Deep Learning Neural Modelling as a Precise Method in the Assessment of the Chronological Age of Children and Adolescents Using Tooth and Bone Parameters

Dental age is one of the most reliable methods for determining a patient’s age. The timing of teething, the period of tooth replacement, or the degree of tooth attrition is an important diagnostic factor in the assessment of an individual’s developmental age. It is used in orthodontics, pediatric dentistry, endocrinology, forensic medicine, and pathomorphology, but also in scenarios regarding international adoptions and illegal immigrants. The methods used to date are time-consuming and not very precise. For this reason, artificial intelligence methods are increasingly used to estimate the age of a patient. The present work is a continuation of the work of Zaborowicz et al. In the presented research, a set of 21 original indicators was used to create deep neural network models. The aim of this study was to verify the ability to generate a more accurate deep neural network model compared to models produced previously. The quality parameters of the produced models were as follows. The MAE error of the produced models, depending on the learning set used, was between 2.34 and 4.61 months, while the RMSE error was between 5.58 and 7.49 months. The correlation coefficient R2 ranged from 0.92 to 0.96.


Introduction
Dental age is one of the most reliable methods for determining the maturity of an organism [1]. It is extremely useful in areas such as orthodontics, pediatric dentistry, endocrinology, anthropology, or forensic medicine [2][3][4][5][6][7][8][9]. It allows us to determine whether the body is developing properly and when a pubertal growth spurt occurs. Moreover, the dental age assessment can be used to determine the age of individuals without identification documents or those suspected of having falsified documents, with memory loss, illegal immigrants, or international adoptions [10,11].
Age determination using pantomographic radiographs is an easy, widely available, and low-cost method. In children, the developmental stages of tooth buds, mineralization of crowns and roots, and the eruption stages of teeth can be assessed [12][13][14]. In the elderly, changes in the dentition are not very noticeable, thus age assessment is much more difficult. However, it is possible to take advantage of the fact that, with age, odontoblasts deposit more and more secondary dentin, causing a reduction in pulp chamber volume. Methods that analyze the alveolar bone level have also been described [15][16][17].
The commonly used methods to determine dental age, such as Demirjian's method, Schour and Massler's method, Ubelaker's method, Moorres', Fanning and Hunt's method, Noll's method, or Gustafson and Koch's method, are methods developed in the previous century [13,[18][19][20][21][22]. The phenomenon of acceleration, or growth spurt, occurring in the the metric age with a quality for the test set of 0.997394 and an error for the test set of 0.036526. On the contrary, the model containing cases of women only had a quality for the test set of 0.963090 and an error for the test set of 0.033634, while the test quality of the model determining the metric age of men was 0.999342 and the error for the test set was 0.039840.
Artificial neural network is an information processing system whose structure and operating principle resemble the information processing system in a human neuron. It is on biological inspiration that artificial neuron schemes and structure are based.
Currently, neural modeling is a method widely used by scientists and in industry. Neural networks are a computer tool that can solve complex problems without prior mathematical formalization.
Neural modelling is very popular method in the biological and medical community [52]. It can be used in many diagnostic aspects [53][54][55][56][57][58][59][60][61][62][63]. Increasingly, deep learning methods are being used to solve scientific problems. One simulator of deep neural networks is the H 2 O program [64][65][66]. The H 2 O software can be obtained for free from the H2O.ai website and used in accordance with the license. The project itself is Open Source. The application can be used via a web browser, e.g., on a local computer where H 2 O simulator is running. In this study, H 2 O simulator and Deep Learning method were used to generate new neural networks determining the metric age of children from 4 to 15 years old. The aim of this study is to check the possibility of creating accurate (as low as possible MAE and RMSE error, high R 2 coefficient) models, which would allow to quickly and effectively determine the metric age of the examined patients on the basis of the provided data.
The present work is a continuation of the work of Zaborowicz et al. In the presented research, a set of 21 original indicators was used to create deep neural network models. The aim of this study was to verify the ability to generate a more accurate deep neural network model compared to models produced previously.
Ethical Statements: The Bioethics Committee of the Medical University of Poznań considered that the research carried out does not have the characteristics of a medical experiment and therefore agreed to carry out the relevant work.

