A Fuzzy Inference Map approach to cope with

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Intelligent Decision Technologies 5 (2011) 1–16 DOI 10.3233/IDT-2011-0108 IOS Press

A Fuzzy Inference Map approach to cope with uncertainty in modeling medical knowledge and making decisions Elpiniki I. Papageorgiou Department of Computer Science & Information Technology, Technological Educational Institute of Lamia, 3rd Old National road Lamia-Athens, 35100 Lamia, Greece E-mail: [email protected]

Abstract. In this work, the Fuzzy Inference Map approach (also known as Fuzzy Cognitive Map) is investigated to handle with the problem of risk analysis and assessment of pulmonary infections during the patient admission into the hospital. A Fuzzy Inference Mapping is an artificial cognitive structure within which the relations between the elements of a mental landscape can be used to assess the impact of these elements. It has the advantageous features of representing medical knowledge in a symbolic manner, giving system’s transparency, interpretability of results and easiness of use by non experts. Fuzzy Cognitive Map (FCM) proved by the literature as an appropriate reasoning tool to explicitly encode the knowledge and experience accumulated on the operation of a complex system. This study presents a first tool for making decisions in medical domain that will help physicians, through the design of the knowledge representation and reasoning using FCM to automate the decision making process in the case of infectious diseases prediction. After drawing the FCM model for pulmonary risk prediction, the Decision Making Trial and Evaluation Laboratory (DEMATEL) method is implemented to analyze the map and outrank the concepts according to their importance for physicians. A number of different scenarios concentrated on the pulmonary infections are examined to demonstrate the application of the proposed methodology and its prediction capabilities. This work proves that FCM can handle efficiently with uncertainty in modeling medical knowledge. Keywords: Fuzzy cognitive maps, knowledge-based systems, modeling, knowledge representation, medical decision making, prediction

1. Introduction During the last years, an enormous number of decision support systems (DSS) for diverse medical problems have been developed. Some of these systems are easy to find on the Internet, and they are freely available for public use. The traditional medical expert systems [10], were equipped with a rule knowledge base supplied by the domain experts (physicians). On the basis of rules inserted in the expert system, it is possible to classify new instances of medical observations by matching symptoms to the conditional part of a rule and then to perform forward and backward reasoning to achieve the diagnosis or construct a therapy plan. In our opinion, one of the main disadvantages for the application of the classic rule-based knowledge repre-

sentation in medical DSS is its limitation of representing some of the more complex associations that may be experienced within the medical data. For example, in a rule-based DSS, the representation of the complex phenomenon of causality [34] is, in fact, left to the interpretation and expertise of the doctor. There is a vast number of knowledge-representation methods that can be considered, in general, as exemplification of the conceptual modeling approach. The best-known of them are ontologies and semantic networks that are able to express concepts and relationships among them. Maybe less known in computer science are fuzzy cognitive maps (FCMs). Fuzzy Cognitive Map is a soft computing technique capable of dealing with situations including uncertain descriptions using similar procedure such as human

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reasoning does [17,18]. Fuzzy Cognitive Maps (FCMs) are originated from cognitive maps and are used to represent the causal relationships among factors in order to outline a decision-making process. They are modeling methods based on knowledge and experience for describing particular domains using concepts (variables, states, inputs, outputs) and the relationships between them. FCM can describe any system using a model having signed causality (that indicates positive or negative relationship), strengths of the causal relationships (that take fuzzy values), and causal links that are dynamic (i.e. the effect of a change in one concept/node affects other nodes, which in turn may affect other nodes). In this work, our attention primarily has focused on the process of making decisions in medical domain. In medical science, patients have symptoms that prompt them to see a doctor. In other words, symptoms are comprised of the observations reported by patients and the observations of doctors while examining patients. The identification of the underlying cause of symptoms is crucial and improves the chance of proper diagnosis of the disease and prescribing the correct treatment. Here, we use FCMs as a first step, to model a physicianexpert’s behavior in the decision making [28]. The behavior to be modeled is centered in the decision making process, whose reasoning implies to reach a predefined goal, coming from one or more initial states. Therefore, the reasoning system will be more efficient when a least number of transitions to reach the final goal are achieved. Thus, increasing the efficiency implies to minimize intermediate states, and that is represented in the organization of the knowledge base. FCM was chosen because of the nature of the application problem. The prediction of infectious diseases in pneumonia is a complex process with sufficient interacting parameters and FCMs have been proved suitable for this kind of problems. They can handle with the available experience and accumulated knowledge from experts. The easy of use and the low time requirement are important features of FCMs. FCMs have been widely applied in computing and decision sciences [6,16,22,30,36–38,45], although other research areas have used these techniques, such as business and management [?], political decisions [42], agriculture and ecological sciences [25,35], engineering [?,21], robotics [44], pattern recognition [33] and medicine [8,11,27–30,39]. The main goal of this work is to present a method based on fuzzy inference map approach that can be applied for the development of an expert system for predicting the risk of pulmonary infection. In upcoming

work, this FCM tool will be enhanced by other knowledge schemes to assess the type and the severity of infectious diseases. No any previous work has been done till today on implementation of FCM technique to handle with the specific problem of defining factors as well as to model their complex cause-effect relationships that affecting infectious diseases and/or adverse events in Intensive Care Unit (ICU).

