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BMC Medical Informatics and Decision Making

Open AccessResearch articleDecision theory applied to image quality control in radiologyPatrícia S Lessa*1,2, Cristofer A Caous1,2, Paula R Arantes1,2, Edson Amaro Jr1,2 and Fernando M Campello de Souza3

Address: 1Instituto Israelita de Ensino e Pesquisa Albert Einstein, São Paulo, SP, Brasil, 2Neuroimagem Funcional, LIM 44, Universidade de São Paulo, FMUSP, São Paulo, SP, Brasil and 3Eletrônica e Sistemas, Universidade Federal de Pernambuco, Recife, PE, Brasil

Email: Patrícia S Lessa* - [email protected]; Cristofer A Caous - [email protected]; Paula R Arantes - [email protected]; Edson Amaro - [email protected]; Fernando M Campello de Souza - [email protected]

* Corresponding author

AbstractBackground: The present work aims at the application of the decision theory to radiologicalimage quality control (QC) in diagnostic routine. The main problem addressed in the frameworkof decision theory is to accept or reject a film lot of a radiology service. The probability of eachdecision of a determined set of variables was obtained from the selected films.

Methods: Based on a radiology service routine a decision probability function was determined foreach considered group of combination characteristics. These characteristics were related to thefilm quality control. These parameters were also framed in a set of 8 possibilities, resulting in 256possible decision rules. In order to determine a general utility application function to access thedecision risk, we have used a simple unique parameter called r. The payoffs chosen were:diagnostic's result (correct/incorrect), cost (high/low), and patient satisfaction (yes/no) resulting ineight possible combinations.

Results: Depending on the value of r, more or less risk will occur related to the decision-making.The utility function was evaluated in order to determine the probability of a decision. The decisionwas made with patients or administrators' opinions from a radiology service center.

Conclusion: The model is a formal quantitative approach to make a decision related to themedical imaging quality, providing an instrument to discriminate what is really necessary to acceptor reject a film or a film lot. The method presented herein can help to access the risk level of anincorrect radiological diagnosis decision.

BackgroundThe X-Ray is the main investigation method for thehuman body in various clinical conditions. Therefore,quality control procedures were implemented to stand-ardize and define the minimum requirements for a goldstandard radiology service. These requirements are world-wide reviewed by health institutions, including Brazilian

National Institute of Health (Secretaria de Estado daSaúde, Centro de Vigilância Sanitária and Ministério daSaúde), as well as international institutions (InternationalCommission on Radiological Protection – ICRP, WorldHealth Organization – WHO, and International Commis-sion of the European Communities – ICRU) monitoringthe radiology practice [1].

Published: 13 November 2008

BMC Medical Informatics and Decision Making 2008, 8:51 doi:10.1186/1472-6947-8-51

Received: 26 May 2008Accepted: 13 November 2008

This article is available from: http://www.biomedcentral.com/1472-6947/8/51

© 2008 Lessa et al; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

BMC Medical Informatics and Decision Making 2008, 8:51 http://www.biomedcentral.com/1472-6947/8/51


The main idea behind the regulations in X-Ray applica-tion is to protect patients from radiation injury. The basicprinciples can be summarized as those described by ICRP73 [2]: ionization radiation should be applied in caseswhere the benefits overcome the potential hazards. Theobjective is to maximize the gains over the potentialharms using procedures to protect against radiation. Theapplications of these two principles are the main goal of aQuality Control (QC) program: apply the minimumrequired radiation to maintain the diagnostic value of theimages. Radiation must be controlled and the radiologicalimages should have always enough quality to provide therequired information.

Excellence in imaging quality standards is achievedthrough published guidelines of radiological societies(e.g. Radiological Society of North America guidelines).There are standard procedures for each exam that the tech-nologist knows and follows when acquiring an image.There are also general rules related to examination stepsand acquisition device settings. Every step is dependent ofthe patient's characteristics, such as height. In addition,some issues are difficult to standardize, although themajor of technical procedures were monitored for aproper quality assurance [3]. These non-measurable fac-tors usually have a negative impact in the standardizationof radiological practice with undesirable effects. A prelim-inary study made between 1992 and 2003 in Brazilshowed that 5.15% of the films were rejected films; result-ing in patient's re-expositions, equipment lifetime reduc-tion and an overall increased operational cost. The maincause of film loss identified in this study was lack of stand-ardization [4].

