**3.1 Bed side equations for equilibrated urea prediction**

Final models were built using the whole patients and using the parameters found in the previous section (for PLS and SVM). We found that the coefficients estimated using the full data set (equations 14 to 15) where similar to the mean of the cross validation coefficients for OLS and SVM. On the contrary, coefficients estimated by PLS where different when using the whole data set compared to those estimated in the cross validation evaluation. In the OLS case the final bed side equation was the following:

0 120 240 0 <sup>ˆ</sup> *Y UU U B* 3.02449 0.01381 0.18576 0.79713 0.00028 0.16252 *W Uf* (14)

In the PLS case and accounting only the first three PLS factors the achieved model is

$$\hat{Y} = 0.84616 + 0.00810 \cdot \text{U}\_0 + 0.20652 \cdot \text{U}\_{120} + 0.75386 \cdot \text{U}\_{240} + 0.06862 \cdot \text{B} \text{W}\_0 - 0.26812 \cdot \text{U} \text{f} \text{ (15)}$$

For the SVM we get

$$\hat{Y} = 4.27754 - 0.03362 \cdot \text{U}\_0 + 0.27904 \cdot \text{U}\_{120} + 0.78921 \cdot \text{U}\_{240} - 0.01210 \cdot \text{B} \text{V}\_0 + 0.02323 \cdot \text{U} \text{f} \text{ (16)}$$

The SVM identify 77 support vectors. This means that the ˆ **β** coefficients were estimated using only %70 of the data base. On the contrary, the other two methods require the full data set to build the solution.

### **4. Discussion**

In this work we show how to build linear models from three different linear regression estimation procedures relying on different optimization algorithms. Ordinary Least squares is based on the minimization of the sum of squared residuals while Partial Least Squares uses maximization of co-variance information by means of repetitive deflation of the input and output matrices based on correlation. Finally, the Support Vector Machine Regression is based on the empirical risk minimization of non-linear loss function. Theoretically, none of the method requires any specific assumption; however, it is known that if the observed variable (the equilibrated urea in this case) follows a normal distribution, the statistical significance of the **β** coefficients estimated by OLS and PLS can be proved.

Even though all the models predict similarly well, they show different estimates not only in value but also in sign for U0, body weight and ultrafiltration. Analyzing the "raw" data relationships between these variables (see Fig. 4) and urea rebound *UU U eq* <sup>240</sup> *eq* it is

possible to see the known [Gotch & Kleen, 2005] slightly inverse relationship (see smooth trend curves) between BW and Uf with urea rebound. This behaviour seems to be capture for Uf by PLS (negative *β5*). The *β5* estimated by OLS method seems to follow the positive linear relationship mostly found in the Uf vs Ueq pairs plot. The SVM finds a solution in between, estimating much smaller values for *β5* than the others two. For the case of body weight coefficient (*β4*), estimations by OLS and SVM are smaller than for PLS, however, SVM method captures the known small tendency between BW and urea rebound. In this sense, PLS is able to capture known biological relationships while still providing broad ranges for the estimation of the Uf coefficient. On the opposite OLS does not reflect the biological effect of Uf. The SVM method provides an in-between solution providing small estimates of the Uf coefficient. Thus, those methods that account for co-linearity (PLS and in some extent SVM) provide better solutions than OLS which do not account for it.

Final models were built using the whole patients and using the parameters found in the previous section (for PLS and SVM). We found that the coefficients estimated using the full data set (equations 14 to 15) where similar to the mean of the cross validation coefficients for OLS and SVM. On the contrary, coefficients estimated by PLS where different when using

*Y UU U B* 3.02449 0.01381 0.18576 0.79713 0.00028 0.16252 *W Uf* (14)

*Y UU U B* 0.84616 0.00810 0.20652 0.75386 0.06862 0.26812 *W Uf* (15)

*Y UU U B* 4.27754 0.03362 0.27904 0.78921 0.01210 0.02323 *W Uf* (16)

using only %70 of the data base. On the contrary, the other two methods require the full

