A Fuzzy Grey Cognitive Maps-based Decision Support System for ...

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Oct 2, 2012 - Email addresses: Corresponding author, [email protected] (Jose L. ... Fuzzy Grey Cognitive Map-based Decision Support System (FGCM-.
A Fuzzy Grey Cognitive Maps-based Decision Support System for Radiotherapy Treatment Planning

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Jose L. Salmerona,, Elpiniki I. Papageorgioub University Pablo de Olavide, School of Engineering, 1st. km. Utrera Road, 41013 Seville, Spain b Technological Educational Institute of Lamia, Department of Informatics and Computer Technology, 3rd km. Old National Road Lamia-Athens, TK 35100 Lamia, Greece

Abstract

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Recently, Fuzzy Grey Cognitive Map (FGCM) has been proposed as a FCM extension. It is based on Grey System Theory, that it is focused on solving problems with high uncertainty, under discrete incomplete and small data sets. The FGCM nodes are variables, representing grey concepts. The relationships between nodes are represented by directed edges. An edge linking two nodes models the grey causal influence of the causal variable on the effect variable. Since FGCMs are hybrid methods mixing Grey Systems and Fuzzy Cognitive Maps, each cause is measured by its grey intensity. An improved construction process of FGCMs is presented in this study, proposing an intensity value to assign the vibration of the grey causal influence, thus to handle the trust of the causal influence on the effect variable initially prescribed by experts’ suggestions. The explored methodology is implemented in a well-known medical decision making problem pertaining to the problem of radiotherapy treatment planning selection, where the FCMs have previously proved their usefulness in decision support. Through the examined medical problem, the FGCMs demonstrate their functioning and dynamic capabilities to approximate better human decision making. Keywords: Fuzzy Grey Cognitive Maps, Fuzzy Cognitive Maps, Knowledge-Based Systems, Decision Support Systems

Email addresses: Corresponding author, [email protected] (Jose L. Salmeron), [email protected] (Elpiniki I. Papageorgiou)

Preprint submitted to Knowledge-Based Systems

October 2, 2012

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Cognitive Maps (Axelrod, 1976) are signed digraphs designed to capture the causal assertions of a person with respect to a certain domain and then use them to analyze the effects of alternatives, e.g.: policies or business decisions in respect to achieving certain goals. A Fuzzy Cognitive Map (FCM) is a graphical representation consisting of nodes indicating the most relevant factors of a decisional environment; and links between these nodes model the relationships between those ones (Kosko, 1986). FCM is a modelling methodology for complex decision systems (Dickerson & Kosko, 1994; Papageorgiou et al., 2006a; Salmeron et al., 2011), which has originated from the combination of fuzzy logic and neural networks. A FCM describes the behavior of a system in terms of concepts; each concept representing an entity, a state, a variable, or a characteristic of the system (De Maio et al., 2011; Salmeron and Lopez, 2011; Xirogiannis & Glykas, 2004; Yaman & Polat, 2009). FCMs constitute neuro-fuzzy systems, which are able to incorporate experts’ knowledge (Konar & Chakraborty, 2005; Kosko, 1986; Lee et al., 2002; Papageorgiou & Groumpos, 2005; Salmeron, 2009). Recently, a FCM extension, called Fuzzy Grey Cognitive Map (FGCM), has been proposed by Salmeron (2010). FGCM is based on Grey System Theory (GST). The improved results obtained with the FGCM in comparison with the conventional FCM approach on an Information Technology application (Salmeron, 2010) motivated us to investigate an enhanced FGCM model for decision support. The model presented in this paper co-evaluates human hesitancy through greyness not only in the definition of the causal relations between the concepts, but also in the definition of the concept values. The proposed methodology of FGCMs is applied to a two-level integrated decision support tool, constructed to handle the complex problem of making decisions in radiation therapy treatment. The tool consists of a clinical treatment simulation tool and a supervisor decision making tool based both on FGCMs, using the construction process. Fuzzy Grey Cognitive Map-based Decision Support System (FGCMDSS) results are meaningful as weight and concepts values are measured by their grey intensity to describe more reliable the causal influences among concepts as well as the concepts steady states and encourage our research towards this type of Decision Support Systems in medicine.

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1. Introduction

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2. Grey Systems Theory

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2.1. Grey uncertainty

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Grey System Theory has been designed for solving high uncertainty problems with discrete small and incomplete data sets (Deng, 1989). GST have been widely applied in domains as hydrology science, agriculture, energy, medicine, industry, military science, business, transportation, meteorology, geology, and so on. GST avoid the drawbacks of conventional statistic methods and just need a small amount of data to estimate the unknown systems’ behavior. GST split systems in three kinds according to the known information’s degree. If the internal structures and features of a system are fully known (whole understanding), the system is called a white one, while the system’s internal structures and features is completely unknown is called a black system. A system with partial information known and partial information unknown is a grey system. GST include fuzziness, because it can flexibly handles it (Li et al, 2007; Liu & Lin, 2006; Yamaguchi et al., 2007). Moreover, fuzzy mathematics need some previous information (usually based on cognitive experiences); while GST handle objective data, it does not require any previous information other than the data to be disposed (Wu et al., 2005). In addition, intension and extension of the analyzed objects are the critical difference between fuzzy and GST concepts. GST deal with objects with ambiguous intension and clear extension, fuzzy theory mostly handles objects with ambiguous extension and clear intension. For instance, a grey concept with a clear extension could be “People attending to the meeting are around 10 to 20 ”. A fuzzy concept with a clear intension and ambiguous extension could be “Old men”. All of us know what “old ” is, but the specific range where men are old is not clear (Wu et al., 2005). Moreover, fuzzy theory has its strength in the study of environments with cognitive uncertainties.

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The outline of this paper is as follow. Section 2 presents briefly the Grey System Theory. Section 3 describes the Fuzzy Grey Cognitive Map technique and its advantages over classical Fuzzy Cognitive Map. Section 4 shows the medical problems and their experimental analysis. In section 5, the results with the discussion follow, and section 6 concludes the paper and discusses the usefulness of the new methodology for FGCMs. Finally, an appendix shows several tables with all the relevant problem’s data.

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2.2. Grey numbers

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One stronger point of GST over fuzzy approach is that GST fits better with multiple meanings or grey environments (Bellman & Zadeh, 1970; Zadeh, 1965). Grey uncertainty emerges due to the lack of accurate values. For instance, as the sentence “The expected costs for a project are between 2.0 and 3.0 millions dollars”, the uncertainty of which is produced by the multiple meanings of the sentence. The costs (multiple meaning or grey variable) could be 2.0 or 2.1 or any value between up to 3.0 millions. That is, we can only know about the costs is its range [2.0, 3.0], but we do not know the accurate costs.