2. Overview of fuzzy cognitive maps Fuzzy cognitive maps offer an alternative knowledge fusion scheme [18,26]. Knowledge is represented in a symbolic manner using states, processes and events. Classic cognitive maps were originally developed to represent concepts and their causal relationship in the domain of political science or psychology [24]. Kosko introduced the FCMs by suggesting the use of fuzzy concepts taking numbers in [0, 1] which linked together by interaction strengths that can vary from −1 to 1, indicating the degree to which one variable influences another in concept maps. The nodes of the graph stand for the concepts that are used to describe the behavior of the system and the weighted interconnections represent the causal relationships that exist between the concepts. The advantage of FCMs is that they are relatively easy to construct and parameterize and are capable of handling the full range of system feedback structure. In Fig. 1, a fuzzy cognitive map designed to model some factors, selectors-measurements, and decisions in medical informatics, is illustrated. This FCM has four generic vertices representing the factors (F1 to F4), three vertices representing the selectors- measurements/examination results (S1 to S3), one output concept representing the decision of the system and the weighted arcs (edges) showing the relationships between these concepts. In this simple fuzzy cognitive map, the relation between two vertices is determined by taking a value in interval [−1, 1]. While −1 corresponds to the strongest negative, +1 corresponds to strongest positive one. The other values express different levels of influence. This model can be presented by a square matrix called adjacency matrix. The main objective of building a fuzzy cognitive map around a problem is to be able to predict the outcome by letting the relevant issues interact with one another. These predictions can be used in a decision support system (DSS) for finding out whether a conclusion arrived at is consistent with the whole collection of stated causal assertions.

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Fig. 1. A generic FCM model for medical decision making with its corresponding adjacency matrix. (Colours are visible in the online version of the article; http://dx.doi.org/10.3233/IDT-2011-0108)

The edge strength between two nodes Ci and Cj may be denoted by Wij , with Wij , taking values within [−1, 1]. Values −1 and 1 represent respectively full negative and full positive causality; zero denotes no causal effects and all other values correspond to different fuzzy levels of causal effects [21]. In general, an FCM is described by an edge or connection matrix W , whose elements are the connection strengths (or weights) Wij . The element in the ith row and jth column of matrix W represents the connection strength of the link directed out of node Ci into Cj . If the value of this link takes on discrete values in the set {−1, 0, 1}, it is called a simple or crisp FCM. The concept values of nodes C1 , C2 , . . . , Cn (where n is the number of concepts in the problem domain) together represent the state vector V. Once constructed, a FCM is then solved numerically to find the equilibrium value of variables (Vi ), given any fixed boundary conditions (e.g. sustained press of a variable) given the matrix of interaction strengths Wij . An FCM state vector at any point in time gives a snapshot of events (concepts) in the scenario being modeled. To let the system evolve, the state vector V is passed repeatedly through the FCM connection matrix W . This involves multiplying V by W , and then transforming the result as follows: V = f (V + V · W)

(1)

or Vi (t + 1) = f (Vi (t) +

N X j6=i j=1

Vj (t) · Wji )

(2)

where Vi (t) is the value of concept Ci at step t, Vj (t) is the value of concept Cj at step t, Wji is the weight of the interconnection from concept Cj to concept Ci and f is the threshold function that squashes the result of the multiplication in the interval [0, 1]. This equation indicates that a FCM is free to interact; at every step of interaction every concept has a new value. Several formulas can be used as threshold function [3], and as the interval of concept is bivalent (i.e., the concepts belong to the interval [0, 1]), we propose the function f (x): f (x) = 1/(1 + exp(−mx))

(3)

where m is a real positive number and x is the value Vi (t) on the equilibrium point. In this work we use m = 5, because this value showed best results in previous works [26]. A concept is turned on or activated by making its vector element 1. New state vectors showing the effect of the activated concept are computed using method of successive substitution, i.e., by iteratively multiplying the previous state vector by the relational matrix using standard matrix multiplication. Figure 2 shows a sample simulation of the model from Fig. 1. The sigmoid transformation function as described in formula (3) is used to carry out the simulations, as it is the most commonly used function [3]. Figure 2. Sample FCM simulation of the model from Fig. 1. The algorithm used to obtain the final vector V f is consisting on the following steps: 1. (1) Definition of the initial vector V that corresponds to the elements identified in Table 1.

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Table 1 Factor and selector concepts coding pulmonary infection Concepts C1: Dyspnea

Description of concepts Dyspnea is a subjective experience of breathing discomfort that is comprised of qualitatively distinct sensations that vary in intesity.

C2: Cough

Cough is a deep inspiration folowed by a strong expiration against a closed glottis, which then opens with an expulsive flow of air , followed by a restorative inspiration.It is a defense mechanism against respiratory infections and can be productive or nonproductive, depending on whether sputum is produced or not. Rigor is the involuntary shaking occurring during a high fever. It is part of an immune response and increase the set point for body temperature in the hypothalamus. The increased set point causes the body temperature to rise (pyrexia), but also makes the patient feel cold-chills until the new set point is reached. Rigor occurs because the patient is effectively shivering in a physiological attempt to increase body temperature to the new set point. Fever is a frequent medical sign that describes an increase in internal body temperature to levels above normal. Usually the initial presentation is acute and it is accompanied by chills in ’typical’ pneumonia rather than the gradual onset of “atypical” pneumonia Hypothermia can also be present in a small group of patient and indicates serious infection and worse prognosis. Loss of appetite is the decreased sensation of appetite, leading to loss of weight if the symptom prolong

