JavaScript needs to be enabled for this application to run correctly
This is the Enhanced Reader view. For maximum accessibility screen reader users should use the HTML format which is available on the article page for most content.
Expand menu
Previous
PDF
Next
PDF
Article start
Copyright
©
IFAC
Control
Systems
Design,
Bratislava,
Slovak
Republic,
2003
IFAC
~
Publications
www.elsevier.com/locate/ifac
NONLINEAR
SYSTEM
MODELS
BASED
ON
INTERVAL
LINEARlZATION:
A
MEDICAL
APPLICATION
Jaroslav
Kultan
Abstract:
The
paper
deals
with
an
application
of
models
based
upon
interval
linearization
in
modelling
human
organism
parameters,
especially
during
long-term
medical
treat-
ments.
The
generated
model
enables
to
predict
patient's
state
as
well
as
the
impact
of
taken
therapeutics
on
it.
Application
of
such
models
brings
about
improvement
of
the
treatment
quality
and
helps
physicians
in
the
decision
making
process.
Copyright
©
2003
IFAC
Keywords:
identification.
interval
linearization,
parameters
of
human
organism,
state
pre-
diction,
non
linear
systems
I. INTRODUCTION
In
the
medical
practice
there
are
often
encountered
diseases
requiring
a long-term
treatment
whereby
it
is
almost
impossible
to
continuously
monitor
the
organism
parameters.
Very
often.
physicians
are
obliged
to
go
repeatedly
through
a huge
amount
of
data
acquired
from
the
blood,
urine
and
other
analy-
ses,
to
be
able
to
take
decisions
based
upon
the
pa-
tient's
state.
This
process
can
be
supported
by
creat-
ing
a model
of
patient's
state,
based
on
which
it
is
possible
to
take
a
correct
decision.
Therefore
we
focused
on
the
identification
of
human
organism
parameters
under
long-term
medical
treatments.
Results
of
blood
tests,
creation
of
their
model
and
its
application
in
the
patient's
state
prediction
have
facilitated
the
decision
making
about
additional
in-
terventions
during
the
treatment.
which
helped
to
improve
the
whole
process.
For
the
identification
and
model
generation
we
have
used
the
interval
lineariza-
tion
method.
2.
NON
LINEAR
SYSTEM
IDENTIFICATION
2.1.
Identification
using
the
intervallineari=ation
method
many
models
as
in
case
of
linearization
in
the
operat-
ing
point,
nor
to
search
intricate
nonlinear
functions
which
implementation
is
often
questionable.
Identification
of
nonlinear
dynamic
systems
by
the
interval
linearization
method
has
been
invented
in
1986.
The
method
has
been
updated
several
times
by
improving
some
of
their
properties
or
the
identifica-
tion
process
itself
[2]
[3].
The
presented
method
has
been
applied
not
only
in
nonlinear
system
modelling
but
also
in
designing
controllers
for
non
linear
sys-
tems
[4].
Identification
of
non
linear
dynamic
systems
consists
in
splitting
the
whole
working
range
of
the
input
and
output
variables
by
the
limits
uk;
and
yk
j
â¢
respec-
tively,
into
several
intervals
(from
where
the
name
of
the
method).
Intersect
of
these
intervals
defines
the
linearization
interval
(Fig.
I).
Generating
a
system
model
using
this
method
re-
quires
availability
of
the
measured
input
and
output
data.
Measurements
are
to
be
carried
out
during
the
system
operation
and
organized
in
a table.
Next.
the
limits
of
individual
intervals
for
the
input
and
output
variables
are
to
be
specified.
Intersect
of
these
inter-
vals
defines
the
linearization
section.
For
every
nonlinear
system
there
can
be
several
such
sections
and
the
identification
yields
model
of
each
lineariza-
tion
section
in
the
following
form
(I)
.I{I)
1/
+
I
q'>{1
-
/7)
+:t
q'
1/(1
-
(.J
-
d
+
1)7)
o
,"'I'
)
..
1
1f}.J
qk
is
vector
of
model
coefficients
in
the
k-th
lineari-
zation
section.
whereby
nu
- number
of
samples
of
the
input
variable
(2)
(3)
(4)
k=1.2
....
kk
nu+ny
y(t)
=
z(t)
qk
where
z(t)=[
I.y(t-I
).....
y(t-ny).u(t-d-I
).....
u(t-d-nu+
I)]
k
k
k
qk=(qo·ql····q
or
concisely
The
interval
linearization
method
is
one
of
relatively
new
methods
for
nonlinear
systems
modelling.
