Elasticity Imaging of Polymeric Media

[+] Author and Article Information
Mallika Sridhar

 University of California, Davis, CA 95616

Jie Liu

 University of Illinois at Urbana-Champaign, Urbana, IL 61801

Michael F. Insana

 University of California, Davis, CA, and University of Illinois at Urbana-Champaign, 405 North Mathews, Room 4247 Urbana, IL 61801mfi@uiuc.edu

J Biomech Eng 129(2), 259-272 (Sep 15, 2006) (14 pages) doi:10.1115/1.2540804 History: Received July 12, 2006; Revised September 15, 2006

Viscoelastic properties of soft tissues and hydropolymers depend on the strength of molecular bonding forces connecting the polymer matrix and surrounding fluids. The basis for diagnostic imaging is that disease processes alter molecular-scale bonding in ways that vary the measurable stiffness and viscosity of the tissues. This paper reviews linear viscoelastic theory as applied to gelatin hydrogels for the purpose of formulating approaches to molecular-scale interpretation of elasticity imaging in soft biological tissues. Comparing measurements acquired under different geometries, we investigate the limitations of viscoelastic parameters acquired under various imaging conditions. Quasi-static (step-and-hold and low-frequency harmonic) stimuli applied to gels during creep and stress relaxation experiments in confined and unconfined geometries reveal continuous, bimodal distributions of respondance times. Within the linear range of responses, gelatin will behave more like a solid or fluid depending on the stimulus magnitude. Gelatin can be described statistically from a few parameters of low-order rheological models that form the basis of viscoelastic imaging. Unbiased estimates of imaging parameters are obtained only if creep data are acquired for greater than twice the highest retardance time constant and any steady-state viscous response has been eliminated. Elastic strain and retardance time images are found to provide the best combination of contrast and signal strength in gelatin. Retardance times indicate average behavior of fast (110s) fluid flows and slow (50400s) matrix restructuring in response to the mechanical stimulus. Insofar as gelatin mimics other polymers, such as soft biological tissues, elasticity imaging can provide unique insights into complex structural and biochemical features of connectives tissues affected by disease.

Copyright © 2007 by American Society of Mechanical Engineers
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Figure 1

(a) Creep curves for a second-order (L=2) Voigt model and a step stress stimulus are illustrated. Curve a is drawn directly from Eq. 8 with finite η0; its slope at t≫T2 is σa∕η0. Curve b is from the same equation where η0=∞. In both cases, ϵ2∕ϵ1=2.5, T1=3s, and T2=100s. (b) The corresponding Fourier spectra D̆(ω) are from Eq. 10. Spectra from a step and 1s ramp stress stimulus are compared.

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Figure 2

(a) Retardation spectra from simulated data. Plotted are L̃(ω)=L(τ)∣τ=1∕ω for comparison with the Fourier spectrum. Creep data were generated from Eq. 12 for ϵ0=σa∕η0=0 assuming a broadband, bimodal input as given by the circle points (Input). Estimated retardation spectra (RS) L̃(k)(ω) for k=1,2,5,6, are compared to the Fourier spectrum (FS) D̆(ω), computed from the same data. (b)L̃(6) estimates without noise in the creep data and with noise (signal-to-noise ratio=32.2dB). A ninth-order polynomial filter was applied to the noisy data before estimation.

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Figure 3

Limitation of L(k)(τ) for representing retardance time distributions. The abscissa is b∕a from the log-normal input distribution L(τ)=exp[−(lnτ−a)2∕2b2]. The ordinate is the full-width-at-half-maximum bandwidth of retardance spectral estimates. Circles denote the exact output bandwidth for the input distribution, while the curves are bandwidths for kth-order estimates using noiseless creep data. Results suggest that the L(6)(τ) represents bandwidths of log-normal distributions above 150s with acceptable bias error.

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Figure 4

Illustration of collagen structures in connective tissue (fibril) and in gelatin (aggregates)

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Figure 5

Illustrations of four viscoelastic experiments. (a) Measurement method A applies uniaxial stress or strain stimuli to unconfined gelatin samples to estimate compressive relaxation modulus E(t) or compressive creep compliance D(t). It is also the ultrasonic strain imaging technique. (b) Method B applies uniaxial strain to estimate the compressive wave modulus M(t) for rigidly confined sample boundaries. (c) Method C is a cone-plate rheometer applied to estimate shear creep compliance J(t). (d) Method D applies an indenter to gelatin samples to estimate the elastic modulus E0. All positions are computer controlled with submicrometer accuracy, and forces are measured with a precision of 0.01g.

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Figure 6

(a) Shear creep measured with applied step stresses of σa′=3 and 30Pa using Method C and Type B gelatin (5.5%). (b) Viscosity estimates (Sec. 2) versus time for creep data at 30Pa. Steady-state values were attained beginning at ∼600s. (c) Example of shear creep recovery curve for Type A gelatin at σa′=100Pa. Values calculated from the creep and recovery phases are reported separately.

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Figure 7

Demonstrations of linearity. (a) Stress-strain curves for stiff (10%) and soft (5.5%) Type A gelatin using unconfined samples and uniaxial harmonic stimuli (Method A). The two stress levels indicated were used in subsequent creep measurements. (b) Shear creep Fourier spectra for Type B gelatin (Method C).

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Figure 8

Poisson’s ratio estimates versus time, i.e., ν(t). Error bars denote one standard deviation computed by propagating displacement measurement errors.

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Figure 9

(a) Dependence of Tℓ on acquisition time, and the effect of eliminating steady-state viscosity (linear term in Eq. 17). T1 and T2 estimates for a third-order Voigt model are shown. (b) Variation of T1 contrast over acquisition time is shown.

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Figure 10

Comparisons of measurements made using different methods. Samples were all type A gelatin aged three days. (a) Elastic modulus, (b) equilibrium compliance, and (c) steady-state viscosity under compression. Error bars are standard deviations that indicate uncertainty between repeated measurements.

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Figure 11

(a) Contrast between 10% and 5.5% homogeneous gelatin samples for seven compliance parameters. (b) Example ϵ0 image for a composite sample consisting of 5.5% gel background with a 10% gel inclusion. (c) Example T1 image.

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Figure 12

Normalized Fourier, retardation, and relaxation spectra. (a) Unconfined type A gelatin samples (aged three days) loaded uniaxially at σa=860Pa are measured for 2000s using Method A. (b) Confined type A gelatin samples (aged 1 day) strained uniaxially at ϵa=0.02 are measured for 2500s using Method B. (c) Type B gelatin samples (aged 1 day) sheared at σa′=3Pa are measured for 3000s in a rheometer using Method C. (d) Unconfined type A gelatin samples (aged three days) strained uniaxially at ϵa=0.08 are measured for 2000s by combining Methods A and B. Arrows indicate frequencies corresponding to the respondance times given in Table 1. Spectral amplitudes are uniformly reduced across the bandwidth as samples age.

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Figure 13

Effects of rest time on viscoelastic estimates. Top: Variation of T1 (left group), T2 (middle group), and T3 (right group) for a third-order Voigt model are shown for baseline measurements (0) and rest times of 1 and 2h. Error bars indicate fitting uncertainties. Bottom: Table showing initial baseline retardance times in seconds and percent biases for rest times of 1 or 2h between measurements.




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