Graph-Grammar Productions for the Modeling of Medical Dilemmas John W. Egar, Mark A. Musen Section on Medical Informatics Stanford University School of Medicine, Stanford, CA
ABSTRACT We introduce graph-grammar production rules, which can guide physicians to construct models for normative decision making. A physician describes a medical decision
problem using standard terminology, and the graph-grammar system matches a graph-manipulation rule to each of the standard terms. With minimal help from the physician, these graph-manipulation rules can construct an appropriate Bayesian probabilistic network. The physician can then assess the necessary probabilities and utilities to arrive at a rational decision. The grammar relies on prototypical forms that we have observed in models of medical dilemmas. We have found graph grammars to be a concise and expressive formalism for describing prototypical forms, and we believe such grammars can greatly facilitate the modeling of medical dilemmas and medical plans.
MEDICAL DILEMMAS Physicians, other health-care givers, and patients are regularly faced with dilemmas: Tests, treatments, and more complex plans all have costs, risks and benefits to consider. Over the past three decades, researchers have worked to provide decision analysts with tools that help them to weigh costs and benefits of difficult decisions. These tools presume that the decision maker has modeled each dilemma using probabilities and utilities. This presumption fails to account for three facts: 1. Physicians have little time to devote to modeling each tough decision they have to make.
2.The toughest decisions often are the most difficult to model, involving more than a half-dozen different considerations. 3.Physicians may be more comfortable describing medical situations using medical terminology than they are using state variables and probabilistic dependencies. For these three reasons, we are developing a graphgrammar production system that can help health-care
0195-4210/92/$5.00 (C1993 AMIA, Inc.
349
workers to construct a decision-theoretic model from a list of standard medical concepts. Because of the complex and highly intertwined relationships among medical concepts, we are using a graph grammar. A graph grammar is a system of replacement rules that operates on graphs, rather than on the strings on which traditional context-free grammars operate. Our present underlying representation is an extension to the influence-diagram notation [1]. This class of semantic-network representation has a formally defined meaning, where nodes represent uncertain parameters of the problem, solid arcs represent probabilistic dependencies, and dashed arcs represent information available at the time of a decision. In our grammar, there is a graph-manipulation rule corresponding to each key factor or altemative that can be expressed as either a concept from SNOMED III [2] or a term from CPT [3]. The graph-grammar system begins with a list of such standard concepts and terms from a textual description of the dilemma. The system accepts each term from this description as a token symbol, and maps it to a graph-grammar production rule that introduces that concept into an evolving influence diagram. Once the requirements for applying a production rule are satisfied, the user selects among the possible ways in which the rule can be applied.
GRAPH GRAMMARS Graph granmmars were first introduced 20 years ago [4, 5]. Although they were used originally for problems in image recognition and datatype definition, they have since evolved, and have been used for visual language specification and knowledge representation [6]. In this research, we are using such grammars to guide the derivation of graphs from an unordered list of node labels. A graph grammar consists of a set of production rules that dictates how a graph can be transformed and rewritten. These production rules are different from the productions used in rule-based expert systems: Graph-grammar rules can specify a wide range of contexts for which they are applicable, and can describe different graph manipulations for those different contexts. A graph grammar speci-
fies a laniguage over a set of symbols that are elements of a graph. We have found use of graph grammars to be an expressive and concise way to represent prototypical forms for modeling dilemmas. We also believe that graph grammnars can provide high-level abstractions that help users to manage complexities.
(a)
(b)
Figure 1. Sample graph-grammar production rule. The production denotes that some node of type X in the host graph is to be replaced by new instances of XX and XZ. VL, VA, VB, and VR (left, above, below, and right) are the four regions of a graph-grammar production rule.