C3: Rigor/chills

C4: Fever

C5: Loss of appetite C6: Debility

C7: Pleuritic pain

C8: Heamoptysis C9:Oxygen requirement

C10: Tachypnea C11: Acoustic characteristics

C12:GCS

C13:Systolic Blood Pressure

Debility is a non specific symptom of pneumonia refereed to the altered general well-being. According to the NYHA classification there are four discrete values describing four progressively more serious states of debility. Pleuritic pain is the result of acute inflammation of the pleural surfaces that covers lungs. It is restricted and tends to be distributed along the intercostals nerve zones. The pain perceived while the patient is breathing quietly, it is typically worsened by taking a deep breath, and coughing or sneezing causes intense distress Hemoptysis or haemoptysis is the expectoration (coughing up) of blood or of blood-stained sputum from the respiratory tract. The severity depends on the extent (small < 20 ml/24 h to massive > 200–600 ml/24 h and life threatening) There are four different states describing four progressively more serious states of oxygen requirement: no need of oxygen, the need of applying nasal kanula (∼ 2– 4lt oxygen) or Ventury mask (∼ 4–15lt oxygen), NIMV (non invasive mechanical ventilation) and MV ( mechanical ventilation) Tachypnea (or “tachypnoea”) is defined as the increase of respiratory rate of > 16 for men and 19 breaths for women breaths per minute. Respiratory rate > 30 breaths per minute is associated with worse prognosis and possible need of MV. Bronchial breath sounds produced when the lung parenhyma is consolidated and the airway leading to this region is parent. It is loud, high-pitched, tubular or whistling. Bronchophony mean that spoken sounds are transmitted with increased intensity and pitch through consolidated lung. Crackles are explosive nonmusical sounds caused by the explosive opening of a series of small airways that had closed owing the process of consolidation. Glasgow Coma Scale (GCS) is a neurological scale which aims to give a reliable, objective way of recording the conscious state of a person, for initial as well as continuing assessment. A patient is assessed against the criteria of the scale, and the resulting points give a patient score between 3 (indicating deep unconsciousness) and 15 for alert individual. Blood pressure is a measurement of the force applied to the walls of the arteries during cardiac cycle The pressure is determined by the force and amount of blood pumped, and the size and flexibility of the arteries. The top number is the systolic blood pressure reading. It represents the maximum pressure exerted when the heart contract.

Type of values Four fuzzy values (no dyspnea, less serious, moderate serious, serious dyspnea state) (discrete values in NYHA are 0,1,2,3). Three fuzzy values (no cough, nonproductive and productive)

Two discrete values (exist or no)

Six Fuzzy values (hypothermia (34– 36◦ ), no fever (36–38, 4◦ ), low grade (38.5–38.9◦ ), moderate, high grade (39.5–40.9◦ ), hyperpyrexia (>41◦ )) Two discrete values (0,1) Four fuzzy values (no, small, moderate, large) Two discrete values (0, 1)

Two discrete values (0, 1)

Four fuzzy values (no need of oxygen, low, medium and high)

Four fuzzy values (normal (12–24), moderate (25–38), severe (35–49) and very severe (>50)) Three fuzzy values (no rales, localized and generalized)

Three fuzzy values: Severe altered mental status, GCS 68 Moderate, GCS 9–12 Minor altered mental status, GCS > 13. Seven fuzzy values (Hypotension < 90 Optimal < 120 Normal < 130 High-normal 130–139 Grade 1 hypertension 140–159

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Table 1, continued Concepts

Description of concepts

C14: Diastolic blood pressure

The bottom number is the diastolic blood pressure reading. It represents the pressure in the arteries when the heart is at rest

C15: Tachycardia

Tachycardia is the increased heart rate greater than 100 beats per minute. It is indicator of the haemodynamic situation of individual and often associated with fever. Pulse should increase by 10 beats per minute per degree Celsius of temperature elevation Alveolar infiltrate: localized in segment, lobe, nodulal or diffuse Interstitial infiltrate; Nodular, reticular, septal, linear pattern. Their presence indicates serious infection worse prognosis and possible need for more aggressive therapeutic techniques. Such complications are: atelectasis, pleural effusion, lung abscess, empyema, ARDS et. other It is an indicator that reflects the effectiveness of mechanism for regulating the acid-base status of the organism. It can be calculated by arterial blood gas analysis.

C16: Radiologic evidence of pneumonia C17: Radiologic evidence of complicated pneumonia C18: pH

C19:pO2

C20: pCO2

C21: sO2%

C22: WBC

C23: Immunocompromise

C24: Comorbidities

C25: Age

C26 (DC): Risk of pulmonary infection

The partial pressure of oxygen in the arterial blood is an early indicator of respiratory failure. Low values of pO2 demand oxygen therapy pO2 = 102–0,33 x age (mmHg) The partial pressure of carbon dioxide is also an indicator of respiratory failure. High values of pO2 demonstrate hypoventilation and possible need for NIMV or MV. Oxygen saturation (sO2%) is the fraction of the hemoglobin molecules in a blood sample that are saturated with oxygen at a given partial pressure of oxygen. Normal saturation is 95%–100%.It is an easy and practical way to detect respiratory failure using pulse oximetry White blood cells.Marked leukocytosis (> 10 × 103 /µl) with leftward shift (increased absolute number of neutrophils > 7,7103 /µl) is more often seen in bacterial pneumonia caused by Streptococcus pneumoniae, Haemophillus influenzae rather than normal number or leukopenia (< 1000/µl) which is usual in atypical and viral pneumonia Immunodeficiency is a condition of altered mechanical and cellular defense mechanism of organism. As a result of that is the inability to fight infection. An immunocompromised person may be particularly vulnerable to opportunistic infections Also in immunocompromised patients the signs and symptoms of pulmonary infection may be muted and overshadowed by nonspecific complaints. Such conditions are:HIV infection, organ transplantation, neoplasms, corticosteroides, chemotherapeutics et Comorbidities include conditions and diseases of individual associated with increased rate of pneumonia. Most frequently they are: COPD, bronchiectasis, asthma, congestive heart disease, diabetes, renal and liver disease et) Patient age is a serious factor for assessment severity of pneumonia according to CURB-65 scale. Age > 65 is connected with greater mortality. Also due to the lack of specific symptoms the diagnosis of pneumonia is frequently delayed in the elderly Risk of the pulmonary infection