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Let U be the universal set. Then a grey set G ∈ U is defined by its both mappings. Note that  µG (x) : x → [0, 1] G= (1) µG (x) : x → [0, 1]

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where µG (x) is the lower membership function, µG (x) is the upper one and µG (x) ≤ µG (x). Also, GST extend fuzzy logic, since the grey set G becomes a fuzzy set when µG (x) = µG (x). A grey number is one whose accurate value is unknown, but it is known the range within its value is included. We denote a grey number as ⊗G, with both a lower limit (G) and an upper limit, it is called an   interval grey number (Liu & Lin, 2006), and it is denoted as ⊗G ∈ G, G |G ≤ G. Both limits are fixed numbers in first order interval grey numbers. A black  number would be ⊗G ∈ (−∞, +∞), and a white number is ⊗G ∈ G, G , G = G. We have not information about black numbers and we have the complete information about white numbers. If the grey number ⊗G has only lower limit is denoted as ⊗G ∈ [G, +∞), and if it has only upper limit is ⊗G ∈ −∞, G . There is another kind of grey numbers that vibrate around a base value (a) and it can be denoted as ⊗G(a) ∈ [a − δa , a + δa ]. Note that the grey number ⊗G is formed with the vibration of the base value a with an intensity δa .

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2.3. Grey operations

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We define the length of a grey number as ` (⊗G) = | G − G |. In that sense, if the length of the grey number is zero (` (⊗G) = 0), it is a white number. In other sense, if ` (⊗G) = ∞ the grey number is not necessarily a black one, because the length of a grey number with only one limit (lower 4

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 or upper limit) is infinite, ⊗G ∈ [G, +∞) or ⊗G ∈ −∞, G , but it is not a black number. In addition, if we have two grey numbers ⊗Gx and ⊗Gy , then the following operations are defined.   ⊗Gx + ⊗Gy ∈ Gx + Gy , Gx + Gy (2)

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Note that − ⊗ G ∈ [−G, −G], then

⊗Gx − ⊗Gy ∈ ⊗Gx + (− ⊗ Gy )

∈ [Gx , Gx ] + [−Gy , −Gy ]

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∈ [Gx − Gy , Gx − Gy ]

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The multiplication of two grey numbers is denoted as  ⊗Gx · ⊗Gy ∈ min(Gx · Gy , Gx · Gy , Gx · Gy , Gx · Gy ),  max(Gx · Gx , Gx · Gy , Gx · Gy , Gx · Gy )

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Note that ⊗G−1 ∈ ⊗Gx ⊗Gy



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1 1 , G G

h i ∈ Gx , Gx · "

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(6) (7)

, then

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  # Gx Gx G x Gx Gx Gx Gx G x min , , , , , max , , G y Gy Gy Gy Gy G y Gb Gy

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Futhermore, the multiplication of a scalar λ and a grey number ⊗G is defined as (10) λ · ⊗G ∈ [λ · G, λ · G] Moreover, the whitenization process converts grey numbers as white ones e of the grey number ⊗G ∈ [G, G] (Liu & Lin, 2006). The whitenization ⊗G computes as e = α · G + (1 − α) · G (11) ⊗G where 0.0 ≤ α ≤ 1.0. If α = 0.5, then it is called equal mean whitenization.

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3. Fuzzy Grey Cognitive Maps

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3.1. Fundamentals

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where i is the pre-synaptic (cause) node and j the post-synaptic (effect) one. Figure 1 shows a FGCM example, and A (⊗) is its adjacency grey matrix (equation 13).

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Fuzzy Grey Cognitive Map (FGCM) is an innovative soft computing technique (Salmeron, 2010). A FGCM represents unstructured knowledge through causalities expressed in imprecise terms and grey relationships between them based on Fuzzy Cognitive Maps (Kosko, 1986, 1996). FGCM, as FCM, represents human tacit knowledge. FGCMs are dynamical systems involving feedback, where the effect of change in a node may affect other nodes, which in turn can affect the node initiating the change. FGCM nodes represent concepts with grey variables. The causal influence of the causal grey variable over the effect one is modelled by an edge linking both nodes. The intensity of each edge is measured by its grey intensity (weight) as follows   ⊗wij ∈ wij , wij | ∀i, j → wij ≤ wij , {wij , wij } ∈ [−1, +1] (12)

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Figure 1: Fuzzy Grey Cognitive Map example

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    A (⊗) =     

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         

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FGCMs dynamics begins with the design of the initial grey vector state ~ 0 at instant t = 0, which represents a proposed initial grey stimuli. We ⊗C denote the initial grey vector state with N nodes as  ~ 0 = ⊗C 0 ⊗ C 0 . . . ⊗ Cn0 ⊗C h 1 i 2h i h i (14) 0 0 0 = C 01 , C 1 C 02 , C 2 ... C 0n , C n

The updated nodes’ states (Salmeron, 2010) are computed in an iterative inference process with an activation function, which is used to map monotonically the grey node value into a normalized range [0, +1] or [−1, +1]. Each single node would be update according to the equation 15.

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0 ⊗w12 ⊗w13 0 0 0 0 0 0 0 ⊗w24 0 0 0 0 0 0 ⊗w34 0 0 0 ⊗w41 0 0 0 ⊗w45 ⊗w46 ⊗w47 0 ⊗w52 0 0 0 ⊗w56 0 0 0 0 0 0 0 ⊗w67 0 0 0 0 0 0 0

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 ⊗Cjt+1 = f ⊗Cjt +

N X



 ⊗wij · ⊗Cit 

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i t∗ = f C t∗ , C h   t∗ i = f C t∗ , f C h i t+1 = C t+1 , C

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The unipolar sigmoid function is the most used one (Bueno & Salmeron, 2009) in FCM and FGCM when the concept value maps in the range [0, 1]. ~ t+1 after If f (·) is a sigmoid, then the component i of the vector state ⊗C the inference is denoted as follows   −1  t∗ −1 t+1 −λ·C t∗ −λ·C ~ i i , 1+e ⊗Ci ∈ 1 + e (16) On the other hand, when the concepts’ states map in the range [−1, +1] the function used would be the hyperbolic tangent. The component i of the 7

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(17)

