Mathematical Models for Dental Materials Research
In the introduction section, the authors described how they feel about mathematics.
“I began to see that mathematics is not only understandable and useful but also intellectually rewarding and beautiful. Especially when freed from the artificial constraints of examinations and competition, mathematics can be an engaging and intellectually satisfying pastime, with unexpected relevance to normal life.”
It’s feel like the subject is being romanticised, I see their points though. When I read just for the sake of reading and pure enjoyment of it, it usually brings different perspectives and new ideas. At times these can be serendipitously useful in many different aspects of life.
The book focuses on how to
1. construct a mathematical model for the behaviour of dental materials by making informed assumptions of the relationship between cause and effect;
2. simplify the model by making appropriate simplifications;
3. calibrate the model by calculating the key parameters using experimental results; and, finally,
4. refine the model when there are discrepancies between predictions and experiments
I find that the way we study nature or scientific method relies heavily on models. It doesn’t really matter what models we are talking about. Mathematical, computer or even clinical trials, we are trying to build a model that can be linked to natural phenomena in some way (e.g. computer model-validation process, clinical trial-random sampling and statistics). Models are, therefore, simple representative of natural phenomena or a real world. We study model because real world is too hard to study, too messy, too chaotic.
Benefits of mathematical models
They are cheap, simple and repeatable. The advantage of non-destructive methods is that the same region of interest can be measured throughout the test, thus minimising the sample size and uncertainties due to inherent differences among biological samples. A more mechanistic theory may be needed in the design and performance optimization of dental restorations, which can account for their complex shapes and stress distributions. The analysis based on mathematical or computer model is essential before committing to extensive laboratory or even clinical studies, which can be expensive and prone to greater uncertainties and variations associated with, for example, those among a group of patients.
Structural failure
The largest flaw is not necessarily the weakest if this flaw is situated in a region or oriented in a direction such that it is not highly stressed (e.g. non-load/stress bearing area). As a result, when testing the load capacity or durability of dental restorations, even when they are anatomically similar, the measured failure loads or lifetimes could exhibit large variations.
A corollary of the weakest-link theory is the so-called size effect, i.e. the bigger the volume of a specimen, the lower its survival probability under the same load intensity. Intuitively, the bigger is a structure, the more likely it will contain a large failure-causing flaw and, hence, fail at a lower load.
The survival probability of the structure can then be determined by integrating the stress-dependent Weibull function over its entire volume. Thus, provided the material properties, including the Weibull parameters, of the restorative materials are known, the survival probability of a dental restoration of any shape and loading can be predicted using the weakest-link theory, with stresses given by the finite element method. This is a powerful approach that allows new designs of dental restorations to be evaluated effectively prior to actual mechanical testing or clinical trials