**Bedside Linear Regression Equations to Estimate Equilibrated Blood Urea**

Elmer A. Fernández1,2, Mónica Balzarini2,3 and Rodolfo Valtuille4

*1Faculty of Engineering, Catholic University of Córdoba 2National Council of Scientific and Technological Research (CONICET) 3Biometry Department, National University of Córdoba 4Fresenius Medical Care Argentina* 

#### **1. Introduction**

Three decades ago Sargent and Gotch established the clinical applicability of Kt/V, a dimensionless ratio which includes clearance of dialyzer (K),duration of treatment(t) and volume of total water of the patient (V), as an index of Hemodialysis (HD) adequacy (Gotch & Keen, 2005). This parameter, derived from single-pool(sp) urea(U) kinetic modelling, has become the gold standard for HD dose monitoring and it is widely used as a predictor of outcome in HD populations (Locatelli et al., 1999; Eknoyan et al., 2002; Locatelli, 2003). However, this spKt/V overestimates the HD dose because it does not take into account the concept of U rebound (UR). UR begins immediately at the end of HD session and it is completed 30-60 minutes after. UR is related to disequilibriums in blood/cell compartments as well as the flow between organs desequilibriums, both produced during HD treatment.

Therefore, equilibrated (Eq) Kt/V is the true HD dose and it requires the measurement of a true eqU when UR is completed. A blood sample to obtain an eqU concentration has several drawbacks that make this option impractical (Gotch and Keen,2005). For this reason in the last decade several formulas were developed to predict the eqU and also (Eq) Kt/V eliminating the need of waiting for a equilibrated urea mesurement. For instance, the "rate formula" (Daurgidas et al., 1995) is the most popular and validated equation. It is based in the prediction of (Eq)Kt/V as a linear function of (sp)Kt/V and the rate of dialysis(K/V). Another approach has been proposed by Tattersall, a robust formula based on double–pool analysis (Smye et al.1999). However, spite this eqU prediction approach is conceptually rigorous, it is not accurate (Gotch, 1990; Guh et al., 1999; Fernandez et al., 2001). Consequently, the availability of a model to predict subject-specific equilibrated concentration will be very helpful.

Although the behaviour of urea is non-linear since its extraction from blood follows some exponential family model as a function of time, we found that prediction of its equilibrated concentration after the end of the treatment session by means of linear models is accurate. In this study, we have shown how to build linear models to predict equilibrated urea based on two statistical procedures and a machine learning method that can be implemented in hemodialysis centres. The fitted model can be used for daily treatment monitoring and is

Bedside Linear Regression Equations to Estimate Equilibrated Blood Urea 3

this study, all patients were dialyzed over 240 min and the flows of blood (QB) and dialysate (QD) were fixed at 300 and 500 ml/min, respectively. It is known that hemodialysis dose is influenced by several factors including dialysis time, hemodialysis schedule and blood and dialysate flow (Daugirdas et al. 1997). In order to decrease the complexity, such variables were handled externally, fixing their values to control their effects on the equilibrated urea

Blood samples were obtained at the mid-week HD session. They were taken from the arterial line at different times to obtain urea determinations: 1) predialysis urea (U0), at the beginning of the procedure; 2) intradialysis urea (U120), in the middle of the HD session (at

For the intradialysis urea (U120) and postdialysis urea (U240), QB was slowed to 50 ml/min and blood was sampled 15 seconds later. At this point, access recirculation ceased and the dialyzer inlet blood reflected the arterial urea concentration. Regarding the protocols for intradialysis samples, it is worth noting that originally Smye et al. 1997 proposed taking them within 60 min from the beginning of the session and at 20 min before its finalization. We, however, decided to take the intradialysis sample 120 min after the beginning of the HD session (U120), which allowed us to compare our results with those reported by Guh et