Basi-
cally.
it
differentiates
from
other
identification
and
modelling
approaches
for
such
systems.
one
of
which
consists
in
substituting
the
real
system
by
a set
of
nonlinear
functions
and
relations
which
describe
processes
in
the
system
(the
human
organism
can
be
considered
as
a system);
another
possibility
is
substi-
tution
of
the
real
system
by
a mathematical
model
acquired
by
its
linearization
in
a
chosen
operating
point,
or
it
is
possible
to
create
abstract
nonlinear
models.
The
interval
linearization
method
offers
a substantially
different
approach.
generating
a linear
model
created
not
only
for
one
operating
point
but
for
a whole
region
called
a linearization
interval
[I].
Such
a model
is
able
to
include
properties
of
the
given
system.
whereby
it
requires
neither
to
create
so
495
Jaroslav
Kultan
ny
-
number
of
samples
of
the
output
variable
t -
time
where
r =
1.2
...
n :
ny+nu+
I
<
n
The
identification
result
is
the
matrix
Q
T -
sampling
time
d -
input
variable
shift
k -
index
of
the
linearization
interval
A
total
number
m
of
linearization
intervals
for
nu
= 1
and
ny=
I
is
m =
mu.my
and
for
nu>
1
and
ny
>
1
the
number
of
linearization
intervals
is given
by
ql
ql
ql
0
I
ml
+
,~r
Q
k
k
k
q
q
q
0
I
nil
+
~\
q"'
q"'
q"'
0
I
I1U
+
f~\'
(9)
kk
=
munu.myny
(5)
I
-
auxiliary
index
For
many systems.
long-term
operation
measure-
ments
of
u(t).
y(t)
proved.
that
in
practice.
the
num-
bcr
of
linearization
intervals
does
not
comply
with
their
theoretical
number
and
strongly
depends
on
the
appropriateness
of
chosen
limits
and
the
volume
of
mcasured
data
a\ailable.
The
procedure
for
determining
the
index
"k"
of
the
linearization
interval
is
as
follows.
If
the
output
\'ari-
able
y(t-i).
i=I.2
ny
is
from
the
amplitude
band
(yk
s.,.yk,>-
s =
1.2
my.
then
this
band
is
dcnoted
by
I,=s.
Similarly.
if
the
variable
u(t-j-d+I).j=1.2
...nu
is
from
the
band
(ukr.,.uk),
r =1.2
.....
mu.
the
corre-
sponding
band
is
denoted
I
=
r.
These
bands
nY"J
notations
are
collected
in
a
vector
1=(11'
12
..
.Inu+ny).
The
linearization
interval
index
is
calculated
as
fol-
lows
8
6
4
input
u(t)
2
input/output
characteristics
16
14
12
=-
10
>;
:;
8
c.
:;
0
6
4
2
Fig.l.
Input/output
characterisics
of
the
system
3.
IDENTIFIC
A
nON
OF
BLOOD
SYSTEM
PARAMETERS
(6)
(7)
(8)
1).8
I
tor
1=
1.2
.....
nu
wherc
k
8
1
=
ml-
I
)
for
1=
1.2
.....
ny
8
-
.n\"
11-11
It-ny
-
m)
..
mu
Calculation
of
the
linear
model
( 1)
or
(2)
for
the
k-th
lincarization
intenal
consists
in
determining
the
\'ector
of
Iinearization
cocrticients
(4)
from
measured
samples
y(t-i)
and
ult-j-d+
1)
tor
i=
1..
...
ny.
j=
l...
..
nu.
\\ithin
the
k-th
ineari7.ation
intcn·al.
The
model
itsclf
is
then
specified
from
the
relation
A"q"=
b"
, from
\\hich
\\e
express
the
\cctor
of
the
model
qk
in
thc
k-th
linearization
section
using
the
least
squares
algorithm.
To
appl)
it.
it is
necessaf)
to
k
se
tup
the
matrix
A
from
measured
\alucs
y(
t-i)
and
u(t-j-d+
I)
and
the
\'cetor
b"
from
measured
\alues
.
"
y(t).
For
thc
k-th
nm
oftht::
matrix
A
holds
a\""O\
= u(t-j-d+
I)
akl
=
I
"
a
u-l
= y(t-i)
for
i=
1.2
ny
tor
j=
1.2
nu
(9)
3.