The nodes’ states evolve along the FGCM dynamics. The FGCM inference process finishes when the stability is reached. The steady grey vector state represents the effect of the initial grey vector state on the state of each FGCM node. It settles down to a fixed pattern of node states, the so-called grey hidden pattern or grey fixed-point attractor. Moreover, the state could to keep cycling between several fixed states, known as a limit grey cycle. Using a continuous activation function, a third state would be a grey chaotic attractor. It happens when, instead of stabilizing, the FGCM continues to produce differents grey vector states for each iteration. Furthermore, FGCM includes greyness as an uncertainty measurement. Higher values of greyness mean that the results have a higher uncertainty degree. It is computed as follows

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~ t+1 after the inference is computed as follows vector state ⊗C ! " t∗ t∗ !# t∗ λ·C i − e−λ·C i λ·C t∗ i − e−λ·C i e e ~ t+1 ∈ ⊗C , t∗ t∗ t∗ t∗ i eλ·C i + e−λ·C i eλ·C i + e−λ·C i

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φ(⊗Ci ) =

|`(⊗Ci )| `(⊗ψ)

(18)

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where |`(⊗Ci )| is the absolute value of the length of grey node ⊗Ci state value, and `(⊗ψ) is the absolute value of the information space range, denoted by ⊗ψ. FGCM maps the nodes’ states within an interval [0, 1] or [−1, +1] if negative values are allowed. In this sense,  1 if {⊗Ci , ⊗wi } ⊆ [0, 1] ∀ ⊗ Ci , ⊗wi `(⊗ψ) = (19) 2 if {⊗Ci , ⊗wi } ⊆ [−1, +1] ∀ ⊗ Ci , ⊗wi

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3.2. Constructing FGCM

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In general terms, there are two main groups of approaches to establish FCMs (in the same sense FGCMs), namely (a) using experts’ knowledge about the problem’s domain (deductive modeling) and (b) using learning algorithms based on historical data (inductive modeling). Our proposal belongs to the deductive approach. The experts’ team determines the number and kind of grey concepts (nodes) that comprise the FGCM. The experts from their experience know the main factors describing the behavior of the system; each of ones is represented by a grey concept within the FGCM model.

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Furthermore, experts know which nodes influence others; for the corresponding nodes they determine the intensity of the influence and its sign (negative or positive). Each expert, indeed, determines the influence of one node on another as negative or positive and then evaluates the degree of influence using a linguistic variable, such as strong influence, medium influence, weak influence, etc. This is the procedure used for FCM (Papageorgiou et al., 2006b). A grey causal weight must to be determined in FGCMs. In this sense, we will use grey numbers that vibrate around a base value ⊗G(a). According to this, ⊗wij (a) ∈ [a − δa , a + δa ]. This kind of grey numbers can be whitenized relatively easily, because the base value can be used as the main whitenization value. Moreover, the value of δa would be related with the uncertainty about the base value a. If the base value has not uncertainty associated (white numbers), then δa = 0. If the base value is completely uncertain (black numbers), then δa = ∞ for the general case, and a ± δa = ±1 in FGCM models. Figure 2 shows the differents options about grey numbers with base value and vibrations as causal grey weights.

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Figure 2: Grey number ⊗G(a) with base value a, and vibration δa

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In our proposal, the base value for wij (a) is determined as the FCM weights (Papageorgiou et al., 2006b). For easier understanding by experts, the weights would be assigned in two stages. The first one is the definition of the base value. It could be assigned as the same way than in FCM. 9

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According to this, FGCM combines Fuzzy and GST. The construction process starts determining a fuzzy weigth (as FCM do), after that a uncertainty measure (vibration) is stablished. As a result we build the grey weights. Note that grey numbers are are not the same than fuzzy ones according to the differences detailed between fuzzy and grey uncertainty (Liu & Lin, 2006).

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Anyway, the base value is located within the FGCM grey weights’ range a ∈ {[0, +1] | [−1, +1]}. The second stage is determining the vibration δa of the base value a. The vibration value δa would be determined with linguistic variables, such very high uncertainty, high uncertainty, medium uncertainty, weak uncertainty, and so on. The experts’ trust on their own judgement about the base value determine the δa value. If the expert has the whole trust on the base value a, then δa = 0. On the other hand, if the expert has not any trust on the base value, the vibration’s value would be wij (a) ± δa = ±1. The equation 20 shows the computation of the ⊗w(a) upper and lower limits.  [a − δa , a + δa ] if (a + δa ≤ +1) ∧ (a − δa ≥ −1)    [a − δa , +1] if (−1 ≤ a − δa ≤ +1) ∧ (a + δa > +1) ⊗w(a) ∈ (20) [−1, a + δa ] if (−1 ≤ a + δa ≤ +1) ∧ (a − δa < −1)    [−1, +1] if (a + δa > +1) ∧ (a − δa < −1)

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3.3. Troubleshooting FCM drawbacks with FGCMs FGCM is able to compute the models’ steady states by handling uncertainty and hesitancy in the initial experts judgements for causal relations among concepts as well as within the initial concepts states. FGCMs are able to assess medical decision support issues, producing meaningful results for the updated weights and steady state concepts. FCM needs measures about the associated uncertainty in weights and concepts. Note that, even if the FCM dynamic analysis would get the same steady vector state than FGCM after the whitenization process, FGCM handles the intrinsic fuzziness and grey uncertainty of medical decision support. FGCMs instead of weights with discrete numerical values, as FCMs, the edges have grey weights including grey uncertainty and fuzziness to better representing the impact between nodes. Also the FGCM concepts have a greyness state to model the degree of uncertainty associated to each one (Papageorgiou and Salmeron, 2011). Furthermore, it is possible to compute different whitenization state values. This research applies the equal mean whitenization with α = 0.5, but 10

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• It is a generalization and can be applied to closer approximate medical decision making. It handles the uncertainty inherent in complex domains (as the medical one) by assessing the nodes and edges’ greyness.

• It allows modeling of the uncertainty and experts hesitancy related to the description of the causal relations between the concepts and its states’ description. • FGCMs are able to represent more kinds of relationships than FCM. For instance, it is possible to model relations where the intensity is not known at all with black weights (⊗wi ∈ [−1, +1]) or just partially known with grey weights (⊗wi ∈ [wi , wi ]| − 1 < {wi , wi } < +1).

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it is possible to calculate an optimistic or pessimistic whitenization with different α values. The whitenization value belongs to the grey number and it vibrates between the limits. The final whitenization value depends of the α parameter. In this sense, lower α values generate higher whitenization values closer to the upper limit. Regarding to this, it is possible to select optimistic or pessimistic whitenization approaches. This is a so worthy characteristic, specially in the medical domain, because the state values could be referred to a critical issues. As an overview, FGCM shows several advantages over the classical FCM:

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• The reasoning process would consider a degree of uncertainty (greyness) expressed in grey values.

