(238c) Pseudo-Component Viscoelastic Model of Soft Tissues

Motivation:

Understanding the response of human tissues to externally imposed stress and strain is critical to improving quality of life and healthcare in diverse aspects.  Examples include:  i) Improvements in automotive safety devices such as air bags and seat belts depend on better knowledge of how human parts, and the integrated whole, respond to the stresses of impact.  ii) Development of biomedical imaging techniques, which observe in vivo tissue and organ responses to stress and strain, could be used to assess whether a tissue is functioning correctly or needs urgent care, if stress-strain patterns were characterized.  iii) Development of synthetic prosthetic devices and in vitro regenerated tissue requires scaffolds that duplicate mechanical properties of native tissues.  iv) Development of simulators that train next-generation physicians and development of robotic procedures both depend on models of tissue stress-strain response. 

Significant effort in developing models of the stress-strain behavior of biological materials reveals that biological tissues show a more complex mechanical behavior than polymers, plastics, and metal films; and cannot use the conventional models developed for those substances.  However, most viscoelastic models of biological materials are grounded in classic approaches from these other disciplines.  Unfortunately, model modifications are not performed in ways that reflect either mechanistic understanding or phenomenological fidelity.  As a result, the models cannot be used to accept or reject molecular mechanisms.

Problems and Novel Solutions:

There are 2 general problems. They are modeling the viscoelastic behavior and the model optimization techniques.

Since tissue stress relaxes in time under constant strain, and since tissue shapes progressively deforms under constant load, and since tissues gradually return toward original structure when external stress or strain is relieved, time-dependent (viscoelastic) models are required.  The time-dependent behavior also depends on the stress-strain history.  Further, since tissues are comprised of several participating structures (cells, matrix, fibrils, etc.), each having individual mechanisms; a multi-component, viscoelastic model is required.  Further, since tissue properties are not constant; nonlinear, multi-component, viscoelastic models are required.  Finally, since many tissue components do not relax fully to the original internal structure, the commonly employed dashpot element (which lets the spring return to zero stress) is inappropriate.  New constitutive relations are required for the nonlinear, multi-component, viscoelastic models.  This work will demonstrate the validity of appropriate nonlinear viscoelastic relations and a pseudo-component approach for modeling the complex tissue structures. Six pseudo-component models types are used to examine the consistency of the model with the experimental data. The constitutive models are developed from a combination of spring and dashpot model along with total relax back to original structure or retaining the original structure.

The second problem is associated with model optimization. Research reports difficulty in optimizing viscoelastic model parameter values to best match the model with experimental stress-strain-time data.  There are multiple local optima, and approaches to the minima are often exasperatingly slow (even with classic best practice nonlinear optimizers such as Levenberg Marquardt).  Further, optimization must include constraints on the parameter values, suggesting that direct search methods may be more appropriate than gradient based methods.  Accordingly, techniques such as best-of-N starts or direct search techniques have been investigated for determining the probable global optimum subject to multiple constraints.  The applicability of emerging optimization techniques for model parameter adjustment is explored along with new regression approaches and the quality of the model.

A new optimization technique called leapfrogging is also explored. This technique starts with a set of players (trial solution) located randomly in the DV space. This technique relocates the position of players by reflecting the player with worst OF value across the player with the best OF value  at each iteration. Test cases on this technique revealed that this technique  gave better optimized values when compared to other techniques with less number of function evaluations.

The validated models obtained from the explorations will be useful for advancement of biomedical applications.

Results:

The figure shows the combination of hyperelastic spring along with retain and reformation models. The 6 pseudo-components developed show various combination of 2 of these 3 factors. The first 3 pseudo-components 1, 2 and 3 were developed based on 2 of the same types. The next 3 constitutive models 4, 5 and 6 were developed based on combinations of using 2 of these 3.

The model is developed using the above explained 6-pseudo components and is compared with the experimental stress strain relationship with time data as shown below. The figure below illustrates 4 stages of strain ramp and hold. The dots represent experimental stress data and the solid curve the model.

A nonlinear instantaneous stress-strain relation for a pseudo-component is commonly modeled as  where the subscript ?i? indicates the i^th pseudo-component. The constraint laid on this equation is A*B > 0 which means both A and B should be of same sign and also the time constant t > 0. The optimizers should reach the minima satisfying these constraints. The presentation in all will reveal details of modeling and constrained nonlinear optimization.