Research Material and Methodology
The source of the analyzed data was the database of patients (children and adolescents aged from 48 to 144 months) of the University Centre of Dentistry and Specialist Medicine in Poznań, Poland. The research material consisted of 619 digital pantomographic images (296 photos of girls and 323 photos of boys). All analyzed cases were verified, and photographs which presented abnormalities or developmental disorders were excluded. Additionally, it should be added that experiments were not performed on children. The Bioethics Committee of the Medical University of Poznań considered that the research carried out does not have the characteristics of a medical experiment and therefore agreed to carry out the relevant work.
The following research methodology was used in this study: 1. Acquisition of research material-pantomographic images of children and adolescents aged 4 to 15 (from 48 to 144 months); 2.
Verification and exclusion of abnormal cases and preparation of a database of selected digital pantomographic images; 3.
Determination of patients' age at the moment of picture taking, expressed in months; 4.
Determination of a set of tooth and bone parameters; 5.
Collection of tooth and bone parameters using ImageJ software; 6.
Definition of a set of indicators, i.e., values of proportions of measured tooth and bone parameters; 7.
Preparation of a learning set for neural modelling; 8.
Verification of the produced models; 10. Comparison of models with models produced in STATISTICA 7.1 simulator.

Methodology for Obtaining Empirical Data-New Tooth and Bone Indicators
In the conducted research, an original and authored set of 21 indicators was used, i.e., distinctive tooth and bone parameters, which were developed in the form of mathematical proportions X01-X21 by Zaborowicz [51] (Figure 1).

Methodology for Obtaining Empirical Data-New Tooth and Bone Indicators
In the conducted research, an original and authored set of 21 indicators was used, i.e., distinctive tooth and bone parameters, which were developed in the form of mathematical proportions X01-X21 by Zaborowicz [51] (Figure 1).

Research Methods
The pantomographic photos used in the research were taken with the Duerr Dental-VistaPano S Ceph camera which was equipped with an X-ray head with 0.5 mm focus and a digital sensor, Cls-CMOS matrix in DICOM 3.0 format supported by DBSWIN [67]. The measurements of tooth and bone parameters were performed in Open Source software ImageJ 1.52a [68]. Additionally, MS Excel 2007 spreadsheet was used to aggregate and structure the data obtained in the process of image processing and analysis, which also enables saving the data in *.csv format [69].
The process of generating a neural model was carried out using H2O.ai. software (version 3.24.0.5) with Deep Learning methods, which allows us to create, validate, and predict artificial neural network models. In this software, it is also possible to perform a sensitivity analysis of variables of the developed models [64][65][66]. Deep learning is a class of machine learning methods for hierarchical (deep) models with nonlinear layers [70]. The idea of deep learning is to pretrain the network, and in the next step to train the network in a supervised manner-this method can combine supervised and unsupervised learning. In order to carry out the learning process properly, a large dataset is usually required; however, this is not necessary due to the deep neural network's performance, which has the ability to redundancy. In brief, it can be said that the network "breaks" data

Research Methods
The pantomographic photos used in the research were taken with the Duerr Dental-VistaPano S Ceph camera which was equipped with an X-ray head with 0.5 mm focus and a digital sensor, Cls-CMOS matrix in DICOM 3.0 format supported by DBSWIN [67]. The measurements of tooth and bone parameters were performed in Open Source software ImageJ 1.52a [68]. Additionally, MS Excel 2007 spreadsheet was used to aggregate and structure the data obtained in the process of image processing and analysis, which also enables saving the data in *.csv format [69].
The process of generating a neural model was carried out using H 2 O.ai. software (version 3.24.0.5) with Deep Learning methods, which allows us to create, validate, and predict artificial neural network models. In this software, it is also possible to perform a sensitivity analysis of variables of the developed models [64][65][66]. Deep learning is a class of machine learning methods for hierarchical (deep) models with nonlinear layers [70]. The idea of deep learning is to pretrain the network, and in the next step to train the network in a supervised manner-this method can combine supervised and unsupervised learning. In order to carry out the learning process properly, a large dataset is usually required; however, this is not necessary due to the deep neural network's performance, which has the ability to redundancy. In brief, it can be said that the network "breaks" data into smaller parts and, on the basis of these smallest elements, aims to generalize the processed information.