Type of values Grade 2 hypertension 160–179 Grade 3 hypertension > / = 180 (British hypertention society) Seven fuzzy values (Hypotension < 60 Optimal < 80 Norma l < 85 High-normal 85–89 Grade 1 hypertension 90–99 Grade 2 hypertension 100–109 Grade 3 hypertension > / = 110 (British hypertention society) Four fuzzy values (low (less than 80 beats/min), normal (90–110), moderate sevre (110–140), severe (> 140)) Two discrete (exist or no) Two fuzzy values (presence, absence) Three fuzzy values Acidosis < 7.35 Normal 7.35–7.45 Alkalosis > 7.45 Two fuzzy value normal 70–100 mmHg hypoxia is every value under normal Three fuzzy values normal 35–45 mmHg hypocapnia 45 mmHg Two fuzzy values normal > 95% hypoxia < 95% Three fuzzy values Normal 4,3–10 × 103 /µl leukocytosis > 10 × 103 /µl leukopenia < 1000/µl Two fuzzy values (presence, absence)

Two discrete values (presence = 1, absence = 0)

Three fuzzy (Young, middle age, older)

Five fuzzy values (very low, low, moderate, high, very high)

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Fig. 2. Sample FCM simulation of the model from Fig. 1. (Colours are visible in the online version of the article; http://dx.doi.org/ 10.3233/IDT-2011-0108)

(2) Multiply the initial vector V and the matrix W defined by experts by the Eq. (2), as indicated by the Table X. (3) The resultant vector is updating using Eqs (1)– (3). (4) This new vector is considered as an initial vector in the next iteration. (5) Steps 2–4 are repeated until Vt − Vt−1 6 e = 0.001. All type of information has numerical values. FCM allows us to perform qualitative and semi-quantitative simulations and experiments with a dynamic model [19]. This type of analysis allows investigating “what-if” scenarios by performing simulations of a given model from different initial state vectors. Simulations offer description of dynamic behavior of the system that can be used to support decision making or predictions about its future states [39]. 2.1. Construction of fuzzy cognitive maps The development and construction method of FCM is of great importance for its potential to sufficiently model a system. Proposed methods are dependent on the group of experts who operate, monitor, supervise the system and they know its behaviour. This methodology extracts the knowledge from the experts and ex-

ploits their experience of the system’s model and behaviour. More specifically, experts from their experience know which elements of the systems influence other elements; they determine the negative or positive effect of one concept on the others, with a fuzzy degree of causation for the corresponding concepts. In this way, an expert transforms his/her knowledge in a dynamic weighted graph, the FCM. Following the developing methodology, experts are forced to think about and describe the existing relationship between the concepts using fuzzy if then rules and assume the following form, where A, B and D are linguistic variables and are determined from the following prescribed membership functions (values in the range [0, 1]): IF value of concept Ci is A THEN value of concept Cj is B and thus the linguistic weight wij is D Thus, each interconnection is described by an expert with a fuzzy linguistic variable from a determined set, which associates the relationship between the two concepts and determines the grade of causality between the two concepts. The causal inter-relationships among concepts are usually declared using the variable Influence which is interpreted as a linguistic variable taking values in the universe U = [−1, 1]. Its term set T(influence) is suggested to comprise twelve variables. Using twelve linguistic variables, an expert can describe in detail the influence of one concept on another

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and can discern between different degrees of influence. The twelve variables used here are: T(influence) = {negatively very strong, negatively strong, negatively medium, negatively weak, negatively very weak, zero, positively very weak, positively weak, positively medium, positively strong, positively very strong, positively very very strong}. Then, the linguistic variables D (from the set T(influence)) proposed by the experts for each interconnection are aggregated using the SUM method and so an overall linguistic weight is produced which is defuzzified with the Centre of Gravity method and finally a numerical weight for Wij is calculated. Using this method, all the weights of the FCM model are inferred. A detailed description of the development of the FCM model is given in [39].

3. Fuzzy Cognitive Map approach to assess pulmonary infections The development and design of the appropriate fuzzy cognitive map for the description of a system requires the contribution of human knowledge. The experts develop fuzzy cognitive maps using an interactive procedure of presenting their knowledge on the operation and behavior of the system. The procedure described here is an extension of the approach presented in a previous work for constructing FCMs [32]. The FCM is suitable technique to cope with complex decision making tasks such as the prediction of infection, the severity of infectious disease and the therapy plan acceptance. The prediction of risk of infectious diseases is a complex process with enough parameters, factors and different conditions [5,7,12,23]. For the problem of pneumonia, a number of typical symptoms are associated including fever (80%) often accompanied by chills or hypothermia in a small group of patients, altered general well-being and respiratory symptoms such as cough (90%), expectoration (66%), dyspnea-shortness of breath (66%), pleuritic pain-a sharp or stabbing pain, experienced during deep breaths or coughs (50%), and hemoptysis-expectoration of blood (15%). The initial presentation is frequently acute, with an intense and unique chill. Productive cough is present and the expectoration is purulent or bloody. Pleuritic pain may be present. Physical examination reveals typical findings of pulmonary consolidation- bronchial breath sounds, bronchophony, crackles, increased fremitus, dullness during percussion, tachypnea-increased respiratory rate,

