Abstract: In this paper, a framework for an intelligent bedside monitor is presented. The monitor derives an abstraction of the current status of a patient by performing fuzzy state transitions on pre-processed input continuously supplied by clinical instrumentation. So far, an implementation called DiaMon-1 has been used for off-line evaluation of data of patients suffering from the adult respiratory distress syndrome (ARDS)

Friedrich Steimann

Klaus-Peter Adlassnig

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Fuzzy Sets and Systems 61 (1994) 37-42 37 
North-Holland 
Clinical monitoring with fuzzy automata 
Friedrich Steimann and Klaus-Peter 
Adlassnig 
Department of Medical Computer Sciences, University of 
Vienna, Vienna, Austria 
Received April 1993 
Revised July 1993 
Abstract: In this paper, a framework for an intelligent 
bedside monitor is presented. The monitor derives an 
abstraction of the current status of a patient by performing 
fuzzy state transitions on pre-processed input continuously 
supplied by clinical instrumentation. So far, an implementa- 
tion called DiaMon-1 has been used for off-line evaluation of 
data of patients suffering from the adult respiratory distress 
syndrome (ARDS). 
Keywords: Fuzzy automata; monitoring; abstraction; inten- 
sive care; ARDS. 
1. Introduction 
As new devices for on-line sampling of 
physiological data become available, intensive 
care monitors display more and more informa- 
tion. However, instead of the clinical staff being 
relieved, they are faced with a different problem, 
that of monitoring the monitor. Perception and 
interpretation of a multitude of time-varying 
parameters is difficult for humans [2, 5, 10], even 
if the parameter values are displayed in an 
ergonomic style. The situation is further 
complicated when parameters interact such that 
only certain constellations provide hints for 
critical conditions, or when the meaning of a 
value depends on the patient history, i.e., on 
what has happened before. Context-specific 
alarms for which no absolute thresholds can be 
established are a good example of this [2, 9]. 
This paper presents a formal framework for 
the design of monitors that exhibit the following 
properties: 
• abstraction from objectively observed (quan- 
Correspondence to: F. Steimann, Department of Medical 
Computer Sciences, University of Vienna, W~ihringer Giirtel 
18-20, A-1090 Vienna, Austria. 
0165-0114/94/$07.00 © 1994---Elsevier Science B.V. All rights 
SSDI: 0165-0114(93)E0182-R 
titative) parameters to (qualitative) stages of a 
disease, 
• early indication of improvement in or 
deterioration of the patient's state by provid- 
ing smooth transitions between stages, and 
• consideration of previous events, i.e., history- 
based interpretation of data. 
2. Definition o f  a state monitor  
A state monitor as defined in the following 
employs concepts that stem from automata 
theory [3]: inputs, states, and transitions. A state 
monitor is an instrument that traces the patient's 
change of state with time, i.e., that records the 
progress of his illness and, hopefully, of his 
improvement. In this context, a state is 
considered to be an abstraction of the patient's 
status which accounts for a specific stage of a 
disease. Transitions provide possible paths from 
one state to the next. They depend on input, 
events that need to occur or conditions that need 
to be satisfied for a transition to take place. The 
input is obtained by processing objectively and 
preferably automatically acquired data. 
The design of a state monitor is dominated by 
the nature of the monitored disease: it is 
intended to be an abstract model of the medical 
knowledge in that specific area. Because medical 
decision making is based on knowledge that has 
to take uncertainty such as physiological 
variability into account, judgement of the 
current state of a patient is often a matter of 
degree [1]. Consequently, transition from one 
stage of a disease to the next is hardly ever 
abrupt but usually smooth. To allow for smooth 
transitions, a state monitor is based on a fuzzy 
automaton rather than on a conventional one. 
Definition 1 (fuzzy automaton). A fuzzy auto- 
maton is a quadruple 
= (Q, qo, 1, 6) 
reserved 
38 F. Steimann, K.-P. Adlassnig / Clinical monitoring with fuzzy automata 
where 
Q is a finite set of states, 
c]0 is a fuzzy subset of Q called the fuzzy initial 
state, 
I is a finite set of input symbols, and 
6 : Q  × I---~ Q is a transition function that maps 
states and inputs onto states. 