(c)
The particular type of graph grammar that we use is a modification of Gottler's operational graph grammar, suggested by Barthelmann [7]. Briefly, a grammar is a collection of production rules. Each production rule describes a legal graph manipulation. We write these productions as graphs divided into four regions (Figure 1), which partition the nodes into four sets: those in the left region, VL; those in the right region, VR; those in the determinate region below, VB; and those in the indeterminate region above, VA. The two sets, VA and VB, are referred to as the embedding part. Basically, the graph manipulation described by such a production is as follows: Find nodes matching the left region, VL, and replace them with nodes matching the right region, VR. The procedure thus consists of these four steps:
Figure 2. Sample application of the graph-grammar mle from Figure 1. (a) The first view of the host graph shows two nodes matching X and Y in the production. (b) The node matching X is removed. (c) Additional nodes XX and XZ are added. 3. Remove from the host graph those nodes that matched nodes within VL. This step corresponds to the deletion of node X and its incident arcs in Figure 2b.
1. Find a region of the host graph where the nodes and arcs match the nodes and arcs of the detenninate (VB) and left (VL) regions of the production. So, in Figure 2a, we search the evolving diagram for a node of type X and a node of type Y such that there is an arc from the forner to the latter.
4. Add to the host graph new nodes and arcs that correspond to those within the right region (VR) of the production, and add to the graph arcs that correspond to those connecting the embedding part (from VB and the matched portion of VA) of the production to the right region (VR). In Figure 2c, we add to the host graph nodes of type XX and type XY. Also, we add arcs from XX to XZ, from XZ to Y, from both E1 and E2 to XX, and from both E1 and E2 to XZ.
2. Find zero or more arcs in that part of the host graph that match arcs between the left (VL) and indeterminate (VA) regions of the production. In Figure 2a, we find an arc from E1 to X and an arc from E2 to X. In the host graph, the node X has no children of type F.
350
Technical details of our graph-grammar formalism can be found in [8].
QUALITATIVE CONTINGENT INFLUENCE DIAGRAMS Many researchers and decision analysts use influence diagrams and decision trees to model medical decisionmaking problems [9]. The graphical knowledge representation that we use here is a modification of the influence diagram, called a qualitative contingent influence diagram (QCID). Because such diagrams are qualitative, it is not necessary to assess probabilities or explicit utility models, although these may be added by the user of a QCID. Instead, arcs are labeled with a "+" or a "-" to denote whether a change in one variable's probability distribution results in a similar or an opposite change in the other's distribution. This lack of specific probabilities makes the prototypical forms identified in our graph grammar more general than are assessed influence diagrams.
Influence diagrams offer an explicit representation of probabilistic independence, denoted by the lack of an arc between two nodes. They provide a compact language to express the same information conveyed by decision trees. Still, researchers sometimes prefer to use trees so that asymmetries are shown explicitly. For example, a drug's toxicity, which plays a role in a decision only if a previous decision was made to administer that drug, can be omitted from one-half of the tree. Because these asymmetries are common, we use contingency nodes, described by Fung and Shachter.* Roughly, we divide a node into several contingent nodes, each with exclusive conditions. Each contingent node is considered relevant to the rest of the diagram only for scenarios where its conditions are met. If a contingent node's conditions are not met, then the node and its incident arcs can be ignored. In a QCID, contingent nodes have a dot at their left side.
not be specifically listed by the user-such as test results and future disease states-are added to the host graph by the systenm If the grammar includes all the input temas, our system produces the appropriate QCID. At this point, the qualitative model is ready for assessment. As an example, suppose that a decision maker must decide whether to advise undergoing angiography to a patient whom she believes might have a cerebral arteriovenous malfonnation (AVM). The decision maker describes her problem to our system by listing the pertinent medical concepts: cerebral arteriovenous malformalion, cerebral angiography, and AVM repair. All the concepts in this clinical problem have corresponding terns from SNOMED III and CPT-specifically, T-A20000; M-246400, CPT-75671, and CPT-61680. In Figure 3 are the three grammnar rules invoked in this example. The rules are automatically triggered as the most specific rules that could be used to incorporate the three medical concepts. Starting with just a utility node, the system can add only the disease node for cerebral AVM, with an arc labeled "-" to the utility node (Figure 3a). Then, with both utility and disease nodes present, the rule in Figure 3b can be applied to add repair AVM and future cerebral AVM. Finally, the requirement denoted in the determinate region (VB) of the last rule-that utility, disease and treatment are all present-is satisfied, so the system can incorporate perform cerebral angiography and angiogram result, leaving us with the QCID shown in Figure 3d.
