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Extruded rods and an orthopedic screw of polylactide/glycolide copolymers with a ceramic ingredientProduct development of orthopedic bioabsorbable implants requires the knowledge of the dynamics of bioabsorption. Bioabsorption is simulated in vitro by hydrolysis experiments. Several variables including shear and bending strength, mass loss, inherent viscosity, etc. are measured at different intervals of time. These factors depend on the composition of the material and process variables like draw ratio. The product development problem consists of finding good values for composition and draw ratio, which will result in a desired degradation profile. If performed through trial and error, this process would require too much effort and would consume a lot of expensive raw materials. Nonlinear models helped shorten this work by a significant factor.

​Polylactides and Polyglycolides
Polylactides/glycolides are the most widely used bioabsorbable polymers for clinical implants in various musculoskeletal applications. The main advantage of these devices is that these materials degrade in vivo and the degradation products exit through metabolic routes, eliminating the need for a second operation. Different compositions have different properties and bioabsorption dynamics and can also be processed in different ways.

Degradation studies are carried out for materials development as well as product development. As a first stage, in vitro hydrolysis experiments are conducted prior to a smaller number of in vivo tests. Several variables like strength, crystallinity, mass loss, and inherent viscosity are followed over a period of time, typically one to two years. Inherent viscosity is a good indicator of the molecular weight and is easier and simpler to measure than molecular weight distributions.

Figure 1: Nonlinear models for materials development or product development

Mathematical Modeling
Mathematical modeling can be performed in various ways, and different methods are suitable for different situations. The models represent knowledge of quantitative effects of relevant variables in a concise and precise form. They can be used instead of experimentation if they are reliable enough. Mathematical models also permit the user to carry out various kinds of calculations, like determining suitable values of variables that will result in desired product properties or characteristics.

Physical or phenomenological modeling is not particularly effective for predicting material behavior. Physical modeling for hydrolysis of polylactide/glycolides requires quantitative knowledge of the various phenomena taking place, which are not well understood. The reactions taking place — mass transfer and crystallization — are poorly known, let alone their rates. Physical modeling usually requires assumptions and simplifications. Even when feasible, solutions of such partial differential equations tend to be unsuitable for product development.

Empirical and semi-empirical modeling, on the other hand, do not need any major assumptions or simplifications. Empirical models simply describe the observed behavior of a process. Empirical modeling is feasible when the relevant variables are measurable, as is often the case. Conventional techniques of empirical modeling, however, are linear statistical techniques. These tend to have limitations because nothing in nature is very linear, and particularly so in materials science. Therefore, it makes sense to use better techniques of empirical and semi-empirical modeling.

Figure 2: Shear strength degradation profiles for different glycolide contents

Nonlinear Modeling
Nonlinear modeling is empirical or semi-empirical modeling that takes at least some nonlinearities into account. The older techniques include polynomial regression, linear regression with nonlinear terms, and nonlinear regression. These techniques have several disadvantages compared to the new techniques of nonlinear modeling based on free-form nonlinearities.

Nonlinear modeling can also be performed with feed-forward neural networks, series of basis functions, multivariate splines, kernel regression, and other techniques. Among these new techniques, feed-forward neural networks have turned out to be particularly valuable in materials science and chemical engineering1. Feed-forward neural networks have several features that make them better tools for nonlinear empirical modeling. Besides their universal approximation capability2, it is usually possible to produce nonlinear models with some extrapolation capabilities3.

Nonlinear Modeling in Materials Science
Nonlinear modeling has been utilized successfully for various materials including plastics, metals, concrete, semiconductors, mineral wools, glass, etc. Different materials have different characteristics — different raw materials, different compositions, and are produced by different processes. The process may be a batch, continuous, or fed-batch. However, some things are common to modeling of various kinds of material behavior. Material properties or product properties depend on composition variables, process variables, and dimension variables (Figure 1).

Figure 3: Shear strength degradation profiles for different initial inherent viscosities

One would like to determine the best values of composition variables (or feed characteristics), process variables, and/or dimension variables such that the resulting material properties or product properties will be within desired limits. Sometimes, the composition variables or feed characteristics might be constants. In more common situations, the process variables may be constant or dependent variables, and the only degrees of freedom in materials development may be the composition of the feed, the amounts of raw materials, and possibly dimension variables.

The problem looks somewhat similar from the modeling point of view for a wide variety of materials. Nonlinear models of the kind shown in Figure 1 make product or materials development more efficient by reducing expensive experimentation.

Nonlinear Model of Biodegradation
ConMed Linvatec Biomaterials had carried out a large number of in vitro hydrolysis experiments in the past few years. A good amount of degradation data was available from those experiments. Some of that had missing measurements of one or more of the input variables, and therefore was not used. A total of 358 observations were found to be usable from that data for model development purposes.

A large number of nonlinear models were attempted using NLS 020 software. The model that was finally taken into use had good error statistics. The rms errors (root mean square error, roughly speaking, the standard deviation of the prediction error) of the models for shear strength and bending strength were a little more than the measurement inaccuracy and lack of repeatability. These models were then implemented in suitable software (called LUMET systems) to allow them to be used easily.

Figure 4: Bending strength degradation profiles for different initial inherent viscosities

Figure 2 shows the shear strength degradation curves for different glycolide contents, while keeping other variables constant. Higher glycolide contents reduce the life of the implant up to a certain extent. The starting strength is also lower with higher glycolide contents. Figure 3 shows shear strength degradation for different inherent viscosities. Higher inherent viscosities mean higher molecular weights and longer implant lives. The starting strength is only slightly better, however.

Figure 4 shows bending strength degradation for different inherent viscosities, which is qualitatively somewhat similar to Figure 3 for shear strength.

The main purpose for developing this kind of model is to speed up product development and reduce the requirement for experimentation. These nonlinear models can now be used to calculate suitable values of independent variables in presence of various constraints, which will result in a desired degradation profile with the mathematical tools implemented in the system.

Conclusion
Product development of medical devices based on bioabsorbable materials can be far more efficient than is usually carried out these days. Instead of a large number of trial and error experiments, a much smaller number of well-planned experiments could produce sufficient data for developing nonlinear models.

With suitable mathematical tools, it becomes easy to determine compositions, process variables, and/or dimensions that lead to desired degradation behavior.

In the example described in this article, nonlinear models of in vitro hydrolysis of polylactide/glycolide materials were developed from existing experimental data. The nonlinear model, due to its reliability, has been very useful in speeding up implant development. We expect that the requirement for expensive in vivo experimentation will also be reduced significantly.

References
1A. Bulsari (ed.), Neural Networks for Chemical Engineers, Elsevier, Amsterdam, 1995.
2K. Hornik, M. Stinchcombe, and H. White, “Multilayer feedforward networks are universal approximators,” Neural Networks, Vol. 2, (1989) 359-366.
3A. Bulsari, “Quality of nonlinear modelling in process industries,” Internal Report NLS/1998/2.

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