Urea (U) determinations were performed in triplicate on each blood sample using autoanalyzers (see Fernandez et al, 2001 for more details). The urea averages were calculated and recorded with an accuracy of 1% for both machines. For information about the pre- and post-treatment status of the patient, we used the pre- and post-dialysis body weights (BW0, BW240). Both variables are commonly used in clinical practice to decide the treatment schedule as well as to calculate the treatment dose. These variables were recorded

The output variable was the equilibrated urea. For the purpose of this study, the patients were retained one hour in the dialysis center and the equilibrated urea levels (Ueq) were extracted 60 min after the end of HD. The summary statistics for the input and output

U0 U120 U240 Bw UF Ueq

Min 59 31 21 45.3 0.0 23 1st Quantile 127 64 40 59.4 2.0 50 Median 149 77 49 71.0 2.7 59 Mean 149 80 53 72.0 2.7 62 3rd Quantile 169 96 62 83.8 3.3 76 Max 221 144 98 119.0 5.5 112

The Ordinary Least Square approach estimates the **β** coefficient vector by minimizing the

in the same dialysis session when the blood samples were taken.

Table 1. Summary statistics of the patient data distribution.

120 min from the beginning); 3) postdialysis urea (U240), at the end of the HD session.

prediction model.

al. 1999.

variables are shown in Table 1

**2.2 Ordinary least squares** 

sum of squared residuals from the data

**2.1.2 The input and output variables** 

easily implemented in common available spreadsheets. A linear model is based on linear combinations of unknown parameters which must be estimated from data. The first step in looking for an appropriate model relies on prior knowledge or basic assumptions about the problem at hand that should be expressed in a hypothesized mathematical structure. The model can be expressed as *E(Y)=f(X,β)* , where *E*(**Y**) is the expected value of the output vector, "f " is a linear function, i.e. 0 11 22 , ..... *Ey f i i ii ip p* **x β** *xx x* , **X** is a matrix of input variables and **β** is a vector of parameters that needs to be estimated. In this way a set of potential mappings has been defined. The second step implies the estimation of the components of the vector **β**. This step includes the selection of a specific mapping (a 'proper' **β**) from the set of possible ones, choosing the parameter vector **β** that performs best according to some optimization criteria. There are several techniques to find a proper ˆ **β** when using a linear model, being ˆ **β** an estimation of **β** vector. Each of them has its own assumptions and requirements. Here we explore three different approaches for the estimation of the parameters of the **β** vector. They are: the Ordinary Least Square (OLS) procedure, based on the minimization of the sum of squared residuals <sup>1</sup> ˆ, *<sup>N</sup> i i <sup>i</sup> y f* **<sup>x</sup> <sup>β</sup>** which assume independence on the **X** matrix columns. The Partial Least Square (PLS) method based on decomposition schema maximizing the estimated covariance between the input and its outputs, and which is able to handle co-linearity or lack of independence among the **X** matrix columns. Finally, we use the Support Vector Machine algorithm (SVM) which is based on the minimization of the empirical risk over ε-sensitive loss functions. In this study, the three regression procedures were used to estimate the **β** coefficients in order to predict the equilibrated urea concentration at the end of the dialysis session. The input variables were the intradialysis urea concentrations (U0, U120, U240), the predialysis body weight and ultrafiltration patient data. Data analysis and modeling requires performing several tasks. In this work we use the Knowledge Discovery in Data Base (KDD) strategy as an ordered analysis framework. In this sense several steps involving different KDD stages such as problem/data understanding, collection, cleaning, pre-processing, analysismodeling and results interpretation were implemented.