To test the robustness of our graph grammar for formulating decision-analytic models, we developed several QCID models from standard terms found in SNOMED and CPT. Our test problems include an example supplied by WelLnian [10] that involves 18 different uncertainties and 5 different decisions. Our program-implemented in Common Lisp-constructed an influence diagram almost identical to that composed by Weilman. We are currently engaged in further tests and refinements in our systemr
GRAMMAR-ASSISTED MODELING To derive a model, we begin with a single utility node. Each term in the list is added according to production rules in our grammar. If the system cannot incorporate a term due to missing contextual requirements specified in the grammar, then the user may need to add terms for the missing context. For example, if the user lists a test and a treatment, but fails to mention any disease, then she is prompted for the relevant disease. Some nodes that might
*. Fung RM and Shachter RD. Personal communication.
DISCUSSION Few programs have been developed that help the user to model a medical decision problem [11, 12], and these programs have been limited to circumscribed domains, where the developer has already composed a range of possible models. Leong has shown how knowledge-based decision modeling might work [13], and others have recently begun work aimed at developing large knowledge bases that prescribe the relationships in an influence diagram [14]. Our approach has been to define prototypical relationships
351
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depends directly or indirectly on the result of that test. In BUNYAN, this rule is expressed as a pattern for which to search. In our graph grammar, this rule is incorporated into one of two graph-grammar rules that introduce testing decisions, such as perform cerebral angiography, into the model (Figure 3c). (The other graph-grammar rule that introduces testing decisions acknowledges that test results may have utility for the decision maker through an inherent value of knowing a diagnosis, and through resulting psychological preparations that are not modeled explicitly.)
(a) \
/ ~~~~
ABSTRACT We introduce graph-grammar production rules, which can guide physicians to construct models for normative decision making. A physician describes a medical decision
problem using standard terminology, and the graph-grammar system matches a graph-manipulation rule to each of the standard terms. With minimal help from the physician, these graph-manipulation rules can construct an appropriate Bayesian probabilistic network. The physician can then assess the necessary probabilities and utilities to arrive at a rational decision. The grammar relies on prototypical forms that we have observed in models of medical dilemmas. We have found graph grammars to be a concise and expressive formalism for describing prototypical forms, and we believe such grammars can greatly facilitate the modeling of medical dilemmas and medical plans.
MEDICAL DILEMMAS Physicians, other health-care givers, and patients are regularly faced with dilemmas: Tests, treatments, and more complex plans all have costs, risks and benefits to consider. Over the past three decades, researchers have worked to provide decision analysts with tools that help them to weigh costs and benefits of difficult decisions. These tools presume that the decision maker has modeled each dilemma using probabilities and utilities. This presumption fails to account for three facts: 1. Physicians have little time to devote to modeling each tough decision they have to make.
2.The toughest decisions often are the most difficult to model, involving more than a half-dozen different considerations. 3.Physicians may be more comfortable describing medical situations using medical terminology than they are using state variables and probabilistic dependencies. For these three reasons, we are developing a graphgrammar production system that can help health-care
0195-4210/92/$5.00 (C1993 AMIA, Inc.
