Leveraging bone age assessment via a novel joint decomposition teacher–student learning paradigm from X-ray images

Figures & data

Table 1. Review of some recent BAA methods.

Reference	Dataset	Method	Advantages	Disadvantages
He and Jiang (2021)	Private – 12,611 Image	SE-ResNe	Presented an image compression module without loss of important features	The model database only included Chinese children
Spampinato et al. (2017)	Public-DHA − 1380 Image	CNN-BoNet	Tested several existing pre-trained CNNs on a dataset	No image preprocessing
Deshmukh and Khaparde (2022)	RSNA – 12,611 Image	FR-CNN, RNN, AF-SFO	Good performance by the proposed Otsu segmentation model and BAA model compared to other machine learning algorithms	The proposed model’s complexity for implementation and medical use
Salim and Ben Hamza (2021)	RSNA – 12,611 Image	Mask R-CNN and VGG-19	The proposed model also performed well on the chest X-ray dataset	Using hand annotation for detecting hand image bounds
An Ding et al. (2020)	Public-DHA − 1380 Image	VGG-19, GoogleNet AlexNet	Segmented images and tested multiple deep learning methods on the data to achieve better results	Lower accuracy than the proposed model with the same dataset
Lu et al. (2021)	DR – 1666 Image	AlexNet	Improved age detection in the [−1, 0, 1] age error interval by up to 96%	Limited by the large number of neural network parameters and low computational resources
Bui et al. (2019)	Public-DHA − 1380 Image	R-CNN and Inception-SVR	The model database includes children of four different races and improves accuracy	This classification method is limited to assessment of six small ROI segments
Chen et al. (2020)	Private – 12,536 Image	R-CNN and ST-Resnet	The proposed model is (ST-Res) based on Res-Net and spatial transformer	The database is specific to Chinese children. The model is limited by low computational resources and big data
Son et al. (2019)	Private – 12,536 Image	R-CNN and VGG-13	Better image segmentation with good accuracy	The model database was specific to Korean children. High computational burden due to the large number of parameters
Wang et al. (2023)	Private – 12,536 Image	Interpretable CNN	Ability to interpret deep architecture used for bone age detection	High computational complexity of building the proposed architecture
Liu et al. (2022)	Private – 12,536 Image	Ensemble CNNs	Applying the idea of collective intelligence to increase the accuracy of bone age diagnosis	The model database was specific to Korean children. High computational burden due to the large number of parameters
Deshmukh and Khaparde (2022)	Private – 12,536 Image	Faster R-CNN	Better image segmentation with good accuracy	No image preprocessing

Figure 1. Steps of six ROIs’ extraction from an X-ray image from DHA database.

Figure 2. Diagram of student’s learning procedure based on hybrid KD model.

