1 INTRODUCTION

The current availability of innovative logical-mathematical algorithms derived from artificial intelligence (AI) has inspired the development of highly detailed prediction models in disparate fields, including ecology, economy, aerospace and biomedicine.1, 2 One of the basic outcomes of AI applied to medicine and biostatistics is the automatic modelling of underlying processes that generate data and allow for the implementation of decision support systems that attempt to anticipate the future health behaviour of patients.3 While data from the past contain information that can be useful in estimating the future, a classic problem in AI is how to extract general universal clinical rules or guidelines on which the patient's condition depends.4, 5

In the specialty of orthodontics, general rules about the progression of skeletal disharmonies often are difficult to formalize, with the challenge unmanageably large and complex. During continued growth and/or treatment, the dentofacial system is put ‘out of balance’ repeatedly. Craniofacial growth is in a state governed by non-linear, non-predictive laws of cumulative occlusal trauma, adaptability, competition between tooth elements for space and dentoalveolar optimization.6-10

When the principles underlying a domain are not well-understood, the rules governing that domain will be imperfect. When the rules that cause the progression of the system themselves become more apparent, disciplines become more complicated and demand complex knowledge structures, such as moving averages and temporal abstraction.4, 5, 11 In these situations, the solution suggested by the specifics of the individual patient may be more accurate than those suggested by a montage of general clinical rules. The individual patient synthesizes and better reflects what really happens in each set of clinical circumstances.11

Case-based reasoning (CBR) offers a novel approach to this issue, providing an opportunity to fill the knowledge gap between the specifics of a single patient and general clinical guidelines or rules. The possibility of drawing conclusions from personalized data allows the operator to reason by specific circumstances and episodes, making it unnecessary for the clinician to decompose his or her patient experiences and generalize individual patient findings into rules.12-15

The theoretical foundations and basic inference mechanisms underlying CBR reside in the concept of similarity, and more particularly on the idea that situations recognized as similar in important characteristics may be similar in other characteristics as well.16-20

It is known that after the completion of a Class III treatment that resulted in a skeletal correction, with continued and unfavourable mandibular growth the malocclusion may worsen, leading to relapse of the corrected incisor relationships, and the reappearance of reverse overjet.21 Thus, early evaluation and approximation of future growth characteristics of these subjects are of utmost importance.

The current investigation evaluates evidence of the ability of CBR models to extract additional prognostic meaningful information from cephalometric data of growing juveniles and adolescents affected by Class III malocclusion, and to determine the ability of the CBR model to approximate future growth characteristics based on estimates of similar patients with known negative growth characteristics.

2 MATERIALS AND METHODS 2.1 Subjects

The sample consisted of semi-longitudinal cephalometric data of 104 Caucasian subjects with untreated Class III malocclusion (56 females, 48 males, mean age at T1 9.4+-3.6 years, range from 6.8 to 20.1 years) collected from the Department of Orthodontics of the University of Florence and from the Graduate Orthodontic Program at the University of Michigan. The same subjects were re-evaluated a second time at T2 (mean age at 12.6+-3.6 years, range from 7.1 to 20.3 years). These subjects were left untreated because they declined treatment or because their cephalometric records were derived from historical samples taken from Growth Center Studies conducted in the United States. These subjects were derived from a database of 144 Class III patients followed longitudinally.24 Of these, 40 subjects were discarded because the time span between T1 and T2 measurements was less than 1 year and 6 months. Although the large difference in age at T1 could have led to the recommendation not to eliminate any subject from the learning set, we felt that a too short interval between T1 and T2 would not allow a reliable judgement on the actual quality of the facial development.

In order to find examples of Class III subjects of different age and gender, and in an attempt to establish possible similarities between longitudinal and cross-sectional subjects, we collected a population of 1263 Class III cross-sectional subjects (7-17 years of age) obtained from the same departments. Within this population, we collected subjects of maximum and minimum Class III horizontal clinical imbalance, for each age and gender (‘prototypes’). In order to provide clinical and algorithmic simplicity, the model was learned only in the horizontal dimension of skeletal imbalance, as expressed by the Wits appraisal. Six examples of female and male prototypes with the worst and the best Wits appraisal at 7, 11 and 17 years of age are reported in Table 3. Subjects who met criteria of maximum and minimum skeletal imbalance, calculated from the better and worse Wits appraisal for each age and gender, were considered prototypes. Cross-sectional and longitudinal subjects were enrolled previously in estimates of craniofacial growth in subjects with Class III malocclusion.22-24

To be included in the current study, both longitudinal and cross-sectional subjects had to satisfy the following criteria: Caucasian ancestry, no orthodontic/orthopaedic treatment prior to the initial cephalogram, no craniofacial syndromes, no congenitally missing or extracted teeth, diagnosis of Class III malocclusion based on accentuated mesial step relationship of the primary second molars, permanent first molar relationship of at least one-half cusp Class III, a negative Wits appraisal (<−2 mm) and ANB angle less than 0 degree.

