Download INTRODUCTION

KINETIC MODELLING OF NON-LINIER PHYSIOLOGY AND CARDIOVASCULAR
CARTOGRAPHY
Kumar RV1, Kishore AGR2, Chandra VS2, Kumar P2, Rekha BR1, Deepti A1, Imre J3, Lakotosh Y3
1
Center for Artificial Intelligence and Non-Linear Studies (CAINS), 2Department of Cardiology, Manipal Heart Foundation,
Bangalore, India, 3ASKIT kft, Budapest, Hungary.
of our screening tests applied to the
INTRODUCTION
"healthy" populations are bound to end up
The era spawned by the advent, and ready
with many false positives, making life
acceptance of the "chaos" model, has lead to
miserable for those unfortunate victims.
an explosion of nascent schools of thought
Similar is the experience with predicting
that challenge hitherto entrenched linear
death based on the left ventricular ejection
statistical
Euclidean
fractions, in the immediate post myocardial
principles in natural sciences . Dynamic
infarction phase. If one goes deep into all
human body does not obey the rules of
this
linear Euclidean mathematics. Stability in
mathematical formulae could be applied to
biology is a myth. It occurs only after death.
any human organ. Take the heart for
We have been predicting the unpredictable
example; is it a triangle, square or an
in medicine all along. What happens to the
oblong? It has no integer measure at all; and
human organism as time evolves depends on
how do we apply the integer measure
the total knowledge of the initial state of the
formulae to calculate the various facets of
organism. Since we cannot know this, with
cardiac function? For want of a better
the help of the reductionist science, we are
measure of the non-integer human heart, we
unable to predict the future. We can only
still hang on to our old formulae. When the
assess the human phenotype, which forms
conventional science has been heading for a
about 30% of the organism and the rest is
crisis with increased specialisation, it is the
made up of our genetype and consciousness.
new science of non-linear mathematics that
We need non-linear mathematics of "Chaos"
will come to our help2. Fractal configuration,
to predict man's future. The new wave
and such other non-linear systems, has made
thinking should be that there might be a
the need for a second look at physiodynamic
rhythm within a chaotic situation.
processes of the human biology in a
Attempts at predicting the future with the
systemic approach.
Newtonian
and
1
help of the exercise ECG test (Bruce) fail to
follow the positive-negative paradigm. Most
one
wonders
how
any
of
our
It has been felt that, it is not adequate to
As for the respiratory and circulatory
approach the differences from reference
systems, the PRESSURE (P), the VOLUME
values considered normal (healthy state) as a
(V), and the TIME (T) clearly characterise
unique entity, and it is also not adequate to
the status and certain changes in the status of
make
such
these systems. In the above respiratory and
approach. In some cases it is possible that
the circulatory system, the correct unit of
the abnormality found is just a compensation
change is the cycle, which can be considered
for latent anomaly, and so the correction of
an ordinary unit of TIME (T).
the change would have a negative effect on
This Paper illustrates the fallacy in such
the SYSTEM rather than a positive one. For
systemic approaches of studying the non-
instance, the residual volume of a patent
linear parameters of the human physiology
with obstructive emphysema is too high
using a specialised complex multivariable
(pathological) but this is the only way for
model developed by this team. The model
these patients to reduce the resistance of the
was tested for it's practical utility in its
corrections
according
to
3
small pulmonary airways . If the residual
ability
in
predicting
Coronary
Artery
volume gets reduced, the ventilation gets
Disease (CAD) in a recently concluded-
worse. The pathological heart - if it can -
blinded study.
get tachycardic, for in this way the load gets
reduced (because as a result of increase in
the heart rate, the pre-load gets reduced). If
MODELLING
the patient is made bradycardic, the patient’s
DYNAMICS OF BLOOD FLOW
OF
HUMAN
condition gets worse. Similarly when the
cerebral blood flow falls, the perfusion is
In
often compensated by the elevation of the
construction of models has been used as a
blood pressure - If in such cases - the blood
means of assisting in understanding of
pressure is normalised, permanent cerebral
natural phenomena. In engineering it is also
emollitions can result.
regular practice to simulate existing or
many
branches
of
science
the
planned engineering devices and systems as
These examples well demonstrate that it is
well as naturally occurring processes. The
not enough to evaluate a single pathologic
techniques of modeling have in fact become
change by itself: It is necessary to consider
so well developed in the engineering
the function of the change in the system as a
environment that this previously specialised
whole.
knowledge is proving useful in other fields
as well.