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In addition, a FGCM-based Decision Support System can adapt its knowledge from available raw data and/or other knowledge sources and assess its associated uncertainty.

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4. Radiotherapy analysis

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A two-level decision making system for radiation therapy based on human knowledge and experience was developed by Papageorgiou et al. (2003b) and evaluated by respective radiotherapy guidelines. The radiation therapy is applied to patients suffering from cancerous diseases (and/or other diseases) and eliminate infected cells, alone or combined with other modalities (Khan, 1994). Its goal is “to design and perform a treatment plan on how to deliver a precisely desired dose of radiation to the defined tumor volume with as minimal damage as possible to the surrounding healthy tissue”, according

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to Khan (1994). The results of a successful treatment using ionizing radiation are the damage of cancer cells, thus high quality of patient life, and prolongation of survival. That system was consisted of a two-level hierarchical structure, where the lower-level FCM modeled the treatment planning, taking into consideration all the factors and treatment variables as well as their influences (namely CTST-FCM) and the upper-level FCM, namely Supervisor-FCM, modeled the procedure of the treatment execution and calculated the optimal final dose for radiation treatment. All the FGCM experiments (CTST-FGCM and Supervisor-FGCM) have been done using the unipolar sigmoid as activation function with λ = {1.0, 3.0, 5.0, 10.0}.

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Figure 3: Unipolar sigmoid with λ = {1.0, 3.0, 5.0, 10.0}

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4.1. Radiotherapy treatment planning

The CTST-FGCM model which represents the radiotherapy treatment planning procedure according to the test packages, protocols and radiotherapists opinions was designed and illustrated in Figure 4. Five factor concepts and eight selector-concepts were selected with discrete and fuzzy values for the determination of the three output decision concepts (DC). Table 2 shows the nodes’ description. Concepts FC1 to FC5 are the Factor-concepts, that represent the depth of tumor (FC1), the size of tumor (FC2), the shape of tumor (FC3), the type of the irradiation (FC4) and the amount of patient thickness irradiated (FC5). Concepts SC1 to SC8 are the Selector-concepts, representing size 12

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of radiation field (SC1), multiple field arrangements (SC2), beam directions (SC3), dose distribution from each field (SC4), stationery vs. rotationisocentric beam therapy, field modification, patient immobilizing and use of 2D or 3D conformal technique, respectively. The concepts outC1 to outC3 are the three output-concepts. Table 2 gathers these respective concepts. The value of the outC1 represents the amount of dose applied to mean Clinical Target Volume (CTV), which have to be larger than the 90% of the amount of prescribed dose to the tumor. The value of concept outC2 represents the amount of the surrounding healthy tissues volume received a dose, which have to be as less as possible, less than the 5% of volume received of the prescribed dose and the value of concept outC3 represents the amount of organs at risk volume received a dose, which have to be less than the 10% of volume received the prescribed dose (Khan, 1994; ICRU report 50, 1993).

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Figure 4: Conformal/Conventional Radiotherapy model

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The Supervisor-FGCM supervises and evaluates the whole radiation therapy process. It is illustrated in Figure 5. The nodes’ description is shown at Table 1 and the grey weights are detailed at Table 7. One objective of the supervisor-FGCM is to keep the amount of Final Dose (FD) which is delivered to the patient between some limits, an upper FDmax and a low limit FDmin. Another objective is to keep the Dose from Treatment Planning (TPD) between maximum value TPDmax and minimum value TPDmin. These objectives are defined at the related AAPM and ICRP protocols (Khan, 1994; Beard et al., 1998; Willoughby et al., 1996), where the accepted dose levels for each organ and region of human body are determined. So, the overall objective for the upper-level, the supervisor-FCM, is to keep the values of corresponding concepts, Final Dose given to the target volume (FD) and Dose prescribed from Treatment Planning (TPD) in the correct range of values. According to Beard et al. (1998) the FD range must be 0.90 ≤ F D ≤ 0.98, and, according to Willoughby et al. (1996), the TPD range must be 0.80 ≤ T P D ≤ 0.95.

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4.2. Radiotherapy treatment supervisor

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Figure 5: Supervisor-FGCM model

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Two case studies for the problem of prostate cancer therapy depicted by CTST-FGCM model, (which consists of 16 concepts and 64 interconnections among concepts), were considered using in order to test the validity of the proposed methodology of FGCM. 14

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With the intention of observing the evolution of an event, the analysis begins with the definition of the initial vector which represents a proposed initial situation. It means that we are starting a therapy plan in the first two cases with all the needed resources, but the acceptance of the treatment planning therapy is not clear. Note that all the elements in both are grey numbers. Using FGCM is possible to design initial vector states (initial grey scenarios) mixing grey and white numbers. Furthermore, it is possible to develop what-if analysis using different initial grey vector states. In the first problem of treatment planning selection, namely CTSTFGCM, we considered the following two case studies, as described by Papageorgiou (2011). In each of the test cases, we have an initial vector ⊗A0 , representing the presented events at a given time of the process, and a grey vector ⊗Af representing the stable state arrived. The vector ⊗Af is the steady vector produced in convergence region and the 14th to 16th values of this vector are the steady values of decision concepts, namely DC1, DC2 and DC3.

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Radiotherapy physicians and medical physicists choose and specified, the fuzzy membership functions for the weights for each scenario as well as the fuzzy rules according to their knowledge for each treatment planning procedure. The linguistic values of fuzzy weights between factor and selector concepts for the CTST-FGCM are given in Table 3, while in forth and fifth column, the grey intensity of weights between factor and selector concepts, as well as between selector and output concepts and among output concepts, are described.

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1. First case study of CTST-FGCM. For the first case study, the conformal radiotherapy was selected. The following initial grey vector is formed for this particular treatment technique ⊗Aconformal =

[0.6, 0.8][0.6, 0.8][0.6, 0.8][0.6, 0.8]

[0.6, 0.9][0.3, 0.5][0.6, 0.8][0.6, 0.8]

[0.4, 0.7][0.6, 0.9][0.6, 0.9][0.6, 0.9]  [0.6, 0.9][0.0, 0.0][0.0, 0.0][0.0, 0.0]

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2. Second case study of CTST-FGCM. In the second case study, the conventional four-field box technique was implemented for the prostate

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cancer treatment. The initial grey vector for this particular treatment technique is designed as follows ⊗Aconventional =

[0.3, 0.6][0.4, 0.7][0.3, 0.5][0.5, 0.6]

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[0.4, 0.7][0.6, 0.8][0.2, 0.5][0.2, 0.5] [0.2, 0.5][0.2, 0.5][0.0, 0.3][0.0, 0.3]  [0.0, 0.3][0.0, 0.0][0.0, 0.0][0.0, 0.0] 365 366 367 368 369 370 371

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1. In the first case study grey values with `(⊗Ci ) = 0, ∀i for all concepts considered.