A QC program is typically a result of multidisciplinaryinteraction, involving several steps: habitual verificationof technical parameters in X-ray equipments, systems reg-istry (films, digital cassettes) and other procedures used todeliver the images for medical interpretation. The defini-tion of QC also includes a series of standardized testsdeveloped to detect changes in X-ray equipments [5,6].The final result of a successful QC should primarily be areduction in the number of re-exposed or over-exposedpatients and a high quality imaging diagnostic. Also, itshould provide minimal financial costs and increase theequipment lifetime. This usually requires a complex ruleof decisions, usually made by the physicists, engineersand radiologists in a joint effort to maximize two princi-ples: improve the image quality and reduce radiationexposure.

Decision Making Theory (DMT) offers an integratedframework to mathematically define a preference and/orprobability model for each step of the QC program. Prior-ity of preference in sets of possible outcomes and states of

information (uncertainty or imprecision) are representedas distribution possibilities [7] and therefore statisticallyanalyzed. This formulation allows a more direct measure-ment of the outcomes from a certain decision and hasbeen implemented successfully in many different con-texts. One of the most used implementations of DMT wasproposed by Von Neumann and Morgenstern [8]. Theseauthors modeled the quantitative utility function basedon a set of axioms that described decision-maker's behav-ior. Various implementations made in medicine werebased on their approach with practical examples [9,10].Their work has demonstrated the importance of this kindof framework according to a decision policy regarding acase of a preoperative patient management care beforeopting for a major elective surgery. All available informa-tion to decision-making, through informed priors, allowthe appropriate decisions to be made [10].

In essence, what the decision theory formalizes is thecommon sense idea that an individual should take thebest action based upon what he or she wants, knows, andcan do [11,12]. In a radiology routine, it is the technicianwho acquires the image and observes it initially. It is oftenup to them – without ever asking the radiologist –whether or not another image must be taken due to poorquality acquisitions. It should also be noted that the tech-nologists often acquire an image with certain parametersbecause they know which radiologists will evaluate theexam and what their personal preferences are. The classi-fication of this "individual" choice depends on direct orindirect aspects. The person who makes this decisioncould be the administrator of the radiology service depart-ment, the radiologist itself, the technician or even thepatient when refusing to perform a second or third examin order to reacquire the image.

The basic parameters of decision theory involve some sets,probabilistic mechanisms, and decision rules [13].

The four main sets are:

1) a payoff set (p) – the offered benefit in a specific situa-tion, consisting of all the possible consequences of theactions to be taken;

2) a set of "states of nature" ( ) – containing all the possi-ble situations of the study; typically unknown parametersin the phenomena (i.e. if a film has a good or bad quality)involved;

3) a set of observations (x) – relating to the states of nature(i.e. spatial resolution, sensitivity of an image);

4) a set of possible courses of action (a) – to reject oraccept the film.



The three main probabilistic mechanisms are:

1) The consequence function – P(pi| j, ak). It shows the dis-tribution probability of the payoffs for each pair ofdefined events – "state of nature" vs. action. It is necessaryto consider the utility function, which represents the pref-erences of the decision maker in situations of uncertainty.In order to reach the model of utility function (u), a proc-ess of eduction is applied to prioritize the good and badactions strategies incorporated into a unique evaluationdimension. This function is a polynomial one with adegree called r. The higher the r value, the most complexwill be the utility function.

2) The distribution probability (likelihood function) – P(xi| j).According to the determined observations of the state ofnature, a film can have a good or bad quality including itsown characteristics (contrast, speed, fog, digital exposi-tion, latitude and other). The distribution probabilityconsiders the correlation between the observations madeand the state of nature.

3) A prior distribution – ( j). Possibly elicited from anexpert, regarding to the state of nature. If the institutionmaintains a database of QC, it is possible to estimate aprior distribution to be applied to the test.

The decision rule (d) associated for each observation (pos-sible actions to be taken):

1) The deterministic choice is the most simple decisionrule for each observation in which an ideal action shouldbe taken.