349
workers to construct a decision-theoretic model from a list of standard medical concepts. Because of the complex and highly intertwined relationships among medical concepts, we are using a graph grammar. A graph grammar is a system of replacement rules that operates on graphs, rather than on the strings on which traditional context-free grammars operate. Our present underlying representation is an extension to the influence-diagram notation [1]. This class of semantic-network representation has a formally defined meaning, where nodes represent uncertain parameters of the problem, solid arcs represent probabilistic dependencies, and dashed arcs represent information available at the time of a decision. In our grammar, there is a graph-manipulation rule corresponding to each key factor or altemative that can be expressed as either a concept from SNOMED III [2] or a term from CPT [3]. The graph-grammar system begins with a list of such standard concepts and terms from a textual description of the dilemma. The system accepts each term from this description as a token symbol, and maps it to a graph-grammar production rule that introduces that concept into an evolving influence diagram. Once the requirements for applying a production rule are satisfied, the user selects among the possible ways in which the rule can be applied.
GRAPH GRAMMARS Graph granmmars were first introduced 20 years ago [4, 5]. Although they were used originally for problems in image recognition and datatype definition, they have since evolved, and have been used for visual language specification and knowledge representation [6]. In this research, we are using such grammars to guide the derivation of graphs from an unordered list of node labels. A graph grammar consists of a set of production rules that dictates how a graph can be transformed and rewritten. These production rules are different from the productions used in rule-based expert systems: Graph-grammar rules can specify a wide range of contexts for which they are applicable, and can describe different graph manipulations for those different contexts. A graph grammar speci-
fies a laniguage over a set of symbols that are elements of a graph. We have found use of graph grammars to be an expressive and concise way to represent prototypical forms for modeling dilemmas. We also believe that graph grammnars can provide high-level abstractions that help users to manage complexities.
(a)
(b)
Figure 1. Sample graph-grammar production rule. The production denotes that some node of type X in the host graph is to be replaced by new instances of XX and XZ. VL, VA, VB, and VR (left, above, below, and right) are the four regions of a graph-grammar production rule.
(c)
The particular type of graph grammar that we use is a modification of Gottler's operational graph grammar, suggested by Barthelmann [7]. Briefly, a grammar is a collection of production rules. Each production rule describes a legal graph manipulation. We write these productions as graphs divided into four regions (Figure 1), which partition the nodes into four sets: those in the left region, VL; those in the right region, VR; those in the determinate region below, VB; and those in the indeterminate region above, VA. The two sets, VA and VB, are referred to as the embedding part. Basically, the graph manipulation described by such a production is as follows: Find nodes matching the left region, VL, and replace them with nodes matching the right region, VR. The procedure thus consists of these four steps:
Figure 2. Sample application of the graph-grammar mle from Figure 1. (a) The first view of the host graph shows two nodes matching X and Y in the production. (b) The node matching X is removed. (c) Additional nodes XX and XZ are added. 3. Remove from the host graph those nodes that matched nodes within VL. This step corresponds to the deletion of node X and its incident arcs in Figure 2b.
1. Find a region of the host graph where the nodes and arcs match the nodes and arcs of the detenninate (VB) and left (VL) regions of the production. So, in Figure 2a, we search the evolving diagram for a node of type X and a node of type Y such that there is an arc from the forner to the latter.
4. Add to the host graph new nodes and arcs that correspond to those within the right region (VR) of the production, and add to the graph arcs that correspond to those connecting the embedding part (from VB and the matched portion of VA) of the production to the right region (VR). In Figure 2c, we add to the host graph nodes of type XX and type XY. Also, we add arcs from XX to XZ, from XZ to Y, from both E1 and E2 to XX, and from both E1 and E2 to XZ.
2. Find zero or more arcs in that part of the host graph that match arcs between the left (VL) and indeterminate (VA) regions of the production. In Figure 2a, we find an arc from E1 to X and an arc from E2 to X. In the host graph, the node X has no children of type F.
350
Technical details of our graph-grammar formalism can be found in [8].
QUALITATIVE CONTINGENT INFLUENCE DIAGRAMS Many researchers and decision analysts use influence diagrams and decision trees to model medical decisionmaking problems [9]. The graphical knowledge representation that we use here is a modification of the influence diagram, called a qualitative contingent influence diagram (QCID). Because such diagrams are qualitative, it is not necessary to assess probabilities or explicit utility models, although these may be added by the user of a QCID. Instead, arcs are labeled with a "+" or a "-" to denote whether a change in one variable's probability distribution results in a similar or an opposite change in the other's distribution. This lack of specific probabilities makes the prototypical forms identified in our graph grammar more general than are assessed influence diagrams.