#### **2. Material and methods**

#### **2.1 Data collection**

#### **2.1.1 Patients**

One hundred and nine stable patients were selected from two dialysis units as follows: sixty one from Unit1 (mean age 563.5 years and mean time on dialysis (MTD) 3212.3 months) and 48 from Unit2 (mean age 5818.0 and MTD of 4223.5). All patients were from Buenos Aires, Argentina, and were subjected to chronic HD treatment for at least 3 months. The selection criteria to include patients in the study were: (1) patients without infection or hospitalization in the previous 30 days; (2) patients with an A-V fistula (70% autologous fistula and 30% prosthetic fistula) with a blood flow rate (QB) of 300 ml/min, and (3) patients having consented to participate in the study. The study protocol complied with the Helsinki Declaration and was approved by the Ethical Committee of the Catholic University of Córdoba. All patients received HD three times a week with current hemodialysis machines using variable bicarbonate and sodium. Hollow-fiber polysulfone and cellulose diacetate dialyzers were used (see Fernandez et al, 2001 for more details). For the purpose of this study, all patients were dialyzed over 240 min and the flows of blood (QB) and dialysate (QD) were fixed at 300 and 500 ml/min, respectively. It is known that hemodialysis dose is influenced by several factors including dialysis time, hemodialysis schedule and blood and dialysate flow (Daugirdas et al. 1997). In order to decrease the complexity, such variables were handled externally, fixing their values to control their effects on the equilibrated urea prediction model.

## **2.1.2 The input and output variables**

2 Technical Problems in Patients on Hemodialysis

easily implemented in common available spreadsheets. A linear model is based on linear combinations of unknown parameters which must be estimated from data. The first step in looking for an appropriate model relies on prior knowledge or basic assumptions about the problem at hand that should be expressed in a hypothesized mathematical structure. The model can be expressed as *E(Y)=f(X,β)* , where *E*(**Y**) is the expected value of the output

matrix of input variables and **β** is a vector of parameters that needs to be estimated. In this way a set of potential mappings has been defined. The second step implies the estimation of the components of the vector **β**. This step includes the selection of a specific mapping (a 'proper' **β**) from the set of possible ones, choosing the parameter vector **β** that performs best according to some optimization criteria. There are several techniques to find a proper ˆ

assumptions and requirements. Here we explore three different approaches for the estimation of the parameters of the **β** vector. They are: the Ordinary Least Square (OLS) procedure, based on the minimization of the sum of squared residuals <sup>1</sup> ˆ, *<sup>N</sup>*

which assume independence on the **X** matrix columns. The Partial Least Square (PLS) method based on decomposition schema maximizing the estimated covariance between the input and its outputs, and which is able to handle co-linearity or lack of independence among the **X** matrix columns. Finally, we use the Support Vector Machine algorithm (SVM) which is based on the minimization of the empirical risk over ε-sensitive loss functions. In this study, the three regression procedures were used to estimate the **β** coefficients in order to predict the equilibrated urea concentration at the end of the dialysis session. The input variables were the intradialysis urea concentrations (U0, U120, U240), the predialysis body weight and ultrafiltration patient data. Data analysis and modeling requires performing several tasks. In this work we use the Knowledge Discovery in Data Base (KDD) strategy as an ordered analysis framework. In this sense several steps involving different KDD stages such as problem/data understanding, collection, cleaning, pre-processing, analysis-

One hundred and nine stable patients were selected from two dialysis units as follows: sixty one from Unit1 (mean age 563.5 years and mean time on dialysis (MTD) 3212.3 months) and 48 from Unit2 (mean age 5818.0 and MTD of 4223.5). All patients were from Buenos Aires, Argentina, and were subjected to chronic HD treatment for at least 3 months. The selection criteria to include patients in the study were: (1) patients without infection or hospitalization in the previous 30 days; (2) patients with an A-V fistula (70% autologous fistula and 30% prosthetic fistula) with a blood flow rate (QB) of 300 ml/min, and (3) patients having consented to participate in the study. The study protocol complied with the Helsinki Declaration and was approved by the Ethical Committee of the Catholic University of Córdoba. All patients received HD three times a week with current hemodialysis machines using variable bicarbonate and sodium. Hollow-fiber polysulfone and cellulose diacetate dialyzers were used (see Fernandez et al, 2001 for more details). For the purpose of

 *xx x* 

**β** an estimation of **β** vector. Each of them has its own

 

*i i <sup>i</sup> y f* **<sup>x</sup> <sup>β</sup>**

, **X** is a

**β**

vector, "f " is a linear function, i.e. 0 11 22 , ..... *Ey f i i ii ip p* **x β**

when using a linear model, being ˆ

modeling and results interpretation were implemented.