Engineers
are
becoming
particularly
Models and Model Based Reasoning have
interested in modeling biological systems,
been used for many years now. For many
often in the context of the functioning of the
kinds of problem solving tasks, it is
human body. In this field the problems are
necessary to model the behavior of some
acute, but the potential rewards are certainly
object or system.
valuable enough to act as a considerable
physical devices, such as say, electronic
stimulus. First of all there is the possibility
circuits or electric motors, it is necessary to
of gaining some knowledge during the
model the behavior of both the correctly
investigation which will shed light upon the
functioning device and some number of ill
reasons for malfunctioning of the human
functioning variants of it.
physiological system and thereby hopefully
potential designs of these devices requires
suggest new means of treatment. It can be
the same capability. When we think about
reasonably expected that the modeling
constructing a model of some entity in the
procedure might result in new clinical tests
real world, the issue of what we mean by a
being proposed in order to promote better
model soon arises. To what extend should
understanding of the biological system. It is
the structure of the model mirror the
also a possibility that investigation of this
structure of the object being modeled?
type may suggest new approaches to
Some representational techniques tend to
engineering problems. The least one can
support models whose structure is very
expect from modeling or simulating a
different from the structure of the object
human
using
being modeled. For example, in predicate
engineering technique is that the medical
logic we write wff's (well-formed formulas)
and engineering professionals involved will
such as x: raven (x)  black (x). In the
be
learning
real world, though, this single fact has no
procedure as the model serves as a
single realisation, such as casual networks,
continuing basis for discussion. There are
in which the physical structure of the world
many forms of model, that can take; - from
is
scaled
representation.
physiological
engaged
in
physical
mathematical
an
function
efficient
reconstruction
equation.
a
in
the
To, evaluate
structure
of
the
particular
There are arguments in favour of both ends
modeling procedure to be discussed here is
of the spectrum (and many points in the
the use of a very specialised multivariable
middle).
model that was designed to study the human
knowledge structure closely matches the
physiological
problem structure, then the frame problem
haemodynamics.
The
to
closely
To diagnose faults in
variability,
particularly
Take a simple example, if the
may be easier to solve. Suppose; that we
have a planning program and we want to
is basically a Multi variable complex Model.
know if we move a table in to another room,
Dynamics of blood flow involves multiple
what other objects also change location. A
changes in multiple parameters.
model that closely matches the structure of
multiple parameters can be grouped as
the world as shown below, (in A) will make
pressure related, volume related, time
answering
related, and flow related, in all we have
this
question
easy,
while
These
alternative representations as shown (in B)
about 25 different parameters.
will not. There are, however, arguments for
parameters are derived from a combination
representations whose structures do not
of
closely
Such
transthoracic bio impedance, obtained non-
representations typically do a better job of
Invasively (one can also use an invasive
capturing generalisations and thus of making
technique to determine the volume related
predictions about some kind of novel
parameters). First of all the model elements
situations.
(or subjects) are to be created. This is done
model
the
world.
ECG,
These
phonocardiography
and
by conducting measurements on a number of
‘A’
subjects who are angiographically normal
(Living room 1;
individuals, who are free of coronary artery
Contains;
disease (in this case) for generating the
(Table 1;
model variables. A specially designed
Made - of: wood
Has - on: (vase 1: made - of glass)
(Lamp 1: …))
(Table 2 :
Has - on (vase 2: …)))
computer
program
would
pickup
25
qualified subjects out of double the quantity
of
measurements
obtained,
individuals
having non-consistent variability will be
rejected
by
the
program
as
subjects
suspected to have other cardiovascular
‘B’
disorders. Twenty-five consistent qualified
in (Table 1, Living room)
subjects are adequate for construction of a
made of (Table 1, wood)
multi-variable dynamic model3.
