Conceptual Lattice: A Unified Model for Medical Inference Processes

1. INTRODUCTION

The history of attempting to use the computer as a tool for clinical decision making goes beyond two decades. Since the beginning there have been two approaches in designing the computer alogrithm for medical decision making: The statistical approach [1]-[3] and the empirical approach [4]-[6], the latter has been customarily called the artificial intelligence (AI) approach by its school.

The statistical approach is successful whenever enough samples can be accumulated for each category of diseases, but quite useless for more complicated differential diagnosis where reliable data samples are small. The AI approach illustrates its strength when the decision categories are extensive but also where sound medical knowledge for reasoning exists. Here the statistical reasoning based on the objective probabilites calculated from data is replaced by a decision criteria based on an expert confidence measure. But these theories seem to show their weaknesses when applied to the areas where empirical knowledge

is neither well organized nor reliable.

Since 1971, the author has been working on the computer diagnosis of heart diseases based on a third approach, called "semiempirical" [7]. In the semiempirical approach, the specialist's diagnostic ~easoning is first analysized and broken down into more elementary processes. Then the decision process is reconstructed by a hierarchy of elementary decision compartments. The logic at each decision compartment may be given empirical knowledge at the beginning, but these empirical logics can gradually be replaced by probabilistic decision logics as statistically reliable data accumulates. The semiempirical approach thus builds a bridge between the statistical theories and the AI theories, and overcomes the limitations of the both approaches.

One of the gists of the computer-based clinical decision making is the modelling of dieases and simulating the expert's thinking processes based on the resultant model. In this paper a representation model for human inference proce­ sses based on a structure disease model is proposed. This model is a refinement and extention of the algorithmic frame, developed in the semiempirical approach. The model is called conceptual lattice model, because it uses relation among medical concepts. The model is not only relevant for the models proposed up to now, but also useful to unify the semiempirical approach, from the statistical to AI approaches.

2. CONCEPTUAL SYSTEM FOR MEDICAL KNOWLEDGE

Medical knowledge, as a conceptual system, has two embedded structures: biostructural (i.e., etiological, pathological, physiological, and anatomical) semantics and hierarchical human information processings. If we consider any disease etiologically, we see casual relations between concepts parallel to the biological hierarchy. But the present biostructural knowle­ dge in medicine is far from complete, and the biostructural conceptual systems cannot directly be modelled into computer programs.

The structure of human information processing for medical decision making is better ordered, and suitable to a multi-stage and multi-level information reduction process model. Usually a patient takes several tests in sequence. At each test the primary information concerning the patient's body or his specimens are reduced into symptoms and signs, and the medical consultant determines probable diseases from the information.

The ideal situation is that of human guessing based on observations going side by side with the biostructural casual reasoning. Conversely, the biostructural conceptual system may suggest the necessity for new observations. However in reality the medical consultant's mind works neither purely by observation nor on a biostruc­ tural basis. His thinking is somewhere in between. The conceptual lattice model thus intends to abstract the conceptual system structure embedded in the human mind.

3. CONCEPTUAL LATTICE

3.1 Definition

Let concepts used in medical decision making be a, b, c., ··· , x, 1J, and z, etc. In the case of necessity we distinctively use a, b , and c. for definite expression with yes/no or with constant values included such as "having cyanosis" or "age less than 5 years old", while the X, y, and z are used to express variables, such as "the m.unber of white blood cells," or "the systolic blood pressure" without specifing their values. In general the a, fl and c. symbols correspond to propositions appearing in medical textbooks. If a concept a appears as an element to explain concept b, we write a -. b , in other words, we need the concept a in order to define the concept b in some knowledge frame in current medicine. We assume a -+ a always nolds.

