MATHEMATICAL
MODELS
IN
THE
HEALTH SCIENCES
A
Computer-Aided
Approach
This page intentionally left blank
MATHEMATICA
L
MODELS
IN THE
HEALTH SCIENCES
A
Computer-Aided
Approach
Eugene Ackerman, Ph.D.
Profess
or
Lael
Cranmer Gatewood, Ph.D.
Associate
Professor
and
Director
Health Computer Sciences
University
of
Minnesota

UNIVERSITY
OF
MINNESOTA PRESS
D
MINNEAPOLIS
Copyright
©
1979
by the
University
of
Minnesota.
All
rights
reserved.
Published
by the
University Minnesota Press,
2037 University Avenue Southeast, Minneapolis, Minnesota
55455
Printed
in the
United States
of
America
at
North Central Publishing Company,
St.
Paul
Library

of
Congress Cataloging
in
Publication Data
Ackerman,
Eugene,
1920-
Mathematical models
in the
health sciences.
Bibliography:
p.
Includes index.
1.
Medicine—Mathematical
models.
2.
Medicine—Data
processing.
I.
Gatewood,
Lael
Cranmer,
joint author.
II.
Title.
R858.A36
610'.28'54
79-9481
ISBN

0-8166-0864-4
The
University
of
Minnesota
is an
equal-opportunity educator
and
employer.
Preface
Mathematical
techniques have long
been
employed
in the
biological,
medical,
and
related health disciplines. Within
the
past
few
decades,
the
frequency
of
such applications
has
increased
significantly,

as can be
seen
by
scanning current literature
in a field
such
as
physiology. This change
has
been
made possible
by the
availability
of
electronic aids
to
computa-
tion
and by the
development
of
appropriate numeric
and
graphic
methodologies.
The
most ubiquitous mathematical techniques
as
applied
to

biomedical
areas have
been
grouped together under
the
title biostatistics. Probably
all
quantitative studies incorporate statistical methodology,
at
least
to a
limited
degree.
Numerous textbooks have
been
written about biostatis-
tics,
its
subdisciplines,
and its
applications
to the
health sciences.
On the
other hand,
there
exist
a
variety
of

mathematical techniques that
are
employed
in the
health sciences
but
that
are not
primarily statistical
in
nature. These
are
called mathematical modeling
and
form
the
basis
for the
various
topics discussed
in
this book.
Computer technology
has
made possible many
of the
applications
of
mathematics
to

biology
and
medicine. Accordingly, computer programs,
graphics
and
tabular output,
and
block diagrams
are
included
in the
illus-
trative material throughout
the
text.
It is
assumed that
the
reader
has had
previous exposure
to
scientific
computing,
but
specific
knowledge
of a
programming language
is not

required. Thus
the
text
is
concerned
explicitly
with selected topics
from
the
biological
and
health sciences
for
which computers have
been
a
natural tool
for
analysis.
v
vi
Preface
One of the first
reactions that
a
knowledgeable reader
may
have when
looking
at the

table
of
contents
is a
sense
of the
incompleteness
of the
topics covered.
The
pedagogic technique followed here
is
sometimes
re-
ferred
to as a
block-and-gap
method.
The
entire
field of
mathematical
modeling
is
divided into
a
group
of
blocks with intervening gaps.
The

blocks
are
discussed
as
fully
as
space permits;
the
topics
in the
gaps
are
simply
omitted.
It is the
intention
to
emphasize
in
this
fashion
the
general
philosophic approach
as
well
as to
present
specific
methodologies

and
applications
whose importance will
not
fade
too
rapidly. Such
a
selection
is
clearly
a
compromise,
but one
that proves
useful
to a
variety
of
types
of
students.
A
text concerned with biomedical applications
of
mathematics must
perforce
refer
to a
variety

of
areas
of
biology
and
medicine.
It
seems
unreasonable
to
assume
that
all
readers
will
be
equally familiar with
all of
the
areas included.
If the
book
is to be
more than
a
collection
of
recipes,
some
knowledge

of the
significance
and
implications
of the
areas
of
appli-
cation
is
necessary. References
are
given
to
allow
the
interested reader
to
pursue each study more thoroughly. However,
it is
hoped that
the
sup-
plemental material presented with each example
is
adequate
in
itself
for
many

