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Research Papers

Identifying a Minimal Rheological Configuration: A Tool for Effective and Efficient Constitutive Modeling of Soft Tissues

[+] Author and Article Information
Petr Jordan, Amy E. Kerdok, Robert D. Howe

 Harvard School of Engineering and Applied Sciences, Cambridge, MA 02138; Harvard-MIT Division of Health Sciences and Technology, Cambridge, MA 02139

Simona Socrate

Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139; Harvard-MIT Division of Health Sciences and Technology, Cambridge, MA 02139

Note that the specific arrangement of viscoelastic elements in Fig. 2, while logical, is somewhat arbitrary, and similar results may be achieved with other arrangements.

J Biomech Eng 133(4), 041006 (Mar 15, 2011) (11 pages) doi:10.1115/1.4003620 History: Received January 17, 2010; Revised January 31, 2011; Posted February 09, 2011; Published March 15, 2011; Online March 15, 2011

We describe a modeling methodology intended as a preliminary step in the identification of appropriate constitutive frameworks for the time-dependent response of biological tissues. The modeling approach comprises a customizable rheological network of viscous and elastic elements governed by user-defined 1D constitutive relationships. The model parameters are identified by iterative nonlinear optimization, minimizing the error between experimental and model-predicted structural (load-displacement) tissue response under a specific mode of deformation. We demonstrate the use of this methodology by determining the minimal rheological arrangement, constitutive relationships, and model parameters for the structural response of various soft tissues, including ex vivo perfused porcine liver in indentation, ex vivo porcine brain cortical tissue in indentation, and ex vivo human cervical tissue in unconfined compression. Our results indicate that the identified rheological configurations provide good agreement with experimental data, including multiple constant strain rate load/unload tests and stress relaxation tests. Our experience suggests that the described modeling framework is an efficient tool for exploring a wide array of constitutive relationships and rheological arrangements, which can subsequently serve as a basis for 3D constitutive model development and finite-element implementations. The proposed approach can also be employed as a self-contained tool to obtain simplified 1D phenomenological models of the structural response of biological tissue to single-axis manipulations for applications in haptic technologies.

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Copyright © 2011 by American Society of Mechanical Engineers
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References

Figures

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Figure 9

Cervical tissue in uniaxial compression (six material parameters): exponential elastic element (2A), linear back stress elastic element (2B), and nonlinear viscous power law dashpot with reptation-limited flow (2C). Material parameters are listed in Table 4.

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Figure 6

Configuration 3 (six material parameters): exponential elastic element (2A), linear back stress elastic element (2B), and nonlinear viscous power law dashpot with reptation-limited flow (2C). Material parameters are listed in Table 3.

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Figure 5

Configuration 2 (four material parameters): exponential elastic element (2A), linear back stress elastic element (2B), and linear dashpot (2C). Material parameters are listed in Table 3.

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Figure 4

Configuration 1 (three material parameters): linear elastic element (2A), linear back stress elastic element (2B), and linear dashpot (2C). Material parameters are listed in Table 3.

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Figure 3

Indentation response of perfused porcine liver in indentation. A continuous segment of cyclic load/unload ramps at four different rates is shown at the top. The corresponding stress-strain response is shown in the middle with individual displacement ramps distinguished by color. The stress relaxation response is shown at the bottom. All data were collected at the same location on the same liver specimen, allowing 30 min of recovery between the cyclic tests and the stress relaxation.

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Figure 2

General rheological arrangement comprising three parallel networks of increasing complexity

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Figure 1

A schematic view of the constitutive model selection process, comprising an “inner loop” for material parameter fitting and an “outer loop” for constitutive law adjustments

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Figure 8

Brain tissue in ex vivo uniaxial compression (eight material parameters): exponential elastic element (3A), nonlinear viscous power law dashpot with reptation-limited flow (3C), and time-dependent back stress in SLS arrangement (3B, 3D, and 3E). Material parameters listed in Table 4.

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Figure 7

Configuration 4 (eight material parameters): exponential elastic element (3A), nonlinear viscous power law dashpot with reptation-limited flow (3C), and time-dependent back stress in SLS arrangement (3B, 3D, and 3E). Material parameters are listed in Table 3.

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