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In the first case, the 3-D conformal technique consisting of six-field arrangement was selected and in the second one the conventional four-field box technique. The results of conformal and conventional techniques are detailed at Tables 5 and 6. These tables includes the FCM results for each technique. In the case of Supervisor-FGCM, the following two cases studies were examined.

⊗A1 =

374 375

[0.75, 0.75][0.8, 0.8][0.3, 0.3][0.6, 0.6][0.7, 0.7][0.5, 0.5][0.65, 0.65]

2. In the second case study, grey values for the initial concept states were applied.  ⊗A2 = [0.6, 1.0][0.7, 1.0][0.0, 0.5][0.3, 0.6][0.5, 0.7][0.4, 0.7][0.0, 0.5]

377

The results of the both Supervisor-FGCM cases are detailed at Tables 8 and 9. These tables includes the FCM results for each case.

378

4.4. Discussion

376

379 380 381 382 383 384 385

386

387

388

In the case of CTST-FGCM examining different lambda values for conventional and conformal radiotherapy case studies, than when the lambda parameter increases from 3.0 to 5.0 and 5.0 to 10.0, for both radiotherapy treatment planning techniques, the values for output concepts come closer to the desired ones. The values of C1 increase to the highest ones, by decreasing the greyness at the same time. For example, in the case study of conventional radiotherapy, when λ = 3, the C1 takes values [0.97334, 0.99997], with greyness 0.01332, when λ = 5, the C1 is within [0.99895, 1.0] with greyness 0.00053 and when λ = 10, the C1 is 1 with greyness 2.73e-07.

16



390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405

DR A

406

The most meaningful results are taken for λ = 3 (see Tables 5 and 6). The values of C2 and C3 decrease to the lowest ones and the greyness also decreases for them. Furthermore, as the lambda value increases the number of iterations decreases to reach the stable state. In all examined scenarios, the results produced from the proposed approach are acceptable for conformal and conventional radiotherapy, with small values of greyness, which means that the whole model tends to reduce uncertainty. The greyness of a grey number to a certain degree reflects how much the medical expert does not know about the behavioral characteristics of the grey system of concern. In this case, higher greyness values would indicates a great uncertainty about the problem. In other words, a low understanding of the problem. Moreover, FGCM is a modelling technique for complex systems. In this sense, the model dynamics could decrease or increase the greyness associated to the model. Obviously, lower greyness is better for supporting decision making. In CTST-FGCM case, the greyness associated to the ouput nodes is extremely low. In the case of Supervisor-FGCM it is observed that for both examined cases studies, with white and grey concepts values respectively, the desired states for acceptance of radiotherapy selection are reached for λ = 3. It means that the system gives a worthy decision support when uncertainty is included in concepts (system variables) and weight values (causal influences) throughout the grey intensity. For higher values of lambda, the FGCM reaches a steady state where the values of treatment planning dose are higher than the expected ones. The whole approach with λ = 3 approximates human decision making by assessing uncertainty in initial concepts states and relationships. The FGCM methodology allows to handle uncertainty and hesitancy present in the initial experts’ judgements for causal influences among concepts as well as within the initial concept values. The proposed methodology is capable to assess radiotherapy planning, producing meaningful results for the updated weights and steady state concepts. Furthermore, it is possible to get differents whitenization values. This proposal have computed the equal mean whitenization with α = 0.5, but it is possible to calculate an optimistic or pessimistic whitenization. Essentially, the main task of this work was to represent a different approach for construction of decision support tools. The produced results from FGCMs, considering the whitenization values, are comparable with the FCM results in all cases for lower and upper level models. Moreover, we consider that the FGCM results are more reasonable in

FT

389

407

408 409 410 411

412

413 414 415

416

417 418 419 420

421

422 423

424

425 426 427

428

17

438

5. Conclusions

430 431 432 433 434 435 436

FT

437

the case of knowledge representation, as it is not usually enough to have only a numeric value to describe the cause and effect relationships among concepts. The greyness embedded in grey weights through the new methodology gives a more reasonable formalization of FCM theory. In addition, human reasoning is better represented in FGCM than FCM, because the first one includes uncertainty in knowledge representation. Future research will focus on alternative approaches to the expression and incorporation of human uncertainty and system fuzziness in the FGCM model, as well as applications on a variety of decision making tasks.

429

461

This study presents the results of a research on the problem of modeling medical knowledge and capturing the system’s behavior for decision support of radiotherapy treatment planning by using the new approach of FGCMs. More specific, this work proposes a decision support tool based on FGCM formalism for assessing radiotherapy decision making by proposing acceptable system behavior through updated weights that include the inherent uncertainty and fuzziness present in the medical domain. Our proposal represent a worthy approach that exhibits several advantages over the FCM one. It enables modeling of the uncertainty and experts’ hesitancy introduced in the description of the causal relations between the concepts of the cognitive map and in the description of the problem’s components. It is more general and approximate human decision making better than FCM. The output of the process includes a greyness measurement representing the uncertainty associated to the solution. Moreover, FGCMs are able to manage new kinds of relationships regarding FCM. The scope of the proposed methodology was not to achieve better accuracies or results compared with the FCM approaches, but to introduce a novel framework based on the FGCM-DSS that enhanced by greyness in concepts and edges values. Through the results, the prospective performance of the decision support framework based on FGCMs is emerged and encourage us continue towards the direction on including greyness in concepts and causal influences among concepts, thus making decisions contributing to more intelligent systems.

462

Acknowledgements

439 440 441 442 443

DR A

444 445

446

447 448 449 450 451 452 453

454

455 456 457

458

459 460

463

464

Prof. Salmeron’s work is gracefully supported by the Spanish Ministry of Science and Innovation (MICINN-ECO2009.12853). 18

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470 471 472 473 474

475 476 477

478

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DR A

479

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D. Yaman & S. Polat, A fuzzy cognitive map approach for effect-based operations: An illustrative case. Information Sciences 179(4) (2009), pp. 382-403

561

L.A. Zadeh, Fuzzy sets, Information and Control 8 (1965), pp. 338-353.

