78 kPa based on previous literature^27,28 and kept fixed in all samples. As the tumors can have a broad range of Young’s moduli,^29,30 we chose four different Young’s modulus values, 40, 50, 97.02, and 163.90 kPa for the homogeneous samples (simulating tumor) and inclusions (simulating tumor) in inclusion-inserted samples. The Poisson’s ratio is reported in the literature to have a range of values between 0.2 and 0.45 for both tumors and normal tissues.^31-33 Poisson’s ratio values of 0.2, 0.3, and 0.4 for the inclusion and background were used. Based on previous literature,³⁴ the tumors can have a broad range of interstitial permeability. We used different values of background-to-inclusion permeability contrast (10, 20, 50, 100, 200) for different inclusion-inserted samples. The mechanical parameters of all the uniform and inclusion-inserted samples used in current study are tabulated in Tables 1 and 2.

Table 1.

Description of the Sample A to D Used in Current Study.

Sample	$E (kPa)$	$k (m^{4} N^{-1} s^{-1})$	$ν$	a (mm)
A	97.02	1.05 × 10⁻¹¹	0.30	25
B	97.02	9.93 × 10⁻¹²	0.20	20
C	163.90	1.37 × 10⁻¹²	0.40	10
D	40	5.87 × 10⁻¹²	0.40	10

Table 2.

Description of the Sample E to I Used in Current Study.

Sample	$E_{b} (kPa)$	$E_{i} (kPa)$	$k_{b} (m^{4} N^{-1} s^{-1})$	$k_{i} (m^{4} N^{-1} s^{- 1})$	$ν_{b}$	$ν_{i}$	a (mm)	b (mm)
E	32.78	40	$3 \times 10^{- 10}$	$6 \times 10^{- 12}$	0.3	0.3	10	20
F	32.78	97.02	$2.5 \times 10^{- 10}$	$1.25 \times 10^{- 12}$	0.2	0.3	8	40
G	32.78	97.02	$3 \times 10^{- 11}$	$1.5 \times 10^{- 12}$	0.2	0.3	10	20
H	32.78	50	$1.5 \times 10^{- 11}$	$1.5 \times 10^{- 12}$	0.4	0.4	10	20
I	32.78	97.02	$1.25 \times 10^{- 10}$	$1.25 \times 10^{- 12}$	0.3	0.3	8	40

Ultrasound Simulations

The simulated pre- and postcompression ultrasound radio frequency RF data were generated using the transducer characteristics and the mechanical displacement data obtained from FEM employing a convolution model.^35,36 In this model, the object field is convolved with the point spread function (PSF) of the transducer to obtain the transducer response. A Gaussian noise of specific signal to noise ratio (SNR), which depends on the measurement and electronic noise, is then added to the data to obtain the precompressed RF data. Here, the object field consists of the randomly distributed speckle amplitudes, which follows a Gaussian distribution with zero mean and unit variance. Bilinear interpolation is performed on the precompressed object field using the pre- and postcompressed coordinates of the speckles to obtain the postcompressed object field. Similar to precompressed RF data, postcompressed RF data are obtained by convolving the postcompressed object field with the PSF and then adding noise of specific SNR. Using the input mechanical displacement data at different time points (obtained from FEM), postcompression RF frames at different time points are created.³⁷

The simulated ultrasound transducer had 128 elements, frequency bandwidth between 5 and 14 MHz, a 6.6 MHz center frequency, and 50% fractional bandwidth at –6 dB. The transducer’s beamwidth was assumed to be dependent on the wavelength and to be approximately 1 mm at 6.6 MHz.²¹ The sampling frequency was set at 40 MHz and Gaussian noise was added to set the SNR at 40 dB. From one simulated precompressed RF data and postcompression RF data at different time points, the method proposed in Islam et al. ³⁸ was used to estimate the axial and lateral strains at these time points. Axial and lateral strains from 50 independent realizations were averaged to obtain axial and lateral strains at each time point.

Parameter Estimation

Poisson’s ratio ( $ν$ ) and the product of aggregate modulus and interstitial permeability ( $H_{A} k$ ) are estimated from the EPR data of both uniform (samples A-D) and inclusion-inserted media (samples E-I) from finite element and ultrasound simulation data utilizing the curve fitting tool in Matlab, Mathworks Inc., Natick, Massachusetts.¹⁵ The EPR data from finite element and ultrasound simulations are time and space varying. The applied strain and the radii of the inclusion and sample are assumed to be known. In all cases, the initial value of the estimated parameter is set to 10% of the true value. A least squared error minimization technique based on “trust-region-reflective” method is used to estimate the desired parameters. Therefore, the objective function for parameter estimation is the least-square value between the response computed from the analytical model (by approximation through single, double, and 50 exponentials) and finite element/ultrasound simulation data. To approximate infinite number of exponentials in curve fitting, we used 50 exponentials. We found out that if the number of exponential is increased beyond 50, no change is observed in the estimated parameters.