2) The randomized decision rule yields two other types:2.1) The randomization upon the non-randomized rulesbefore any observation is made.

2.2) The behavioral randomized rule b(aj|xi), in which afirst observation is made. After this observation, a rand-omization of all possible actions is considered. Thebehavioral rule is more powerful, in the sense that it canaccess any risk; this is not the case regarding the rand-omized rule.

The risk function (Rd) compares the decision rules and theconsequence function is defined by the process for obtain-ing the probability of a certain result related to the "stateof nature" (in our case, a good or bad image). Based onthe consequence function the decision-maker chooses, acertain proceeding or action is taken. This probabilisticmechanism should be better understood when analyzedtogether with the decision theory techniques. It can beinferred from an available database, from the elicitationsof an expert or from both.

Herein, we suggest a general guideline, based on the deci-sion-making paradigm, as proposed by the report fromICRU in 1996 [1]. According to this report, the techniquesof statistical DMT should be used as a standard procedureto treat problems of image detection, edge detection,noise and sharpness. The parameters referred by thepresent work are: inadequate use, inadequate sensibilityand/or inadequate spatial resolution related to the radio-logical chain process. The application of the decisionmaking theory was restricted to acquired images, not con-sidering other parameters that could interfere before theimage detection. The final outcome considered was toaccept of reject a film lot. With this new approach, thereare good consequences like: the patient's satisfaction, acorrect diagnosis and stable financial costs.

This work does not aim on a new method proposition toimprove the sensibility or specificity of the data acquisi-tion system. The approach also applied a very well knownconcept of the radiology practice: the Receiver OperatingCharacteristics, also known as the ROC curve. This curveis determined according to a set of probabilities: true pos-itive = P (positive test/positive for disease) vs. false posi-tive = P (positive test/negative for disease). In other words,sensibility vs. a false positive or P (altered exam/has thedisease) vs. P (altered exam/has not the disease) parame-ters that are part of most papers published in radiologyjournals, in which the accuracy of a diagnostic imagingtest is verified.

This is the TPF (function of true positive probability) andFPF (function of false positive probability) ROC curve. Ithas a purely inferential character, and is a very particularcase of a more general ROC curve, where the loss functionis the ideal observer loss function (i.e., the negative of theutility function). We have four possibilities: if the film isbad, we reject it; if the film is good, we accept it; if the filmis bad and we accept it, and finally, if the film is good andwe reject it. In the first two cases, the target was reached.In the latest the target was missed, and they correspond toerrors. In mathematical terms, the loss function is givenby:

L( i, aj) = -u( i, aj)

The ideal observer loss function is given by:

L( i, aj) = 0 if ai = j; L( j, ai) = 1 if aj i, i, j = 0, 1.

This means that if we reach the target, we have no fine (theloss is null). If we miss the target, we pay a fine equal toone (the loss is equal to 1). But this rather peculiar utilityfunction does not reflect the preferences of a decision-maker, since the negative of the ideal observer loss func-tion is just an inference criterion.



When the ideal loss function is applied, the risks were cal-culated by:

Rd( j) = - P(xi| j)u( j, d(xi))

These probabilities correspond to the specificity, sensitiv-ity, false positive and false negative conditions. In theseinferential contexts, the classical TPF and FPF ROC curveis the typically presented as the true positive versus thefalse positive conditions, with a 0–100% scale on bothaxes (concave function). In the current work, since we aredealing with preferences, the risks will not be consideredas probabilities. The axes of the ROC curves will have ascale corresponding to the applied utility function, whichshould be educed from the decision-maker and representshis preferences (convex curve). It is the locus of all possi-ble Neyman-Pearson Rules and not a TPF- FPF ROC curve.

The ROC curve axis reflects the preference dimension ofthe decision-maker regarding to the data acquisition sys-tem (i.e. technology or radiography type) and will implyin a likelihood function. This construction is independentfrom a considered database and the probability will usu-ally be very similar.

In order to accept or reject a film lot according to this pro-cedure is necessary to obtain the correct diagnosis, the costoptimization and the client's satisfaction. The client's sat-isfaction will be measured by the utility function, which isdifferent from the one that corresponds to the idealobserver loss function. All values of probable observa-tions were proposed based on previous information fromreal datasets and hypothetical predictions applications.The investigation researched some possibilities accordingto the dependency values related to the image qualityassurance. In summary, this is a case in which the decisiontheory is used to establish a framework for radiographicfilm quality control protocol; this is done by studying theconsequence of an action as a function of a specific prob-ability distribution over possible consequences.