558

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21

FT

Table 1: Supervisor nodes’ description

Node SC1

DR A

SC2

Description Tumor Localization. It is dependent on patient contour, sensitive critical organs and tumor volume. It embodies the value and influence of these three Factor-concepts that are concepts of first-level. Dose prescribed from Treatment Planning (TPD). This concept describes the delivered doses to target volume, normal tissues and critical organs that are calculated at the treatment planning model of the first level. Machine factors. This concept describes the equipment characteristics. Human factors. A general concept describing the experience and knowledge of medical staff. Patient positioning and immobilization. This concept describes the cooperation of the patient with the doctors and the potential of follow instructions. Quality Assurance (QA). Quality assurance includes demands on staff, the therapeutic procedures and the technical systems for complying with the preset standards. Final Dose given to the target volume (FD). A measurement of the radiation dose received by the target tumor.

SC3

SC4

SC5

SC6

SC7

22

Appendix

Table 2: CTST-FGCM radiotherapy nodes’ description

sc7 sc8 outC1 outC2 outC3

Description Accuracy of depth of tumor Size of tumor Shape of tumor Type of irradiated tissues-presence of inhomogeneities Amount of patient thickness irradiated Size of radiation field Single or multiple field arrangements Beam direction(s) (angles of beam orientation) Dose distribution from individual field Stationery vs. rotation-isocentric beam therapy Field modification (no field modification, blocks, wedges, filters and multileaf-collimator shaping blocks) Patient immobilization Use of 2D or 3D conformal technique Dose given to treatment volume Amount of irradiated volume of healthy tissues Amount of irradiated volume of sensitive organs (organs at risk)

FT

Node fc1 fc2 fc3 fc4 fc5 sc1 sc2 sc3 sc4 sc5 sc6

DR A

562

23

Table 3: Base and vibration values for CTST-FGCM conformal radiotherapy Vibration δa 0.25 0.20 0.25 0.25 0.20 0.20 0.20 0.15 0.10 0.20 0.25 0.25 0.25 0.10 0.20 0.15 0.20 0.25 0.20 0.20 0.20 0.10 0.25 0.10 0.20 0.20 0.25 0.20 0.20 0.20 0.20 0.20 0.25 0.10

DR A

fc1-sc1 fc1-sc2 fc1-sc3 fc1-sc4 fc1-sc5 fc1-sc6 fc1-sc7 fc2-sc1 fc2-sc2 fc2-sc3 fc2-sc4 fc2-sc5 fc2-sc6 fc2-sc7 fc2-sc8 fc3-sc1 fc3-sc2 fc3-sc3 fc3-sc4 fc3-sc5 fc3-sc8 fc4-sc1 fc4-sc2 fc4-sc3 fc4-sc4 fc4-sc5 fc4-sc6 fc4-sc8 fc5-sc1 fc5-sc2 fc5-sc3 fc5-sc4 fc5-sc5 fc5-sc6

Base value wij (a) 0.70 0.75 0.30 0.40 0.60 0.60 0.20 0.70 0.60 0.20 0.53 0.55 0.50 0.60 0.50 0.60 0.65 0.45 0.00 0.40 0.75 0.20 0.60 0.50 0.55 0.45 0.50 0.60 0.50 0.60 0.60 0.60 0.20 0.50

Grey weight [a − δa , a + δa ] [0.45, 0.95] [0.55, 0.95] [0.05, 0.55] [0.15, 0.65] [0.40, 0.80] [0.40, 0.80] [0.00, 0.40] [0.55, 0.85] [0.50, 0.70] [0.00, 0.40] [0.28, 0.78] [0.30, 0.80] [0.25, 0.75] [0.50, 0.70] [0.30, 0.70] [0.45, 0.75] [0.45, 0.85] [0.20, 0.65] [−0.20, 0.20] [0.20, 0.60] [0.55, 0.95] [0.10, 0.30] [0.35, 0.85] [0.40, 0.60] [0.35, 0.75] [0.25, 0.65] [0.25, 0.75] [0.40, 0.80] [0.30, 0.70] [0.40, 0.80] [0.40, 0.80] [0.40, 0.80] [−0.05, 0.45] [0.40, 0.60]

FT

Edge

24

Vibration 0.10 0.20 0.10 0.10 0.20 0.25 0.20 0.25 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.10 0.15 0.15 0.10 0.25 0.25 0.20 0.25 0.25 0.10 0.20 0.20 0.25 0.25 0.10 0.10

Grey weight [0.40, 0.60] [0.30, 0.70] [0.60, 0.80] [0.50, 0.70] [0.20, 0.60] [−0.65, −0.15] [−0.60, −0.20] [0.05, 0.55] [−0.70, −0.30] [−0.60, −0.20] [0.20, 0.60] [−0.50, −0.10] [−0.50, −0.10] [0.20, 0.60] [−0.60, −0.20] [−0.60, −0.20] [0.20, 0.40] [−0.45, −0.15] [−0.45, −0.15] [0.30, 0.50] [−0.65, −0.15] [−0.65, −0.15] [0.30, 0.70] [−0.75, −0.25] [−0.75, −0.25] [0.50, 0.70] [−0.70, −0.30] [−0.70, −0.30] [−0.85, −0.45] [−0.75, −0.25] [−0.80, −0.60] [−0.50, −0.70]

FT

Base value 0.50 0.50 0.70 0.60 0.40 −0.40 −0.40 0.30 −0.50 −0.40 0.40 −0.30 −0.30 0.40 −0.40 −0.40 0.30 −0.30 −0.30 0.40 −0.40 −0.40 0.50 −0.50 −0.50 0.60 −0.50 −0.50 −0.60 −0.50 −0.70 −0.60

DR A

Edge fc5-sc7 sc2-sc4 sc5-sc6 sc6-sc5 sc1-outC1 sc1-outC2 sc1-outC3 sc2-outC1 sc2-outC2 sc2-outC3 sc3-outC1 sc3-outC2 sc3-outC3 sc4-outC1 sc4-outC2 sc4-outC3 sc5-outC1 sc5-outC2 sc5-outC3 sc6-outC1 sc6-outC2 sc6-outC3 sc7-outC1 sc7-outC2 sc7-outC3 sc8-outC1 sc8-outC2 sc8-outC3 outC1-outC2 outC1-outC3 outC2-outC1 outC3-outC1

25

Table 4: Base and vibration values for CTST-FGCM conventional radiotherapy Vibration δa 0.25 0.20 0.25 0.25 0.20 0.20 0.20 0.15 0.10 0.20 0.25 0.25 0.25 0.10 0.20 0.15 0.20 0.25 0.20 0.20 0.20 0.10 0.25 0.10 0.10 0.10 0.25 0.20 0.20 0.20 0.20 0.20 0.25 0.10