Results
Three deep neural network models were generated during the study: one for the learning set of women and men, and one each for the learning set of women and the learning set of men. During the modeling process, all 21 new indicators and the gender indicator were used [51]. After each model was generated, predictions were made for each entire learning set. The learning set of women and men contained 619 samples; the learning set of women contained 296 samples; and the learning set of men contained 323 samples. A sensitivity analysis of the variables was also conducted for each of the models that were generated.
The models were characterized by the following parameters: MSE (Mean Squared Error) Equation (1)

Model to Determine Metric Age for Men and Women
The parameters representing the quality of the generated models for the learning set of male and female are presented in Table 1. This means that the mean MAE prediction error was 4.61 months. Additionally, MAPE and RMSPE parameters were calculated, respectively, as 4.10% and 6.36%.
The network learning process is in Figure 2. The sensitivity analysis is shown in Table 2 and Figure 3.
The network learning process is in Figure 2. The sensitivity analysis is shown in Table  2 and Figure 3.

Model to Determine Metric Age for Women
The parameters representing the quality of the generated models for the learning set of male and female are presented in Table 3. Table 3. Parameters of the generated model-age assessment for women. This means that the mean MAE prediction error was 3.85 months. Additionally, MAPE and RMSPE parameters were calculated, were, respectively: 3.48% and 6.86%.

Output-Training Metrics Output-Validation Metrics
The network learning process is in Figure 4. The sensitivity analysis is shown in Table  4 and Figure 5.

Model to Determine Metric Age for Women
The parameters representing the quality of the generated models for the learning set of male and female are presented in Table 3. This means that the mean MAE prediction error was 3.85 months. Additionally, MAPE and RMSPE parameters were calculated, were, respectively: 3.48% and 6.86%.
The network learning process is in Figure 4. The sensitivity analysis is shown in Table 4 and Figure 5.

Model to Determine Metric Age for Men
The parameters representing the quality of the generated models for the learning set of male and female are presented in Table 5.
This means that the mean MAE prediction error was 2.34 months. Additionally, MAPE and RMSPE parameters were calculated, were, respectively: 2.04% and 4.83%.
The network learning process is in Figure 6. The sensitivity analysis is shown in Table 6 and Figure 7.

Model to Determine Metric Age for Men
The parameters representing the quality of the generated models for the learning set of male and female are presented in Table 5. This means that the mean MAE prediction error was 2.34 months. Additionally, MAPE and RMSPE parameters were calculated, were, respectively: 2.04% and 4.83%.
The network learning process is in Figure 6. The sensitivity analysis is shown in Table  6 and Figure 7.

Discussion
The results obtained with the generated deep neural network models indicate the possibility of using this type of machine learning in solving such scientific problems. The network determining the metric age of boys had the lowest prediction errors. MSE error was 31.13, RMSE 5.58, and MAE 2.34. The MAE error means that, in this case, the metric age estimate for boys has an error of 2.34 months. The network assessing boys' age also had the highest R 2 coefficient. A detailed summary of the parameters is shown in Table 7. Table 7. Parameters of the generated models-prediction of age assessment.