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tachycardia-high heart rate (pulse should increase by 10 beats per minute per degree Celsius of temperature elevation) or a low oxygen saturation, which is the amount of oxygen in the blood as indicated by either pulse oximetry or blood gas analysis. In elderly and immunocompromised patients, the signs and symptoms of pulmonary infection may be muted and overshadowed by nonspecific complaints. Laboratory studies should be performed that include blood cell counts, serum glucose, transaminases, urea, creatinine and electrolyte measurements. From these lab tests only the white blood cells (WBC) have been considered as the most important one to increase mainly the risk of infection [5]. These data provide a logical basis for evaluation the risk of infection and the need for intensive care [12,20]. Three physicians-experts, two physicians from the General Hospital of Lamia, and one from the University General Hospital of Patras, Greece, were pooled to define the number and type of parameters-factors affecting the problem of pulmonary infection. These parameters (concepts) are listed in Table 1 and are well documented in bibliography. They represent the main variables that play an important role in the final diagnostic decision about pulmonary infectious disease. For this application, concept values take either two, three, four or five possible discrete or fuzzy values, as shown in Table 1. Thus, experts designed a FCM model, following the previously described construction methodology, which consists of twenty six concepts (Table 1). The twenty six concepts are the factor and selector concepts representing the main variables that physicians in ICU usually take into consideration in assigning the existent and the grade of the infection. The Decision Concept (DC) represents the risk of pulmonary infection in percentage and takes five fuzzy values (very low, low, moderate, high, very high). After the description of FCM concepts, the design of FCM model continues with the determination of membership functions. Determination of membership functions in terms of shape and boundary has a clear effect on the result of decision making performed by FCM. This situation greatly depends on experience and knowledge. Finding the right shape and the boundaries for the membership function will increase the accuracy of the FCM application. There are two different ways to define fuzzy sets for each one concept variable: (a) to define linguistic values based on variable behavior (through historical data where it’s possible to determine the number and the shape of the sets); (b) or to

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E.I. Papageorgiou / A Fuzzy Inference Map approach to cope with uncertainty in modeling medical knowledge and making decisions Table 2 Fuzzy sets describing some concept variables and the decision concept of risk infection

define linguistic values based on experts’ knowledge into a range between zero and one. In our case, the corresponding membership functions for each variable have been defined with a brief interpretation of experts’ knowledge and literature review. In the following Table 2, the fuzzy sets for the three input variables (dyspnea, fever, pH) as well as the fuzzy sets of the decision concept (DC)-(risk of infection) are prescribed. After the determination of fuzzy sets, each expert was asked to define the degree of influence among the concepts and to describe their interrelationship using an IF–THEN rule, assuming the following statement where Ci and Cj are all the ordered pair of concepts: IF a {no, very small, small, medium, large, very large} change occurs in the value of concept Ci THEN a {no, very small, small, medium, large, very large} change in value of concept Cj is caused. THUS the influence of concept Ci on concept Cj is T(influence). Then, experts inferred a linguistic weight to describe the cause and effect relationship between every pair of concepts. To illustrate how numerical values of weights are produced, the three experts’ suggestions on how to indicate the interconnection between concepts C22 (number of white blood cells) and DC (risk of pulmonary infection) are shown below:

1st expert: IF a small change occurs in the value of concept C22, THEN a medium change in value of concept DC is caused. Infer: The influence from C22 to DC is positive medium. 2nd expert: IF a small change occurs in the value of concept C22, THEN a large change in value of concept DC is caused. Infer: The influence from C22 to DC is positive high. 3rd expert: IF a very small change occurs in the value of concept C22, THEN a large change in value of concept DC is caused. Infer: The influence from C22 to DC is positive very high. These linguistic variables (medium, positive strong and positive very strong) are summed and an overall linguistic weight is produced, which with the defuzzification method of centroid is transformed into the numerical value of W22−D1 = 0.617. The twenty six identified concepts (Table 1) keep relations with each other, in order to characterize the process of predicting the risk of pulmonary infectious diseases and to provide a first front-end decision tool

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Fig. 3. The FCM model for assessing the risk of pulmonary infection. (Colours are visible in the online version of the article; http://dx.doi. org/10.3233/IDT-2011-0108)

about the prediction of pulmonary infection. All the corresponding relations between concepts are given in Table A in Appendix. The causal influences between the input-parameter concepts are given. There is one influence from concept C16 -(describe the radiology evidence) towards concept C4 (representing “fever”). This influence from C16 to C4 is medium. Also there are influences from C25 (Age) to C12 (GCS), from C4 to C22 (WBC), from C4 to C5 (loss of appetite), from C4 to C15 , from C21 to C1 , from C21 to C10 , from C21 to C9 , from C21 to C19 , from C19 to C1 , from C19 to C10 , from C19 to C9 , from C20 to C1 , from C20 to C18 . The influence from C25 (Age) to C12 (GCS) is negatively weak (numerical weight W25−12 = − 0.4). The influence from C4 to C22 is strong (numerical weight W4−22 = 0.7). The influence from C4 to C5 is low and the produced numerical weight is equal to 0.3. The influence from C4 to C15 is low (produced numerical weight equal to 0.3), the influence from C21 to C1 is negatively medium (numerical weight is W21−1 = − 0.4), the influence from C21 to C10 is negatively medium (W21−10 = − 0.4), the influence from C21 to C9 is negatively low (numerical weight W21−9 = − 0.3), the influence from C21 to C19 is low (W21−19 = 0.3), the influence from C19 to C1 is negatively low (W19−1 = − 0.3), the influence from C19 to C10 is negatively low (W19−10 = − 0.3), the influence from C19 to C9 is negatively low (W19−9 = −