Table

Algorithm 1: Student Model Training Procedure with Single ROI
1: $\mathbf{Input}\mathbf{:}\mathit{D}=\left\{\mathit{X},\mathit{Y}\right\}\in {\mathbb{R}}^{I_1\times I_2\times I_3}//\mathrm{single}\mathrm{ROI}$
2: $\mathrm{with}:$
3: $M_t//\mathrm{Pre}-\mathrm{trained}\mathrm{teacher}\mathrm{model}$
4: $\mathbf{TraintheStudentModel}\mathit{\hspace{0.17em}}{M}_s:$
5: for $\mathit{i}=1$ to len$(\mathit{X})$ do
6: $p^{\mathcal{T}}=\mathcal{T}(\mathit{X}[\mathit{i}])$
7: $p^{\mathcal{S}}=\mathcal{S}(\mathit{X}[\mathit{i}])$
8: Compute $L_{\mathit{Soft}}$ using Equation 5(5) $L_{\mathit{Soft}}=-\sum _{\mathit{c}=1}^N{p}_c^{T}log{p}_c^S$(5)
9: Compute $L_{\mathit{Res}}$ using Equation 4(4) $L_{\mathit{Res}}=\mathit{\alpha }{L}_{\mathit{Soft}}+(1-\mathit{\alpha })L_{\mathit{Hard}}$(4)
10: Choose four feature maps from teacher $(\mathcal{T})$ and student $(\mathcal{S})$
11: for $\mathit{e}=1$ to $4$ do
12: Convert feature map tensors from previous step to matrices by applying mode-n unfolding ${ }^{⟨\mathit{e}⟩}{\mathcal{T}}_{(\mathit{n})}$ and ${ }^{⟨\mathit{e}⟩}{\mathcal{S}}_{(\mathit{n})}$
13: Compute Euclidean norm $\left(L_2\left({ }^{⟨\mathit{e}⟩}{\mathcal{T}}_{(\mathit{n})}{,}^{⟨\mathit{e}⟩}{\mathcal{S}}_{(\mathit{n})}\right)\right)$ with Equation 18(18) $\begin{array}{c}\mathit{Los}{s}_{\mathit{IF}}=\sum _{\mathit{g}=1}^G{L}_2\left({ }^{⟨\mathit{g}⟩}\mathcal{T},{}^{⟨\mathit{g}⟩}\mathcal{S}\right)\end{array}$(18)
14: Calculate $L_{\mathit{dist},\mathit{e}}\left({ }^{⟨\mathit{e}⟩}{\mathcal{T}}_{(\mathit{n})}{,}^{⟨\mathit{e}⟩}{\mathcal{S}}_{(\mathit{n})}\right)$ according to Equation 17(17) ${ }^{〈\mathit{g}〉}{K}_{k_1,\cdots ,k_N}={\displaystyle \sum _{r_1=1}^{I_1}{\displaystyle \sum _{r_2=1}^{I_2}\cdots }}{\displaystyle \sum }_{r_N=1}^{I_N}〈\mathit{g}〉S_{k_1,\cdots ,k_N}{ }^{〈\mathit{g}〉}{D}_{\left(k_1,r_1\right)}^{\left(1\right)}〈\mathit{g}〉D_{\left(k_2,r_2\right)}^{\left(2\right)}{\cdots }^{〈\mathit{g}〉}{D}_{\left(k_N,r_N\right)}^{\left(\mathit{N}\right)}$(17)
15: end for
16: Compute $(L_{\mathit{Feat}})$ using Equation 13(13) ${ }^{⟨\mathit{g}⟩}\mathcal{J}{\cong }^{⟨\mathit{g}⟩}\mathcal{T}{{\times }_1}^{⟨\mathit{g}⟩}{\mathbf{V}}^{(1)}{{\times }_2}^{⟨\mathit{g}⟩}{\mathbf{V}}^{(2)}\cdots {{\times }_n}^{⟨\mathit{g}⟩}{\mathbf{V}}^{(\mathit{n})}\cdots {{\times }_N}^{⟨\mathit{g}⟩}{\mathbf{V}}^{(\mathit{N})}=\left[\left[{ }^{⟨\mathit{g}⟩}\mathcal{T}{,}^{⟨\mathit{g}⟩}{\mathbf{V}}^{(1)}{,}^{⟨\mathit{g}⟩}{\mathbf{V}}^{(2)}\cdots {,}^{⟨\mathit{g}⟩}{\mathbf{V}}^{(\mathit{n})},\cdots {,}^{⟨\mathit{g}⟩}{\mathbf{V}}^{(\mathit{N})}\right]\right]$(13)
17: Use Equation 3(3) $L_{\mathit{total}}=\mathit{\gamma }{L}_{\mathit{Feat}}+(1-\mathit{\gamma })L_{\mathit{Res}}$(3) to obtain total loss $(L_{\mathit{total}})$ for the student model $M_s$
18: Execute Backward propagation for $M_s$
19: end for

Table 2. Evaluation results of six students and the teacher model.

Model	ROI	Accuracy	Precision	Recall	F-Score	Specificity	MAE
Student model	Student-1	0.84 $\pm$ 0.00	0.53 $\pm$ 0.47	0.55 $\pm$ 0.00	0.53 $\pm$ 0.98	0.86 $\pm$ 0.25	0.034
	Student-2	0.85 $\pm$ 0.56	0.55 $\pm$ 0.67	0.54 $\pm$ 0.12	0.54 $\pm$ 0.50	0.90 $\pm$ 0.69	0.398
	Student-3	0.84 $\pm$ 0.58	0.54 $\pm$ 0.80	0.54 $\pm$ 0.76	0.54 $\pm$ 0.00	0.90 $\pm$ 0.84	0.345
	Student-4	0.81 $\pm$ 0.07	0.54 $\pm$ 0.14	0.55 $\pm$ 0.87	0.54 $\pm$ 0.50	0.88 $\pm$ 0.85	0.030
	Student-5	0.83 $\pm$ 0.96	0.55 $\pm$ 0.86	0.54 $\pm$ 0.87	0.54 $\pm$ 0.50	0.88 $\pm$ 0.47	0.031
	Student-6	0.95 $\pm$ 0.54	0.65 $\pm$ 0.01	0.65 $\pm$ 0.69	0.65 $\pm$ 0.00	0.87 $\pm$ 0.58	0.022
	Average	0.85 $\pm$ 0.45	0.56 $\pm$ 0.49	0.56 $\pm$ 0.55	0.56 $\pm$ 0.00	0.88 $\pm$ 0.61	0.143
Teacher model	All six ROIs	0.97 $\pm$ 0.03	0.76 $\pm$ 0.42	0.76 $\pm$ 0.54	0.76 $\pm$ 0.00	0.98 $\pm$ 0.87	0.013

Table 3. Comparison of the student-6 and the teacher regarding complexity and size of network.