Error Assessment

The percentage errors in single and double exponential approximations of the EPR and fluid pressure are calculated using the following rule of normalized absolute error in percent³⁹

E_{N} (r, t) = 100 \times \frac{| q_{N} (r, t) - q_{50} (r, t) |}{q_{50} (r {,0}^{+})},

(18)

where $q$ is the variable of interest (EPR or fluid pressure) and N = 1, 2; $q_{50}$ is the estimated $q$ with 50 exponentials. Here we assume that $q_{50}$ can represent the expression of the EPR or fluid pressure involving infinite number of exponentials. It should be noted that we did not observe any change in the computed values of EPR and fluid pressure for number of exponential more than 50.

The root mean squared error in percent (RMSE) in estimated parameters is defined as⁴⁰

RMSE = \sqrt{\frac{\sum_{r}^{R} \sum_{c}^{C} {(ρ_{e} (r, c) - ρ_{f} (r, c))}^{2}}{R \times C}} \times \frac{100 \times R \times C}{\sum_{r}^{R} \sum_{c}^{C} ρ_{f} \leq (r, c)},

(19)

where $ρ_{e}$ is the estimated parameter (Poisson’s ratio or product of aggregate modulus and interstitial permeability) and $ρ_{f}$ is the true value of the parameter. R and C are the number of rows and columns in the estimated parameter image.

Results

Finite Element Simulations

Uniform medium

The EPR distributions for the uniform sample (sample A) are shown in Figure 2(a) to (d) for approximations with single, double, and 50 exponentials and the FEM prediction. In all these figures, we see that the EPR starts at 0.5 at $t {= 0}^{+}$ and goes to the Poisson’s ratio of the material at t = 448 s. From a visual comparison among these results, we first note that the FEM results and the results from 50 exponential approximation are practically identical, indicating that the approximation by 50 exponentials is essentially equivalent to the infinite exponentials’ case. For single and double exponential approximations, we see that the difference with FEM is somewhat prominent only in the early time responses. As time progresses, the error becomes almost negligible visually—Figure 2(a3) to (a5), (b3) to (b5), and (c3) to (c5). The fluid pressure for the uniform sample is shown in Figure 3(a) to (d). The fluid pressure starts at a value slightly higher than $μ ε_{0}$ at $t {= 0}^{+}$ and goes to 0 at t = 448 s. These observations are consistent with works of Armstrong et al.⁸ and Berry et al.¹⁵ From a qualitative point of view, we can make similar considerations as the ones for the EPR results. The plots for the estimated errors associated with single and double exponential approximations when compared with the 50 exponentials at different positions are shown in Figure 4 for the EPR and in Figure 5 for fluid pressure. From these results, we see that the errors in approximating the EPR and fluid pressure with single and double exponential can be significant at the early time responses and that the errors in such approximations depend on the spatial position. Generally, the error increases close to the center and the outer boundary of the sample.

Figure 2.

EPRs at different time points of 4 s, 40 s, 100 s, 250 s, and 448 s for single exponential approximation are shown in (a1), (a2), (a3), (a4), (a5); for double exponential approximation shown in (b1), (b2), (b3), (b4), (b5); for 50 exponentials in (c1), (c2), (c3), (c4), (c5); and from FEM in (d1), (d2), (d3), (d4), (d5) for sample A. FEM = finite element modeling.

Figure 3.

Fluid pressures (Pa) at different time points of 4 s, 40 s, 100 s, 250 s, and 448 s for single exponential approximation are shown in (a1), (a2), (a3), (a4), (a5); for double exponential approximation shown in (b1), (b2), (b3), (b4), (b5); for 50 exponentials in (c1), (c2), (c3), (c4), (c5); and from FEM in (d1), (d2), (d3), (d4), (d5) for sample A. FEM = finite element modeling.

Figure 4.

Error in approximating the EPR with single and double exponential at different positions of uniform sample A at (a) r = 5 mm, (b) r = 10 mm, and (c) r = 15 mm. EPR = effective Poisson’s ratio.