I.
Data
meaSllremel71
The
considered
method
\\as
\eriticd
in
modelling
leucocytes
parameters
for
one
year.
During
a
one-
year
chemothcrapeutic
treatment
blood
samples
\\cre
taken
sporadically
and
individual
parameters
\\'ere
measured.
Values
of
indi\idual
parameters
are
organ-
ized
in
a
tablt::
(Tab.
1).
Input
\'ariables
arc
\alues
of
used
therapeutics
(V
- vincristin.
Cl'
-
C)
splatinum).
or
of
certain
supporting
preparations
(S).
applied
in
cases
when
the
organism
parameters
ha\
e
decreased
considerabl).
Output
parameters
are
HU.
Ery.
HKU.
LE.
TR.
Medicine
intake
is denoted
by
..
1"
in
the
day
of
application.
k
The
r-th
entf)
of
the
n~ctor
b
satisfies
b\
=
y(t-O)
(10)
496
CP
I
0.04
v
I
0.3
A.
D
Table
2 summarizes
values
of
measured.
and
interpo-
lated
(linearly
and
quadratically)
values.
Measured
values
of
individual
variables
are
shown
in
Fig.
3.
Comparison
of
linearly
and
quadratically
interpolated
values
is
depicted
in
Fig.
4.
In
measurement
instants
values
of
individual
functions
coincide.
Leucoeytes
and
input
variable
values
used
for
modelling
are
shown
in
Fig.
5
The
simplest
interpolation
method
is
the
linear
inter-
polation.
For
values
in
individual
days
it
substitutes
the
values
obtained
by
a uniform
distribution
of
the
states
between
two
measured
values.
The
quadratic
interpolation
method
is
a more
advan-
tageous
one.
using
a
quadratic
approximation
of
values
between
two
measurements.
Though
it
is
quite
computationally
demanding
it
follows
better
con-
tinuous
changes
of
individual
parameters.
Input
variables
have
been
substituted
by
exponentials
reaching
their
maximum
at
application
time
of
basic
therapeutics
when
their
concentration
In
organism
decreases
successively.
The
exponential
lorgetting
factor
changes
with
each
considered
variable.
N
Tab.
I.
Measured
values
of
blood
parameters
during
the
initial
period
INPUTS
OUTPUTS
HK
Cyk
\"
CP
S
HG
[I").
G
LE
TR
Date
14.07.99
I
I
118
3.70.29
2.1
186
20.07.99
6
7
124
4.4
0.37
1.6
225
22.7.99
2
9
117
4.090.37
1.8
205
29.7.99
7
16
93
3.260.37152
143
317.99
2
18
98
3.37038
1.2
149
3.8.99
3
21
64
2.32
0.1
I
110
5.8.99
2
23
82
2.860.37
1.24
143
25.8.99
20
43
2
126
413035
29
224
2999
8
51
113
3.70.37158
184
6.9.99
4
55
95
3.090.38
1.04
108
9.9.99
3
58
95
3410.27
1.3
145
21.9.99
12
70
109
3490381.33
117
27.9.99
6
76
104
340.38
1.2
123
61099
9
85
113
3.740.361.08
281
13.1099
7
92
117
3760371.65
243
2.11.99
5
112
96
3.180.380.74
138
5.11.99
3
115
94
33
0.27
0.8
149
9.11.99
4
119
101
3.350.36
1.1187
A
graph
of
some
monitored
quantities
is
in
Fig.2
whereby
the
leucocytes.
which
characterize
patienrs
immunity
are
the
most
interesting
parameter.
5 -
4.5
days
4
0
118
3.7
0.29
2.1
I
19
3.73
0.29
2.08
3.5
2
0.7408
0%08
0
119
3.82
0.3
2.02
119
3.73
0.29
2.08
3
3
05488
0.9231
0
120
3.93
0.32
1.93
12~
4.01
0.31
1.89
2.5
4
0.4060
0.8869
0
121
4.05
0.33
1.85
125
4.~~
0.33
1.74
2
5
0.3012
0.8521
0
122
4.17
0.34
1.77
126
437
0.35
1.64
1.5
6
0.2231
0.8187
0
123
4.28
0.36
1.68
In
4.44
0.36
158
0.5
7
01653
0.7866
D
124
4.4
037
1.6
126
4.46
037
1.57
0
8
0.1225
0.7558
0
121
4.25
0.37
1.7
124
4.4
0.37
1.61
(])
(])
(])
(])
(])
Cl
Cl
0
0
0
0
0
Cl
Cl
(])
(])
(])
(])
Cl
0
0
0
0
0
9
10907
0.7261
0
117
409
0.37
18
I
~O
4.24
0.37
1.72
.....