**2. Material and methods** 

**2.1 Data collection 2.1.1 Patients** 

Blood samples were obtained at the mid-week HD session. They were taken from the arterial line at different times to obtain urea determinations: 1) predialysis urea (U0), at the beginning of the procedure; 2) intradialysis urea (U120), in the middle of the HD session (at 120 min from the beginning); 3) postdialysis urea (U240), at the end of the HD session.

For the intradialysis urea (U120) and postdialysis urea (U240), QB was slowed to 50 ml/min and blood was sampled 15 seconds later. At this point, access recirculation ceased and the dialyzer inlet blood reflected the arterial urea concentration. Regarding the protocols for intradialysis samples, it is worth noting that originally Smye et al. 1997 proposed taking them within 60 min from the beginning of the session and at 20 min before its finalization. We, however, decided to take the intradialysis sample 120 min after the beginning of the HD session (U120), which allowed us to compare our results with those reported by Guh et al. 1999.

Urea (U) determinations were performed in triplicate on each blood sample using autoanalyzers (see Fernandez et al, 2001 for more details). The urea averages were calculated and recorded with an accuracy of 1% for both machines. For information about the pre- and post-treatment status of the patient, we used the pre- and post-dialysis body weights (BW0, BW240). Both variables are commonly used in clinical practice to decide the treatment schedule as well as to calculate the treatment dose. These variables were recorded in the same dialysis session when the blood samples were taken.

The output variable was the equilibrated urea. For the purpose of this study, the patients were retained one hour in the dialysis center and the equilibrated urea levels (Ueq) were extracted 60 min after the end of HD. The summary statistics for the input and output variables are shown in Table 1


Table 1. Summary statistics of the patient data distribution.

#### **2.2 Ordinary least squares**

The Ordinary Least Square approach estimates the **β** coefficient vector by minimizing the sum of squared residuals from the data

Bedside Linear Regression Equations to Estimate Equilibrated Blood Urea 5

ˆ <sup>1</sup> <sup>1</sup> *<sup>t</sup> <sup>p</sup> p A <sup>A</sup>*

In the PLS algorithm the input and output data are centered prior to calculate the different matrices. In addition the input training matrix **X** could be scaled dividing each column by its standard deviation. Thus, regression coefficients estimated by means of equation (7) lives in the scaled **X** domain. The values of the **β** coefficients in the raw data domain are calculated

<sup>0</sup> ˆˆ ˆ <sup>ˆ</sup> *PLS raw PLS raw <sup>Y</sup>*

column of **X** and **X** is the vector of columns means from **X**. *Y* is the mean of the response

In previous cases, the sum of squared deviation of the data can be viewed as a loss function measuring the amount of loss associated with the particular estimation of **β**. In the Support Vector Machine framework (Vapnik, 2000), the loss function only provides information on

, , max 0, *p p pL yf y f y f*

*emp i i i R y f x N*

**β**

*i i*

**β**

 

 0

, ' 0 ; 1...

0

 

*y x i N*

' symbols represent slack variables for those points above or below the target in

. This minimization problem can be rewritten in terms of Lagrange

*ii i*

*i N*

*i ii*

 1

constrained to <sup>2</sup> **<sup>β</sup>** *<sup>C</sup>* where *C* is a user defined constant, playing a role of regularization

<sup>1</sup> , *<sup>N</sup> <sup>p</sup>*

<sup>1</sup> ' *<sup>N</sup> <sup>p</sup> <sup>p</sup> i i <sup>i</sup> C* 

**<sup>β</sup>**

*x y iN*

 

0 ˆ ˆ 

those data points from which the loss is beyond a threshold *ε* yielding to

with *p=1* or *2*. Then the algorithm try to minimize an empirical risk defined as

**Y** is the estimated Ueq, **V** is a diagonal matrix of standard deviations for each

**<sup>β</sup> W C** (7)

*raw raw* **Y V <sup>β</sup> X V <sup>β</sup> <sup>X</sup> <sup>β</sup> <sup>X</sup>** (8)

*PLS raw <sup>Y</sup>* **<sup>V</sup> <sup>β</sup> <sup>X</sup>** is the intercept.