on (vase 1, Table 2)
K-model has a structure, which can be
made of (Vase 1, glass)
discussed in simple terms as follows:
on (vase 2, Table 2)
Consider a variable z that can take 2 extreme
on (lamp 1, Table 1)
values Qmax and Qmin. Qmax is the highest
Applications of the present day modeling are
value z can assume and Qmin is the lowest
more complex: K- model (for kinetic model)
value the variable z can assume:
Now z as a variable does not exist beyond
Since we are talking about a set of variables,
Qmax and Qmin and these extremes are named
all of them continuously varying either
Q1 and Q2. When the variable z-tends to
dependent or Independent, linearly or non-
move from Q1 to Q2 it crosses a plane of
linearly, this factor of deviation due to
optimal values Oopt irrespective of the
direction of change.
Now consider that; we have ‘n’ number of
variables, each variable z1, z2, z3 ………. zn .
(All are tending to max from min or min to
max within the frame Q1, Q2). At the core
of Q1, Q2 exists O1 and O2.
When our
variable z transverses through the frame Q1
and Q2 it has to transverse through the
natural forces called N1 and N2 has to be
taken into consideration. Natural forces can
be
any
element
that
can
effect
the
circulatory status in this case. However the
deviations due to natural forces N1 and N2
should always exists in the neighborhood of
the optimal plane O1 and O2.
The behavior of the variable due to
abnormal forces such as a diseased condition
can go beyond the deviation due to natural
forces, but within Q1 and Q2. Figure 2
optimal plane and as long as this point is
within this plane it assumes an optimal or
near perfect behavior. Figure 1 illustrates the
illustrates the behavior of z in a Cartesian
coordinate for a normal subject (zn) and
diseased subject (zd).
behavior of z within the K-model in normal
subject and diseased subject.
When the model for a given population is
Constructed, each of the model subject’s
Such behavior may be ideal, this is because
of the deviations due to natural forces.
data are sampled, to determine if all the data
sets are having the same pattern of change
(Variability), the one’s that don’t fit to the
contour curve f (x, y) = constant in the xy-
pattern of the majority of the subject group
plane.
is rejected as possible abnormality, the
rejected subject is then replaced with a new
This interpretation is illustrated in figure 3
subject, till such time the model is
and 4 for the example
completely constructed (this is done by a
computer Programme specially designed for
z=100-x2-y2………………..(2)
the purpose).
Determination of Model Optimal (O)
The
parameter
z
(figure
2)
can
be
determined in terms of a number of other
parameters. For simplicity, consider a
function of two independent variables x and
y and denote the dependant variable by z.
the equation
z= f(x,y) …………..(1)
In figure 3, the surface is illustrated, which
is a paraboloid of revolution, and indicating
may be interpreted as representing an
elevation of points on a hill above the plane
a cutting plane z=75. The corresponding
level curve of the circle
z=0.
Drawing
“level curves" then yields
“contour lines” in the xy-plane. In this
interpretation, we imagine a base region G
in the xy-plane and at every point in G we
imagine a marker bearing the z value
associated, by (1), with that point. If we
connect the values in G which have the same
z values, z = constant, then we have a
x2+y2 = 25…………………(3)
in the xy-plane. This is the circle which, in
figure 4, carries the marker z = 75.
Equation
(3)
may
represent
any
haemodynamic Parameter. For example, z
may represent the stroke volume z at each
point (x, y) taking x and y as time and
pressure respectively.
z = fy(x,y) = lim f(x,y+y) - f(x,y)… . (5)
y
y0
y
The rate of change of any parameter z with
respect to x and y may be calculated using
equation (4) and (5).
For example equation (2)
z=100-x2-y2
can represent the stroke volume (z=SV, in
ml), where x is the time (here, time
represents the inter-beat interval, essentially
the rate of change of RR-Interval between
subsequent beats in msec) and y is the interbeat pressure difference( in mm hg).
Applying equation (4) and (5) we have
z = -2x ……………….. (6)
x
z = -2y
y
Suppose, now, that z is a function of x and
y, defined for values of (x, y) in some region
G of the xy-plane. Let do(xo,yo) and d1(x1,y1)
be two points of G (figure 5). Then from
partial differential equations, we have:
z = fx(x,y) = lim f(x+x,y) - f(x,y) …(4)
x
x0
x
………..…… (7)
Suppose, for example; lets consider that
there is a change in time of 3msec (say; beat
1= 875ms and beat 2= 878ms, IBI= -3ms).