6 is extended to fuzzy arguments by the 
extension principle [4, 12] such that 
qt+l = (~(qt, /) 
= ~ min(/%(q), I~i)) /6(q,  i) 
Qxl 
where for each q ~ Q 
[ max min(/u,~,(q'),/,,,(i)), 
lzo,÷,( q ) = ~ ¢5(q,.i)=q (1) 
L0 if b-'(q)=fl. 
If ~(A) denotes the set of all fuzzy subsets of A, 
6: ~(Q)  x ~(I)--> ~(Q).  A fuzzy state ~] is said 
to be included in the fuzzy state c]' (~] __ ~]') if 
Vq ~ Q: /~#(q)~</z,v(q). ~ is said to be strictly 
included in c]' (c]c c]') if c] ~_ c]' A C] =/: C~'. A state 
~] is said to be empty if Vq ~ Q: /~,~(q)= 0. A 
sequence of fuzzy states is denoted by (c],) and 
said to be increasing if 4,+1 -~ ~t for all t. 
A set of final states that can usually be found 
in the definition of automata has been omitted 
on purpose, as monitoring is a continuous 
process that is rather terminated on the 
exhaustion of input data than on arrival at a 
certain state. As opposed to other definitions of 
fuzzy automata [3, 4, 8, 11], the transition 
function is not itself fuzzy; instead, the 
uncertainty expressed in a fuzzy input alone 
results in a partial transition from one state to 
another, without any possibility for a general 
statement about how strongly two states can at 
most be related. Thus, the design of a fuzzy 
automaton is not itself fuzzy: apart from the 
fuzzy initial state the fuzzification of the 
automaton is completely covered by the 
extension principle. Non-fuzzy tabular and 
graphical forms of defining an automaton can 
still be used - all the fuzziness is brought into the 
system through fuzzy inputs (compare Figure 2). 
Despite its very similar definition, a fuzzy 
automaton exhibits properties quite different 
from those of its crisp counterpart. Firstly, being 
a fuzzy set, the current state is a distribution 
over several crisp states. Consequently, the 
automaton can perform different (partial) 
transitions simultaneously and therefore track 
parallel paths. Secondly, while crisp automata 
report an error on input not accounted for at the 
current state, a fuzzy automaton reacts on low or 
zero membership grades in the fuzzy input with 
continuously decreasing membership grades in 
its current state as obtained by (1) and depicted 
in Figure 3(b), a depletion of certainty that 
seems coherent with all repeated applications of 
fuzzy set operations. 
However, a fuzzy automaton alone is not very 
appropriate to perform monitoring: firstly, 
automatically acquired data is generally precise 
and hence no source of fuzzy input as required 
by the automaton defined above. Secondly, if 
every single parameter value acquired repre- 
sented an input on its own, the automaton would 
(a) explode because of an excessive number of 
possible input symbols all needing to be 
accounted for, and (b) continuously change its 
state in order to react to a certain input (if it did 
not, the input would remain unconsidered and 
thus become lost). The data is therefore 
pre-processed by a function that abstracts from 
single input parameters by generating fuzzy 
events that are passed on to the automaton. 
Definition 2 (state monitor). If A = (Q, q,,, I, 6) 
is a fuzzy automaton as defined above, n is the 
number of parameters observed, R1 through Rn 
are the parameter ranges, P =RI  x • .- × Rn is 
the parameter value space, and f:P---~ ~(I)  is a 
function that maps parameter tuples to fuzzy 
subsets of the input alphabet of A, then 3) = (A, 
P, f )  is a state monitor. 
Technically, ~(I)  specifies the interface 
between pre-processing of data through f and 
interpretation of input through /{. f can 
therefore be replaced by any computable 
method that yields a fuzzy set i suitable for input 
to /i, regardless of being a function or some 
other evaluation such as trend analysis, integra- 
tion, or any other. 
3. Peak hold 
The definition of A and the extension of 6 in 
(1) guarantee that state membership values 
F. Steimann, K.-P. Adlassnig / Clinical monitoring with fuzzy automata 39 
patient & 
instrumentation state monitor 
clinical 
staff 
i pre-processing interpretation 
digitized data fuzzy input fuzzy states 
Fig. 1. Environment, components, and data flow of a state monitor. 
I~o,(q ) other than those of q_0 can only be 
introduced through fuzzy input i. If the set of all 
fuzzy inputs fed to a fuzzy automaton is finite, 
then the set of fuzzy states it can take on is also 
finite. Particularly: 
Lelnma 1. I f  a fuzzy automaton .A is repeatedly 
fed with constant fuzzy input [, the set of fuzzy 
states it transitions between is finite. 