0.3), the influence from C20 to C1 is low (W20−1 = − 0.3), and the influence from C20 to C18 is negatively low to medium (numerical weight is W20−18 = − 0.35). The same approach was used to determine all the weights of the FCM. A weight matrix W gathering the initially suggested weights of all the interconnections among the concepts of the FCM model was produced. The weight matrix is illustrated in Table 3. Figure 3 illustrates the FCM model for predicting the risk of pulmonary infection with the assigned numerical values of weights. The proposed method based on FCMs for predicting pulmonary infection provides a framework within which physicians evaluate a series of traditional diagnostic concepts (symptoms, signs, laboratory tests, chest x-rays, risk and other factors). The way the FCM prediction model is designed increases the objectivity of the diagnostic process by taking into account the different physicians’ opinions regarding the interplay of factor and selector variables in the diagnostic output/decision. Using these variables, the FCM model predicts the risk percentage of the pulmonary infection. The FCM model gathers all the related parameters contributed to the decision about the risk of pulmonary infection and its prediction degree. If a pulmonary infection is predicted then an immediate assessment is needed from the physicians. An empirical antibiotic therapy have to be suggested on patient based on risk

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −0.3 −0.3 −0.4 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.5 0 0 0 0 0 0 0 0 0 0

0 0 0 0.3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Table 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −0.3 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −0.3 0 −0.4 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.4 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0.3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −0.3 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0.7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0.311 0.25 0.345 0.448 0.2 0.5 0.2 0.691 0.345 0.2 0.2 −0.455 0.584 0.5 0.4 0.584 0.74 0.2 0.2 0.2 0.345 0.617 0.345 0.5 0.4 0

10

The weight matrix W of the proposed FCM decision making tool

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Table 4 Direct and indirect influences of FCM model and concepts ranking from most important to least important one C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 C15 C16 C17 C18 C19 C20 C21 C22 C23 C24 C25 C26

Direct influence 0.03165 0.02045 0.03895 0.33629 0.01309 0.08182 0.01309 0.15627 0.03895 0.01309 0.01309 0.06775 0.11162 0.08182 0.05236 0.23707 0.17922 0.01309 0.06626 0.06626 0.15325 0.12459 0.03895 0.08182 0.11836 0

Indirect influence 0.1113 0 0 0.0818 0.0295 0 0 0 0.0295 0.0818 0 0.0524 0 0 0.0295 0 0 0.0295 0 0 0 0.1604 0 0 0 1.5437

Total influence 0.1429 0.0205 0.039 0.4181 0.0425 0.0818 0.0131 0.1563 0.0684 0.0949 0.0131 0.1201 0.1116 0.0818 0.0818 0.2371 0.1792 0.0425 0.0663 0.0663 0.1532 0.285 0.039 0.0818 0.1184 1.5437

factors and further examinations about microbiological cultures and laboratory tests is important to be investigated. After 48–72 hours, the related results form cultures and antibiograms will be available to the physician and in this time point, a new FCM tool is constructed, which is based in most of the part of the initial FCM tool and of the new measurement parameters of cultures, pathogens, antibiotic resistances, urinary antigen tests and any other related factors. FCM is simple, no time consuming and exploits experience and accumulated knowledge from experts.

4. Implementation of DEMATEL method The DEMATEL method is implemented in our FCM model which consists of 26 concepts for ranking these concepts according to their importance for the physicians-experts. Using the DEMATEL method, as it is described in [13,43], the following four steps are described and used to analyze FCM model. These steps are: Step 1: Generating the direct-relation matrix. As the result of the experts’ suggestions is the defuzzified values of the interaction strengths (weights) among concepts obtaining the direct-relation matrix that is a

Normalized total 0.0333 0.0048 0.0091 0.0973 0.0099 0.019 0.003 0.0364 0.0159 0.0221 0.003 0.0279 0.026 0.019 0.019 0.0552 0.0417 0.0099 0.0154 0.0154 0.0357 0.0663 0.0091 0.019 0.0275 0.3591

Important concept C26 C4 C22 C16 C17 C8 C21 C1 C12 C25 C13 C10 C6 C14 C15 C24 C9 C19 C20 C5 C18 C3 C23 C2 C7 C11

Normalized total influence 0.359 0.097 0.066 0.055 0.042 0.036 0.036 0.033 0.028 0.028 0.026 0.022 0.019 0.019 0.019 0.019 0.016 0.015 0.015 0.01 0.01 0.009 0.009 0.005 0.003 0.003

n · n matrix W , in which Wij is denoted as the degree to which the concept i affects the concept j. Step 2: Normalizing the direct-relation matrix. On the base of the direct-relation matrix W, the normalized direct-relation matrix M can be obtained through formulas: [wij ]n×n M = W new = [wij ]new (4) n×n = n P max wij i

j=1

Step 3: Attaining the total-relation matrix. Once the normalized direct-relation matrix M is obtained, the total relation matrix T can be acquired by using formula (5), in which the I is denoted as the identity matrix: T = M (I − M )−1

(5)

Step 4: Obtaining the inner dependence matrix. In this step, the sum of each row in total-relation matrix gives the direct influences of concepts, and then the indirect influences of each concept can be acquired by the sum of each related column. Thus the direct and indirect influences are produced. After drawing the FCM model for pulmonary prediction, the Dematel is used to analyze the map and outrank the concepts according to their importance for experts. The weight matrix of FCM model (W ) is

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E.I. Papageorgiou / A Fuzzy Inference Map approach to cope with uncertainty in modeling medical knowledge and making decisions Table 5 Initial and final concept values for First Scenario Initial activation levels – vector V1