Figure 5.

Error in approximating the fluid pressure with single and double exponential at different positions of uniform sample for sample A at (a) r = 5 mm, (b) r = 10 mm, and (c) r = 15 mm.

Inclusion-inserted medium

For the inclusion-inserted medium (sample E), the EPRs at 4 s, 40 s, 100 s, 250 s, and 448 s for single exponential approximation are shown in Figure 6(a1) to (a5), for double exponential approximation shown in (b1) to (b5), for approximation with 50 exponentials in (c1) to (c5) and from FEM in (d1) to (d5). In these figures, we see that, inside the inclusion, the EPR starts at 0.5 at $t {= 0}^{+}$ and goes to the Poisson’s ratio of the material at t = 448 s. However, the EPR outside the inclusion starts at a lower value at $t {= 0}^{+}$ and reaches the Poisson’s ratio of the background material at t = 448 s. The fluid pressures at different time points for this sample are shown in Figure 7(a) to (d). Inside the inclusion region, the fluid pressure starts at a value slightly higher than $μ_{i} ε_{0}$ at $t {= 0}^{+}$ and goes to 0 at t = 448 s. These observations are consistent with work of Islam et al.,¹¹ and discussed in great details in this reference. The plots for the errors associated with EPRs and fluid pressures inside the inclusion for approximation with single, double, and 50 exponentials at different positions are shown in Figure 8 and Figure 9. The errors incurred in the approximations of the EPRs and fluid pressure outside the inclusion with different number of exponentials are shown in Figure 10 and Figure 11. Several important observations can be derived from these four figures. The first one is the dependence of the error with the spatial position. The error increases at positions closer to the center of the sample and at the interface between inclusion and background. Because, at a position close to the center of the sample and/or at the interface between the inclusion and background, the nonlinearity of strains increases, which requires a higher number of exponentials to model the strains. Thus, at these positions, if a low number of exponentials is used, the error becomes higher than in other positions inside the sample. In all four figures (Figures 8 -11), the errors are higher at early time samples as for the uniform sample case A.

Figure 6.

EPR at different time points of 4 s, 40 s, 100 s, 250 s, and 448 s for single exponential approximation are shown in (a1), (a2), (a3), (a4), (a5); for double exponential approximation shown in (b1), (b2), (b3), (b4), (b5); for 50 exponentials in (c1), (c2), (c3), (c4), (c5); and from FEM in (d1), (d2), (d3), (d4), (d5) for sample E (inclusion-inserted medium). EPR = effective Poisson’s ratio; FEM = finite element modeling.

Figure 7.

Figure 8.

Error in approximating the EPR inside the inclusion at different positions with single and double exponential for sample E at (a) r = 2 mm, (b) r = 4 mm, and (c) r = 6 mm. EPR = effective Poisson’s ratio.

Figure 9.

Error in approximating the fluid pressure inside the inclusion with single and double exponential at different positions for sample E at (a) r = 2 mm, (b) r = 4 mm, and (c) r = 6 mm.

Figure 10.

Error in approximating the EPR outside the inclusion with single and double exponential at different positions for sample E at (a) r = 13 mm, (b) r = 15 mm, and (c) r = 17 mm. EPR = effective Poisson’s ratio.

Figure 11.

Error in approximating the fluid pressure outside the inclusion with single and double exponential at different positions for sample E at (a) r = 13 mm, (b) r = 15 mm, and (c) r = 17 mm.

Error in parameter estimation

The RMSEs in estimated parameters of samples A to I from the FEM data are shown in Tables 3 to 5. From Table 3, we see that for case of uniform medium, the RMSEs in estimated parameters from all samples for single, double, and 50 exponentials are very low and well below 2% both in cases of $ν$ and $H_{A} k .$ From Table 4, we see that in the inclusion region, the RMSEs in estimated product of aggregate modulus and interstitial permeability in case of 50 exponentials are lower than single and double exponential but the differences are less than 3% in most of the cases. In the samples where the Poisson’s ratio of the inclusion and background are not equal, the RMSE in estimated Poisson’s ratio increases in both inclusion and background regions. As the RMSEs in the estimated product of aggregate modulus and interstitial permeability outside the inclusion are found to be large for single, double, and 50 exponential cases (>40% in many cases), we did not report those results.

Table 3.

RMSEs in Estimated Parameters of the Sample A to D from Finite Element Simulation Data.