Phase	Model	Number of parameters	Computation (GFLOPs)	GPU memory	Compression rate
Train	Student-6	$\sim$ 2.34 $\mathit{M}$	$\sim$ 0.43	0.08 $\mathit{G}$	$\times$124.34
	Teacher (All six ROIs)	$\sim$ 9.46 $\mathit{M}$	$\sim$ 2.57	1.43 $\mathit{G}$	$\times$32.01
Test	Student-6	$\sim$ 0.09 $\mathit{M}$	$\sim$ 0.01	0.01 $\mathit{G}$	$\times$66.12
	Teacher (All six ROIs)	$\sim$ 0.65 $\mathit{M}$	$\sim$ 0.12	0.12 $\mathit{G}$	$\times$3.09

Figure 3. The confusion matrices for age classes of (a) teacher model and (b) student-6 model.

Table 4. Performance of the student-6 model for all 19 bone age classes.

Category	Accuracy	Precision	Recall	F-Score	Specificity	MAE
Age 0	0.96 $\pm$ 0.36	0.34 $\pm$ 0.63	0.32 $\pm$ 0.28	0.32 $\pm$ 0.97	0.98 $\pm$ 0.79	0.011
Age 1	0.96 $\pm$ 0.68	0.62 $\pm$ 0.89	0.53 $\pm$ 0.69	0.57 $\pm$ 0.15	0.98 $\pm$ 0.94	0.110
Age 2	0.96 $\pm$ 0.59	0.60 $\pm$ 0.08	0.54 $\pm$ 0.55	0.56 $\pm$ 0.84	0.98 $\pm$ 0.68	0.009
Age 3	0.95 $\pm$ 0.78	0.57 $\pm$ 0.07	0.44 $\pm$ 0.39	0.49 $\pm$ 0.66	0.98 $\pm$ 0.13	0.012
Age 4	0.96 $\pm$ 0.36	0.60 $\pm$ 0.90	0.61 $\pm$ 0.61	0.60 $\pm$ 0.50	0.98 $\pm$ 0.72	0.014
Age 5	0.95 $\pm$ 0.20	0.66 $\pm$ 0.53	0.71 $\pm$ 0.78	0.68 $\pm$ 0.41	0.97 $\pm$ 0.11	0.009
Age 6	0.97 $\pm$ 0.08	0.72 $\pm$ 0.10	0.80 $\pm$ 0.33	0.75 $\pm$ 0.79	0.98 $\pm$ 0.11	0.009
Age 7	0.95 $\pm$ 0.77	0.63 $\pm$ 0.82	0.65 $\pm$ 0.60	0.63 $\pm$ 0.98	0.97 $\pm$ 0.64	0.009
Age 8	0.96 $\pm$ 0.20	0.71 $\pm$ 0.33	0.74 $\pm$ 0.74	0.72 $\pm$ 0.47	0.97 $\pm$ 0.32	0.015
Age 9	0.95 $\pm$ 0.38	0.67 $\pm$ 0.29	0.64 $\pm$ 0.10	0.65 $\pm$ 0.47	0.97 $\pm$ 0.65	0.017
Age 10	0.93 $\pm$ 0.55	0.70 $\pm$ 0.74	0.71 $\pm$ 0.12	0.70 $\pm$ 0.50	0.96 $\pm$ 0.74	0.017
Age 11	0.93 $\pm$ 0.22	0.67 $\pm$ 0.01	0.72 $\pm$ 0.54	0.69 $\pm$ 0.41	0.96 $\pm$ 0.58	0.011
Age 12	0.92 $\pm$ 0.64	0.68 $\pm$ 0.04	0.70 $\pm$ 0.48	0.68 $\pm$ 0.99	0.95 $\pm$ 0.74	0.014
Age 13	0.92 $\pm$ 0.48	0.70 $\pm$ 0.66	0.65 $\pm$ 0.89	0.67 $\pm$ 0.41	0.96 $\pm$ 0.23	0.019
Age 14	0.93 $\pm$ 0.15	0.75 $\pm$ 0.60	0.66 $\pm$ 0.79	0.70 $\pm$ 0.21	0.97 $\pm$ 0.73	0.008
Age 15	0.93 $\pm$ 0.78	0.67 $\pm$ 0.52	0.60 $\pm$ 0.73	0.63 $\pm$ 0.31	0.97 $\pm$ 0.97	0.091
Age 16	0.95 $\pm$ 0.10	0.68 $\pm$ 0.72	0.69 $\pm$ 0.05	0.68 $\pm$ 0.50	0.97 $\pm$ 0.86	0.017
Age 17	0.95 $\pm$ 0.29	0.63 $\pm$ 0.70	0.76 $\pm$ 0.07	0.68 $\pm$ 0.89	0.96 $\pm$ 0.08	0.019
Age 18	0.96 $\pm$ 0.23	0.70 $\pm$ 0.78	0.87 $\pm$ 0.08	0.77 $\pm$ 0.58	0.97 $\pm$ 0.36	0.014