C1 0 C14 0

C2 0 C15 0

C3 0 C16 0

C4 0 C17 0

C5 0 C18 0

C6 0 C19 0

C7 0 C20 0

C8 0.8 C21 0

C9 0 C22 0

C10 0 C23 0

C11 0.4 C24 0

C12 0 C25 0.8

C13 0 C26 0

Final concepts values after 11 iterations – V1 f

C1 0 C14 0

C2 0 C15 0

C3 0 C16 0

C4 0 C17 0

C5 0 C18 0

C6 0 C19 0

C7 0 C20 0

C8 0.8 C21 0

C9 0 C22 0

C10 0 C23 0

C11 0.5596 C24 0

C12 0 C25 0.8

C13 0 C26 0.7483

the adjacency matrix used for the determination of key concepts according to Dematel method. Following the steps of Dematel method, as they have been presented in a recently work [1], the adjacency matrix W (as the direct relation matrix presented in Table 3) is normalized by the Eq. (4), thus producing the normalized weight matrix M. Next, the total relation matrix T is calculated by formula (5). Table 4 shows both “direct influence” and “indirect influence” which are calculated from the T matrix and the concepts ranking from most important to least important one Thus, according to DEMATEL method, the most important concepts to the least one are depicted. In our case study the concept of fever (concept C4 ) is the most important concept after the decision concept DC of pulmonary infection which is the system output and gathers the interactions of all other input concepts. The next three strong concepts are WBC (C22 ), radiological evidence of pneumonia (C16 ) and radiological evidence of complicated pneumonia (C17 ). This importance has come from all experts’ knowledge.

5. Simulations and results After construction of FCM tool for the first step of predicting pulmonary infection, a number of scenarios have been introduced and the decision making capabilities of the technique are presented by simulating these scenarios and finding the predicted outcomes according to the available data. In each of the test scenarios we have an initial vector V, representing the presented events at a given time of the process, and a final vector V f, representing the last state that can be arrived at. For the interpretation of the results, an average only for the output value of the decision concept DC is computed according to the following criteria: 0, x 6 0.5 R(x) = x−0.5 (6) × 100%, 0.5 < x 6 1 0.5

where 0 represents the characteristic of the represented process by the concept is null, and 1 represents, the characteristic of the process represented by the concept is present 100%. For the specific approach, the function R(x) gives the risk of pulmonary infection in percentage. When R(value of DC) = 1, then the risk is 100%. The final value of decision concept DC applying this criterion is denoted by DC f. Thus DC f = R(V f(26)). This criterion can be modified according with the expert judgment. The final vector V f is the last vector produced in convergence region and the 26th value of this vector is the DC f, the final value of decision concept. The FCM performance is illustrated by means of simulation of some representative scenarios in case of medium risk, high risk and very high risk of pulmonary infection. 5.1. First scenario For this scenario, an old patient (A25 = 0.8) has been considered, with altered mental status (A12 = 0.4), with high oxygen requirements (A9 = 0.8), and normal number of leukocytes-WBC (A22 = 0). In order to assess the pulmonary risk, the following initial activation levels are considered and the FCM reasoning approach consisted on the above five steps, is followed. Table 5 gathers the initial and final calculated values of concepts for this scenario. Following the FCM simulation process, the system converges in a steady state with the final decision concept to be V2 f(26) = 0.7483. Figure 4 illustrates the subsequent values of calculated concepts after 9 steps. According to the above criterion, this value of decision concept corresponds to the 49.66% of risk, thus means that the risk of pulmonary infection is medium according to the related fuzzy regions in concept description. This result clearly states that the above symptoms and examination results contribute to a medium risk of pulmonary severity. This result is in congestion with the physicians’ opinions.

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13

Fig. 4. Variation of concepts values for the FCM tool for the first case study after nine simulation steps. (Colours are visible in the online version of the article; http://dx.doi.org/10.3233/IDT-2011-0108)

5.2. Second scenario According to the results derived from DEMATEL method, we can see that the six strongest concepts are the Fever (C4 ), the WBC (C22 ), the Radiologic evidence of pneumonia (C16 ), the radiologic evidence of complicated pneumonia (C17 ), the heamoptysis (C8 ) and the sO2 (C21 ). If we consider in the following scenario, that the first four of the strongest concepts, as they have been ranked using DEMATEL, are activated, the following initial vector is constructed to be: V3 = [0 0 0 0.7 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0.8 0 0 0 0] Following the FCM simulation process, the system converges in a steady state (Fig. 9), which defined form the following concept vector: V3 f = [ 0.7209 0 0 0.7830 0.7226 0 0 0 0 0 0 0 0 0 0.7226 1.0000 1.0000 0 0 0.7597 0 0.8000 0 0 0 0.9813] Assessing the value of decision concept using the criterion function R(x), the probability of risk is assessed. In this case, the risk is 96.25%. This means that the risk of pulmonary infection is very high according

to the related fuzzy regions. The result is clearly in congestion with the physicians’ information. 5.3. Third scenario For this scenario, the six strongest concepts, produced by the implementation of DEMATEL methods, are considered. These six concepts are the Fever (C4 ), the WBC (C22 ), the Radiologic evidence of pneumonia (C16 ), the radiologic evidence of complicated pneumonia (C17 ), the heamoptysis (C8 ) and the sO2 (C21 ). Table 6 gathers the initial and final calculated values of concepts for this scenario. Following the FCM simulation process, the system converges in a steady state with the final decision concept to be V4 f(26) = 0.9938. In this case, the risk is 98.75%. Thus considering that the six strongest concepts are activated initially then the predicted risk is again very high. Three representative scenarios were considered to assess the process of predicting risk of pulmonary infection, and the results of this approach seem reliable about the degree of the pulmonary infection. From the above scenarios, it is clearly that if more than the four most important symptoms (which are fever, WBC and radiologic evidences of pneumonia and complicated pneumonia) are emerged in a patient then the risk