In this work we show how to build linear models from three different linear regression estimation procedures relying on different optimization algorithms. Ordinary Least squares is based on the minimization of the sum of squared residuals while Partial Least Squares uses maximization of co-variance information by means of repetitive deflation of the input and output matrices based on correlation. Finally, the Support Vector Machine Regression is based on the empirical risk minimization of non-linear loss function. Theoretically, none of the method requires any specific assumption; however, it is known that if the observed variable (the equilibrated urea in this case) follows a normal distribution, the statistical

Even though all the models predict similarly well, they show different estimates not only in value but also in sign for U0, body weight and ultrafiltration. Analyzing the "raw" data relationships between these variables (see Fig. 4) and urea rebound *UU U eq* <sup>240</sup> *eq* it is possible to see the known [Gotch & Kleen, 2005] slightly inverse relationship (see smooth trend curves) between BW and Uf with urea rebound. This behaviour seems to be capture for Uf by PLS (negative *β5*). The *β5* estimated by OLS method seems to follow the positive linear relationship mostly found in the Uf vs Ueq pairs plot. The SVM finds a solution in between, estimating much smaller values for *β5* than the others two. For the case of body weight coefficient (*β4*), estimations by OLS and SVM are smaller than for PLS, however, SVM method captures the known small tendency between BW and urea rebound. In this sense, PLS is able to capture known biological relationships while still providing broad ranges for the estimation of the Uf coefficient. On the opposite OLS does not reflect the biological effect of Uf. The SVM method provides an in-between solution providing small estimates of the Uf coefficient. Thus, those methods that account for co-linearity (PLS and in

**β** coefficients were estimated

the whole data set compared to those estimated in the cross validation evaluation.

0 120 240 0 <sup>ˆ</sup>

<sup>0</sup> <sup>120</sup> <sup>240</sup> <sup>0</sup> <sup>ˆ</sup>

<sup>0</sup> <sup>120</sup> <sup>240</sup> <sup>0</sup> <sup>ˆ</sup>

significance of the **β** coefficients estimated by OLS and PLS can be proved.

some extent SVM) provide better solutions than OLS which do not account for it.

In the PLS case and accounting only the first three PLS factors the achieved model is

**3.1 Bed side equations for equilibrated urea prediction** 

In the OLS case the final bed side equation was the following:

The SVM identify 77 support vectors. This means that the ˆ

For the SVM we get

**4. Discussion** 

data set to build the solution.

Fig. 4. Pairs plots and correlation coefficients between *U240, BW0 Uf, Ueq* and urea rebound.

We showed that by means of linear models we were able to build bedside equations that can be easily implemented in any calculator or electronic spreadsheet such as Excel®.

All the presented methods performed better than traditional methods (Smye et al, 1999) over the same data (Fernández et al, 2001) suggesting the appropriateness of the simple linear approaches. In addition, each hemodialysis centre can build its own predictor based on its own patient population by following the described process or implementing the accompanying source code (see appendix).

In this work we show that the use of an intradialysis sample (U120) provided valuable information to predict the equilibrated urea. Smye et al. (1999) were the first to use an intradialysis sample to model Ueq. In clinical practice the extraction of an additional blood urea sample could be very problematic. In a recent publication (Fernandez et al, 2008) we showed that a linear model built without this urea sample can also provide accurate Ueq estimation. Future challenges for Ueq prediction by linear models are emerging with the implementation of different HD schedule proposals based on the variation of session time and/or weekly frequency.

#### **5. Appendix: R source code for OLS, PLS and SVM linear models for estimate equilibrated urea**

In order to apply the R (www.r-project.org) algorithm to build the linear models presented in this work, we assume that the patient data base is stored in a comma separated values (CSV) file as follows (any electronic spreadsheet program allows to save CSV files).