MethodsThe decision making theory was implemented for specificcomputational tools. The data was stored in a Quattro Prov. 9.0 (Borland Inc – USA) and in a Statistica v. 5.3(StatSoft – USA). All data was obtained from densitome-try data of the radiographic films of a phantom obtainedby a digital densitometer (Model 07–443, Nuclear Associ-ates Victoreen 07–443 Clamshell, United States). The val-ues of the film characteristics were calculated from 15different measures of radiation intensity.

The images were made in a conventional X-Ray, Medicor500 mA and 125 kV, from the Hospital das Clínicas daUniversidade Federal de Pernambuco (HC-UFPE), Brazil.

The values were obtained from data analysis of the radio-graphic film (characteristic curve, modulated transferfunction and the Wiener spectrum as exposed in ICRU[1]). Attributes such as contrast, latitude, speed and fogwere determined according to our observations from eachfilm condition (bad quality or good quality).

In case of two possible actions (accept/reject) and no priorprobability, we applied the behavioral randomized [7]and Neyman-Pearson rules. In case of two possibleactions, the observation sample space was partitionedinto two regions, one of rejection and the other of accept-ance. The model is very simple in compliance with Ock-ham's razor principle. The consequence function wascalculated based on the subjective preference of the per-son making the decision.

ResultsThe first approach will consider a dichotomy in each ofthe three attributes: diagnosis, cost and client satisfaction,where eight possible payoffs cases (see figure 1) will beconsidered:

p0 = (incorrect diagnostic, high cost, low client satisfaction)

p1 = (incorrect diagnostic, low cost, low client satisfaction)

p2 = (incorrect diagnostic, high cost, high client satisfaction)

p3 = (incorrect diagnostic, low cost, high client satisfaction)

p4 = (correct diagnostic, high cost, low client satisfaction)

p5 = (correct diagnostic, low cost, low client satisfaction)

p6 = (correct diagnostic, high cost, high client satisfaction)

p7 = (correct diagnostic, low cost, high client satisfaction)

Based on the parameters described above, a set of "statesof nature" was established. A film lot is considered goodif the percentage of bad units from a film box is below acertain pre-specified value. There is a natural doubts ofwhich of these states reflect a true condition scenario.

This uncertainty establishes the decision problem. Where represents the real "state of nature" of the film, we have:

0 = the film is a bad quality one;

1 = the film is a good quality one.

The space or set of observations should consider a finitenumber of possibilities. We considered three groups of



attributes: film usage, film sensibility and spatial resolu-tion.

For each situation of film usage, film sensibility and spa-tial resolution, a dichotomy was considered, based on thelimits of the values dependency. Another set (eight possi-bilities) was originated for the following observations:

x0 = (inadequate for use, inadequate sensibility, inadequatespatial resolution)

x1 = (inadequate for use, inadequate sensibility, adequate spa-tial resolution)

x2 = (inadequate for use, adequate sensibility, inadequate spa-tial resolution)

x3 = (inadequate for use, adequate sensibility, adequate spatialresolution)

x4 = (adequate for use, inadequate sensibility, inadequate spa-tial resolution)

x5 = (adequate for use, inadequate sensibility, adequate spatialresolution)

x6 = (adequate for use, adequate sensibility, inadequate spatialresolution)

x7 = (adequate for use, adequate sensibility, adequate spatialresolution)

The problem considered in this case had only two possi-bilities of action:

Schematic representation of the Decision Making Theory applied to the radiographic film quality control parameterFigure 1Schematic representation of the Decision Making Theory applied to the radiographic film quality control parameter.

Probabilistic

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Neyman-Pearson decision rules

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act ions: A��

Main set

��Risk set :

))(,()7

0|()( ixdjuji i

xPjdR

1,0j



a0 = to reject the film lot;

a1 = to accept the film lot.

The effective action related to the condition of the state ofnature reflects the probability of the feasible outcomes.