<xi
oi
(])
(])
Cl
Cl
0
0
0
0
0
~ ~
N
N
~ ~
N
Cl
«)
~
0
~
~
N
N
M
'V
N
N
N
~
~ ~
~
lCi
10
0.808
06977
0
114
3.97
D37
1.76
116
4.07
D37
1.81
co
M
0
.....
lCi
N
oi
ci
N
N
N
N
"
05986
06703
0
110
3.85
0.37
1.72
109
H4
0.37
1.85
--Ery
--LE
I~
04435
0.644
0
107
3.73
0.37
1.68
103
3.65
0.37
I
8t>
Fig.2
Measured
values
of
some
monitored
13
0.3285
06188
0
103
3.6~
037
164
98.t>
3.:'
U.37
I
85
quantities
14
0.2434
05945
o
99.9
3.5
037
1.6
95.~
3.38
0.37
1.8
3.1.
Processing
olmeasl/red
data
15
0.1803
o
571~
0%.4
3.38
lU7
1.56
9'
,
3.3
037
1.74
.'
-
As
the
input
data
were
measured
In
various
time
16
1.1336
0.5488
0
93
3.~t>
D.37
I 52
9~
4
3.2t>
037
1.64
intcl\als.
they
need
to
be
pre-processed
and
appro-
priately
for
17
0.8398
0.5273
o
955
' "
037
Ut>
955
3.33
039
1.5
adapted
next
computations.
Computed
:J.:J_
inll~l\als
between
individual
measurements
and
miss-
18
0.62~1
0.5066
0
98
337
0.38
1.2
99.3
343
0.4
1.33
ing
\alues
have
heen
interpolated
so
as
to
reflect
in
the
best
way
thc
\alues
expected
in
the considered
time.
497
5
4.5
respect
to
the
possible
application
of
chemotherapeu-
tics.
Even
the
state
of
leucoc)tes
(along
with
other
parameters.
of
course)
determines
whether
it
is
pos-
sible
to
give
the
patient
the
therapeutics
or
not.
4
3.5
I
5 -
3
+---.
----tt---~---;----__ft
4.5
Fig.
3
Graphical
representation
of
the
linear
approximation
of
selected
organism
parameters
Ery
--
HKG
--
LE
--
400
300
200
100
O....:...-ll..L~.l1.....ll-...l....:...:>.L..lo.-'-'--.L.....l"----
o
-0.5
4
1
.5
--++-+--+-1
3.5
0.5
3
2
2.5
----ft---fl
------tt---
300
200
100
0.5
o
o
-0.5
--v--cP
2
2.5
1.5
5
--v
--CP
--5
--LE
4.5
Fig.
5 Parameters
of
the
input
therapeutics
and
the
response
of
the
organism
- leucoc)tes
3.3.
Search
o/Iineari=ation
intervals
A
scrious
drawback
of
the
described
method
is
the
search
for
lincarization
intervals.
i.e.
specification
of
limits
tor
individual
intervals.
In
practicc
this
prob-
lem
is
bcing
solved
in
various
ways.
As
medical
processes
are
rclatively
slow
a stepwise
changing
the
limits
of
domains
and
ranges
of
indi\'idual
\uriables
ean
sohe
this
problem.
Next.
a model
is
calculated
for
all
obtained
intenals.
and
a simulation
is
carried
out
with
the
already
a\ail-
able
samples.
Individual
models
are
compared
using
the
classical
method
of
summing
squared
diftcrenees
betwcen
the
measured
and
the
calculated
\·alues.
Using
a graphical
representation
of
indi\idual
input
\'ariabks
and
modi
tied
\alues
of
squared
error
sums
f<lr
each
model
we
can
find
the
intenals
with
the
least
difference
bet\\een
the
model
outputs
and
the
real
\ alues.
Table
3
shows
\alucs
of
some
limits
(hrl.
hr2.
hr3.
hr~
and
thc
"sqc"
-
sum
of
squared
errors).