readers.
The
book
has
been
written with
the
hope
that
it
will
be
used
as a
text
for
courses
at the
graduate level.
The
emphasis
has
been placed
on the
mathematical techniques rather than
on
detailed derivations.
The
latter
are the

logical
justification
for the
techniques discussed.
On the
other
hand,
a
text
on
biomedical applications must assume that
the
interested
reader will have mathematical books available that develop
the
underly-
ing
proofs
to the
degree
of
rigor that
is
desired. Such knowledge will
augment
the
understanding
of
readers
with more mathematical

interests,
but
others whose
training
and
research emphasize practical applications
should
find the
methodologies
as
presented here
to be
sufficient
in
them-
selves.
The
primary audience
for
whom this text
has
been written
are
students
in
the
program
of
Biometry
and

Health Information Systems
at the
Uni-
versity
of
Minnesota
who are
working toward
an
M.S.
or
Ph.D. degree.
They have
had
graduate courses
in
biostatistics,
biomedical
computing,
and at
least
one
area
of
biology,
as
well
as an
interest
in

quantitative,
analytical
approaches
to
biomedical studies.
For
such students this course
provides
an
introduction
to a
different
set of
mathematical
and
computer
methodologies applied
to the
health sciences.
The
book should also prove
useful
for
those working
in
other health-
related
and
biomedical sciences. Essentially, what
is

required
as
prereq-
Preface
vn
uisites
are
mathematics through calculus
and
advanced training
in
some
health science
or
biomedical
field. A
knowledge
of
biostatistics
and
com-
puter programming
may be
useful
in
following
some
of the
detailed
examples.

Readers
may find
some parts
of the
text overly
simplified
and
redundant, other parts
too far
from
their area
of
interest. However,
for
one
interested
in
quantitative approaches
to
biology
and
medicine most
of
the
text should prove
useful.
Mathematical
Models
in the
Health Sciences

may
also
be of
value
to
graduate
and
postdoctoral students
in
mathematics, computer science,
the
physical sciences,
and
engineering. They
may
have
been
exposed
to
thorough developments
of
mathematical
and
computer techniques
but
may
find
their
biological background
requires

more supplementation than
is
provided
in
this text. Nonetheless,
if
they wish
to
expand their knowl-
edge
of
biomedical applications
of
mathematics, this book
and its
refer-
ences should help
to
meet their needs.
All
of the
types
of
students described
in the
preceding paragraphs have
been included
in
courses entitled "Mathematical Biology"
and

taught
as
part
of the
graduate program
at the
University
of
Minnesota. Each
time
the
course
has
been
offered
student preparation
and
interests have
been
different.
Attempts were made
to
vary
the
content
and
even
the
emphasis
of

the
course
to
meet
the
perceived needs
of the
class
as
well
as to
include
some
of the
current interests
of the
instructors.
In
addition
to the
formal
lectures,
the
course
at the
University
of
Min-
nesota included individual
reports

and
homework assignments.
These
reports, presented both orally
and in
writing, have encouraged greater
library utilization.
The
other
out-of-class
assignments have included
computer-based problems that increased
familiarity
with
the
locally avail-
able computer resources. Topics
for
reports
and
problems were obtained
from
current references similar
to
those presented
at the end of
each
chapter.
These classes
in

Mathematical Biology have resulted
in
extensive stu-
dent participation
and
interaction. Although this varied
from
one
person
to the
next,
all
contributed
in
some
fashion
to the
selection
of
applications
and
examples.
The
authors
gratefully
acknowledge their help
and
advice.
Numerous
of the

authors' colleagues have also provided assistance
in one
fashion
or
another. Particularly deserving
of
acknowledgment
is Dr.
Lynda
Ellis,
who
originally suggested including
the
material
in
Chapter
13,
leading
to a
major
revision
in the
selected
chapters.
Several groups have supported
in
part
the
preparation
of

this text.
These include
the
Northwest Area Foundation
as
well
as the
Biotechnol-
ogy
Research Resource Facility,
the
College
of
Pharmacy
and the De-
viii
Preface
partment
of
Laboratory Medicine
and
Pathology
of the
University
of
Minnesota.
In
order
to
complete this text,

the
senior author spent
a
year
on
sabbatical leave
at the
University
of
Washington's Department
of
Lab-
oratory Medicine.
The
help
of the
latter
faculty
is
also
gratefully
acknowl-
edged.
The
text would
not
have
been
possible without
the

typing
and
editorial support provided
by
Mrs. Margie Henry,
Ms.
Kathy
Seidl,
and
Dr.
Margaret Ewing.
E.A.
L.C.G.
Contents
Preface
v
INTRODUCTION
1
Chapter
1
Models
and
Goals
3
A.
Origins
and
Definitions
3
B.