This probabilistic mechanism depends on the technologyof the data acquisition system (sensors, digitizers and oth-ers). A plausible set of probabilities for this problem wasshown in table 1. These values correspond to an estimateddistribution of the combined information mentionedabove obtained from radiologists and physicists involvedin the QC program.

The likelihood function P(xi| j) consists of two values, inwhich i corresponds to the observed value and j to the truevalue. The value of xi (a set of observations) representswhat is obtained from the equipment, and j (a set of"states of nature") is the "true" value: a good or bad filmquality.

The consequence function is the probability of obtainingcertain payoff regarding the "state of nature" and a specificaction of the decision-maker. Table 2 shows the probabil-ities for each pair ( i, aj) of a consequence function. Thesevalues where calculated with data from Table 1. Forinstance, regarding xo("inadequate for use, inadequate sensi-bility, inadequate spatial resolution") the probability of eachconsequence (p0 - p7) if the action is to reject (a0) the filmcan be obtained from Table 2. If the film is bad ( o), thehighest probability is p4 (0.300), which means that thereis a high chance of making the right diagnosis. A high costis involved in this case due to the necessity of anotherfilm, trial procedure and professional staff time. As a con-sequence, the client satisfaction in the process is compro-mised since the patient will probably repeat theprocedure, and be re-exposed to the X-rays.

The decision rulesThe "state of nature" and the problem solution wereobtained by the rule of Neyman-Pearson. The decision-maker utility function enunciated the level of toleratedrisk when 0 was considered the true "state of nature".

We have applied two r values to simulate the decision-maker's influence or preference upon the decision rule toadopt an optimized action (ideal). The r value is actuallythe polynomial grade used in the mathematical formula-tion. We chose r = 2 and r = 4, in order to have the classicalquadratic loss function (from Theory of Decision Making)and a more complex formulation. This was done in orderto check whether the results would be the same, regardlessof the decision-making opinion.

The observations were considered from a set of 8 possibil-ities as shown in Table 1. Thus, we have 256 possible deci-sion rules (2 actions with 8 established observations). Inorder to determine a general utility function to access thedecision risk, we have used just one parameter – r – inwhich the utility function considered was u(p) = pr.

Two values of r were used: r = 2 and r = 4. The higher val-ues of r reflect an increased risk of the decision-maker. Ifthe patient had to decide about the rejection or accept-ance, he or she would tend to be more adverse to risk. Wechose r = 2 as a conservative risk behavior simulation tothe patient choice, so that, the values of P(xi| j) = proba-bility values of the signal/noise ratio were relayed onimaging equipments and its components (see Table 1)and P(pi| j, ak) = once the nature condition of the chosendata 0(bad film) or 1(good film) regarding to a0 = to rejector a1 = to accept action (data not shown). The consequencefunction is defined as pi (payoffs), under probability ofP(pi| j, ak) (see Table 2).

The values shown in figures 2 and 3 are a simple plot ofthe ROC obtained from all possible Neyman-Pearsondecision rules results. The observation of those figuresdepicts the low risk level of the d16 decision rule. In otherwords, if the decisor-maker chooses d16, a low probabilitysituation (related to the risk) of rejecting a good film oraccepting a bad one would be the plausible decision. Thedecision rule d16 suggests that the decision-maker willrefuse all observations including the "inadequate foruse"(x0 - x3) parameter.

Simulations with the modelThe main purpose of the simulation is to analyze theinfluence of the utility function on the decision rule to

Table 1: The film usage, film sensibility and spatial resolution attributes provide the likelihood function (P(xi| j)) of the analyzed images related to the values dependency.

P(xi| j) x0 x1 x2 x3 x4 x5 x6 x7

0 0.41177 0.16807 0.14706 0.11345 0.07563 0.05042 0.02521 0.0084

1 0.00977 0.01303 0.03257 0.06515 0.09772 0.13029 0.16287 0.4886

These values correspond to an estimated distribution of the combined information obtained from radiologists and physicists involved in the quality control program.



accept or reject the film lot. There are several possibilitiesconcerning the emphasis and accuracy on various aspectsof the possible outcomes, contemplating the quality con-trol protocol. In particular, the characteristic of risk aver-sion was evaluated. The probability of x0 = (inadequate foruse, inadequate sensibility, inadequate spatial resolution)when ("states of nature"), 0 (bad film) is 0.41177 (worst

scenario) – case in which the report is incorrect and thepatient re-exposure will be necessary – the probability ofx0 is 0.00977, if ("states of nature") is 1 (good film). Onthe other hand, the probability of when ("states ofnature"), 0 (bad film) is 0.0084 makes the probability ofx7 be 0.4888 of 1 (good film).