Than the resulting rate of change in SV for a
corresponding change in x = -3 ms is then
z = -2 x -3 = 6 ml………….
x
from (7)
Similarly the rate of change in stroke
volume for a corresponding change in
pressure can be calculated. Here the
derivative of z = f(x, y), which is defined in
same magnitude. In either of these cases, the
(4) and (5). But the rate of change in z does
partial derivative fx would not exist. If,
not depend only on pressure and time, it also
however, fx does exist, then it gives the
depends
and
directional derivative to the right, while -fx
Contractility4 . Thus we need to define the
gives the directional derivative to the left
variable z (in this case stroke volume) as the
(the change in sign is due to the fact that
function of pressure, time, preload, afterload
(x)-1/2 =-x if x is negative). That is if fx
and contractility and z = f (x, y, p, a, c)
exists at the point, then both the right and
which can be defined in the similar way as
left directional derivatives exist at that point
pressure and time, and we will than have the
and have the same magnitude but opposite
collective change in z, that can be
signs.
on
Preload,
Afterload
considered as the optimal value z can attain,
that is, a value in the rate of change of z that
CARTOGRAPHY CONSTRUCTION
has no external forces acting on it. In this
way we derive the optimal values of all the
The next stage is to super impose the
other variables with respect to its dependent
suspected patient’s parameters on the model
variables.
to generate a resultant Cardiovascular
Cartogram.
In assigning to the limit in eqn (4) and (5) it
it understood that x may be either positive
or negative. If, on the other hand, we
calculate the directional derivative in the
direction of the positive x-axis, then x is
Cardiovascular Cartography is a process of
converting the relationship of many variants
from its original form to a more useful one.
The simplest form of cartography is to deal
with is one to one (i.e. each different
restricted to positive values x0+x, x>0,
statement
y=0. The directional derivative and the
representation that is different from that
partial derivative fx, differ in that in the
arising from any other statement).
directional
derivative
the
point
maps
to
a
single
target
d1
approaches d0 always from the same side,
Although one-to-one cartograms are, in
while in fx, d1 may approach d0 either from
general, the simplest to perform, they are
the left or from the right. In certain
rare in interesting input systems for several
"pathological" cases, a function may have a
reasons.
directional derivative from the right but not
many domains, inputs must be interpreted
from the left or may both directional
not absolutely, but relatively (like in
derivatives but the two may fail to have the
Cardiovascular cartography), with respect to
One important reason is that in
some reference model (in this case K -
clusters extending outward from the null
model).
For example, when images are
cluster has positive value and the clusters
being interpreted, size and perspective will
extending inwards has a negative value. The
change as a function of disease and to the
center of the cartogram has the lowest
extent of the disease.
negative value and the outmost cluster has
A
second
reason
that
many-to-one
the highest positive value of deviation.
cartograms are used, is that free variations is
The
often allowed, either because of the physical
Cartography are obtained on patients at rest
limitations of the system that produces the
in supine position. Six measurements are
inputs or because such variation simply
taken in the interval of one minute (this
make the task of generating the inputs
protocol is used simply to make the large
manageable.
amount
Both these factors help to
measurements
of
data
for
Cardiovascular
manageable).
Each
explain why such physiodynamic variables
measurement constitutes one sub-cartogram
require many-to-one cartography. At micro
and these are super imposed again on one
level no two people’s disease is exactly
another, thus the resultant cartogram has
identical, but at macro levels they are,
different colours. Darkest colours represent
identical disease influences the physiology
changes that occurred in the initial part of
in a characteristic manner, this is why the
the measurement and lighter colours are
interpreter
cartograms
later part of the measurement, showing a
require to know all the ways that a target
pattern of change. The dark line and dot
representation can be made. As a result, it is
combination shows the end point or the last
important, that analysis of these cartograms
measurement taken.
typically require a structured analysis of the
averaged situation.
input, rather than a simple, exact pattern
Along the circumference of the outermost
match.
cluster, we have all the 24 parameters
The Cardiovascular cartogram is a set of
designated along 24 axes, dividing the
concentric circles called "clusters". There
concentric circles into slices of 15 degrees.
are 9 clusters and each of these clusters has
When a patch of colour points outwardly
5 minor circles called "sectors".
