*How nonlinear models of crystallinity and strength of polylactide/glycolide implants speed up product development by Abhay Bulsari, Nonlinear Solutions, Mikko Huttunen, Kimmo Lähteenkorva and Esa Suokas, Conmed Linvatec Biomaterials *

*Figure 1. Products of Conmed Linvatec Biomaterials like SmartPin are produced from extruded rods, like the ones shown here with (white) and without (transparent) ceramic components*

Although a variety of bioabsorbable materials are available, polylactides are the most commonly used bioabsorbable polymer materials for clinical implants. By producing copolymers with L-lactide, D-lactide and glycolide in different proportions, their degradation dynamics as well as mechanical properties can be tailored to suit the application.

Most of the implant development work today is done by trial and error experimentation which is inefficient and expensive. Raw material suppliers offer only a limited variety of materials, and the implant developers meekly select from that. Conmed Linvatec Biomaterials instead worked with the suppliers and tested non-standard polylactide/glycolide compositions extruded under different conditions. Since experimentation was quite expensive and time-consuming, there was no sense in carrying out a lot of trial and error experiments. Instead, nonlinear models of shear strength, bending (flexural) strength and crystallinity were developed from limited experimental data combined with production data.

**Mathematical modelling**

Mathematical models represent knowledge of quantitative effects of relevant variables in a concise and precise form. Mathematical modelling can be performed in various ways and different ways are suitable in different situations. Physical or phenomenological modelling is not particularly suitable for predicting material behaviour. Physical modelling usually requires assumptions and simplifications.

Empirical and semi-empirical modelling do not need any major assumptions or simplifications. They simply describe the observed behaviour of a material or a process. It is feasible when the relevant variables are measurable, as is often the case. Conventional techniques of empirical modelling are linear statistical techniques. These tend to have limitations because nothing in nature is very linear and particularly so in materials science. It therefore makes sense to use better techniques which take nonlinearities into account.

**Nonlinear modelling **

Nonlinear modelling can be performed with several techniques including feed-forward neural networks. They have turned out to be particularly valuable in materials science and chemical engineering [1]. Besides their universal approximation capability [2], it is usually possible to produce nonlinear models with some extrapolation capabilities [3].

**Nonlinear modelling in materials science**

Nonlinear modelling has been utilised successfully for various materials [4] including plastics, metals, concrete, mineral wools, semiconductors, glass, etc. Some things are common to modelling material behaviour. Material or product properties depend on composition variables, process variables and dimension variables, as summarised in figure 2.

Figure 2. Material properties depend on composition variables, process variables and dimension variables

**Nonlinear models of strength and crystallinity**

After some amount of pre-processing and data analysis, a large number of nonlinear models in the form of feed-forward neural networks were attempted using the NLS 020 software from the available data for shear strength, bending strength and crystallinity. One or more parameters of the models were fixed or restricted. A fairly large amount of data was available for shear strength from several years. 330 observations were usable for model development. The nonlinear model which was taken into use had the following prediction error characteristics.

Figure 3 shows a comparison of the measured values of shear strength and the values predicted by the nonlinear model. Unlike linear regression models, this model will never predict negative values of shear strength.

For bending strength, only 67 observations were available, but the quality of the data seemed to be better. As a result, the models were also better. The nonlinear model for bending strength which was taken into use had the following prediction error characteristics.

Figure 4 shows a comparison of the measured values of bending strength and the values predicted by the nonlinear model.

Figure 3. Comparison of measured values with values of shear strength predicted by the nonlinear model

Figure 4. Comparison of measured values with values of bending strength predicted by the nonlinear model

For crystallinity, only 63 observations were available. The nonlinear model for crystallinity which was taken into use had the following prediction error characteristics.

The nonlinear models were then implemented in a LUMET system, which is a set of software components for efficient use of nonlinear models. It was then used to see the effects of various input variables on strengths and crystallinity. Figure 5 shows the effect of D-lactide content on shear strength for different draw ratios, while keeping other input variables constant. Figure 6 shows the effect of draw ratio on shear strength at different area to volume ratios. Figure 7 shows the effect of glycolide content on crystallinity for different inherent viscosities.

Figure 5. Effect of D-lactide content on shear strength for different draw ratios

Figure 6. Effect of draw ratio on shear strength at different area to volume ratios

Figure 7. Effect of glycolide content on crystallinity for different inherent viscosities

It is now easy to determine compositions which will lead to a desired combination of shear strength, bending strength and crystallinity. More variables can be added when suitable data becomes available.

**Conclusion**

Implant development work is expensive and time-consuming. It usually aims at achieving a desired combination of a few material properties. Those material properties depend on the composition of the material, process variables of extrusion and dimension variables. The relations between these variables tend to be complicated and nonlinear. New techniques of nonlinear modelling are very effective for describing material behaviour.

Implant development work can be speeded up significantly by nonlinear models. It results in savings on experimentation costs as well as development time. The final result is better than what is usually achieved by trial and error experimentation.

*References*

*[1] A. Bulsari (ed.), Neural Networks for Chemical Engineers, Elsevier, Amsterdam, 1995*

*[2] Hornik, K., Stinchcombe, M. and White, H., “Multilayer feedforward networks are universal approximators,” Neural Networks, Vol. 2, (1989) 359-366*

*[3] A. Bulsari, ”Quality of nonlinear modelling in process industries”, Internal Report NLS/1998/2*

*[4] http://www.nonlinear-solutions-oy.com/articles.html*