2.2 Cephalometric analysis

A cephalometric analysis comprising 15 variables was performed (Table 1). A standardized enlargement factor of 8% was applied to all linear cephalometric measurements.

TABLE 1. Cephalometric variables S-N (mm) Anterior cranial base SNA (deg.) Antero-posterior position of the maxilla to the anterior cranial base SNB (deg.) Antero-posterior position of the mandible to the anterior cranial base ANB (deg.) Angle between points A and B Wits (mm) Wits appraisal (distance between the projections of points A and B along the functional occlusal plane) SN-PP (deg.) Palatal plane to anterior cranial base angle PP-MP (deg.) Palatal plane to mandibular plane angle ArGoMe (deg.) Gonial angle (Articulare-Gonion-Menton) Co-A (mm) Maxillary length from condylion to point A Co-Gn (mm) Total mandibular length from condylion to gnathion Co-Go (mm) Length of the mandibular ramus from condylion to gonion N-Me (mm) Total anterior face height Overjet (mm) Distance measured along the occlusal plane from the incisal edge of the maxillary central incisor to the most facial aspect in the incisal third of the mandible central incisor Overbite (mm) Vertical distance between incisal edges of the maxillary and mandibular central incisors U1-PP (deg.) Axis of the upper central incisor to the palatal plane

The first step in the protocol was to delineate appropriate subsets of patients among the untreated Class III subjects followed longitudinally. Among the available 104 longitudinal subjects, 14 gave evidence of craniofacial growth that tended to become substantially worse with respect to the following 3 criteria of maxillomandibular imbalance (very serious growing subjects, VS). The remaining 90 subjects experienced the usual slight worsening typical of the Class III malocclusion (mild subjects, M).

The criteria that distinguished VS from M patients were as follows: The worsening of ANB angle >−0.35 degrees/y. The worsening of the Wits appraisal >−0.4 mm/y. Co-Gn (T2-T1)/Co-A (T2-T1) >=1.30

Subjects that fulfilled these three requirements were considered VS. The average time between T1 and T2 was 2.9 years for VS subjects, and 3.1 years for M subjects.

2.3 Method error

The method error for the cephalometric measurements was evaluated by repeating the measures in 30 randomly selected cephalograms (Dahlberg's formula). Error was on average 0.8 degrees for angular measures and 0.9 mm for linear measures. The current study was exempted from review by the Medical School Institutional Review Board of the University of Michigan (HUM00143467).

2.4 Estimation of future craniofacial growth

For the prediction of the quality of growth and the feature importance in the logistic regression, the following 4 parameters were added to the panel of 15 variables listed in Table 1: age (years/months), gender, min_dist and max_dist. These two new variables (max_dist and min_dist) were created for each subject. These variables were computed as the Mahalanobis distance between each subject and the corresponding prototypes of maximum and minimum skeletal imbalance (as calculated from the better and worse Wits appraisal between 1263 cross-sectional data). The Mahalanobis distance is a distance function like the better-known Euclidean distance, but it is more suited from a mathematical point of view to measure the distance between points in high-dimensional spaces and/or between points with coordinates with different unit of measurement. It is also less prone to be influenced by outliers in some coordinates (2,18). The obtained Mahalanobis distances are used to define and codify the proximity to the most imbalanced and least imbalanced Class III subjects in the cross-sectional data.

The following 15 variables (Table 1) were added to max_dist and min_dist for the distance calculation: S-N, SNA, SNB, ArGoMe, Co-Gn, Co-Go, ANB, Co-A, Wits appraisal, N-Me, overjet (OJ), overbite (OB), PP-SN, PP-MP and U1-PP.

The fitted logistic regression, with the new variables given by the Mahalanobis distances to the prototypes incorporating the knowledge obtained through the 1263 cross-sectional subjects, allows to forecast the risk of adverse growth. An analysis of the obtained model can be found in the Results section.

2.5 Case-based reasoning

Case-based reasoning can be used to best characterize the information that the individual case contains, to define the representation of that information and to select effective useful information from the available data.17, 18 The first step in CBR is to determine which patients are similar and which features of the current patients are relevant.

In the present study, a prototype can be defined as an ordered set of clinical and/or cephalometric entities representing the typical signs of the severity of a given malocclusion. As mentioned above, we evaluated a data base of 1263 cross-sectional untreated Class III subjects derived from the same centres collecting patients followed over time (Table 2) in a search for the most appropriate prototypes matched by age and gender. From 7 to 17 years of age, we picked the maximally skeletal imbalanced patients (for the sake of simplicity, for each age and gender, the patient with the worst Wits appraisal score) and the least imbalanced patients (the patient with the best Wits appraisal score, Table 3).

