1.3 Mathematical Modeling in Biology
5
1.3.3 The Art of Modeling
Developing a realistic model for a biological system is, in many ways, an art. In this
regard, it is helpful to follow advice often attributed to Einstein:
Everything must be made as simple as possible, but no simpler.
Although complex models may be more realistic, models that include unnecessary detail can make it considerably more difficult to gain insight into fundamental
behavior. One systematic way to follow Einstein’s advice is to begin with a relatively
simple model and add complexities as needed, one step at a time. This strategy
makes it easier to determine the importance of each added factor. If a new feature
has a relatively small effect on the results, it can be omitted at least until the end,
when its effects should be checked again.
But which factors need to be included before a model is good enough? That
depends on the purpose of the model. For example, if we want to compute wall stress
in an artery shortly after a sudden pressure change, there is no need to consider how
the artery remodels in response to the altered loading. Remodeling takes time. If the
pressure change occurs within seconds, elasticity may be a reasonable assumption,
but if the pressure increases over dozens of minutes, viscoelastic effects may be
significant.
These issues are related to the time scale. The spatial scale is just as important.
For instance, if we want to compute the average stress in the artery wall, a thinwalled membrane approximation is fine. However, computing stress distributions
across the wall requires a thick-walled model. Unless we need to compute the
stresses experienced by individual cells, there is no need to include details at the
cell level.
Geometry is another important consideration. With the power of today’s computers, there is a trend toward developing patient-specific models. This is useful, for
example, if the goal is to diagnose disease or predict the effects of specific treatments
in a given individual. Still, we must always be aware of limitations. Even the most
geometrically accurate model can yield unreliable results if material properties or
other important information is unknown.
On the other hand, if our intent is to understand fundamental behavior, then realistic geometry may just complicate matters unnecessarily. In this case, qualitative
trends may be more important than quantitative accuracy. In fact, relatively simple
geometries sometimes produce surprisingly accurate results with considerably less
effort.
Finally, as already mentioned, the inherent complexity and variability of biological systems usually make precise agreement between numerical and experimental
results an unrealistic goal. In general, we can consider a model to be a useful
representation of the actual system if it yields the correct qualitative trends for all
experimental tests.
Incidentally, it is a misconception that a mathematical model always can be
made to match any set of data if it is complex enough and appropriate parameter
5
1.3.3 The Art of Modeling
Developing a realistic model for a biological system is, in many ways, an art. In this
regard, it is helpful to follow advice often attributed to Einstein:
Everything must be made as simple as possible, but no simpler.
Although complex models may be more realistic, models that include unnecessary detail can make it considerably more difficult to gain insight into fundamental
behavior. One systematic way to follow Einstein’s advice is to begin with a relatively
simple model and add complexities as needed, one step at a time. This strategy
makes it easier to determine the importance of each added factor. If a new feature
has a relatively small effect on the results, it can be omitted at least until the end,
when its effects should be checked again.
But which factors need to be included before a model is good enough? That
depends on the purpose of the model. For example, if we want to compute wall stress
in an artery shortly after a sudden pressure change, there is no need to consider how
the artery remodels in response to the altered loading. Remodeling takes time. If the
pressure change occurs within seconds, elasticity may be a reasonable assumption,
but if the pressure increases over dozens of minutes, viscoelastic effects may be
significant.
These issues are related to the time scale. The spatial scale is just as important.
For instance, if we want to compute the average stress in the artery wall, a thinwalled membrane approximation is fine. However, computing stress distributions
across the wall requires a thick-walled model. Unless we need to compute the
stresses experienced by individual cells, there is no need to include details at the
cell level.
Geometry is another important consideration. With the power of today’s computers, there is a trend toward developing patient-specific models. This is useful, for
example, if the goal is to diagnose disease or predict the effects of specific treatments
in a given individual. Still, we must always be aware of limitations. Even the most
geometrically accurate model can yield unreliable results if material properties or
other important information is unknown.
On the other hand, if our intent is to understand fundamental behavior, then realistic geometry may just complicate matters unnecessarily. In this case, qualitative
trends may be more important than quantitative accuracy. In fact, relatively simple
geometries sometimes produce surprisingly accurate results with considerably less
effort.
Finally, as already mentioned, the inherent complexity and variability of biological systems usually make precise agreement between numerical and experimental
results an unrealistic goal. In general, we can consider a model to be a useful
representation of the actual system if it yields the correct qualitative trends for all
experimental tests.
Incidentally, it is a misconception that a mathematical model always can be
made to match any set of data if it is complex enough and appropriate parameter