Automated Computational
Aids
5
C.
Deterministic
and
Stochastic Models
6
D.
Inverse
Solutions
7
E.
Model Conformation
and
Parameter
Estimation
8
F.
Health-Related Goals
11
G.
Notation Used
in
Text
13
H.
Summary
14
DETERMINISTIC MODELS

17
Chapter
2
Compartmental
Analysis
19
A.
Illustrative
Examples
19
B.
Compartmental Analysis
24
C.
Single Compartment Models
27
D.
Parameter Estimation
32
E.
Multicompartment Models
34
ix
x
Contents
F.
Computer Simulation
38
G.
Non-Linear Parameter Estimation

41
H.
Model Selection
and
Validation
45
I.
Summary
49
Chapter
3
Modified Compartmental Analysis
53
A.
Extensions
of
Compartmental Analysis
53
B.
Blood Glucose Regulation
54
C.
Ceruloplasmin Synthesis
64
D.
Dye
Dilution Curves
68
E.
Lung Models

69
F.
Summary
72
Chapter
4
Enzyme Kinetics
76
A.
Enzymes
and
Biology
76
B.
Proteins
and
Amino Acids
77
C.
Prosthetic Groups, Cofactors,
and
Coenzymes
80
D.
Molecular Conformation
and
Chemical
Reactions
82
E.

Michaelis-Menten
Kinetics
85
F.
Estimation
of
Michaelis-Menten Parameters
88
G.
Catalase
and
Peroxidase Reactions
92
H.
Enzyme Kinetics
and
Mathematical Biology
96
Chapter
5
Enzyme Systems
99
A.
Transient Kinetics
99
B.
Perturbation Kinetics
100
C.
King-Altman

Patterns
104
D.
Metabolic Pathways
107
E.
Oxidative
Phosphorylation
109
F.
Simulation
of
Multienzyme Systems
113
G.
Summary
121
TIME
SERIES
123
Chapter
6
Discrete
Time Series
125
A.
Introduction
125
B.
Analog

to
Digital Signal Conversion
126
Contents
xi
C.
Fourier Transforms
128
D.
Discrete Fourier Transforms
138
E.
Fast Fourier Transforms
142
F.
Laplace Transforms
148
G.
Sampling Theorems
150
H.
Summary
155
Chapter
7
Transforms
and
Transfer Functions
157
A.

Transfer Functions
157
B.
Convolution Integrals
159
C.
Compartmental Analysis
164
D.
Dye
Dilution Curves
169
E.
Fast Walsh Transforms
172
F.
Applications
175
Chapter
8
Electrocardiographic Interpretation
178
A.
Physiological Basis
178
B.
EKG
Characteristics
182
C. VKG

Patterns
185
D.
Abnormalities
189
E.
Simulation
and the
Inverse Problem
191
F.
Automated Interpretation
of the EKG 197
G.
Automated Aids
to
Clinical Diagnosis
200
H.
Summary
202
Chapter
9
Electroencephalographic
Analyses
206
A.
Central Nervous System
206
B.

EEC
Characteristics
209
C.
Applications
of
EEC
Patterns
213
D.
Sleep Stages
214
E.
Spectral Analyses
216
F.
Compressed Spectral
and
Other
Analyses
220
G.
Spatial Analyses
224
H.
Evoked Response Averages
227
I.
Automation
and the

EEC
229
INFORMATION
AND
SIMULATION
233
xii
Contents
Chapter
10
Information
Theory
235
A.
Basic Concepts
235
B.
Messages
and
Entropy
238
C.
Redundancy
239
D.
Continuous Signals
240
E.
Analog Digitization
243

F.
Discrete
Systems
244
G.
Health Sciences Applications
248
Chapter
11
Genetic Transfer
of
Information
250
A.
Genes
and
Chromosomes
250
B.
Cell Replication
and
Division
252
C.
Molecular
Basis
of
Genetics
253
D.

