Mathematical Biology II Spatial Models and Biomedical Applications
Preface to the Third Edition
In the thirteen years since the first edition of this book appeared the growth of mathematical
biology and the diversity of applications has been astonishing. Its establishment
as a distinct discipline is no longer in question. One pragmatic indication is the increasing
number of advertised positions in academia, medicine and industry around the
world; another is the burgeoning membership of societies. People working in the field
now number in the thousands. Mathematical modelling is being applied in every major
discipline in the biomedical sciences. A very different application, and surprisingly
successful, is in psychology such as modelling various human interactions, escalation
to date rape and predicting divorce.
The field has become so large that, inevitably, specialised areas have developed
which are, in effect, separate disciplines such as biofluid mechanics, theoretical ecology
and so on. It is relevant therefore to ask why I felt there was a case for a new edition of
a book called simply Mathematical Biology. It is unrealistic to think that a single book
could cover even a significant part of each subdiscipline and this new edition certainly
does not even try to do this. I feel, however, that there is still justification for a book
which can demonstrate to the uninitiated some of the exciting problems that arise in
biology and give some indication of the wide spectrum of topics that modelling can
address.
In many areas the basics are more or less unchanged but the developments during
the past thirteen years have made it impossible to give as comprehensive a picture of the
current approaches in and the state of the field as was possible in the late 1980s. Even
then important areas were not included such as stochastic modelling, biofluid mechanics
and others. Accordingly in this new edition only some of the basic modelling concepts
are discussed—such as in ecology and to a lesser extent epidemiology—but references
are provided for further reading. In other areas recent advances are discussed together
with some new applications of modelling such as in marital interaction (Volume I),
growth of cancer tumours (Volume II), temperature-dependent sex determination (Volume
I) and wolf territoriality (Volume II). There have been many new and fascinating
developments that I would have liked to include but practical space limitations made
it impossible and necessitated difficult choices. I have tried to give some idea of the
diversity of new developments but the choice is inevitably prejudiced.
As to general approach, if anything it is even more practical in that more emphasis
is given to the close connection many of the models have with experiment, clinical
data and in estimating real parameter values. In several of the chapters it is not yet
possible to relate the mathematical models to specific experiments or even biological
entities. Nevertheless such an approach has spawned numerous experiments based as
much on the modelling approach as on the actual mechanism studied. Some of the more
mathematical parts in which the biological connection was less immediate have been
excised while others that have been kept have a mathematical and technical pedagogical
aim but all within the context of their application to biomedical problems. I feel even
more strongly about the philosophy of mathematical modelling espoused in the original
preface as regards what constitutes good mathematical biology. One of the most exciting
aspects regarding the new chapters has been their genuine interdisciplinary collaborative
character. Mathematical or theoretical biology is unquestionably an interdisciplinary
science par excellence.
The unifying aim of theoretical modelling and experimental investigation in the
biomedical sciences is the elucidation of the underlying biological processes that result
in a particular observed phenomenon, whether it is pattern formation in development,
the dynamics of interacting populations in epidemiology, neuronal connectivity
and information processing, the growth of tumours, marital interaction and so on. I
must stress, however, that mathematical descriptions of biological phenomena are not
biological explanations. The principal use of any theory is in its predictions and, even
though different models might be able to create similar spatiotemporal behaviours, they
are mainly distinguished by the different experiments they suggest and, of course, how
closely they relate to the real biology. There are numerous examples in the book.
Why use mathematics to study something as intrinsically complicated and ill understood
as development, angiogenesis, wound healing, interacting population dynamics,
regulatory networks, marital interaction and so on? We suggest that mathematics,
rather theoretical modelling, must be used if we ever hope to genuinely and realistically
convert an understanding of the underlying mechanisms into a predictive science. Mathematics
is required to bridge the gap between the level on which most of our knowledge
is accumulating (in developmental biology it is cellular and below) and the macroscopic
level of the patterns we see. In wound healing and scar formation, for example, a mathematical
approach lets us explore the logic of the repair process. Even if the mechanisms
were well understood (and they certainly are far from it at this stage) mathematics would
be required to explore the consequences of manipulating the various parameters associated
with any particular scenario. In the case of such things as wound healing and
cancer growth—and now in angiogensesis with its relation to possible cancer therapy—
the number of options that are fast becoming available to wound and cancer managers
will become overwhelming unless we can find a way to simulate particular treatment
protocols before applying them in practice. The latter has been already of use in understanding
the efficacy of various treatment scenarios with brain tumours (glioblastomas)
and new two step regimes for skin cancer.
The aim in all these applications is not to derive a mathematical model that takes
into account every single process because, even if this were possible, the resulting model
would yield little or no insight on the crucial interactions within the system. Rather the
goal is to develop models which capture the essence of various interactions allowing
their outcome to be more fully understood. As more data emerge from the biological
system, the models become more sophisticated and the mathematics increasingly challenging.
