Multiple Laplacian Regularized RBF Neural Network for Assessing Dry Weight of Patients With End-Stage Renal Disease

Dry weight (DW) is an important dialysis index for patients with end-stage renal disease. It can guide clinical hemodialysis. Brain natriuretic peptide, chest computed tomography image, ultrasound, and bioelectrical impedance analysis are key indicators (multisource information) for assessing DW. By these approaches, a trial-and-error method (traditional measurement method) is employed to assess DW. The assessment of clinician is time-consuming. In this study, we developed a method based on artificial intelligence technology to estimate patient DW. Based on the conventional radial basis function neural (RBFN) network, we propose a multiple Laplacian-regularized RBFN (MLapRBFN) model to predict DW of patient. Compared with other model and body composition monitor, our method achieves the lowest value (1.3226) of root mean square error. In Bland-Altman analysis of MLapRBFN, the number of out agreement interval is least (17 samples). MLapRBFN integrates multiple Laplace regularization terms, and employs an efficient iterative algorithm to solve the model. The ratio of out agreement interval is 3.57%, which is lower than 5%. Therefore, our method can be tentatively applied for clinical evaluation of DW in hemodialysis patients.


INTRODUCTION
Dry weight (DW) refers to a patient's target weight after the end of dialysis (Grassmann et al., 2000;Wabel et al., 2009). After removing excess water from the body, the patient had no facial swelling, wheezing or sitting breathing, edema of both lower limbs, and distended jugular vein (Alexiadis et al., 2016). The patient's blood pressure, heart rate, breathing, and other vital signs are stable. There are individual differences in the specific value of DW. Good dry weight control can effectively reduce adverse reactions during dialysis. At present, the DW of hemodialysis patients is mainly evaluated by clinical means. This method is labor-intensive and time-consuming, and requires repeated use of various clinical instruments and biological indicators to complete the evaluation. In the past 10 years, a measuring instrument based on human bioelectrical impedance analysis (BIA), called body composition monitor (BCM) (Jiang et al., 2017), has been accurately determining the DW of patients. The above methods require professionals and cannot be processed on a large scale.
In this study, we proposed a novel predictive model based on radial basis function neural network (RBFN). Different from RBFN, multiple Laplacian-regularized RBFN (MLapRBFN) is a multi-view Laplacian regularized model with L 2 , 1 -norm, which introduces multiple graph regular items. The Laplacian regular items consider the topological relationship between each patient.

Materials
The data set of this study came from the hemodialysis center of Wuxi and the northern Jiangsu People's Hospital. Our study was approved by the ethics committees (Nos. 2018KY-001 and KYLLKS201813). There are a total of 476 hemodialysis patients. These patients meet the following conditions: over 18 years old; more than 3 months end-stage renal disease (ESRD) and maintenance hemodialysis; diseases such as metal implants, infections, heart failure, disability, pregnancy, and edema, do not appear in the patient population. DW is determined by clinical scoring, which is based on brain natriuretic peptide (BNP), chest computed tomography (CT) image, ultrasound, and bioelectrical impedance analysis (BIA). In addition, age, gender, diastolic blood pressure (DBP), systolic blood pressure (SBP), years of dialysis (YD), heart rate (HR), and body mass index (BMI) are employed to construct our  model. The summary of information is shown in Table 1.
This study is a retrospective study, and feature of patient is easy to obtain. We hope to provide patients with a non-invasive dry weight assessment method through machine learning models.

Radial Basis Function Network
The RBF neural network (Park and Sandberg, 1990) is composed of an input layer, a hidden layer, and an output layer, and is shown in Figure 1. The feature of the hemodialysis patient can be fed into the RBF network for processing. Suppose that there are N samples containing d variables x i , y i , i = 1, 2, . . . , N . The output isy i ∈ R 1×q and input data x i ∈ R 1×d . In Figure 1, the RBF network has three layers, which are the input, hidden, and output layers. The transformation from the input space to the hidden layer space is non-linear, and the transformation between the hidden layer and the output layer is linear. The fundamental of the RBFN is the RBF is the "base" of the hidden unit. The vector of input layer can be mapped to the space of the hidden layer without weight connection. The mapping relationship is determined with center point of hidden unit. In Figure 1, the number of hidden layer nodes is p (p center points). The activation function of the RBF neural network can be represented as: where x i is a feature vector of sample, c j , j ∈ 1, 2, . . . , p is the vector of j-th center point, and σ is the width parameter of the function, which controls the radial range of the function. RBFN can be represented as: where matrix ∈ R N×p is the output of the hidden nodes. W ∈ R p×q is the weight matrix between output and hidden nodes. Y ∈ R N×q is the matrix of dependent variable.
∈ R N×p and W ∈ R p×q can be represented as: Frontiers in Physiology | www.frontiersin.org To train RBFN, 3 parameters should be solved: the center points of the basis function c j ∈ R 1×d , j ∈ 1, 2, . . . , p , width (σ), and weight matrix between the output and hidden layer W ∈ R p×q . In most cases, the self-organized selection center learning method is used: (1) unsupervised learning process, solving the center points and width of RBF, (2) supervised learning process, solving the weights W ∈ R p×q . First, select p centers for k-means clustering (p clusters). For the radial basis of the Gaussian kernel function, the width is solved by the formula: where σ max is the maximum distance between the selected center points. For W ∈ R p×q , the RBF network can be represented as: The gradient of Eq. 6 can be set as 0: For a new test sample x new , we can estimate y * new as:

Proposed Model of Multiple Laplacian-Regularized RBF Network
To further improve the generalization performance of the RBF network, a multi-view Laplacian-regularized RBF network is proposed. The unsupervised process of the first stage remains unchanged; we mainly improve the model in the second part. The objective function of Eq. 6 is revised as: where λ 1 and λ 2 denote the parameters of Laplacian and L 2 , 1norm term. L 2 , 1 -norm can be used to obtain a sparser solution during the training process, which makes the model more robust. ρ > 1, which is used to prevent the extreme situations of η v = 0 (or η v = 1). L v ∈ R N×N is the Laplacian matrix, which is employed to represent the manifold of samples. V is the number of views. L v is built by heat kernel matrix S v ∈ R N×N : In our study, we employ the following functions to construct the heat kernel matrix: where γ is a constant, and we set it as 1.

Optimization
The third term of ||W|| 2 2,1 cannot be diversified, so Eq. 9 is converted into: where G ∈ R p×p is a diagonal matrix, and i-th is an element: Since there are multiple variables (W, η v ) that need to be optimized, we first fix η v and optimize W. We initialize η We obtain the derivative of formula (14) for variable W: Then, we fix the variant W and optimize η v , v = 1, 2, . . . , V, which is related to: The above formula can be converted to a Lagrange function: We set η v and ξ to 0: Frontiers in Physiology | www.frontiersin.org   .

RESULTS AND DISCUSSION
In this section, we perform 10-fold cross-validation (10-CV) to test the predictive performance of different models, the BCM device, multiple kernel support vector regression (MKSVR), linear regression (LR), back propagation-based artificial neural network [ANN (BP)], and multi-kernel ridge regression (MKRR).

Evaluation Measurements
Some traditional assessment methods include correlation coefficient (R), R square, root mean square error (RMSE), empirical cumulative distribution plot (ECDP), and Bland-Altman analysis. In particular, The Bland-Altman analysis usually can evaluate the agreement between two methods, and determines whether the two methods can be replaced with each other.

Selection of Optimal Parameters
We obtain the optimal model parameters, such as the number of hidden layer nodes (the number of clusters), regularization parameters, and number of optimization iterations. First, we fix the number of iterations (tmax = 10) and regularization coefficients (λ 1 = 2 0 , λ 2 = 2 0 ), and adjust the number of hidden layer neurons. In Figure 3A, we test the number of nodes from 10 to 250 with step of 10. After adding 140 neuron nodes, RMSE tends to be flat. In addition, the RMSE also tends to remain unchanged (minimum) after 10 iterations ( Figure 3B). Finally, we get the number of hidden layer nodes as 140 and times of iteration as 10, and adjust the regularization coefficient. The results are shown in Figure 4, and the optimal coefficients are λ 1 = 2 −3 , λ 2 = 2 −3 .

Comparison With Other Existing Models
We compare our model with other existing machine learning methods , which include multi-kernel ridge regression (MKRR), multiple kernel support vector regression (MKSVR), artificial neural network (ANN), linear regression (LR), and BCM measuring instrument. The gold standard is clinical dry weight. In Table 2

Bland-Altman Analysis
In this section, we employ a Bland-Altman plot to further evaluate the regression error of different models. In Figure 6 and Table 3, several models evaluate the agreement with clinical DW. MLapRBFN achieves the smallest range of -0.2413 to 0.1601 (95% confidence interval). What is more, the number (ratio) of out agreement interval is the key indicator to evaluate whether the two methods are equivalent. For the predictive models, the number (ratio) should be all less than 24 (5%). In Table 3, all the predictive models meet this standard. In particular, our method obtains the least number (ratio) of 17 (3.57%). If 95% of the samples are in agreement range, the predictive models are clinically acceptable. It can be seen that our method can replace clinical methods.

CONCLUSION
The limitations of BCM and clinical are time-consuming and laborious. In our study, a MLapRBFN method is developed to predict the DW of hemodialysis patients. Different from standard RBFN, our method contains multiple Laplace regularization terms, and uses an efficient iterative algorithm to solve the model. MKRR, LR, MKSVR, and ANN are compared with our model. Bland-Altman analysis and RMSE are the main evaluation methods. In the Bland-Altman analysis of MLapRBFN, the number of out agreement interval is the least (17 samples).

DATA AVAILABILITY STATEMENT
The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding authors.

ETHICS STATEMENT
The studies involving human participants were reviewed and approved by the experimental protocol was established and approved by the Human Ethics Committee (Wuxi and Northern Jiangsu People's Hospital Ethics Committee). The ethical approval numbers are 2018KY-001 and KYLLKS201813. The patients/participants provided their written informed consent to participate in this study.

AUTHOR CONTRIBUTIONS
XG and WZ did the experiments and wrote the manuscript. YD, CL, and XG designed the method. YY, YZ, AD, QL, and YC revised the manuscript. All authors have read and approved the final manuscript.