Information Content
of DNA 255
E.
Types
of
Genes
259
F.
RNA
and
Protein Synthesis
262
G.
Information Theory
and
Evolution
265
H.
Genetic Models
and
Evolution
267
Chapter
12
Simulation
of
Epidemics
271
A.
Epidemics

and
Epidemic Theory
271
B.
Simulation
of
Stochastic Models
274
C.
Simplest Stochastic Models
276
D.
Competition
and
Vaccination
282
E.
Structured Populations
289
F.
Influenza
Epidemic Model
293
G.
Overview
300
Chapter
13
Population, Ecology,
and the

World System
304
A.
Introduction: Population Models
304
B.
Exponential Growth
306
C.
Logistic
Growth
309
D.
Competition
and
Predator-Prey Interactions
312
E.
Other
Ecology Models
317
F.
World Systems Models
320
G.
Simulation
and
Prediction
325
H.

Summary
330
Contents
xiii
OVERVI
EW

333
Chapter
14
Mathematical Models
in the
Health Sciences .335
A.
Summary
of
Text
335
B.
Other
Areas
of
Mathematical Biology
337
C.
Other
Health
Science
Applications
339

D.
Health Computer Sciences
341
E.
Future Implications
342
Index
347
This page intentionally left blank
INTRODUCTION
Chapter
1 on
models
and
goals provides
an
overview
of the
philosophic approach taken
in
the
text.
It is
hoped that this chapter will
be
read
first and
then
reread
several times while

the
text
is
being used.
The
scientific
setting
of
the
text, references
to the
biomedical
litera-
ture,
and an
explanation
of the
notational
scheme used throughout
the
text
are
pre-
sented
here.
This page intentionally left blank
CHAPTER
1
Models
and

Goals
A.
Origins
and
Definitions
For
centuries scientists have used mathematical
functions
to
describe
the
observable world,
but the
early records
of
applications
of
mathematics
to
biological phenomena
are
difficult
to find. The
types
of
applications
selected
for
presentation
in

this text have
been
developed since
the
nineteenth century
by a
diverse group
of
scientists working
in
many
fields.
As
recently
as
1850
it was
possible
for one
person
to
acquire
the
skills
of a
physician, surgeon, physicist,
and
mathematician
as
exemplified

by
von
Helmholtz.
Until
the
introduction
of
digital computers,
the
studies
of
these scientists, individually
and in
groups,
were
usually
in the
areas
now
called biophysics. Examples include
von
Helmholtz's
and
Rayleigh's
studies
of
hearing
and
Einthoven's analyses
of

electrocardio-
grams. Rashevsky's group
at the
University
of
Chicago chose
the
term
mathematical biophysics
for
their
studies
of
diffusion,
permeability,
growth, metabolism,
and
neurobiology.
From perhaps 1900 activities
of
this nature grew
at an
exponential rate
but
with
a
long time constant.
Many
biologists
and

most clinicians
regarded
this growth
as an
oddity,
having
little
to do
with biology
or
medicine. However,
a
discipline
de-
scribed
as
mathematical biology began
to
emerge
as a
separate
field of
study
and
research although frequently
as
part
of
programs
still

called
biostatistics
or
biophysics.
The
introduction
of the
digital computer
and the
consequent technolog-
ical developments such
as
operating systems, high-level programming
3
4
Models
and
Goals
languages,
and
special simulation languages, caused
a
rapid change
in the
use of
mathematical models
for all
health sciences.
In the
1970s,

the
question
of
whether
a
separate
or
integrated discipline
devoted
to
mathematical
modeling
exists
is
competitively discussed
and
debated.
This text discusses selected applications
of
mathematics
to
biology,
to
medicine,
and to
other health-related disciplines
in
which
the
analyses

are
neither overly simplistic
nor
primarily biostatistical. These
qualifiers
imply
considerable personal judgment
by the
authors
as
influenced
by
their colleagues
and
students.
The use of
quantitative analytic techniques including mathematical
models
in
biology
and
medicine
is
often
termed mathematical biology.
However, many
different
concepts
or
relationships

are
suggested
by
this
term. Mathematical biology
and
biostatistics
are
often
combined
and
called
biomathematics,
and if
biomedical
computing
is
incorporated,
the
combination
is
sometimes called biometry. Some reserve
the
last word
for
biostatistics
per se.
Mathematical modeling
as
presented

