1.3 Mathematical Modeling in Biology
3
bution, mechanobiology remained relatively dormant until the 1980s, when Fung’s
ideas on tissue engineering 2 and residual stress 3 in arteries helped reinvigorate the
discipline (Fung 1991). The advent of molecular biology also has had a major
impact on the field.
Both fields have flourished since the 1980s. Compared to the relatively mature
field of biomechanics, however, interest in mechanobiology has intensified considerably during the last two decades.
1.3 Mathematical Modeling in Biology
This book focuses primarily on mathematical modeling. While experimentation
is indispensable in scientific research, mathematical models provide a valuable
complement to experiments, as they help us understand, not just describe, the
behavior of a system. In fact, mathematical modeling has been associated with
biomechanics for centuries. While Aristotle’s ideas concerning animal motion can
be considered qualitative models, da Vinci and Galileo brought mathematics into
the field.
During the last 20 years, the use of mathematical models in mechanobiology has
become increasingly widespread. Finite-element methods make solving even the
most complex problems possible. Significant difficulties remain, however.
1.3.1 Some of the Challenges
Numerous factors make modeling in biomechanics and mechanobiology a challenging endeavor. On the physical side, complex 3D geometry, large deformation, timedependent material properties, active force generation (e.g., contraction), multiscale
effects, and instabilities can test even the most sophisticated finite-element codes.
In addition, some problems involve interactions between mechanical, biochemical,
electrical, and other factors.
On the biology side, other significant complications enter the picture. Here, we
mention three. First, the results from experiments can be highly variable, often
making precise quantitative agreement between theory and experiment an unreasonable goal. Second, experimental perturbations used to test model predictions may
trigger an adaptive response that alters behavior through mechanisms not included
in a model. Third, biological systems typically contain large numbers of unknown
parameters, many of which cannot be measured directly. Because of the assumptions
2 Interestingly, tissue engineering has been around since at least the mid-1900s (Hayashi and
Boehm 1952)
3 Residual stress is the stress remaining in a body when all external loads are removed.
3
bution, mechanobiology remained relatively dormant until the 1980s, when Fung’s
ideas on tissue engineering 2 and residual stress 3 in arteries helped reinvigorate the
discipline (Fung 1991). The advent of molecular biology also has had a major
impact on the field.
Both fields have flourished since the 1980s. Compared to the relatively mature
field of biomechanics, however, interest in mechanobiology has intensified considerably during the last two decades.
1.3 Mathematical Modeling in Biology
This book focuses primarily on mathematical modeling. While experimentation
is indispensable in scientific research, mathematical models provide a valuable
complement to experiments, as they help us understand, not just describe, the
behavior of a system. In fact, mathematical modeling has been associated with
biomechanics for centuries. While Aristotle’s ideas concerning animal motion can
be considered qualitative models, da Vinci and Galileo brought mathematics into
the field.
During the last 20 years, the use of mathematical models in mechanobiology has
become increasingly widespread. Finite-element methods make solving even the
most complex problems possible. Significant difficulties remain, however.
1.3.1 Some of the Challenges
Numerous factors make modeling in biomechanics and mechanobiology a challenging endeavor. On the physical side, complex 3D geometry, large deformation, timedependent material properties, active force generation (e.g., contraction), multiscale
effects, and instabilities can test even the most sophisticated finite-element codes.
In addition, some problems involve interactions between mechanical, biochemical,
electrical, and other factors.
On the biology side, other significant complications enter the picture. Here, we
mention three. First, the results from experiments can be highly variable, often
making precise quantitative agreement between theory and experiment an unreasonable goal. Second, experimental perturbations used to test model predictions may
trigger an adaptive response that alters behavior through mechanisms not included
in a model. Third, biological systems typically contain large numbers of unknown
parameters, many of which cannot be measured directly. Because of the assumptions
2 Interestingly, tissue engineering has been around since at least the mid-1900s (Hayashi and
Boehm 1952)
3 Residual stress is the stress remaining in a body when all external loads are removed.