TABLE 2. Descriptive statistics of means and standard deviations of cephalometric values from 1263 cross-sectional Class III subjects between 7 and 17 y 7 y (n = 114) 8 y (n = 183) 9 y (n = 160) 10 y (n = 116) 11 y (n = 90) 12 y (n = 114) 13 y (n = 119) 14 y (n = 104) 15 y (n = 78) 16 y (n = 48) 17 y (n = 137) Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD ANB 0.8 2.2 0.7 2.2 0.5 2.2 0.2 2.3 0.1 2.2 0.2 2.5 0.0 2.7 −0.1 2.4 −0.9 3.1 −0.9 3.2 −1.1 3.0 ArGoMe 130.5 5.4 129.7 6.0 129.4 6.1 130.0 6.2 129.9 5.0 129.9 6.8 129.2 6.3 128.5 7.1 126.8 7.0 129.0 7.3 128.6 7.4 Co-A 80.5 4.6 82.9 4.9 83.3 4.7 84.8 5.2 86.0 5.7 88.2 6.4 88.7 6.0 90.8 5.6 90.1 6.7 93.1 6.8 93.6 6.7 Co-Gn 104.1 6.2 107.8 6.2 109.0 6.4 112.1 6.7 115.6 6.7 119.0 7.8 120.9 7.4 125.0 6.6 125.6 7.7 129.3 8.2 133.4 10.3 Co-Go 46.3 4.0 48.6 4.2 49.1 4.4 50.0 4.9 52.3 5.1 54.2 6.3 55.4 5.6 57.7 5.7 59.3 6.0 61.4 6.7 63.9 7.5 N-Me 103.9 6.5 108.2 6.8 109.6 7.1 111.8 6.6 116.1 7.8 118.7 8.5 121.1 8.6 125.2 8.8 123.7 8.4 127.1 9.8 132.3 10.6 OB 0.2 1.9 1.0 1.9 1.2 2.1 1.2 1.9 1.3 1.9 1.6 1.8 1.1 1.7 1.1 1.9 1.5 2.0 0.7 1.8 1.0 2.0 OJ −0.6 1.7 −0.5 1.8 −0.4 1.9 0.0 1.8 −0.1 2.0 0.5 2.2 0.6 2.2 0.5 2.6 0.2 2.8 −0.1 3.0 −0.8 3.1 PP-MP 26.9 4.7 26.7 5.1 26.6 4.9 26.4 5.0 27.7 5.7 26.9 6.0 26.7 4.8 26.6 5.6 25.1 6.0 25.1 6.8 25.0 6.3 PP-SN 8.1 3.6 8.3 3.3 8.3 3.1 8.0 3.0 8.4 3.0 8.9 3.6 8.8 2.9 8.7 3.6 8.4 3.3 8.7 3.7 9.4 4.0 SNA 80.2 3.5 79.7 3.6 80.0 3.4 80.6 4.1 80.0 3.9 80.4 4.0 80.6 3.5 80.7 4.2 80.8 4.2 80.9 3.9 80.6 4.0 SNB 79.4 3.4 79.0 3.5 79.5 3.3 80.4 3.6 79.9 3.4 80.2 3.7 80.6 3.1 80.9 4.3 81.7 4.1 81.8 4.3 81.7 4.1 U1-PP 104.3 5.6 104.9 6.2 106.2 6.7 108.7 6.3 108.6 6.6 110.6 6.8 114.5 7.6 114.7 7.2 116.3 7.0 116.1 6.8 118.3 7.1 Wits −4.4 2.1 −4.5 3.0 −4.7 2.5 −4.8 2.4 −5.3 2.2 −5.1 3.1 −5.3 3.1 −5.2 3.4 −5.5 4.0 −5.0 4.5 −6.2 4.6 TABLE 3. Prototypes. First row: maximum imbalancement. Second row: minimum imbalancement ANB ArGoMe Co-A Co-Gn Co-Go N-Me OB OJ PP-MP PP-SN S-N SNA SNB U1-PP Wits Female 7 y of age 0.0 131.3 82.8 111.2 49.6 110.7 0.4 −0.1 28.8 7.5 68.9 79.7 79.6 110.0 −6.6 2.5 133.0 83.0 101.0 44.4 103.0 0.6 0.9 24.6 11.5 67.0 78.9 76.4 104.3 −1.2 Female 11 y of age −0.1 132.0 84.0 115.6 52.0 117.4 1.3 0.1 29.3 7.6 69.0 79.7 79.8 111.6 −8.7 0.41 131.0 90.4 90.6 53.8 118.1 −2.4 2.0 27.0 8.5 72.2 80.7 80.3 117.0 −2.1 Female 17 y of age −1.7 125.9 90.4 127.5 60.5 122.1 1.9 −2.1 22.2 9.2 72.8 82.2 84.5 115.0 −12.3 −0.6 128.2 92.0 129.3 60.1 129.8 0.6 0.0 26.0 8.6 74.0