Table 2: The probabilities are shown for each pair ( i, aj) of a consequence function.

P(xi| j, ak) p0 p1 p2 p3 p4 p5 p6 p7

( 0, a0) 0.150 0.110 0.070 0.050 0.300 0.200 0.080 0.040( 0, a1) 0.300 0.250 0.150 0.100 0.060 0.040 0.080 0.020( 1, a0) 0.100 0.060 0.100 0.150 0.100 0.050 0.250 0.190( 1, a1) 0.050 0.060 0.090 0.060 0.110 0.130 0.200 0.300

All values where calculated with the data obtained from table 1. The highest probability of a correct diagnosis with an increased cost and low client satisfaction was represented by P(p4| 0,a0) = 0.30.

The risk set for the case r = 2Figure 2The risk set for the case r = 2.Rd( 0) – Risk set for a bad film quality and Rd( 1) -Risk set for a good film quality. u(p) = p2 – Utility function to use just one parameter (r = 2).

RISK SETRISK SETRISK SETRISK SET



Table 2 demonstrates all considered consequence func-tions. The highest probability of a correct diagnosis withan increased cost and low client satisfaction was repre-sented by P(p4 | 0, a0) = 0.30 (Table 2).

Consider the utility function

and

. In order to calcu-

late the risk function Rd( 0) and Rd( 1) corresponding to

each utility function (polynomial degree) r, we used

and

, presented as Rd( 0) ×

Rd( 1) in figure 2, with r = 2 and in figure 3 with r = 4.

u a u p P p aor k k ork

( , ) ( ) ( | , )0 0 1 0 0 10

7

==∑

u a u p P p aor k k ork

( , ) ( ) ( | , )1 0 1 1 0 10

7

==∑

R P x u ad i ori

( ) ( | ) ( , )0 0 0 0 10

7= −

=∑

R P x u ad i ori

( ) ( | ) ( , )1 1 1 0 10

7= −

=∑

The risk set when r = 4Figure 3The risk set when r = 4.Rd( 0) – Risk set for a bad film quality and Rd( 1) – Risk set for a good film quality. u(p) = p4 – Utility function to use just one parameter (r = 4).

RISK SETRISK SETRISK SETRISK SET

Table 3: Representation of the results accomplished with the Neyman-Pearson decision rules for two values of r.

r Regra x0 x1 x2 x3 x4 x5 x6 x7 Rd( 0) Rd( 1)

2 d16 a0 a0 a0 a0 a1 a1 a1 a1 -0.31 -0.442 d192 a1 a0 a1 a1 a1 a1 a1 a1 -0.18 -0.444 d16 a0 a0 a0 a0 a1 a1 a1 a1 -0.18 -0.364 d192 a1 a0 a1 a1 a1 a1 a1 a1 -0.10 -0.37

Results of the simulation for a parameter (r) to a determined decision rule (d); Rd( 0) – Risk set for a bad film quality; Rd( 1) – Risk set for a good film quality.



Tables 1 and 2 correspond to the likelihood and conse-quence functions, respectively. In addition, Table 3 repre-sents the results obtained with the Neyman-Pearsondecision rules for two values of r. Simulated values wereused to calculate the risks from decision rules for eachobservation and also to determine a ROC analysis for eachrisk set.

In figure 3, the decision rule d16 corresponds to the vertexof the curve when the risk Rd( 0) is set to r = 4. This rulesays that if the observation was inadequate for use, thefilm must be rejected. All other options were accepted (seeTable 3). This rule corresponds to Rd( 0) = -0.18 andRd( 1) = -0.36. When r = 2 for the same value, we haveRd( 0) = -0.18 and the minimum value in the curve corre-sponds to d192 (Rd( 1) = -0.44). This rule expresses that theonly rejection condition for the film is when usage andsensibility are inadequate and spatial resolution is ade-quate (Figure 2).