DR A

fc1-sc1 fc1-sc2 fc1-sc3 fc1-sc4 fc1-sc5 fc1-sc6 fc1-sc7 fc2-sc1 fc2-sc2 fc2-sc3 fc2-sc4 fc2-sc5 fc2-sc6 fc2-sc7 fc2-sc8 fc3-sc1 fc3-sc2 fc3-sc3 fc3-sc4 fc3-sc5 fc3-sc8 fc4-sc1 fc4-sc2 fc4-sc3 fc4-sc4 fc4-sc5 fc4-sc6 fc4-sc8 fc5-sc1 fc5-sc2 fc5-sc3 fc5-sc4 fc5-sc5 fc5-sc6

Base value wij (a) 0.70 0.75 0.30 0.40 0.60 0.60 0.00 0.75 0.60 0.00 0.60 0.55 0.50 0.60 0.50 0.60 0.65 0.45 0.20 0.40 0.75 0.20 0.60 0.50 0.50 0.40 0.50 0.40 0.50 0.60 0.60 0.50 0.20 0.50

Grey weight [a − δa , a + δa ] [0.45, 0.95] [0.55, 0.95] [0.05, 0.55] [0.15, 0.65] [0.40, 0.80] [0.40, 0.80] [−0.20, 0.20] [0.60, 0.85] [0.50, 0.70] [0.00, 0.40] [0.35, 0.85] [0.30, 0.80] [0.25, 0.75] [0.50, 0.70] [0.30, 0.70] [0.45, 0.75] [0.45, 0.85] [0.20, 0.65] [0.00, 0.40] [0.20, 0.60] [0.55, 0.95] [0.10, 0.30] [0.35, 0.85] [0.40, 0.60] [0.40, 0.60] [0.30, 0.50] [0.25, 0.75] [0.20, 0.60] [0.30, 0.70] [0.40, 0.80] [0.40, 0.80] [0.30, 0.70] [−0.05, 0.45] [0.40, 0.60]

FT

Edge

26

Vibration 0.10 0.20 0.10 0.10 0.10 0.25 0.20 0.15 0.20 0.20 0.10 0.20 0.20 0.20 0.10 0.10 0.10 0.15 0.15 0.10 0.15 0.15 0.20 0.25 0.25 0.10 0.20 0.20 0.20 0.20 0.10 0.10

Grey weight [0.50, 0.70] [0.30, 0.70] [0.60, 0.80] [0.50, 0.70] [0.20, 0.40] [−0.65, −0.15] [−0.60, −0.20] [0.10, 0.40] [−0.70, −0.30] [−0.60, −0.20] [0.20, 0.40] [−0.50, −0.10] [−0.50, −0.10] [0.05, 0.45] [−0.30, −0.10] [−0.30, −0.10] [0.20, 0.40] [−0.45, −0.15] [−0.45, −0.15] [0.30, 0.50] [−0.15, 0.15] [−0.15, 0.15] [0.20, 0.60] [−0.55, −0.05] [−0.55, −0.05] [0.30, 0.50] [−0.60, −0.20] [−0.60, −0.20] [−0.60, −0.20] [−0.60, −0.20] [−0.80, −0.60] [−0.50, −0.70]

FT

Base value 0.60 0.50 0.70 0.60 0.30 −0.40 −0.30 0.25 −0.50 −0.40 0.30 −0.30 −0.30 0.25 −0.20 −0.20 0.30 −0.30 −0.30 0.20 0.00 0.00 0.40 −0.30 −0.30 0.40 −0.40 −0.40 −0.40 −0.40 −0.70 −0.60

DR A

Edge fc5-sc7 sc2-sc4 sc5-sc6 sc6-sc5 sc1-outC1 sc1-outC2 sc1-outC3 sc2-outC1 sc2-outC2 sc2-outC3 sc3-outC1 sc3-outC2 sc3-outC3 sc4-outC1 sc4-outC2 sc4-outC3 sc5-outC1 sc5-outC2 sc5-outC3 sc6-outC1 sc6-outC2 sc6-outC3 sc7-outC1 sc7-outC2 sc7-outC3 sc8-outC1 sc8-outC2 sc8-outC3 outC1-outC2 outC1-outC3 outC2-outC1 outC3-outC1

27

Table 5: CTST-FGCM conformal radiotherapy results

FCM

1.0

3.0

5.0

0.9786 0.0386 0.0453 0.9991 0.0027 0.0041 1.0000 0.0000 0.0000 1.0000 0.0000 0.0000

Stable state [0.90792, 0.99554] [0.00369, 0.17928] [0.00471, 0.20458] [0.9945, 1.0] [3.06e − 08, 0.00459] [6.48e − 08, 0.00716] [0.99992, 1.0] [2.11e − 13, 7.51e − 05] [7.36e − 13, 0.00016] [1.0, 1.0] [3.93e − 26, 3.54e − 09] [4.79e − 25, 1.59e − 08]

DR A

10.0

Node outC1 outC2 outC3 outC1 outC2 outC3 outC1 outC2 outC3 outC1 outC2 outC3

Whitenization 0.95173 0.09148 0.10465 0.99724 0.00229 0.00358 0.99996 3.75e-05 7.95e-05 1.0 1.77e-09 7.94e-09

FT

λ

FGCM Length Greyness 0.08762 0.04381 0.17559 0.08779 0,19987 0.09993 0.00551 0.00276 0.00459 0.00229 0.00716 0.00358 8.31e-05 4.15e-05 7.51e-05 3.75e-05 0.00016 7.95e-05 3.59e-09 1.80e-09 3.54e-09 1.77e-09 1.59e-08 7.94e-09

28

Table 6: CTST-FGCM conventional radiotherapy results

FCM

1.0

3.0

5.0

0.8072 0.2348 0.2437 0.9860 0.0020 0.0039 0.0039 0.9999 0.0003 1.0000 7.30e-06 4.27e-05

Stable state [0.63039, 0.94722] [0.02262, 0.32743] [0.02796, 0.35245] [0.97334, 0.99997] [1.79e − 06, 0.04789] [3.79e − 06, 0.07190] [0.99895, 1.00000] [1.82e − 10, 0.00483] [6.35e − 10, 0.01014] [1.00000, 1.00000] [2.87e − 20, 1.73e − 05] [3.49e − 19, 7.74e − 05]

DR A

10.0

Node outC1 outC2 outC3 outC1 outC2 outC3 outC1 outC2 outC3 outC1 outC2 outC3

Whitenization 0.78881 0.17502 0.19021 0.98665 0.02395 0.03595 0.99947 0.00242 0.00507 1.00000 8.64e-06 3.87e-05