Table 5. Performance of the teacher model per each 19 bone age classes.

Category	Accuracy	Precision	Recall	F-Score	Specificity	MAE
Age 0	0.97 $\pm$ 0.67	0.52 $\pm$ 0.25	0.46 $\pm$ 0.17	0.48 $\pm$ 0.82	0.98 $\pm$ 0.10	0.009
Age 1	0.97 $\pm$ 0.69	0.77 $\pm$ 0.29	0.64 $\pm$ 0.42	0.69 $\pm$ 0.90	0.99 $\pm$ 0.61	0.010
Age 2	0.97 $\pm$ 0.06	0.72 $\pm$ 0.61	0.67 $\pm$ 0.09	0.69 $\pm$ 0.41	0.98 $\pm$ 0.77	0.008
Age 3	0.97 $\pm$ 0.25	0.70 $\pm$ 0.26	0.70 $\pm$ 0.59	0.70 $\pm$ 0.00	0.98 $\pm$ 0.42	0.010
Age 4	0.97 $\pm$ 0.22	0.75 $\pm$ 0.82	0.66 $\pm$ 0.47	0.70 $\pm$ 0.21	0.99 $\pm$ 0.09	0.010
Age 5	0.96 $\pm$ 0.66	0.73 $\pm$ 0.98	0.77 $\pm$ 0.69	0.74 $\pm$ 0.95	0.98 $\pm$ 0.26	0.007
Age 6	0.98 $\pm$ 0.84	0.82 $\pm$ 0.73	0.89 $\pm$ 0.69	0.85 $\pm$ 0.36	0.98 $\pm$ 0.15	0.007
Age 7	0.96 $\pm$ 0.34	0.73 $\pm$ 0.34	0.73 $\pm$ 0.63	0.73 $\pm$ 0.00	0.98 $\pm$ 0.28	0.006
Age 8	0.97 $\pm$ 0.78	0.82 $\pm$ 0.58	0.84 $\pm$ 0.03	0.82 $\pm$ 0.99	0.98 $\pm$ 0.44	0.012
Age 9	0.96 $\pm$ 0.67	0.73 $\pm$ 0.10	0.72 $\pm$ 0.06	0.72 $\pm$ 0.50	0.98 $\pm$ 0.52	0.014
Age 10	0.96 $\pm$ 0.00	0.77 $\pm$ 0.90	0.81 $\pm$ 0.31	0.78 $\pm$ 0.95	0.97 $\pm$ 0.45	0.013
Age 11	0.96 $\pm$ 0.60	0.76 $\pm$ 0.87	0.83 $\pm$ 0.53	0.79 $\pm$ 0.35	0.97 $\pm$ 0.87	0.009
Age 12	0.95 $\pm$ 0.38	0.81 $\pm$ 0.81	0.76 $\pm$ 0.65	0.78 $\pm$ 0.42	0.98 $\pm$ 0.51	0.009
Age 13	0.96 $\pm$ 0.91	0.83 $\pm$ 0.26	0.82 $\pm$ 0.40	0.82 $\pm$ 0.50	0.98 $\pm$ 0.94	0.004
Age 14	0.96 $\pm$ 0.00	0.84 $\pm$ 0.59	0.75 $\pm$ 0.81	0.79 $\pm$ 0.25	0.98 $\pm$ 0.63	0.008
Age 15	0.96 $\pm$ 0.46	0.78 $\pm$ 0.02	0.70 $\pm$ 0.71	0.73 $\pm$ 0.78	0.98 $\pm$ 0.95	0.091
Age 16	0.97 $\pm$ 0.42	0.81 $\pm$ 0.42	0.83 $\pm$ 0.96	0.81 $\pm$ 0.99	0.98 $\pm$ 0.24	0.011
Age 17	0.97 $\pm$ 0.46	0.77 $\pm$ 0.31	0.87 $\pm$ 0.53	0.81 $\pm$ 0.70	0.98 $\pm$ 0.67	0.009
Age 18	0.97 $\pm$ 0.77	0.77 $\pm$ 0.16	0.93 $\pm$ 0.32	0.84 $\pm$ 0.25	0.98 $\pm$ 0.28	0.003