in
this text
can
be
considered
an
essential part
of a
program
in
health computer sciences.
The
modeling techniques included
in
mathematical biology
are
inti-
mately
involved
in
many
other
interdisciplinary areas, such
as
physiology,
biophysics,
biochemistry, medical physics,
and
biomedical engineering.
Many

of the
topics discussed
in the
following
chapters
are
included
in
courses
in
these disciplines.
In
addition models have
been
used
in
many
other
health-related
areas,
including
epidemiology,
basic
health
sciences,
and
health services. Many hospitals
and
clinics
use

techniques derived
from
modeling studies
in
laboratory instruments, radiological treatment
planning,
resource allocation
and
scheduling,
and
other
facets
of
health
care
delivery.
Quantitation
in the
health sciences
is
dependent
on the use of
mathematical models. This approach
is
natural
to the
physicist,
the
chemist,
and the

engineer; they
often
do not
note
the
extent
to
which
they
use
models
or
abstractions
of
reality.
The
biological
and
health sci-
ences have
been
so
dominated
by
descriptive methodologies that
the use
of
mathematics requires
the
explicit

definition
of a
model. Biomedical
scientists,
often
unfamiliar
with this approach, sometimes
tend
to
expect
far
too
much
or to
accept
far too
little
of
what
a
study based
on a
mathematical model
can
offer.
Consequently
it is
important
in
mathemat-

ical
modeling
to
define
the
uses, goals,
and
validation
of
models.
The
remainder
of
this
chapter
is a
general
discussion
of
various
types
of
mod-
els,
as
these
bear
on the
goals
of

mathematical modeling
in the
health
sciences.
Models
and
Goals
5
B.
Automated
Computational
Aids
Before
the
introduction
of
computer technology,
it was
necessary
in
working
with mathematical models
of
biomedical systems
either
to
over-
simplify
and
approximate

to an
unacceptable
degree
or to
perform labori-
ous
numerical calculations
by
hand
or
with
a
desk calculator;
the
labor
cost
was
often
prohibitively high. Thus computer representation
has be-
come
a
necessary part
of
many mathematical models.
The
following
dis-
cussion explains this relationship
in

more
detail
by
considering
how
mathematical
models
are
used.
First,
a
quantitative representation
is
hypothesized
for the
relationship
among
variables within
the
model.
The
internal variables
may
involve,
for
example,
concentrations
and
their time derivatives
or

factory
output
and
pollution
indices. Customarily
the
model
is
then solved
to
describe
rela-
tionships
that
can be
observed experimentally, such
as the
plasma con-
centration
of one or
more tracers
as a
function
of
time
or
age-specific
attack rates during
an
epidemic.

These
examples
of
such
use are
discussed
in
other chapters.
The
solution
may
involve integrating
differential
equa-
tions,
but, depending
on the
model,
need
not be of
that
form.
Given specific details
for the
mathematical model,
the
solutions that
are
obtained
can

generally
be
represented
as
tables
of
numbers.
People
find it
difficult
to
recognize
the
information
contained
in
such
lists
of
numbers,
whereas they
can
quickly grasp
the
form
and
message
of a
well-con-
structed

graph.
If
many solutions
for
different
forms
of the
model
and
different
initial values
of
conditions
are
desired, numerous graphs
may be
needed.
The
computer allows
the
preparation
of
graphic displays
of
data
in
a
form
that
is

easier
to
modify
and is far
less expensive than
a
hand-
drawn presentation.
However,
the
frequent
use of
numeric calculations creates
a
basic
need
for
automated computational techniques.
The
models with which
it is
simplest
to
deal, namely, those that permit
a
closed solution, nevertheless
require calculations
to
express
the

solution
in a
form
that
can be
compared
with experimental results.
If
solutions
for
several
different
sets
of
initial
values
or for
several sets
of
pseudorandom numbers
are
desired,
the
manual
calculation task
may
become prohibitively expensive.
In
some
applications

the
model system
can be
solved only
by
numeric techniques.
In
others
it may
prove more convenient
to
solve
the
model
by
numeric
analysis
than
to
derive
and use a
closed-form solution.
Both analog computers
that
deal
with
continuous signals
and
digital
computers

that
deal with discrete numbers have
been
used
to aid in
numeric computation.
In the
early
1950s
many scientists preferred
the
6
Models
and
Goals
analog
computer because
of its
speed
and
accuracy, which
was
similar
to
that
of
experimental methodology. Subsequent experience
and
develop-
ment