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E.I. Papageorgiou / A Fuzzy Inference Map approach to cope with uncertainty in modeling medical knowledge and making decisions Table 6 Initial and final concept values for Third Scenario

Initial activation levels – vector V1

Final concepts values after 11 iterations – V1 f

C1 0 C14 0

C2 0 C15 0

C3 0 C16 1

C4 0.7 C17 1

C5 0 C18 0

C6 0 C19 0

C7 0 C20 0

C8 0.8 C21 0.8

C9 0 C22 0

C10 0 C23 0

C11 0 C24 0

C12 0 C25 0

C13 0 C26 0

C1 0.6369 C14 0

C2 0 C15 0.7226

C3 0 C16 1

C4 0.7830 C17 1

C5 0.7226 C18 0

C6 0 C19 0

C7 0 C20 0.7597

C8 0.8 C21 0.7568

C9 0.5897 C22 0.8

C10 0.5653 C23 0

C11 0 C24 0

C12 0 C25 0

C13 0 C26 0.9938

Fig. 5. Variation of values of ten concepts for the FCM tool for the second case study after eleven simulation steps. (Colours are visible in the online version of the article; http://dx.doi.org/10.3233/IDT-2011-0108)

is very high. If more concepts (symptoms), add to the above six concepts, are activated initially then the tool always predict a very high risk of infection. All the scenarios were illustrated to prove the functionality of the proposed FCM approach to assess the risk on pulmonary infections. As it is known from the literature, it appears crucial for the reliable diagnose, to predict an infection, as well as its type and severity of infectious disease. On the other hand, there are known in medicine science diseases expressed by a mixture of two or few causally independent symptoms. The analysis of these cases leads to the sophisticated evaluation of diverse possible causes and to the assumption of preferences within the space of concepts. The experiments show that it is much easier for a doctor to find even few mutually

independent causes but strongly associated with the observed symptoms, than a single common cause of all symptoms. This is enhanced by the use of FCMs. We have tested our system using data taken from real patient cases as they have been prescribed by the physicians. It was difficult to present the results of all cases and the evaluations due to paper strengthen. The presentation of the entire FCM decision support module fall beyond the scope of this paper. This is a first approach in the development of an expert system that will help in the decision making process, through the design of the knowledge representation and the design of reasoning with FCM to automate the decision making process. Therefore we decided to construct and simulate only a part of it as in the Fig. 5. The strongest point of the methodology is the insight it can provide

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on the role of key feedbacks in the system. These feedbacks can remain hidden and can be uncovered by applying a tool such fuzzy cognitive mapping. The tool can provide a lucid representation that, especially when used in combination with the theory of resilience, can help understanding short-term and long-term dynamics. The presented solution has been raised by some of the requirements imposed by the targeted application: the causal association of disease symptoms, signs, laboratory tests, that seem to be crucial for the correct medical diagnosis.

6. Conclusions In this work, the problem of modeling and predicting the risk of pulmonary infections using the FCM approach was studied. An alternative approach to consider the level vagueness and uncertainty management in medicine was proposed, especially when the disease has multiple symptoms based on the idea that the FCMs are suitable to model complex systems and to make decisions easily and efficiently. The FCM tool as a part of decision support was designed with a view to help medical and nursing personnel to assess patient status, assist in making a diagnosis, and facilitate finally, after the investigation of decision support tool, the selection of a course of antibiotic therapy. The produced model was tested in a number of patient cases showing its functioning and demonstrating that the use of the FCM as dynamic models is reliable and good.

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Acknowledgment [15]

The research was supported in part by the European Commission’s Seventh Framework Information Society Technologies (IST) Program, Unit ICT for Health; project DEBUGIT (no. 217139). The author wants to thank the three medical doctors, namely G. Karagianni, D. Sfyras and N. Papandrianos, who gave their useful knowledge and opinions for the construction and evaluation of FCM model for pulmonary risk prediction.

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Appendix

Table A: Linguistic weights assigned by each one of three experts and the produced numerical weight

C1-Dyspnea C2-Cough C3-Rigor C4-Fever C5-appetite C6-debility C7-chest pain C8- hemoptysis C9-ox.req. C10-tachypnea C11- rales C12- GCS C13-systolic C14-diastolic C15-heart rate C16-infiltrate, consolidation C17-radiology complicated evidence C18-pH C19-PO2 C20-pCO2 C21-sO2 C22-WBC C23-immunoc C24-comorbid C25-Age

D1 First expert Very weak weak weak med v. weak Med v. weak strong weak v.weak weak Neg.med med weak weak med strong weak v.weak v.weak weak strong weak weak med

D1 Second expert medium med med weak weak Med med Very strong med v.weak weak Neg.weak strong strong med strong v. strong v.weak v.weak v.weak med v.strong weak strong med

D1 Third expert weak Very weak weak strong v. weak med v. weak strong weak weak v.weak Neg.med strong strong med strong v. strong v.weak weak weak weak med med strong weak

Numerical weight 0.311 0.25 0.345 0.448 0.20 0.50 0.20 0.691 0.345 0.20 0.20 −0.455 0.584 0.50 0.40 0.584 0.740 0.20 0.20 0.20 0.345 0.617 0.345 0.50 0.40

17