**xx x** (9)

**β β** (10)

; 1... ' ; 1... (11)

*PLS*

^ 1 1

variable from the training data set, and <sup>1</sup>

constant, a trade-off between complexity and losses.

The optimization problem, in primal form, can be defined as follows

minimize <sup>2</sup>

subject to

**2.4 Support vector machine** 

as follows:

where ^

The and 

more than ε and ' 0 *i i*

multipliers (dual form) as

 

$$L\left(\mathfrak{P}\right) = \sum\_{i=1}^{N} \left( y\_i - f\left(\tilde{\mathbf{x}}\_i, \mathfrak{P}\right) \right)^2 \tag{1}$$

where 1, *i i x x* with *<sup>i</sup> x* the "*i-th*" row of the input matrix **X**. The algorithm looks for the **β** that minimize (1). This is achieved taking derivatives of equation 1 and setting them to zero, yielding the following closed solution:

$$
\hat{\boldsymbol{\mathfrak{g}}}\_{OLS} = \left(\tilde{\mathbf{x}}^{l} \cdot \tilde{\mathbf{x}}\right)^{-1} \cdot \tilde{\mathbf{x}}^{l} \cdot \mathbf{Y} \tag{2}
$$

where "*t*" means "transpose" and *<sup>t</sup>* **X X** is a singular matrix with **<sup>X</sup>** the extended input matrix holding *x x i i* 1, in each row.

#### **2.3 Partial least squares**

Partial Least Squares not only generalizes but also combines features from regression and Principal Component Analysis, to deal with correlated explanatory variables in linear models (abdi, 2003, Shawe-Taylor & Cristianini, 2005). It is particularly useful when one or several dependent variables (outputs) must be predicted from a large and potentially highly correlated set of independent variables (inputs). In the PLS algorithm (Wood et al., 2001), **X** and **Y** are expressed as:

$$\mathbf{X}^{N \times p} = \mathbf{T}^{N \times A} \left(\mathbf{P}^{p \times A}\right)^{t} + \mathbf{H}^{N \times p} \tag{3}$$

$$\mathbf{Y}^{N \times p} = \mathbf{U}^{N \times A} \left( \mathbf{C}^{1 \times A} \right)^{t} + \mathbf{R}^{N \times p} \tag{4}$$

where A is the number of PLS factors (A p) and **H** and **R** are error matrices. The columns of **T** and **U** ("score" matrices) provide a new representation of the **X** and **Y** variables in an orthogonal space. The matrices **P** and **C** are the projections ("loadings") of the **X** and **Y** columns into the new set of variables in **T** and **U**. The **T** matrix is calculated as **T=XW** where **W=U(P´U)-1**. In the PLS algorithm, **U** and **P** are built iteratively (Wood et al.,2001) by means of matrix products between consecutive deflations of the original matrices **X** and **Y**. Thus, the **T** matrix is also a good estimator of **Y**, so

$$\mathbf{Y}^{N \times p} = \mathbf{T}^{N \times A} \left( \mathbf{C}^{1 \times A} \right)^{t} + \mathbf{E}^{N \times p} \tag{5}$$

where **C**1xA is the "loadings" matrix of **Y** that projects it over the new space represented by **T**. The error term in **E** represents the deviations between the observed and predicted responses. Replacing **T** in the above equation yields:

$$\mathbf{Y} = \mathbf{X} \cdot \mathbf{W} \cdot \mathbf{C}^{\ell} + \mathbf{E} = \hat{\mathbf{\hat{p}}}\_{PLS} \cdot \mathbf{X} + \mathbf{E} = \hat{\mathbf{Y}} + \mathbf{E} \tag{6}$$

where **Ŷ** is the predicted output.