of the
digital computer have proved that
the
latter
is
easier
to use for
most
purposes. Special digital computer languages that mimic analog
computers have made
the
advantages
of
both types
of
computers available
in
one.
Analog
computer techniques
are
still used
to
preprocess
continu-
ous
signals
from
biological systems. Except
for

that role,
the
digital com-
puter
is
today
the
necessary
and
essential apparatus
for a
health scientist.
C.
Deterministic
and
Stochastic Models
The
models used
in the
health
sciences
can be
classified
in
several
fashions.
One
system
differentiates
between deterministic

and
stochastic
models.
A
deterministic model
is one
that has, given
the
initial condi-
tions,
an
exact, determined solution that relates
the
dependent
variables
of
the
model
to
each other
and to the
independent variable
(or
variables).
In
contrast,
a
stochastic model
and its
solution involve probablistic con-

siderations.
Classical
physics
and
chemistry dealt almost exclusively with
deter-
ministic
models. This type
of
model
is
also popular
in
biomedical
studies.
Most
uses
of
tracers
are
based
on an
explicit
or
implicit deterministic
model. Enzyme kinetic models,
hydrodynamic
models
of the
cardiovascu-

lar
system,
and
other physiological models using physical
and
engineering
analogies
are,
by and
large, deterministic. Models
of
medical diagnosis
that have
a
dendritic pattern with definitive decisions
at
each
node
are
also
deterministic.
On the
other
hand modern quantum physics
and
chemistry have
turned
to
models that
are

stochastic
and
provide only
the
probability
of an
event occurring rather than
a
statement that
it
will
or
will
not
occur.
Biostatistical
models
are by
definition
stochastic,
and
information
theory,
another tool
of the
health scientist, deals with stochastic processes.
To-
day's
approaches
to

epidemic
simulation
and to
analysis
of
electrocardio-
grams also contain
major
stochastic elements. Thus both deterministic
and
stochastic models
are
used
in
applying mathematics
in the
health sci-
ences.
Although
the
dichotomy between deterministic
and
stochastic models
is
intellectually pleasing,
in
actual practice
it is
simplistic.
All

determinis-
tic
models that
are
intended
to
represent real, measurable quantities
must
be
used recognizing
the
limits
of
precision
of the
measurements.
These
limits introduce
an
uncertainty
and
hence
a
probabilistic element,
Models
and
Goals
7
into both
the

initial conditions used
in the
model
and the
values
of the
observables predicted
by the
model.
Stochastic models
may be
reduced
in a
trivial
fashion
to
deterministic
ones under some circumstances.
For
example,
if the
number
of
molecules
or
persons involved
is so
large
that
the

random stochastic events cannot
be
observed,
the
model leads
to
deterministic predictions even though
the
underlying process
is
stochastic.
In
addition many stochastic models,
perhaps all, contain some deterministic elements.
Because
the
distinction between
these
models,
as
defined,
is not
always
clear,
a
revised definition
is
perhaps
needed.
Models

are
deterministic
if
their principal features lead
to
definitive
predictions, albeit modulated
by
recognized uncertainties.
On the
other hand, models
are
stochastic
if
their more important parts
depend
on
probabilistic
or
chance consid-
erations, even though
the
model also contains deterministic elements.
D.
Inverse
Solutions
There
is
frequently
a

major
difference
between
model applications
in
the
physical
and the
engineering sciences
on one
hand
and the
biomedical
disciplines
on the
other.
The
physicist
and
engineer
often
can
design
and
build systems
to
predetermined specifications. Accordingly they
often
use
a

model
to
predict
how a
given system will behave. This
type
of
solution
of
the
mathematical model, whether performed analytically
or
numerically,
is
referred
to as a
direct
or
forward solution.
The
design
of
health
care
delivery systems also
may
involve such forward solutions
of
mathematical
models.