Table 6. Comparing analysis of student-6 results with the student model from three other KD methods.

Methods	(%)	Precision (%)	Recall (%)	F-Score (%)	Specificity (%)
O-KD Hinton et al. (2015)	74.67 $\pm$ 0.78	59.56 $\pm$ 0.90	58.67 $\pm$ 0.47	59.11 $\pm$ 0.17	78.56 $\pm$ 0.17
FitNet Romero et al. (2014)	78.35 $\pm$ 0.99	58.69 $\pm$ 0.85	57.54 $\pm$ 0.08	58.10 $\pm$ 0.93	77.03 $\pm$ 0.99
T-KD Mirzadeh et al. (2020)	89.53 $\pm$ 0.08	61.99 $\pm$ 0.70	60.89 $\pm$ 0.43	61.43 $\pm$ 0.51	89.86 $\pm$ 0.84
Proposed KD	95.16 $\pm$ 0.54	65.16 $\pm$ 0.01	65.34 $\pm$ 0.69	65.24 $\pm$ 0.99	87.36 $\pm$ 0.48

Table 7. Results of comparison the paired t-test of the proposed method and three other KD approaches.

Criteria	Methods	${\mathit{T}}_{\mathit{obs}}$	${\mathit{T}}_{\mathit{critical}}$	$\mathit{p}\mathbf{\_}\mathit{Value}$
Precision	O-KD Hinton et al. (2015)	−4.5685	4.5844	0.0065
	FitNet Romero et al. (2014)	−4.8540	1.3737	0.0173
	T-KD Mirzadeh et al. (2020)	−2.6867	1.5854	0.0455
Sensitivity	O-KD Hinton et al. (2015)	−4.4857	3.8885	0.0068
	FitNet Romero et al. (2014)	−5.4757	2.3722	0.0185
	T-KD Mirzadeh et al. (2020)	−0.4746	1.0004	0.0365
F-Score	O-KD Hinton et al. (2015)	−6.4740	3.5833	0.0055
	FitNet Romero et al. (2014)	−2.3731	1.0830	0.0056
	T-KD Mirzadeh et al. (2020)	−2.4742	1.5730	0.0301
Specificity	O-KD Hinton et al. (2015)	−6.4573	2.6664	0.0036
	FitNet Romero et al. (2014)	−3.6733	0.8522	0.0687
	T-KD Mirzadeh et al. (2020)	−0.8883	1.7842	0.0343
Accuracy	O-KD Hinton et al. (2015)	−8.5329	3.5855	0.0074
	FitNet Romero et al. (2014)	−6.4742	2.9433	0.0890
	T-KD Mirzadeh et al. (2020)	−1.0459	1.6750	0.0315

Table 8. Performance evaluation of the student-6 and some state-of-the-art studies.

Reference	Method	No. of samples	Age	Accuracy (%)	MAE
Kashif et al. (2016)	SVM	1100-DHA	0–18	–	0.605
Gertych et al. (2007)	Fuzzy Classifiers	1400-DHA	0–18	79	–
Mansourvar et al. (2012)	Histogram technique	1100-DHA	0–18	–	0.170
Spampinato et al. (2017)	CNN	1391-DHA	0–18	79	–
An Ding et al. (2020)	CNN	1400-DHA	0–18	–	0.590
Bui et al. (2019)	CNN + SVR	1400-DHA	0–18	67	–
Wang et al. (2023)	Interpretable CNN	1400-DHA	0–18	82.5	0341
Liu et al. (2022)	Ensemble CNNs	1400-DHA	0–18	83.8	–
Deshmukh and Khaparde (2022)	Faster R-CNN	1400-DHA	0–18	89.1	0.24
Proposed Method	Knowledge Distillation	1400-DHA	0–18	95.16	0.022

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