Sample	RMSE in Estimation of $ν$ (%)			RMSE in Estimation of $H_{A} k$ (%)
Sample	Fifty Exponentials	Double Exponential	Single Exponential	Fifty Exponentials	Double Exponential	Single Exponential
A	0.03	0.03	0.03	0.53	1.12	1.12
B	0.13	0.13	0.13	0.44	0.90	0.90
C	0.05	0.05	0.05	0.82	1.22	1.22
D	0.09	0.09	0.09	0.59	1.43	1.43

RMSE = root mean squared error.

Table 4.

RMSEs in Estimated Parameters of the Inclusion in Sample E to I from Finite Element Simulation Data.

Sample	RMSE in Estimation of $ν_{i}$ (%)			RMSE in Estimation of $H_{A i} k_{i}$ (%)
Sample	Fifty Exponentials	Double Exponential	Single Exponential	Fifty Exponentials	Double Exponential	Single Exponential
E	0.43	0.43	0.43	2.69	4.58	4.59
F	0.63	0.64	0.64	7.37	7.57	7.57
G	2.49	2.49	2.49	6.32	9.87	9.87
H	0.29	0.29	0.29	18.21	21.94	21.94
I	0.17	0.17	0.17	21.23	26.25	26.25

RMSE = root mean squared error.

Table 5.

RMSEs in Estimated Parameters of the Background in Sample E to I from Finite Element Simulation Data.

Sample	RMSE in Estimation of $ν_{b}$ (%)
Sample	Fifty Exponentials	Double Exponential	Single Exponential
E	0.067	0.067	0.067
F	6.17	6.17	6.17
G	2.73	2.73	2.73
H	0.017	0.017	0.017
I	0.067	0.067	0.067

RMSE = root mean squared error.

Ultrasound Simulations

Uniform and inclusion-inserted medium

The EPR estimated at five time points from ultrasound simulated data of samples A and E are shown in Figure 12(a1) to (a5) and (b1) to (b5), respectively. By comparing this figure with Figure 2(d1) to (d5) and Figure 6(d1) to (d5), we understand that the EPR estimated from the ultrasound simulated data and the EPR from finite element simulation data match well.

Figure 12.

Estimated EPR at different time points of 4 s, 40 s, 100 s, 250 s, and 448 s from ultrasound simulated data of sample A (uniform medium) are shown in (a1) to (a5) and estimated EPR at the same time points from ultrasound simulated data of sample E (inclusion-inserted medium) are shown in (b1) to (b5). EPR = effective Poisson’s ratio.

Error in parameter estimation

The RMSEs in estimated parameters, that is, Poisson’s ratio and the product of aggregate modulus and interstitial permeability of the uniform and inclusion-inserted samples (samples A-I) from the ultrasound simulated data are shown in Tables 6 to 8. From Table 6, we observe that for uniform samples, the RMSEs in estimated $ν$ and $H_{A} k$ are higher in case of ultrasound simulated data in comparison to the FEM data because of the inherent noise in ultrasound simulated data. However, the RMSE is always below 4% in case of $ν$ and below 8% in case of $H_{A} k .$ For the inclusion-inserted media (samples E-I), the RMSEs are also higher in case of ultrasound simulated data in comparison to FEM data as shown in Tables 7 and 8. Similar to FEM data, the RMSEs in estimated Poisson’s ratio from ultrasound simulation data are same for single, double, and 50 exponentials. The RMSEs in the estimated product of aggregate modulus and permeability are higher in case of single and double exponential by less than 3% in comparison to 50 exponentials inside the inclusion in most of the cases. The RMSEs in estimation of the Poisson’s ratio by single, double, and 50 exponential fittings are higher in the background region in comparison to the inclusion region.

Table 6.

RMSEs in Estimated Parameters of the Sample A to D from Ultrasound Simulation Data.

Sample	RMSE in Estimation of ν (%)			RMSE in Estimation of $H_{A i} k_{i}$ (%)
Sample	Fifty Exponentials	Double Exponential	Single Exponential	Fifty Exponentials	Double Exponential	Single Exponential
A	2.43	2.43	2.43	3.53	5.81	5.81
B	3.23	3.23	3.23	3.14	6.63	6.63
C	2.45	2.45	2.45	4.92	6.12	6.12
D	2.19	2.19	2.19	4.59	7.18	7.18

Table 7.

RMSEs in Estimated Parameters of the Inclusion in Sample E to I from Ultrasound Simulation Data.