By
contrast
the
biomedical scientist usually cannot design
the
system
to
be
studied
but can
observe
the
behavior
of the
system.
In
this case
a
goal
of
model study
is
often
to find
characteristics
by
which
the
system
can be

described.
For
this purpose
the
model's forward solution
is
compared
with
observed behavior
and
some
form
of an
objective
function
is
com-
puted.
The
objective
function
provides
a
suitably weighted measure
of
the
agreement
(or
lack thereof)
between

the
forward solution
and the
actual system's behavior.
It is
then possible
to
seek parameters that
will
optimize this agreement.
These
parameters
are
referred
to as the
inverse
solution,
which
can
then
be
used
to
characterize
the
individual system.
Engineering technology
often
faces
a

similar problem. Suppose
a
trial
system
has
been
designed,
a
suitable mathematical model described,
and
a
forward solution found.
If
this system
is to
perform
a
preassigned task,
one may ask how
well
the
model predicts that these objectives will
be
8
Models
and
Goals
met.
To
answer this question quantitatively

an
objective
function
is
needed.
The
technologist then must seek alternate
forms
for the
model
or
perhaps alternate parameters within
the
model, which will
be
used
to
bring
the
performance
of the
system
closer
to its
objectives.
Such
a
design
process
is

called system optimization.
The
objectives
of a
biomedical
scientist
in
seeking
an
inverse solution
may
differ
from
those
of an
engineer attempting
to
optimize
a
system.
Nonetheless
the
mathematical
and
computer-based techniques
are
quite
similar.
Therefore, some biomedical scientists adopt engineering ter-
minology

and
speak
of
system optimization
as
though
it
were
equivalent
to finding an
inverse solution.
E.
Model Conformation
and
Parameter
Estimation
In one
area
of the
physical sciences, namely,
X-ray
crystallography,
inverse solutions
of the
type used
in the
health sciences
are
essential.
Given

a set of
X-ray
diffraction
spots
(an
X-ray
diffraction
pattern),
the
problem
is to
select locations
and
bond angles
for the
atoms
or
atomic
groups
within
the
crystal.
The
solution
of
this problem
is
particularly
important
in

studying crystals
of
large molecules such
as
occur
in
biologi-
cal
systems.
The
process
is
closely analogous
to the
system optimization
of
the
engineer although
different
computer
and
mathematical techniques
are
used.
Crystallographers
call their process refinement;
in
effect
it
con-

sists
of
iteratively selecting
the
atom locations, bond angles,
and
arrange-
ments
to find
forward
solutions that conform increasingly well
to the
requirements
of the
X-ray
diffraction
pattern.
In
mathematical modeling
the
iterative process
of
refining
an
inverse solution
is
sometimes called
model
conformation.
Inverse solutions

are
often
developed
by
biostatisticians
who
call this
process parameter estimation. Unbiased estimates
are
sought that will
provide closer correspondence
to
reality
as
more data
are
examined.
By
and
large
the
biostatistician seeks estimates that
in
some sense optimize
an
objective
function.
Some measure
of
uncertainty

of
these
estimates
is
desirable.
This
procedure
works
best
when
the
parameters
to be
esti-
mated appear
in a
linear
fashion
in the
solution
of the
model. Linear
parameter estimation
is
discussed
in
statistical texts
on
linear models
and

linear regression analysis.
It is
well
to
note that most models discussed
in
this text
are
nonlinear
by
the
biostatistician's
definition.
In
other words,
the
parameters
to be
estimated
do not
appear
in a
linear
fashion
in the
analytical solution
to the
Models
and
Goals

9
model.
The
word nonlinear
is the
source
of
much
confusion
because
it is
often
used
in two
different
fashions
by
scientists
and
technologists.
Essen-
tially,
technologists
use
linearity
to
refer
to the
differential
(or

other)
relationships
between
the
variables
in the
model rather than
to the oc-
currence
of the
parameters
to be
estimated
in the
analytical solution.
In
the
succeeding
chapters
most
of the
examples
presented
are
related
to
specific
biomedical
applications. However,
to

emphasize
the two
senses
in
which linear
is
used,
four
abstract examples
are
presented
in an ac-
companying
table. Mathematical models
are
presented
in the
table both
as
differential
equations
and as
their analytic solutions. Arbitrary decisions
concerning integration constants have
been
introduced.
The
variables
are
labeled

y
and
t,
and the
parameters
to be
estimated
as a,
b,
and c. The
notation
is
explained
in
Section
G of
this chapter.
Linear
for
Linear
for
Differ
ential