Let tx, 13 , y, be the aggregate of concepts, we introduce lattice structure between the aggregate of concepts, which are called "concep­ tual lattices". For example, we may extract a partially ordered set of concepts from any given conceptual system having the relation ~.

a disjoint set of subsets (aggregate of concepts) a, 8, ... , and introduce a new realtion ~ w hich is defined by " a · 8 iff for any a 1 £ a there exists a b.£ 8 such that b 1 ....., a 1 ana for any ai E a there Jxists a flE 8 such tnat b ... ai", we again obtain a newllattice under cjrtain aaditional conditions. The operations A and V can be introduced as l.u.b.(Q-,8) and g.l.b. (a,8 ) respectively, as usual [8].

To illustrate that the conceptual lattice model originates from a common language in science, few examples of conceptual lattices will be given. From an extremely reductionistic view point a patient is considered a physical system, whose state may either be represented by a boolean propositional lattice (classical case) or a weak modular lattice (quantum case) [9]. In usual pattern recognition language, a patient is represented by a n-tuplet observation, and the diseases correspond either to regions or subspaces in then-dimensional space. These pictures are again examples of a boolean and a modular lattice system [10].

3.2 Inference Compartments

Between any two conceptual lattice elements Cl = (a 1 ,a 2 , ... , oin,) and 8= (b 1 ,b 2 , ... , bn) which satisfy a * 8, we say that Ehere exists an inter-nodal logic, which is a mapping between the two element concepts of the lattices. Since there may be several other lattice elements a', a·~ ·. , for the element~ with the same relation + , the values of b i cannot uniquely be specified unless giveing a mapping which determines the values of bi uniquely based on the outputs of the above mapping. This new input-output relation within the lattice element, when specified by an explicit function relation, is called the compartment logics.

When compartment logics are specified ·· the conceptual lattice is called a state lattice. Elements of the state lattice are called state nodes. Concepts included in the state nodes are called state variables. Thus a state lattice which corresponds to a computer algorithm of diagnosis, is the model of physicians inference process for diagnosis. However a conceptual lattice only gives an algorithmic frame, and

not the algorithm itself.

In order to specify a compartment logic one needs an empirical logic. Usually these logics are given by "If ... , then ... " sentences written by a medical consultant [4]. But for checking logical consistency, and for improving and up dating the logic, a logic table is preferable, because the size of the logics is usually very large. Moreover table logic can easily be transformed into a probabilistic logic, such as linear discriminant function, as data is gathered. Thus a translator, which generates a table logic matrix from conversational "If. .. then ... " type sentences becomes useful [7].

3.3 Models for Progonosis

The applicability of the conceptual lattice model for progonosis problems is briefly explained. Prognosis is the prediction of patient state transitions. In our model, state transitions are represented by transitions between two state lattices. Symbolically we represent a patient's state at time t by a state lattice 'l'(t) havingX 1 (t), x 2 (t), ... as its nodes. As was explained, these nodes may either be the concept (physiological variables) or an aggregate of concepts. We then denote the transition probability between two states 'I' (ti) and 'l'(tj) by

We may also consider the two probabilities

and

If we take ~s in (3) to be a same physiological parameter such as blood pressure, (3) predicts its change. However such predictive probability may not be known in general.

4. DISCUSSIONS

As we have emphasized the conceptual lattice model is not an algorithm for compuer-based decision making, but it is a methodology to generate an algorithmic "frame" with which one can draw in the body of logic which is "squee­ zed" from human empirical thinking. One of the salient feature of the above method­ logy is that the computer alogrithm accepts some incoml·ete data, which a usual probabi­ listic logic can not handle. Another big advantage of the conceptual lattice model is its evolutional feature, which uaual AI approachs

do not emphasize.

Despite such partial success the model is still in its development stage. Several applications of the model has been attempted for heart, metabolic, and. hematological diseases.

ACKNOWLEDGEMENTS

The author owes his medical knowledge to Dr.s K. Machii, Y. Yahata, and S. Kurashina. He also would like to thank Mr. I. Suzuki for his helpful discussions.