FT

λ

FGCM Length Greyness 0.31683 0.15841 0.30481 0.15240 0.32449 0.19275 0.02663 0.01332 0.04789 0.02394 0.07189 0.03595 0.00105 0.00053 0.00483 0.00241 0.01014 0.00507 5.47e-07 2.73e-07 1.73e-05 8.64e-06 7.74e-05 3.87e-05

29

Table 7: Base and vibration values for Supervisor-FGCM

Vibration 0.10 0.25 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.25 0.25 0.25 0.25 0.10

Grey weight [0.40, 0.60] [0.05, 0.55] [0.50, 0.70] [−0.40, −0.20] [−0.35, −0.15] [−0.50, −0.30] [−0.40, −0.20] [−0.40, −0.20] [0.50, 0.70] [0.30, 0.80] [0.25, 0.75] [0.05, 0.55] [0.45, 0.95] [0.45, 0.65]

FT

Base value 0.50 0.30 0.60 −0.30 −0.25 −0.40 −0.30 −0.30 0.60 0.55 0.50 0.30 0.70 0.55

DR A

Edge sc1-sc7 sc2-sc1 sc2-sc7 sc3-sc2 sc3-sc7 sc4-sc5 sc4-sc7 sc5-sc4 sc5-sc7 sc6-sc2 sc6-sc7 sc7-sc1 sc7-sc2 sc7-sc5

30

Table 8: Supervisor-FCM 1st scenario results

FCM

1.0

0.7861 0.8375 0.3000 0.5926 0.7258 0.5000 0.8814 0.9916 0.9965 0.3000 0.8369 0.9724 0.5000 0.9991 0.9997 0.9999 0.3000 0.9991 0.9999 0.5000 1.0000 0.9999 0.9999 0.3000 0.9991 0.9999 0.5000 1.0000

Stable state [0.68092, 0.86815] [0.73869, 0.89995] [0.65883, 0.65883] [0.56321, 0.61834] [0.67526, 0.77176] [0.65889, 0.65889] [0.81456, 0.94892] [0.56099, 0.96299] [0.74176, 0.97690] [0.5, 0.5] [0.26585, 0.40438] [0.64538, 0.84646] [0.5, 0.5] [0.89271, 0.99817] [0.61469, 0.99591] [0.87840, 0.99807] [0.5, 0.5] [0.12897, 0.30762] [0.81125, 0.95506] [0.5, 0.5] [0.98993, 0.99999] [0.72929, 0.99998] [0.98201, 1.0] [0.5, 0.5] [0.01810, 0.12351] [0.97980, 0.99841] [0.5, 0.5] [0.99999, 1.0]

DR A

3.0

Node sc1 sc2 sc3 sc4 sc5 sc6 sc7 sc1 sc2 sc3 sc4 sc5 sc6 sc7 sc1 sc2 sc3 sc4 sc5 sc6 sc7 sc1 sc2 sc3 sc4 sc5 sc6 sc7

5.0

10.0

Whitenization 0.77453 0.81932 0.65883 0.59078 0.72351 0.65889 0.88174 0.76199 0.85933 0.5 0.33512 0.74592 0.5 0.94544 0.80530 0.93823 0.5 0.21830 0.88315 0.5 0.99496 0.86463 0.99100 0.5 0.07080 0.98911 0.5 1.0

FT

λ

FGCM Length Greyness 0.18723 0.09362 0.16126 0.08063 0.0 0.0 0.05513 0.02757 0.09650 0.04825 0.0 0.0 0.13436 0.06718 0.40200 0.20100 0.23514 0.11757 0.0 0.0 0.13853 0.06926 0.20108 0.10054 0.0 0.0 0.10546 0.05273 0.38121 0.19061 0.11967 0.05984 0.0 0.0 0.17866 0.08933 0.14382 0.07191 0.0 0.0 0.01005 0.00503 0.27070 0.13535 0.01798 0.00899 0.0 0.0 0.10541 0.05270 0.01861 0.00930 0.0 0.0 8.03e-06 4.01e-06

31

Table 9: Supervisor-FCM 2nd scenario results

FCM

1.0

0.7867 0.8432 0.2000 0.5926 0.7263 0.4000 0.8846 0.9916 0.9968 0.2000 0.8369 0.9724 0.4000 0.9992 0.9997 0.9999 0.2000 0.9655 0.9970 0.4000 0.9999 0.9999 0.9999 0.2000 0.9991 0.9999 0.4000 1.0000

Stable state [0.68113, 0.86818] [0.73884, 0.89996] [0.65883, 0.65883] [0.56340, 0.61855] [0.67526, 0.77176] [0.65889, 0.65889] [0.81456, 0.94892] [0.56098, 0.96299] [0.74175, 0.97690] [0.5, 0.5] [0.26585, 0.40441] [0.64522, 0.84647] [0.5, 0.5] [0.89267, 0.99817] [0.61469, 0.99591] [0.87840, 0.99807] [0.5, 0.5] [0.12897, 0.30764] [0.81118, 0.95506] [0.5, 0.5] [0.98993, 0.99999] [0.72929, 0.99998] [0.98201, 1.0] [0.5, 0.5] [0.01810, 0.12351] [0.97981, 0.99841] [0.5, 0.5] [0.99999, 1.0]

FGCM Length Greyness 0.18705 0.09352 0.16112 0.08056 0.0 0.0 0.05515 0.02758 0.09650 0.04825 0.0 0.0 0.13436 0.06718 0.40201 0.20100 0.2351 0.11758 0.0 0.0 0.13855 0.06928 0.20124 0.10062 0.0 0.0 0.10551 0.05275 0.38121 0.19061 0.11968 0.05984 0.0 0.0 0.17867 0.08934 0.14388 0.07194 0.0 0.0 0.01006 0.00503 0.27070 0.13535 0.01798 0.00899 0.0 0.0 0.10541 0.05270 0.01860 0.00930 0.0 0.0 8.03e-06 4.01e-06

DR A

3.0

Node sc1 sc2 sc3 sc4 sc5 sc6 sc7 sc1 sc2 sc3 sc4 sc5 sc6 sc7 sc1 sc2 sc3 sc4 sc5 sc6 sc7 sc1 sc2 sc3 sc4 sc5 sc6 sc7

5.0

10.0

Whitenization 0.77466 0.81940 0.65883 0.59097 0.72351 0.65889 0.88174 0.76199 0.85933 0.5 0.33513 0.74585 0.5 0.94542 0.80530 0.93823 0.5 0.21830 0.88312 0.5 0.99496 0.86463 0.99100 0.5 0.07080 0.98911 0.5 1.0

FT

λ

32