Influence diagrams offer an explicit representation of probabilistic independence, denoted by the lack of an arc between two nodes. They provide a compact language to express the same information conveyed by decision trees. Still, researchers sometimes prefer to use trees so that asymmetries are shown explicitly. For example, a drug's toxicity, which plays a role in a decision only if a previous decision was made to administer that drug, can be omitted from one-half of the tree. Because these asymmetries are common, we use contingency nodes, described by Fung and Shachter.* Roughly, we divide a node into several contingent nodes, each with exclusive conditions. Each contingent node is considered relevant to the rest of the diagram only for scenarios where its conditions are met. If a contingent node's conditions are not met, then the node and its incident arcs can be ignored. In a QCID, contingent nodes have a dot at their left side.
not be specifically listed by the user-such as test results and future disease states-are added to the host graph by the systenm If the grammar includes all the input temas, our system produces the appropriate QCID. At this point, the qualitative model is ready for assessment. As an example, suppose that a decision maker must decide whether to advise undergoing angiography to a patient whom she believes might have a cerebral arteriovenous malfonnation (AVM). The decision maker describes her problem to our system by listing the pertinent medical concepts: cerebral arteriovenous malformalion, cerebral angiography, and AVM repair. All the concepts in this clinical problem have corresponding terns from SNOMED III and CPT-specifically, T-A20000; M-246400, CPT-75671, and CPT-61680. In Figure 3 are the three grammnar rules invoked in this example. The rules are automatically triggered as the most specific rules that could be used to incorporate the three medical concepts. Starting with just a utility node, the system can add only the disease node for cerebral AVM, with an arc labeled "-" to the utility node (Figure 3a). Then, with both utility and disease nodes present, the rule in Figure 3b can be applied to add repair AVM and future cerebral AVM. Finally, the requirement denoted in the determinate region (VB) of the last rule-that utility, disease and treatment are all present-is satisfied, so the system can incorporate perform cerebral angiography and angiogram result, leaving us with the QCID shown in Figure 3d.
To test the robustness of our graph grammar for formulating decision-analytic models, we developed several QCID models from standard terms found in SNOMED and CPT. Our test problems include an example supplied by WelLnian [10] that involves 18 different uncertainties and 5 different decisions. Our program-implemented in Common Lisp-constructed an influence diagram almost identical to that composed by Weilman. We are currently engaged in further tests and refinements in our systemr
GRAMMAR-ASSISTED MODELING To derive a model, we begin with a single utility node. Each term in the list is added according to production rules in our grammar. If the system cannot incorporate a term due to missing contextual requirements specified in the grammar, then the user may need to add terms for the missing context. For example, if the user lists a test and a treatment, but fails to mention any disease, then she is prompted for the relevant disease. Some nodes that might
*. Fung RM and Shachter RD. Personal communication.
DISCUSSION Few programs have been developed that help the user to model a medical decision problem [11, 12], and these programs have been limited to circumscribed domains, where the developer has already composed a range of possible models. Leong has shown how knowledge-based decision modeling might work [13], and others have recently begun work aimed at developing large knowledge bases that prescribe the relationships in an influence diagram [14]. Our approach has been to define prototypical relationships
351
K/
depends directly or indirectly on the result of that test. In BUNYAN, this rule is expressed as a pattern for which to search. In our graph grammar, this rule is incorporated into one of two graph-grammar rules that introduce testing decisions, such as perform cerebral angiography, into the model (Figure 3c). (The other graph-grammar rule that introduces testing decisions acknowledges that test results may have utility for the decision maker through an inherent value of knowing a diagnosis, and through resulting psychological preparations that are not modeled explicitly.)
(a) \
/ ~~~~