Bedside Linear Regression Equations to Estimate Equilibrated Blood Urea 13


Table 4. Data base in comma separated file format. The R code assumes this file for processing (PP: Body weight)

PatientID U0 U120 U240 BW0 Uf Ueq 1 121 63 47 94.5 2.9 51 2 166 87 68 59.4 1.4 71 3 196 68 40 61.6 1.9 42 4 167 73 43 45.7 2.6 43 5 128 64 46 54.8 1.1 46 6 127 77 50 72.6 1.8 56 7 139 49 28 45.3 2.5 32 … … … … … … … … … … … … … …

Table 4. Data base in comma separated file format. The R code assumes this file for

processing (PP: Body weight)

**6. References** 

Cambridge University Press,.

*Renal Replacement Therapy*. 2(4) 295-304,

Depner, T.A. (1999) History of dialysis quantification. *Sem. Dial*. 12:S1:14-19

in maintenance hemodialysis. *New Engl J Med* 347: 2010–2019

Abdi H. (2003) Partial Least Squares (PLS) Regression. In: Lewis-Beck M, Bryman A, Futting T (Eds). *Encyclopedia of Social Sciences Research Methods*. Thousand Oaks, CA. Cristianini, N., Shawe-Taylor, J.,(2000) *An Introduction to Support Vector Machines*,

Daugirdas J. (1995) Simplified equations for monitoring kt/v, pcrn, ekt/v and epcrn. *Adv. in* 

Eknoyan, G.; Beck, G.J.; Cheung, A.K. et al (2002). Effect of dialysis dose and membrane flux

### **6. References**


Depner, T.A. (1999) History of dialysis quantification. *Sem. Dial*. 12:S1:14-19

Eknoyan, G.; Beck, G.J.; Cheung, A.K. et al (2002). Effect of dialysis dose and membrane flux in maintenance hemodialysis. *New Engl J Med* 347: 2010–2019

**2** 

Mary Hammes *University of Chicago* 

*United States* 

**Hemodialysis Access: The Fistula** 

The primary aim of this chapter is to understand the importance of placement and maintenance of arteriovenous fistula (AVF) in the patients with advanced renal failure prior to the need for dialysis. Vascular access complications contribute significantly to the morbidity and mortality associated with end-stage renal disease patients on hemodialysis. The major concern this publication will address is that with the recommendation for an increased number of AV fistulas, we are faced with the fact that many fistulas fail, with limited data to understand complications of AVF specifically stenosis and thrombosis. Attempts to understand underlying mechanisms of stenosis and thrombosis will aide in

The care and outcome of the patient with end-stage renal failure (ESRD) on chronic hemodialysis is dependent on their access. Although a variety of techniques have been developed for providing hemodialysis access, there have been no major advances in the past three decades. This contributes to the fact that hemodialysis access dysfunction is one of the most important causes of morbidity and mortality in the hemodialysis population. In addition, the expense of providing ESRD care in the US is a significant portion of the Medicare budget, totaling \$23.9 billion in 2007, of which a significant portion is spent on

The fistula provides the best outcome and can be placed with the least expense and complication rate when compared to a catheter or graft. Therefore, regional and network indicators promote the placement of AVF. Several recent initiatives have focused on vascular access and ways to improve outcomes. The National Foundation for Kidney-Dialysis Outcomes Quality Initiative (K-DQOL), End Stage Renal Disease Clinical Performance Measures (CPM) and Fistula First Initiative (FFI) have provided guidelines that mandate fistula access in patients on hemodialysis (Vasquez, 2009). FFI, developed to promote fistula placement, had an initial goal of 40% of prevalent patients with fistula access. This goal was achieved in 2005, with a goal of 66% set for 2009. Nationwide, however, there are only 54.4% of prevalent hemodialysis patients with fistula access as of November, 2009, with the number of fistula access placements falling for the first time in

New insights into the care and maintenance of fistula access will help to ensure duration of long term access patency. With national initiatives to place more fistulas, the number of fistulas has and will continue to increase. There are gaps in knowledge as to surveillance, maturation, cannulation techniques and mechanism and treatment of stenosis and thrombosis.

The following chapter on fistula access for hemodialysis will help to fill these voids.

access design, treatment options, and hence improve morbidity and mortality.

placement and maintenance of vascular access (USRDS, 2009).

**1. Introduction** 

2007 (USRDS, 2009).