These
along a given parameter, it means that this
clusters are scaled depending on the
parameter has been deviated by a percentage
parametric deviation from the K - model.
(in
This scale is named K-scale and has no
(Example;
units. The 5th cluster is designated as ‘0’
projecting outwardly up to the 2nd cluster
and that is called the null cluster.
from the null cluster and if the k-scale is 10
of
many-to-one
The
time
Yellow shows the
domain)
from
if
(stroke
SV
the
k-model
volume)
is
than SV has deviated by +20 on k-scale
factors that affect these changes. In such a
away from the Model). If the deviation is
continuously changing situation it is very
inward, then the value is negative.
difficult to make any meaningful diagnosis
Cardiovascular cartography is hence the
out
collective representation of PRESSURE,
haemodynamic
VOLUME, TIME, FLOW, PRELOAD,
analysis
AFTERLOAD
haemodynamic parameters, when mapped
thus
the
and
fluid
CONTRACTILITY,
mechanics
generally
of
obtained
discrete
parameters.
Collective
carefully
acquired
blood
against a multivariable mathematical model
circulation. It is a technique that is truly a
representing a set of carefully selected
systemic
normal
approach.
At
of
of
the
moment
subjects,
resultant
cartogram
studied with respect to coronary artery
haemodynamic variability and deviations
disease only, where the ability of the heart to
(from the model), that has significant
meet the basic demands of the body at rest is
applications in studying the haemodynamic
analysed and this is seen to be different in
variability in patients with cardiovascular
patients with and without coronary artery
disorders. The technique was used in a
disease. Unlike in stress ECG, where
blinded multicentred study for reliable
electrical changes can occur only in elevated
detection of coronary artery diseases.
body
50 South Asian male and female (36:14)
in
cardiovascular
occur at the basic body demands at rest.
Angiographically normal were recruited.
Stress ECG looks at one single parameter,
Haemodynamic parameters (Stroke volume,
the electrical variation, thus a large scope of
Cardiac
errors. But in cardiovascular cartographic
vascular resistance and index and systolic
study, it is the variations of multi-parameter,
time intervals) were carefully obtained in
which is why the sensitivity and specificity
supine position at rest, from the recruited
are above 90%. We are sure future
subjects, using a non-invasive impedance
researchers will work on other areas of
cardiogram (ASKIT: ICG-M-501) that was
cardiovascular
precisely time related with phonocardiogram
using
this
technique.
and
54,
who
the
subjects
Output
age:
indicates
cartographic study haemodynamic changes
disorders
mean
clearly
a
variability of haemodynamics at rest is
demands,
that
produces
Index,
were
Systemic
and electrocardiogram. From these acquired
parameters a model was constructed using a
specially designed computer program, which
Methodology
Haemodynamics
and
circulatory
status
generates the model extremities (Q1 and Q2),
changes continuously and there are many
model core (O1 and O2) and natural
deviations (N1 and N2).
These model
including
beat-to-beat
stroke
volume,
variables are then loaded into a cartography
systolic time intervals and blood pressure
computer program (Scalene: CVM-ver 4.5).
were obtained for 6 minutes at rest and in
The acquired haemodynamic data from a
supine position in 273 patients (43 females;
patient, who needs to be studied, is
mean age 46 years) scheduled for coronary
superimposed
angiography.
cartography
on
the
computer
model
by
program
the
and
a
was
found
that
the
were
made
using Impedance Cardiography, that was
precisely time related with simultaneously
resultant cartogram is obtained.
It
Measurements
pattern
of
haemodynamic variability on the resultant
cartogram was similar in patients with the
similar type of coronary artery disease.
Cartograms of normal individuals had more
harmonious and logical haemodynamic
changes on the cartogram pattern and could
be easily distinguished.
obtained
phonocardiographic,
electro-
cardiographic and non-invasive arterial blood
pressure data. These data were superimposed
on the model and an integrated CCG was
obtained
for
each
patient.