The number of factors chosen impacts the estimation of the regression coefficients. In a model with "A" factors, the **β** coefficients are calculated as follows:

$$\hat{\boldsymbol{\mathfrak{g}}}\_{PLS}^{p \times 1} = \mathbf{W}^{p \times A} \left[ \mathbf{C}^{1 \times A} \right]^t \tag{7}$$

In the PLS algorithm the input and output data are centered prior to calculate the different matrices. In addition the input training matrix **X** could be scaled dividing each column by its standard deviation. Thus, regression coefficients estimated by means of equation (7) lives in the scaled **X** domain. The values of the **β** coefficients in the raw data domain are calculated as follows:

$$\hat{\mathbf{Y}} = \mathbf{V}^{-1} \hat{\mathbf{\hat{\beta}}\_{PLS}} \mathbf{X}\_{nnw} + \mathbf{V}^{-1} \hat{\mathbf{\hat{\beta}}\_{PLS}} \overline{\mathbf{X}}\_{nnw} + \overline{Y} = \hat{\boldsymbol{\beta}}\_0 + \hat{\mathbf{\hat{\beta}}}\_{nnw} \mathbf{X}\_{nnw} \tag{8}$$

where ^ **Y** is the estimated Ueq, **V** is a diagonal matrix of standard deviations for each column of **X** and **X** is the vector of columns means from **X**. *Y* is the mean of the response variable from the training data set, and <sup>1</sup> 0 ˆ ˆ *PLS raw <sup>Y</sup>* **<sup>V</sup> <sup>β</sup> <sup>X</sup>** is the intercept.

#### **2.4 Support vector machine**

4 Technical Problems in Patients on Hemodialysis

<sup>2</sup>

where 1, *i i x x* with *<sup>i</sup> x* the "*i-th*" row of the input matrix **X**. The algorithm looks for the **β** that minimize (1). This is achieved taking derivatives of equation 1 and setting them to zero,

<sup>1</sup> <sup>ˆ</sup> *t t*

Partial Least Squares not only generalizes but also combines features from regression and Principal Component Analysis, to deal with correlated explanatory variables in linear models (abdi, 2003, Shawe-Taylor & Cristianini, 2005). It is particularly useful when one or several dependent variables (outputs) must be predicted from a large and potentially highly correlated set of independent variables (inputs). In the PLS algorithm (Wood et al., 2001), **X**

where A is the number of PLS factors (A p) and **H** and **R** are error matrices. The columns of **T** and **U** ("score" matrices) provide a new representation of the **X** and **Y** variables in an orthogonal space. The matrices **P** and **C** are the projections ("loadings") of the **X** and **Y** columns into the new set of variables in **T** and **U**. The **T** matrix is calculated as **T=XW** where **W=U(P´U)-1**. In the PLS algorithm, **U** and **P** are built iteratively (Wood et al.,2001) by means of matrix products between consecutive deflations of the original matrices **X** and **Y**.

where **C**1xA is the "loadings" matrix of **Y** that projects it over the new space represented by **T**. The error term in **E** represents the deviations between the observed and predicted

The number of factors chosen impacts the estimation of the regression coefficients. In a

*OLS*

where "*t*" means "transpose" and *<sup>t</sup>* **X X** is a singular matrix with **<sup>X</sup>**

*i i*

,

**β β** (1)

**β XX XY** (2)

*<sup>t</sup> <sup>N</sup>p p N A A N <sup>p</sup>* **X TP H** (3)

<sup>1</sup> *<sup>t</sup> <sup>N</sup><sup>p</sup> NA A <sup>N</sup><sup>p</sup>* **Y UC R** (4)

<sup>1</sup> *<sup>t</sup> <sup>N</sup><sup>p</sup> NA A <sup>N</sup><sup>p</sup>* **Y TC E** (5)

*<sup>t</sup>* <sup>ˆ</sup> <sup>ˆ</sup> *PLS* **Y XWC E <sup>β</sup> XEYE** (6)

the extended input

1

*i L y fx* 

yielding the following closed solution:

matrix holding *x x i i* 1, in each row.