However, when r = 4 (d192) the risk of accepting a bad filmis increased (Rd( 0) = -0.18) and the risk of accepting agood film increases as well (compared to d16). This rulewas not the ideal one since it rejects the finest option: ade-quate for use, sensibility and spatial resolution.

DiscussionThe problem of accepting or rejecting a radiographicimage based on the decision theory method was modeledhere. The decision theory was applied on a hypotheticalmodel. The ROC curve axis represent risks levels based ona preference concept or choice determined by the educ-tion of the utility function von Neumann-Morgenstern –this curve represents how a set of decision rules behaverelated to the risk of a consequence set. The ROC analysisfor each risk set is a graphic representation, common toradiologists, and provides a manner to decide how a deci-sion rule could be used, according to the false positive orsensitivity rate (Neyman-Pearson decision rule [14]).

In modern radiological services, it is extremely difficult tomanage quality control processes. The mathematicalapproach can help, dealing with complex decisions suchas reject or accept a radiographic image. The main resultobtained in this modeling exercise was provided by a deci-sion rules set that minimizes the risk of an undesired con-sequence. In addition, the solution proposed at this pointcan be applied in a practical radiology routine.

Residents and non-professionals staff of the radiologyfield have to learn how to make decisions, generally fromgathering information from other more experienced pro-fessionals and from literature information. Decision The-ory offers a way to arrange decisions in a rational manner,thus, avoiding many risks that may affect the patient

health quality. If one radiologist considers an imageappropriate for radiological appraisal, many possibilitiesmay be influenced by this decision. Besides, the level ofconsequences depends on the type of the clinical ques-tion. If the radiography is performed to diagnose a brokenarm, the presence of an insignificant spot in the image isgenerally not relevant as in the case of a slight spot inmammography exam performed for tumoral screening.The presence of micro-calcifications with specific mor-phological aspects has a direct impact on the next step,which may include or not additional invasive procedures.Therefore, the probability of a correct diagnosis could bedirectly affected by the quality of a simple X-ray image.

The expense of an image production is increased if a filmdoes not have a good quality. X-ray re-exposure will beinevitable and the patient will certainly be unsatisfied atthis point. This behavior variable should not be mis-guided with a simple level of comfort. The client content-ment must be contemplated, not only by the correctdiagnosis, but also by the number of necessary radio-graphic trials in the service. As a presumable consequence,the service could lose a representative number of clients,which would represent a substantial loss in profits.

All radiology services have an administrative design forquality control (QC) of the radiographic film. This isemphasized by the ICRU and is generally based on goldstandard guidelines and limits of technical parametersamong the used values to access the film quality.Although there are similar procedures reported in the lit-erature [15], none of them is based on a formal quantita-tive model. Generally, the result of a QC program ismeasured by the number of exams correctly performed,costs involved and the patient's satisfaction. These varia-bles are not easily measured beyond descriptive statistics.The decision is commonly not settle on a formal quanti-tative model, but in personal experience and historicalbackground of the institution. The management team isresponsible to decide upon the equipment choices variety,imaging procedures, questionnaires and personal trainingprograms.

The use of Bayesian statistics allows one to give a closerlook at the real based management information[14,16,17]. Historical information was considered andused for a prior distribution of the probabilities as a meanto avoid fake or unlikely occurrences [18,19]. Thisapproach can be applied in distinct radiological scenarios.This concept or perception adapted to other settings ofvariables could include another image modality, clinicalcontext and/or strategies for a perspective research of deci-sions or administrative managing.



The main challenge concerning the decision theory imple-ment is to define the state of nature (good or bad qualityof the radiographic film). Therefore, a given action canlead to a favorable payoff of the decision-maker whateverthe state of nature. If the state of nature is a good one, thepreferred result would be more likely to occur. Even so, adetermined action for a situation could probably unfoldas a complete disaster in the case of a very unfavorablepayoff, while the true state of nature is completely differ-ent or wrong. Note that in two other cases, still with a lesssignificant extent, positive probabilities have appeared.The consequence of an action must be establishedthrough a numeric scale according to the decision-makerpreferences [13].