A
single
investigator blinded to the angiographic data
interpreted these maps.
Results: The CCG was positive for CAD in
204 patients and negative in 69 patients.
Angiographically, CAD was present in 199
CLINICAL STUDY
Background:
multi-variable
sensitivity, specificity, positive predictive
mathematical model was designed using
accuracy (PPA) and negative predictive
Haemodynamic variability data obtained
accuracy (NPA) of this technique for
from 50 control patients with normal
detecting CAD respectively was 92%, 92%,
coronary angiograms and left ventricular
98% and 75%.
functions.
Conclusion:
By
The
patients and absent in 55 patients. The
superimposing
the
data
The
technique
of
obtained from other patients on the model, a
Cardiovascular Cartography is a reliable
pattern, cardiovascular cartography (CCG)
non-invasive tool to detect the presence of
could be generated. In a pilot study it was
and assess the severity of CAD. Preliminary
observed that Coronary Artery Disease
results
(CAD) characteristically altered the CCG
available non-invasive tests to detect CAD.
pattern. This study was designed to assess
Future Trends
the feasibility of using such modeling and
The availability of more powerful computer
cartography technique to detect the presence
workstations will lead to more detailed analysis
of and assess the severity of CAD.
of the large amount of data collected. Of
Methods:
particular promise is the use of the knowledge in
Haemodynamic
measurements
compare
favorably
with
other
the cardiovascular cartograms to virtually image
the coronaries. It remains to be seen whether this
Chaos Theory, Bangalore, India,
promise will be fulfilled in the very near future.
1996.
2
Acknowledgments
The authors wish to thank Dr. Devi Prasad
B.M,
"Science
Common
Sense",
Address,
Proceedings
National
Shetty, Chairman, Manipal Heart Foundation, for
and
Keynote
of
the
Conference
on
Biomedical Engineering, Manipal,
granting permission for conducting the clinical
India.
studies at the heart foundation and Dr.Alok Roy,
Project Director, Manipal Heart Foundation for
Hegde
3
Attila
Nashlady,
Cardiologia
Hungatia, supplement, editorial.
his encouragement and great motivation. The
authors are grateful to Sir. Athila Nashlady, Dr.
Sabastin, Dr. Tripati, Dr. Gabor Vareski and Dr.
4
Flowers NC, Horan LG, Johnson
Murali Mohan for their valuable suggestions and
JC. Anterior infarctional changes
Ideas.
They are also thankful to the staff of
occurring during mid and late
Manipal Heart foundation, Scalene Health,
ventricular activation detectable by
ASKIT kft., for their excellent co-operation
surface
during the study. The authors are thankful to
McGraw-Hill Book Co., for permission to reuse
parts of materials from their publications. And
mapping
techniques.
Circulation. 1976; 54: 906
5
Huesman RH, Reutter BW, Zeng
GL and Gullberg GT, "Kinetic
lastly they wish to thank all the patients, who
parameter estimation from SPECT
willfully co-operated during the study.
cone
beam
projection
measurements" Phys. Med. Biol.
Plates
Vol43, pp. 973-982, 1998.
Plates shown explain the different types of
6
Carson
RE,
Lange
K.
"The
cardiovascular cartograms and their association
Parametric image reconstruction
with the coronary artery disease. Note the
algorithm," Amer. Ststist. Assoc.,
similarity of these cartograms in patients with
vol. 80, pp. 20-22, 1985.
Identical diseases.
References
1
Kumar A. Hegde B.M and Prabhu.
"The chaos theory, cardiology and
electrocardiograms", Proceedings
of the national conference on
Plates
Plate 1. Cartogram of normal subjects
Plate 2. Cartogram of two patients with RCA 100% Stenosis (left) and RCA 60% (right)
Plate 3. Cartograms of two patients with LAD 100% Stenosis (Left) and LAD 80% (Right)
Plate 4. Cartogram of two patients with LCX disease (90% left, 100% right)
Plate 5. Cartogram of patients with two vessel CAD on RCA and LAD branches.
Plate 6. Cartogram of patients with disease in all three vessel branches.
Note: A 20 hour physician training was required for interpreting and diagnosing these cartograms in our
experience.