Thus, the **T** matrix is also a good estimator of **Y**, so

responses. Replacing **T** in the above equation yields:

model with "A" factors, the **β** coefficients are calculated as follows:

where **Ŷ** is the predicted output.

**2.3 Partial least squares** 

and **Y** are expressed as:

*N*

In previous cases, the sum of squared deviation of the data can be viewed as a loss function measuring the amount of loss associated with the particular estimation of **β**. In the Support Vector Machine framework (Vapnik, 2000), the loss function only provides information on those data points from which the loss is beyond a threshold *ε* yielding to

$$\left. \Pi\_{\varepsilon}^{p} \left( \mathbf{x}, y, f \right) \right| = \left| y - f \left( \mathbf{x} \right) \right|\_{\varepsilon}^{p} = \max \left( 0, \left| y - f \left( \mathbf{x} \right) \right|^{p} - \varepsilon \right) \tag{9}$$

with *p=1* or *2*. Then the algorithm try to minimize an empirical risk defined as

$$\mathcal{R}\_{emp}\left(\mathfrak{P}\right) = \frac{1}{N} \sum\_{i=1}^{N} \left( \left| y\_i - f\left(\mathbf{x}\_i, \mathfrak{P}\right) \right|\_{\mathcal{E}}^p \right) \tag{10}$$

constrained to <sup>2</sup> **<sup>β</sup>** *<sup>C</sup>* where *C* is a user defined constant, playing a role of regularization constant, a trade-off between complexity and losses.

The optimization problem, in primal form, can be defined as follows

$$\begin{aligned} \text{minimize } \left\| \mathbf{J} \right\|^2 + \mathcal{C} \sum\_{i=1}^N \left( \tilde{\boldsymbol{\varphi}}\_i^p + \tilde{\boldsymbol{\varphi}}\_i^p \right) \\\\ \text{subject to } \begin{cases} \left( \left\langle \mathbf{J} \cdot \mathbf{x}\_i \right\rangle + \beta\_0 \right) - y\_i \le \boldsymbol{\varepsilon} + \boldsymbol{\xi}\_i \; \boldsymbol{i} = 1 \dots N \\\ y\_i - \left( \left\langle \boldsymbol{\theta} \cdot \mathbf{x}\_i \right\rangle + \beta\_0 \right) \le \boldsymbol{\varepsilon} + \boldsymbol{\xi}\_i^\prime \; \boldsymbol{i} = 1 \dots N \end{cases} \end{aligned} \tag{11}$$

The and ' symbols represent slack variables for those points above or below the target in more than ε and ' 0 *i i* . This minimization problem can be rewritten in terms of Lagrange multipliers (dual form) as

Bedside Linear Regression Equations to Estimate Equilibrated Blood Urea 7

predictive ability of the fitted models was evaluated using a 20 fold cross-validation strategy over the whole data set. The data set was split in 20 consecutive sets of equal size and 19 were alternatively used for **β** estimation and one for prediction from the estimated model.

In table 2, cross validation statistics for PLS models with different number of factors is shown. Table 2 summarizes mean and standard deviation of Mean Prediction Error (RMPE) and mean and standard deviation of correlations between estimated and measured Ueq (*R*). It is possible to see that a PLS model with 3 or 4 components are very competitive. We chose a linear fit with 3 Factors because it yields the lowest RMPE with a parsimonious model

Table 2. Expected prediction error for PLS model with different number of factors.

In Fig.1 the achieved RMPE of the SVM models are shown for each *C*×*ε* grid point. The

Fig. 1. Cross-validation MSE for each *C*×*ε* combination in the SVMR algorithm. The best *C*×*ε*

combination pair is indicated with a filled circle