Sample	RMSE in Estimation of $ν_{i}$ (%)			RMSE in Estimation of $H_{A i} k_{i}$ (%)
Sample	Fifty Exponentials	Double Exponential	Single Exponential	Fifty Exponentials	Double Exponential	Single Exponential
E	2.67	2.67	2.67	4.16	5.64	5.64
F	2.98	2.98	2.98	8.13	9.37	9.37
G	5.95	5.95	5.95	15.22	16.81	16.81
H	3.25	3.25	3.25	26.59	29.74	29.74
I	2.23	2.23	2.23	27.19	31.23	31.23

RMSE = root mean squared error.

Table 8.

RMSEs in Estimated Parameters of the Background in Sample E to I from Ultrasound Simulation Data.

Sample	RMSE in Estimation of $ν_{b}$ (%)
Sample	Fifty Exponentials	Double Exponential	Single Exponential
E	2.16	2.16	2.16
F	9.14	9.14	9.14
G	4.76	4.76	4.76
H	3.53	3.53	3.53
I	3.29	3.29	3.29

RMSE = root mean squared error.

Discussion

The analytical models developed by Armstrong et al.,⁸ Berry et al.,¹⁵ and Islam et al.^11,12 are important to understand the response of tissues that can be modeled as poroelastic materials and for better interpretation of medical imaging results. These theoretical models demonstrate the impact of various material properties on the strains, stresses, and fluid pressure behavior inside the tissues. However, these models contain sums of an infinite number of exponentials. Therefore, they have limited applicability to clinical data. In parameter estimation methods, which are necessary to extract the underlying mechanical properties of tissues, every exponential is treated as a nonlinear parameter, and the complexity as well as number of computations increase dramatically with increasing number of exponentials. In this paper, we analyze the error that occurs when these poroelastic models are approximated with a single exponential model or a double exponential model. Two field variables are considered in this analysis, the EPR and fluid pressure. These field variables were chosen due to their importance in poroelasticity imaging techniques, such as ultrasound poroelastography and MRI poroelastography.

From the results reported in this paper, we observe that the error associated to the single and double exponential approximations of the theoretical poroelastic models is higher for time samples immediately following compression. The exact number of these time samples depends on the properties of the material, but, in general, it is a small fraction of the overall time constant of the sample. For the uniform sample analyzed in this paper, we found an error greater than 5% only for the first five time samples of the EPR when using the single exponential model. In this particular sample, the first five samples span a time of 16 s, which corresponds to 14.04% of the first time constant. For the double exponential approximation case, the error for the EPR estimation was found to be always below 5% and to go below 2% after only two time samples (corresponding to approximately 7 s) as shown in Figure 4. Similar considerations apply to the inclusion-inserted medium when estimating the EPR inside the inclusion as the nature of the error inside the inclusion is similar to the one in the uniform sample while estimation of EPR outside the inclusion seems to be less affected in both the single and double exponential approximation cases. These observations are also applicable to the error associated to the single and double exponential approximations in the case of the fluid pressure both in uniform and inclusion-inserted media. It should be noted that the errors associated to the approximated models (both for EPR and fluid pressure) are also dependent on the boundaries and interfaces. We found out that, in proximity of boundary or interfaces between two materials, these approximations can lead to higher amounts of error. In nonuniform materials with multiple tissue interfaces, the error due the approximation of the analytical models with single/double exponential models may be higher than in the cases studied in this paper. Although the error analysis in this paper has been performed for the EPR and fluid pressure only, the same conclusions hold for circumferential strain, radial stress, and axial stress as their theoretical expressions also contain an infinite number of exponentials similarly to EPR and fluid pressure.¹⁵

It has been reported that, in stress relaxation applications, the total analysis time span should theoretically be at least nine times the time constant of the first exponential of the strains in tissue sample $(e^{- 9} = 0.0000123 \approx 0) .$ ⁸ In practice, our group has found that total acquisition time should be greater than at least 70% of the strain time constant.⁴¹ Therefore, the time points affected by errors due to single and double exponential approximations correspond to a relatively short time interval after compression and may have no effect on the actual time constant estimation. In some applications, however, the early time response of the EPR and fluid pressure is important especially if long data acquisitions are not experimentally possible.⁴² Furthermore, if the acquired data are affected by a large amount of noise, early time response is essential for a correct parameter estimation. However, based on our analysis, the error in estimating the Poisson’s ratio both in uniform and nonuniform samples because of single and double exponential approximations is very low. More caution should be taken when estimating the product of the aggregate modulus and interstitial permeability because the error in the estimation of this parameter outside the inclusion may be underestimated due to limitations inherent to the analytical model itself, regardless of the number of exponentials used in the model.