Proof. Follows directly from (1). 
Note: If the range of membership grades is 
discrete, the set of fuzzy states of fi, is also finite 
(proof omitted). 
The rain in the definition of t5 further implies 
that no state can become more evident than the 
most evident of its predecessors and is further 
limited by the fuzzy input, as depicted in Figure 
3(b). The sequence (hgt(~t)), where hgt(~/)= 
maxq i~o,(q ) is called the height of the current 
state, is therefore decreasing reflecting a 
continuous loss of certainty in the automaton. In 
practice the situation is further aggravated when 
the monitor is provided input in rapid succes- 
sion, as the height can fall rapidly even if the 
input does not change, and once the current 
state is the empty set, it can never recover. In 
fact, if the automaton does not contain any 
feedback loops, i.e., does not provide circular 
transitions, it will arrive at the empty state after 
at most as many steps as there are states, as 
demonstrated in Figure 2(b). This is clearly not a 
desired property of a state monitor. However, 
instead of leaving the responsibility for providing 
appropriate feedback loops to the designer of 
the automaton, the following defines a property 
that overcomes this inadequate behaviour. 
Definition 3 (peak hold). A fuzzy automaton is 
said to provide a peak hold if there is a transition 
for every state to itself on every input that leads 
to the state, i.e., if 
Vq', i, q: 6(q' ,  i) = q--> 6(q, i) -- q. (2) 
The condition implies that no state can be 
entered and left on the same input, otherwise 6 
would no longer be deterministic. Semantically, 
the peak hold guarantees that the maximum 
evidence for a state provided by its predecessors 
is memorized and held as long as input of 
ingoing transitions can support it, as shown in 
Figure 3(c). However, because a state does not 
remember its predecessor, the peak hold may 
also be sustained by an input other than the one 
that initially led to that state, and consequently 
the grade of membership can unintentionally 
remain high. Careful design of the state monitor 
is therefore necessary. 
A positive side effect of the peak hold is that 
the automaton cannot oscillate on constant input 
40 F. Steimann, K.-P. Adlassnig / Clinical monitoring with fuzzy automata 
(a) 
(c) 
• i2 
il 
(b) 
i 1 I - "  ~ ,  
(d) 
Fig. 2. (a) Fuzzy input [ supported by two (crisp) input 
symbols i t and i 2, (b) an automaton that is empty after at 
most three transitions on any input, (c) circular transitions 
that can lead to oscillations on repeated inputs of t, (d) 
automaton with peak hold that cannot oscillate. 
[11] (compare Figures 2(c) and (d)), a property 
that would clearly not be acceptable in the 
clinical context, as stable (i.e., not changing) 
input should be reflected in stable output. 
Instead, the following theorem holds• 
(2) (indirect) Lemma 1 implies that there is no 
infinite sequence of fuzzy states (~,) such that 
q,+l = 6(q,, D ^ q,+l ~ qt- Therefore, there has 
to be a step r after which ~,+r ~ -- 4,+r-~. Because 
(~,) is increasing, q,÷r must equal q,+r-," 
In particular, the proof shows that (qt) does not 
converge to the empty state. The reason why it 
can take several steps until ,4 is stable is basically 
the fuzzy input [ which, when supported by more 
than one input symbol, can cause a propagation 
of higher grades of membership along a 
sequence of transitions. Note that non-fuzzy 
deterministic automata with peak hold are stable 
after one step. This is yet another example of 
how fuzzification yields more general results• 
4.  A c t i v e  s ta tes  
T h e o r e m  1 (stability). The f u z z y  state o f  a f u z zy  
automaton with peak hold always becomes stable 
after a finite number o f  repeats o f  the same input• 
Proof. The proof is performed by showing that 
there is a step r such that 
= . . . .  ( 3 )  
in two steps: 
(1) (#,) is increasing, i.e., #t ~- #,+l - ' "  ", 
(2) :lr: ~(qt+r, D = #,+r. For all subsequent 
states (3) follows from t5 being a function. 
(1) For every fuzzy state qt following the initial 
state qo and every input t (1) implies: For every 
state q there is a transition that determines its 
membership value, i.e. 