Possible consequences were summarized based on thequality of an image in three distinct aspects: diagnosis(correct or incorrect), cost (high or low) and the patientsatisfaction (high or low). In summary, the proposeddecision problem is: having one of the eight possibilities(x) what decision should be made regarding the value of( ) so that it can be considered true? The person responsi-ble for the radiology department or the radiology manage-ment team (usually the decision-maker) will adopt acriterion for the decision process in order to guide thechoice that will minimize the risk of having an undesiredconsequence. The choice of the vector x (a set of observa-tions) and of the vector (a set of "states of nature") willdepend on the considered precision degree as well as onthe reliable technology. The consequence function P(p | ,a) was estimated from the data obtained from the experi-ments or from a historical service database. It includesmore probabilistic mechanisms than those found in afilm. The utility function's eduction was not determinedby any decision-maker. It is important to mention that theproposal of probabilistic mechanism in the likelihoodfunction can be adjusted to any information provided. Asa consequence the data consistency will depend on thesource confidence. A consequence (p – a set of payoff), asdescribed in the results, was related to the set of payoffsinvolved in a right or wrong diagnosis, a higher or lowercost and higher or lower client satisfaction, independentlyof the technology or of the radiologist that made the diag-nosis. Each pair ( , a) has a specific risk correlation to thepreferences of the decision-maker. The decision-makerpreferences, per se, were dictated by a behavior parameterin a risky circumstance. Decision theory application esti-mates the risks for all possible rational choices underuncertainty and allows a rational decision for all com-puted information during a specific procedure (Bayesiantheory).

Again, the choice of the best decision rule can be found atthe vertex region of the curve in the risk set Rd( 0) with r =4 which is a result of the 256 possible combinations, as

described previously. The corresponding graphic (Figure3) information corresponds to d16, which means that theobservation was inadequate for use (bad film), and thefilm was rejected (Table 3). The decision-maker optionwas to accept it. When adopting the criterion mentionedabove, we found that the minimum risk combination ofRd( 0) × Rd( 1) corresponds to (Figure 3) Rd( 0) = -0.18when the state of nature of the film was bad ( 0), and therisk of the film was considered good ( 1) with r = 4 corre-sponding to Rd( 1) = 0.36. In the case of r = 2, if the deci-sion-maker had chosen the same value Rd( 0) = -0.18, thedecision rule would be different: the film is rejected whenthe state of nature and its technical quality are bad -d192.Thus, the minimum risk combination Rd( 0) × Rd( 1) isnot the best decision rule for this case. This model can berefined by constructing p, , x, values which may assumeany mathematical (finite or infinite, discrete or continu-ous, scale or vectorial) representations. There are norestrictions for the current adopted classification.

ConclusionThe proposed model allows a cost-utility analysis of theradiographic film quality control. The decision theoryaddressed to this quality control issues established a sys-tematic procedure, a well defined protocol approach andsuggested a better use of the available imaging technolo-gies acquisitions. Moreover, depending on the utilityfunction, it allows us to analyze how the patient choicesor the radiographic service decision-making preferencesaffect the decision rule. It constitutes a paradigm not onlyfor a film quality control, but also for the radiographicservice quality control in an integrated framework.

Competing interestsThe authors declare that they have no competing interests.

Authors' contributionsPSL developed this work as a part of her Ph.D. thesis; shedesigned the study, analyzed the data and is responsiblefor the manuscript preparation. CAC and PA contributedto ideas, to the organization and to the writing of themanuscript with PSL. EAJ contributed to the discussionfor application in radiological images and to the overallmanuscript preparation. FMCS was PSL's supervisor andmentor during her PhD program, suggesting the subject ofuse of image analysis in radiology in the context of deci-sion making theory.

AcknowledgementsThe first author was sponsored by a scholarship from Fundação de Amparo à Ciência e Tecnologia do Estado de Pernambuco (FACEPE).

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Pre-publication historyThe pre-publication history for this paper can be accessedhere:

http://www.biomedcentral.com/1472-6947/8/51/prepub

AbstractBackgroundMethodsResultsConclusion

BackgroundMethodsResultsThe decision rulesSimulations with the model

DiscussionConclusionCompeting interestsAuthors' contributionsAcknowledgementsReferencesPre-publication history

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