Analytical
Biostatistician?
Engineer?
Equation
Solution
Yes

Yes
d
2
y/dt
2
= a
y
=
a-t
L>
/2
+
b-t
+
c
(1-1)
Yes
No
a-dy/dt
=
y'
2
y = -
a/t
(1-2)
No
Yes
dy/dt
= -
a-y

y -
b-exp
(-
a-t)
(1-3)
No
No
dy/dt
=
a-y
-
b-y
2
y
=
c/[b-c/a
+ exp (-
a-t)]
(1-1)
The first
example (Equation
1—1)
has
been
chosen
to
emphasize that even
though
the
differential

equation
may be
linear
and the
parameters
to be
estimated
may
appear only
in
linear
fashions,
the
resultant analytical
solution
need
not be the
equation
of a
straight line.
The
second example
has
been
included
for
completeness only. However, models similar
in
their linearity
to

Equations
1—3
and
1-4
form
the
bases
for
several models
discussed
in
this text.
The
specific
example
in
Equation
1-3
is
used
in
Chapter
2 and the one
illustrated
in
Equation
1-4
appeals
in
Chapter

13.
Although
all
real biological systems
can be
shown
to be
nonlinear
in the
engineering sense, nonetheless many
can be
adequately approximated
by
models that
are
based
on
linear
differential
relationships between
the
variables
but
involve parameters
in a
nonlinear
fashion
in
their solution.
One

property
of
linear
differential
equations should
be
noted, namely,
if
there
are two or
more solutions known,
the sum of
these solutions
or
any
linear combination thereof
is
also
a
solution. This
is
sometimes
re-
ferred
to as the
superposition theorem.
It
implies that
in a
model with

several inputs
(or
initial conditions),
one may
solve repeatedly allowing
only
one
input
(or
initial condition)
at a
time
to be
nonzero
and
then
add
these partial solutions
to find the
general solution.
By the
same reasoning
multiplying
all the
inputs
and
initial conditions
by a fixed
constant results
in

multiplying
the
general solution
by the
same constant.
In
some cases
it
is
convenient
to use
experimental tests
of the
superposition theorem
to
10
Models
and
Goals
judge whether
a
mathematical model
is
linear
in the
engineering sense.
No
matter what
the
decision, however,

finding
inverse solutions
to the
model
usually
involves nonlinear parameter estimates.
Sometimes
nonlinear estimation
can be
avoided
by
transforming
the
analytical
solution into
a
form
in
which
new
parameters
can be
defined
that
are
linear
in the
biostatistical
sense. Thus taking
the

logarithm
of
both
sides
of the
solution
to
Equation
1-3
leads
to
When
a
transformation
of
this nature
is
possible, statisticians call solutions
of
the
form
of
Equation
1—3
pseudononlinear.
It
should
be
noted
that

in
most cases estimates
of
a
and b
based
on
Equation
1-5
differ
from
ones
based directly
on
Equation
1-3.
The
problems
of
nonlinear parameter estimation
are far
more compli-
cated
than
of
linear
parameter
estimation.
The
latter

can be
done
exactly,
whereas nonlinear parameter estimation always requires
an
iterative,
trial
and
retrial approach.
Various
schemes have been developed
for au-
tomated computation
of
nonlinear parameter estimates.
Many
computer
centers have several packaged programs
for
this purpose because
no one
program
is
ideal
for
all
models.
Nonlinear
parameter estimation
is

also
difficult
in a
number
of
other
ways.
Usually
there
are not
suitable data
to
determine whether
the
esti-
mate
is
biased. Worse, there
is
usually
a
large coupling between
different
parameters, which some biostatisticians describe
as
very large covariance
terms.
Accordingly estimates
of
uncertainty

in the
nonlinearly estimated
parameters become questionable
in
meaning.
A
better
approach seems
to
be to
seek combinations
of
parameters that
are
relatively insensitive
to
experimental error. (See Chapter
2 for
further
discussion.)
Use
of
many nonlinear parameter estimation routines requires knowl-
edge
of the
numerical values
of
partial derivatives.
By and
large

methods
that
do not
require derivatives
are
easier
to use
because analytical speci-
fication
of
the
partial derivatives
of the
objective
function
is not
needed.
Some
so-called derivative-free methods actually approximate
the
deriva-
tives numerically within
the
routines, whereas others
use
directly
the
values
of the
function

itself
at
various trial points.
Any
iterative method
of
parameter estimation
may end at a
local
minimum
of the
objective
function.
There
is no way to
guard against this
eventuality. Moreover,
in a
search
for the
best
set of
parameters
the
where