Download
http://s18.alxa.net/s18/srvs2/02/003...plications.rar
Preface to the Third Edition
In the thirteen years since the first edition of this book appeared the growth of mathematical
biology and the diversity of applications has been astonishing. Its establishment
as a distinct discipline is no longer in question. One pragmatic indication is the increasing
number of advertised positions in academia, medicine and industry around the
world; another is the burgeoning membership of societies. People working in the field
now number in the thousands. Mathematical modelling is being applied in every major
discipline in the biomedical sciences. A very different application, and surprisingly
successful, is in psychology such as modelling various human interactions, escalation
to date rape and predicting divorce.
The field has become so large that, inevitably, specialised areas have developed
which are, in effect, separate disciplines such as biofluid mechanics, theoretical ecology
and so on. It is relevant therefore to ask why I felt there was a case for a new edition of
a book called simply Mathematical Biology. It is unrealistic to think that a single book
could cover even a significant part of each subdiscipline and this new edition certainly
does not even try to do this. I feel, however, that there is still justification for a book
which can demonstrate to the uninitiated some of the exciting problems that arise in
biology and give some indication of the wide spectrum of topics that modelling can
address.
In many areas the basics are more or less unchanged but the developments during
the past thirteen years have made it impossible to give as comprehensive a picture of the
current approaches in and the state of the field as was possible in the late 1980s. Even
then important areas were not included such as stochastic modelling, biofluid mechanics
and others. Accordingly in this new edition only some of the basic modelling concepts
are discussed—such as in ecology and to a lesser extent epidemiology—but references
are provided for further reading. In other areas recent advances are discussed together
with some new applications of modelling such as in marital interaction (Volume I),
growth of cancer tumours (Volume II), temperature-dependent sex determination (Volume
I) and wolf territoriality (Volume II). There have been many new and fascinating
developments that I would have liked to include but practical space limitations made
it impossible and necessitated difficult choices. I have tried to give some idea of the
diversity of new developments but the choice is inevitably prejudiced.
As to general approach, if anything it is even more practical in that more emphasis
is given to the close connection many of the models have with experiment, clinical
data and in estimating real parameter values. In several of the chapters it is not yet
possible to relate the mathematical models to specific experiments or even biological
entities. Nevertheless such an approach has spawned numerous experiments based as
much on the modelling approach as on the actual mechanism studied. Some of the more
mathematical parts in which the biological connection was less immediate have been
excised while others that have been kept have a mathematical and technical pedagogical
aim but all within the context of their application to biomedical problems. I feel even
more strongly about the philosophy of mathematical modelling espoused in the original
preface as regards what constitutes good mathematical biology. One of the most exciting
aspects regarding the new chapters has been their genuine interdisciplinary collaborative
character. Mathematical or theoretical biology is unquestionably an interdisciplinary
science par excellence.
The unifying aim of theoretical modelling and experimental investigation in the
biomedical sciences is the elucidation of the underlying biological processes that result
in a particular observed phenomenon, whether it is pattern formation in development,
the dynamics of interacting populations in epidemiology, neuronal connectivity
and information processing, the growth of tumours, marital interaction and so on. I
must stress, however, that mathematical descriptions of biological phenomena are not
biological explanations. The principal use of any theory is in its predictions and, even
though different models might be able to create similar spatiotemporal behaviours, they
are mainly distinguished by the different experiments they suggest and, of course, how
closely they relate to the real biology. There are numerous examples in the book.
Why use mathematics to study something as intrinsically complicated and ill understood
as development, angiogenesis, wound healing, interacting population dynamics,
regulatory networks, marital interaction and so on? We suggest that mathematics,
rather theoretical modelling, must be used if we ever hope to genuinely and realistically
convert an understanding of the underlying mechanisms into a predictive science. Mathematics
is required to bridge the gap between the level on which most of our knowledge
is accumulating (in developmental biology it is cellular and below) and the macroscopic
level of the patterns we see. In wound healing and scar formation, for example, a mathematical
approach lets us explore the logic of the repair process. Even if the mechanisms
were well understood (and they certainly are far from it at this stage) mathematics would
be required to explore the consequences of manipulating the various parameters associated
with any particular scenario. In the case of such things as wound healing and
cancer growth—and now in angiogensesis with its relation to possible cancer therapy—
the number of options that are fast becoming available to wound and cancer managers
will become overwhelming unless we can find a way to simulate particular treatment
protocols before applying them in practice. The latter has been already of use in understanding
the efficacy of various treatment scenarios with brain tumours (glioblastomas)
and new two step regimes for skin cancer.
The aim in all these applications is not to derive a mathematical model that takes
into account every single process because, even if this were possible, the resulting model
would yield little or no insight on the crucial interactions within the system. Rather the
goal is to develop models which capture the essence of various interactions allowing
their outcome to be more fully understood. As more data emerge from the biological
system, the models become more sophisticated and the mathematics increasingly challenging.
Download
http://s18.alxa.net/s18/srvs2/02/003...plications.rar