Based on the error analyses reported in this paper, the single and double exponential approximation can be “safely” used in most applications. For determination of strains, displacements, EPR, and fluid pressure, when material properties are known beforehand, the double exponential approximation model may be the preferred one because of the error associated to this model is low and the model is still computationally efficient. The single exponential approximation model is preferred for estimation of mechanical parameters because it is computationally efficient and has a similar performance to the double and infinite exponential models. Single exponential approximations have been used for elastographic time constant imaging.^42-44

In general, human soft tissues are characterized by both interstitial permeability and vascular permeability.²⁸ When analyzing the strain response to applied compression of poroelastic samples with both interstitial permeability and vascular permeability, three cases may be considered: (a) when interstitial permeability is dominant over vascular permeability, (b) when vascular permeability is dominant, and (c) when interstitial permeability and vascular permeability are comparable.²⁷ In this paper, we refer to the first case, that is, when interstitial permeability is dominant over vascular permeability. In the second case, the strain response of a uniform poroelastic sample can be expressed using a single exponential model. So, this case does not require error analysis.²⁷ In the third case, the strain response of the uniform poroelastic sample is very similar to that of the first case except that it is multiplied by an exponential term containing vascular permeability and aggregate modulus.²⁷ Therefore, in the third case, the error due to single or double exponential approximations would be lower than in the first case as the exponential term containing the vascular permeability increases the weight of the first exponential on the overall strain response. As our analysis is for the interstitial permeability dominant case, it represents the worst case scenario, the one for which the error due to the approximation with single/double exponential models is expected to be the highest among all three cases.

In the field of soil mechanics and poromechanics, the permeability ( $k$ ) and hydraulic conductivity ( $Ω$ ) are often used “interchangeably” or “loosely.” The actual relationship between these two parameters can be written as¹⁵

Ω = k γ_{f} .

(20)

In the literature, the hydraulic conductivity over $γ_{f}$ is often replaced by the interstitial permeability in the equations of poroelasticity. As $γ_{f}$ is just a scaling factor between the interstitial permeability and hydraulic conductivity, if the value of interstitial permeability of a poroelastic material (such as tissue) is taken as reported in the literature, it does not matter what the value of $γ_{f}$ is used in the simulation or model. We fixed $γ_{f} = 1 {Nm}^{- 3}$ so that “hydraulic conductivity” and “interstitial permeability” could be used interchangeably in our models.¹⁵

This study has been designed for application in ultrasound poroelastography. Experimental validation of the simulation study included in this paper was not possible due to the lack of suitable phantoms with known poroelastic mechanical properties. Furthermore, since in experiments the actual tissue mechanical properties are unknown, the errors associated to the estimated mechanical parameters (Poisson’s ratio, product of aggregate modulus and interstitial permeability, etc.) cannot be experimentally measured. As the ultrasound simulations simulate a practical poroelastography experiment, the error associated to the parameters estimated from the ultrasound simulation data when the infinite exponentials are approximated with single/double exponentials represents a lower bound for the error that may be incurred when using practical experimental data.

As a final remark, the availability of single and double exponential models that approximate theoretical poroelastic models containing an infinite number of exponentials can be useful in a number of ways. The first one is the use of the approximated equations to estimate the material properties by using inverse formulation. The estimation of material properties becomes much easier to perform using the single or double exponential models rather than the general model with an infinite number of exponentials. Another possible use of the approximated models is in the determination and analysis of the time-dependent stress, strain, and pore pressure behavior in the material when the material properties are known a priori. The computationally efficient approximated analytical solutions can provide an easy and fast tool to understand the effect of different material properties inside normal and diseased tissues.^45-47

Conclusion

In this paper, we analyze the error incurred in the estimation of the EPR and fluid pressure in uniform and nonuniform poroelastic samples when the theoretical models containing an infinite number of exponentials are approximated with a single exponential model or a double exponential model. It is demonstrated that the error for such approximations is significant only for early time samples of the field variable under investigation. It is also demonstrated that such approximations introduce insignificant error in the reconstruction of mechanical parameters such as Poisson’s ratio and the product of aggregate modulus and interstitial permeability. These findings can be useful to assess the suitability of simpler and more computationally efficient models to understand the performance of medical imaging methods focused on the estimation of the poroelastic response of tissues.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Raffaella Righetti

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