F
300
100
200
--v
--CP
-
Ery
--
HKG
--
LE
4
~\
'Y'
3.5
3
J
2.5
w
2
1.5
0.5
o
o
-0.5
Fig.
~.
Quadratic
approximation
of
selected
parameters
of
the
organism
For
modelling
blood
paramcters
we
ha\c
chosen
one
of
its
most
important
representatives.
namely
the
numbcr
of
leucoc)tes
indicating
patient's
statc
with
498
Tab.)
Sum
ofs9uared
errors
(Sge)
as
a
function
of
limits
4.
APPLICATION
OF
GENERATED
MODELS
For
the
next
analysis
a model
with
the
limits
(hr
I.
hr2.
hr3.hr4)
= (
0.3:0.3:0.7:0.5)
has
been
chosen.
Mathematical
model
and
simulated
data
are
in
Fig.
6.
Fig.6
Simulation
of
LE
parameters
--v--cp
--8
pc
hr1
1
0.1
2
0.1
3
0.1
4
0.1
5
0.1
6
0.3
7
0.1
8
01
9
0.1
10
0.1
11
0.1
13
0.1
14
0.1
15
0.1
16
0.1
17
0.1
18
0.1
19
0.1
Based
on
the
analysis
of
measured
data
and
simula-
tion
of
the
possible
patient's
state
(Fig. 6)
it
is
possi-
blc
to
assume
that
the
patient's
state
will
remain
on
a
low
level
fix
a long-time
and
he
cannot
be
gi\en
the
relevant
therapeutics
due
to
it.
In
a
standard
treatment
the
doctor
takes
the
blood
samples
several
times
and
after
hm'ing
gathered
long-term
results
he
decides
to
skip
one
cycle.
Some
of
the
following
facts
underlie
such
a decision:
- on
checking
parameters
he
supposed
that
the
pa-
tient's
state
will
improve.
however
this
did
not
hap-
pen:
- there
was
a possibility
of
helping
by
other
means.
howe\er
relati\ely
mueh
time
has
passed
since
then
and
therdore
such
a stimulation
is
already
inappro-
priate.
Of
course.
there
are
a lot
of
other
faets
to
be
consid-
ered
in
a physician's
decision
making.
Simulation
of
patient's
state
before
starting
a
new
curing
cycle
could
be
one
possible
tool
for
helping
his
decision
taking.
The
obtained
model
shows
that
the
organism
parameters
arc
relati\ely
low
and
the
patient
is
not
able
to
create
a sufficient
amount
of
necessary
su~
-/.2.
Simulation
o/the
medical
treatment
-/.1.
Problems
0/
patient's
treatment.
Application
of
some
therapeutics
may
bring
about
a
considerable
worsening
of
some
human
organism
parameters
and
thus
their
intake
is
possible
only
under
the
assumption
that
the
patient
tolerates
such
a
treatment.
One
of
such
procedures
is
the
application
of
cytostatics
in
curing
oncogenous
diseases.
Appli-
cation
of
therapeutics
is
possible
only
if
the
leuco-
cytes
parameters
have
reached
some
specified
value
(2000).
To
find
out
the
patient's
actual
state
it
is
necessary
to
take
his
blood.
evaluate
the
parameters
and
to
continue
the
treatment
under
the
assumption
that
the
state
is
sufficiently
good.
If
there
are
low
leucocytes.
the
blood
taking
has
to
be
repeated
after
1-
3
days.
Such
frequent
blood
takings
disturb
the
patient.
and
make
him
suffer.
Taking
into
account
his
overall
state.
such
a
large
number
of
blood
with-
drawals
cannot
contribute
to
an
improved
recovery.
On
the
other
hand
frequent
blood
takings
are
impor-
tant
because
it
is
necessary
to
apply
the
therapeutics
in
the
specified
time
(necessity
to
keep
the
curative
procedures)
or
with
a least
possible
delay.
It
use
to
happen
that
despite
numerous
tests.
the
pa-
tient's
state
does
not
improve
and
in
order
to
keep
certain
curative
procedures
it
is
necessary
to
apply
supporting
and
stimulating
pharmaceuticals.
It
is
especially
the
decision
making
about
whether
to
make
the
patient
suffer
or
stimulate
the
organism
all
the
same.
which
can
be
facilitated
by
the
proposed
parameter
model.
From
Fig.