Discussion
The results obtained with the generated deep neural network models indicate the possibility of using this type of machine learning in solving such scientific problems. The network determining the metric age of boys had the lowest prediction errors. MSE error was 31.13, RMSE 5.58, and MAE 2.34. The MAE error means that, in this case, the metric age estimate for boys has an error of 2.34 months. The network assessing boys' age also had the highest R 2 coefficient. A detailed summary of the parameters is shown in Table 7. It should be noted that the first stage of the study produced RBF (Radial Basis Function) networks and did not use all of the developed indicators. Both the first study and the current analysis show that the neural model generated from the learning set determining the tooth and bone parameters of men has a higher accuracy. There is greater inaccuracy in the model determining the metric age of women (Table 8). All prepared, original indicators were used to generate the models. None of the indicators had less than 0.5 significance. It should be noted that variable X02, X04, and X15 had a large variation compared to other indicators (Table 9). In the future, it is recommended to omit these variables from the network learning process. A summary and characterization of the indicators can be found in Table 10.  The models presented in the study are characterized by high accuracy. Compared with the work of Kim and co-authors [46], the quality of the model determining the age of men and women was 9 percentage points higher. The R 2 coefficient of the produced model was 0.93; Kim's model had a quality level of accuracy of 0.84. On the other hand, the difference between the accuracy of the model produced by Farhadian et al. [47] is much higher. The MAE error presented in this team's study was 4.12 years, while the RMSE error was 4.4 years. The error of the models produced in this work varies depending on the learning set within: MAE from 2.34 to 4.61 months, and RMSE error from 5.58 to 7.45 months. However, it is important to note the difference in the age range of the study subjects, which may have translated into network quality. In Farhadian's study, the range was between 14 and 60 years of age, whereas in the research presented here, the range was between 4 and 15 years. In turn, Banjšak et al. [48] used convolutional networks to estimate the age of found skulls. This team's model works with an accuracy of 73%. It should be noted that this team could not know the precise metrical age. Very high accuracy of the produced models was presented in their works by Milošević et al. [49] and Kahaki et al. [50]. However, despite the high values of the indicators defining the networks, the error was measured in years rather than individual months.
Compared to the work of our team [51], it can be seen that the quality of deep neural models is comparable, with an indication for deep learning methods. Table 11 shows the network quality and RMPSE error for each learning set. The neural model developed in this study is applicable to assess the metric age of only children and adolescents in the age range of 4-15 years. Pantomographic radiographs of patients without systemic diseases and with normal development of the dental buds were used for the study. All images of persons with root canal treatment or extensive fillings in their teeth were also excluded. This is a strong advantage from the point of view of network creation and function. However, from the point of view of diagnostics, the collection should take into account a whole range of cases including anomalies. In addition, the number of teaching cases should increase. The strengths of the paper are the fairly large scope of the dataset and the well-defined cases. The plus side of the research conducted is the use of proprietary indicators that allowed for the development of a new method and the use of neuronal modeling methods. Additionally, note that the artificial neural network simulator used is publicly available under an open license. On the other hand, the disadvantage of works comparing the effect of different technologies is the divergence of quality indicators-different simulators have different measures, and special attention should be given to this.

Conclusions
The conducted research indicates that neural modeling methods are an appropriate tool for determining the metric age based on the developed proprietary tooth and bone indices. The indicated issue of metric age assessment belongs to the area of medical, biological, and natural sciences and is a highly nonlinear problem. The MAE error of the produced models, depending on the learning set used, is between 2.34 and 4.61 months, while the RMSE error is between 5.58 and 7.49 months. The correlation coefficient R 2 ranges from 0.92 to 0.96. The produced deep neural models have higher quality already in the first iteration of learning the network using all the developed metrics. It is recommended to prepare deep neural networks based on the set of indicators used in the first stage of the research. ratio between section |C13C43| and section |C15C45| X02 ratio between section |C13C43| and section |C16C46| X03 ratio between section |C13C43| and section |C17C47| X04 ratio between section |C15C45| and section |C16C46| X05 ratio between section |C15C45| and section |C17C47| X06 ratio between section |C16C46| and section |C17C47| X07 ratio between section |C43A43| and section |P43A43| X08 ratio between section |C45A45| and section |P45A45| X09 ratio between section |C46A46| and section |P46A46| X10 ratio between section |C47A47| and section |P47A47| X11 ratio between section |CeM43CeD43| and section |PCeM43PCeD43| X12 ratio between section |CeM45CeD45| and section |PCeM45PCeD45| X13 ratio between section |CeM46CeD46| and section |PCeM46PCeD46| X14 ratio between section |CeM47CeD47| and section |PCeM47PCeD47| X15 ratio between section |C43M43| and section |A43M43| X16 ratio between section |C45M45| and section |A45M45| X17 ratio between section |C46M46| and section |A46M46| X18 ratio between section |C47M47| and section |A47M47| X19 ratio between section |A43M43| and section |A45M45| X20 ratio between section |A43M43| and section |A46M46| X21 ratio between section |A45M45| and section |A46M46|