Vq, 6-1(q) :#0: 3q ' ,  i: 
6(q' ,  i) = q A /re(q) = min(#4, ,(q'), /~i)) .  
Thus, It4,(q)<~#i-(i) and following min(~#,(q), 
/*i(i))=po,(q). Repeated input of l and 
6(q, i) = q then implies 
kt4,+,(q) = max(/~,~,(q), max min(/~,~,(q'),/~i,(i))), 
b(q',i)=q 
~4,+,(q) >I la¢(q), 
which is justifying the term peak hold, 
consequently 
4,+, =- q,. 
and 
Despite the peak hold property, the height of 
the current state of a fuzzy automaton is still 
decreasing, as high grades of membership cannot 
be regained once they are lost• A particular 
source of loss is a situation where the grade of 
membership of one state decays while its 
successor's rises, as depicted in Figure 3(c). 
This behaviour does not model the natural 
decision process correctly: once a decision has 
been made, it is usually pursued rather 
uncritically until there is sufficient evidence for 
another decision to be made. 
The state monitor can be modified to adopt 
this kind of inertia in its behaviour: if a state is 
said to be active when its grade of membership 
in the current state exceeds a certain threshold 
re, then an active state is defined to remain 
active until there is a transition that induces 
activity of one of its successors, i.e., 
f/~4,(q) i f /~,(q)  >/o¢ A ~=lq', i: 
/*~,+,(q) 6(q, i) = q'  A IZ~,(i) >i re, 
(1) else• 
(4) 
In other words, once a state has gained a 
certain grade of membership, it keeps it until a 
transition can pass it on to one of its successors, 
a behaviour that is illustrated in Figure 3(d). It 
implies that the height of the current state is 
F. Steimann, K.-P. Adlassnig / Clinical monitoring with fuzzy automata 41 
(a) ' l ~ i 2 .  (b) ql 
~tl01 ~ P'I ..... q2 
• 0 / ~  • 
t t 
(C) B10[ q ~  'd' i ]: q ~  q2~ 
t t 
Fig. 3. (a) Membership course of input signals il and i~, (b) 
membership grades of ql and q2 where /t(qo)-=l, 6(qo, 
it) =ql, and 6(qt, i2)=q2, (c) same as (b) with peak hold, 
and, (d) same as (b) with peak hold and active state level tx. 
always greater than tr, a certain level of certainty 
thus always being maintained. 
For ( r = l  (4) implies that q, is always 
normalized, i.e,, there is always at least one 
state q such that #4,(q)  = 1. This accounts for the 
fact that the patient is considered to be at least 
in one state at a time, even if only no successor 
with more evident support could yet be 
determined. 
Note that the proof of Theorem 1 is only 
slightly affected by (4): 
(1) (~,) is still increasing, as the peak hold 
also works for i~ , (q)>1 (r and /~,7,(q) can only 
once drop below o:, namely on the first input of 
[, and 
(2) still holds because Lemma 1 is not 
affected. 
Also note that (4) without peak hold, although 
keeping the height above (r, cannot prevent the 
automaton from oscillation. 
Because automata do not provide adequate 
means of performing complex mathematical 
operations, trend and artefact detection as well 
as evaluation of derived parameters are best 
placed in a function that pre-processes input 
before it is fed to the automaton. This work 
clearly separates the idea of state-based inter- 
pretation of events from rather general problems 
such as trend analysis quite common to other 
medical expert systems by introducing a layered 
architecture as suggested in [2, 5]. 
Originally, a diagnostic monitor DiaMon-1 
very similar to the one introduced, with peak 
hold but with a restriction to only one active 
state was developed and implemented to 
retrospectively analyze data of patients suffering 
from the adult respiratory distress syndrome 
(ARDS) [7]. The idea arose from the demand 
for standardized criteria for the different stages 
of ARDS together with an objective evaluation 
technique, as widely differing mortality rates 
were considered to be due to different definitions 
of the syndrome itself as well as varying entry 
criteria for its possible therapies. The monitor is 
currently used in a multi-centre study with the 
aim of standardizing the A R D S  criteria of 
different clinical centres. Independently, the 
concept of fuzzy events resulting in smooth 
transitions between states was judged to model 
human decision making naturally, and prospec- 
tive operating of the monitor promises to be of 
valuable help to clinical staff. 
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A formal framework has been presented that 
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straightforward to implement. It competes 
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Clinical monitoring with fuzzy automata

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