6
it
is
evident
that
in
the
last
250
-300
days
of
curing.
the
leucocytes
(LE)
number
drops
and
at
the
time
of
therapeutics
application
it
even
does
not
reach
the
lower
limit
2.
o
LE
--Lem
0.41
0.61
0.83
1.12
1.47
hr4
sko
2.4
0.39
2.4
0.58
1.4
0.71
0.4
1.08
0.8
1.38
0.2
2.44
2
25
0.4
97.3
0.4
124
1.2
282
0.6
340
0.6
388
1.2
2.6
0.4
1.2
1.2
2
hr2
hr3
0.5
0.7
0.9
0.4
0.5
0.7
0.1
1
0.5
1
0.1
0.4
0.5
0.7
0.3
0.1
0.7
0.1
0.3
1
0.1
1
0.7
1
0.1
1
0.9
0.7
0.7
0.7
0.3
0.7
0.7
1
0.5
0.1
J
j
V
V
~
-
.\
[\
l\
l\
1\
\
1\
l\
~
\.
"-
\
.
\
100
200
300
4
-0.5
o
0.5
2.5
1.5
4.5
2
4
3
5
3.5
499
stances.
Should
these
facts
be
confinned
by
the
re-
sults
from
the
first
blood
test.
the
physician
can
con-
sider
application
of
stimulating
therapeutics.
suppon-
ing
this
decision
by
a simulation
on
the
models.
In
a real
situation
we
have
first
simulated
the
impact
of
the
blood
transfusion
on
other
parameters.
Trans-
fusion
was
carried
out
already
at
about
the
IOOth
day
of
the
curing
procedure.
Simulation
results
under
stimulator
application
are
depicted
in
Fig.
7.
Based
on
obtained
results
(growth
of
the
LE
value
above
3)
correctness
of
such
a decision
could
have
been
an-
ticipated.
5
4.5
4
3.5
3
2.5
2
1.5
0.5
o
-0.5
~.
I
-
l
--
J
V
I
I
I
'"
IV
V
~
l\
l\
1\
l\
[\
.
....--
l\
\
\.
\.
100
200
300
4
o
Cenainly.
the
interval
linearization
method
is
not
the
only
suitable
one
but
hopefully
it
is
not
the
last
one
applied
in
medicine.
and
its
implementation
will
stimulate
application
of
other
methods
as
well.
REFERENCES
S.A.
Bilings
and
W.S.F.
Yoon:
Piecell'ise
linear
non-
linear
system
identification.
Internation
Journal
of
Control,
1'01-16.,\'.1.1987
Harsanyi
L..
Kultan
J.:
IdentifikiJcia
nelinearnych
sys/(?mo\'
metodou
inten'alm'ej
lineari=acie.
Journal
of
Electrical
Engineering.
1-'01.-11
..
.\'0
11,
1990.
str.
825-836.
Harsanyi
L..
Kultan
J.:
Alethod
~f
selectil'e
forgelling
for
nonlinear
system
identification
Journal
({f
Electrical
Engineering.,
/'01
-13,
,Vo
7.
207-210.
Bratislava,
1992
Yesel~'
Y
..
Kultan
J.:
Regulator
synthesis
for
nonlin-
ear
systems.
Technical
report
EF
SI"ST.
De-
partment
of
Automatic
Control
S)'stems.
1992
--v--cp
--5
LE
--Lem
Fig.7
Simulation
of
LE
parameters
aner
application
of
a stimulator
Then
the
transfusion
was
completed
and
the
w'hole
curati\e
cycle
was
completed
successfully.
Of
course.
the
e.'\perience
of
physicians
played
an
imponant
role
in
the
whole
treatment
process
as
\\ell.
and
the
proposed
model
has
sef\ed
only
as
one
supporting
tool
for
the
decision-making.
CONCLlJSIO
The
aim
of
this
paper
was
to
present
a practical
ap-
plication
of
one
of
the
dynamic
systems
identifica-
tion
methods
in
medicine.
At
the
same
time
\Ie
wanted
to
emphasize
that
implementation
of
identifi-
cation.
modelling.
simulation
and
control
methods
in
this
field
not
only
imprmes
the
treatment
process
but
can
rem\we
lot
of
ache
in
this
treatment
as
\Iell.
and
enables
a
more
e.'\act
decision
making.
onen
e\'en
saying
the
patient's
life.
500
Previous
PDF
Next
PDF
To print this document, select the Print icon
or use the keyboard shortcut,
Ctrl + P
.