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Cyclosporine Concentration Prediction using G. Camps-Valls∗, E. Soria-Olivas∗, J. J. P´ erez-Cruz‡, A. R. Figueiras-Vidal‡, A. Art´ ∗ Dept. Enginyeria Electr`onica. Universitat de Val`encia, Spain.
C/ Dr. Moliner, 50, 46100 - Burjassot (Val` † Pharmacy Service. Dr. Peset University Hospital. Val`encia, Spain∗.
‡ Dpto. Teor´ıa de la Se˜nal y las Comunicaciones. Universidad Carlos III de Madrid.
This paper proposes a combined strategy of clustering and Support Vector Regression (SVR) methods to predict Cyclosporine A (CyA) con- centration in renal transplant recipients.
variability and non-stationarity of the time series and reports knowledge gain in the problem. The SVR outperforms other classical neural net- ∗J. J. P´erez-Ruixo is also with the Global Clinical Pharmacokinetics and Clinical Pharma- codynamics Department. Janssen Research Foundation (Belgium).
Despite progress with newer agents, Cyclosporine A (CyA) is still the corner- stone of immunosuppression in patients who have undergone kidney transplan- tation. However, CyA is generally considered to be a critical dose drug. Under- dosing may result in graft loss and overdosing causes kidney damage, increases opportunistic infections, systolic and diastolic pressure, and cholesterol. More- over, the pharmacokinetic behavior of CyA presents a substantial inter- and intra-individual variability which appears to be particularly evident in the ear- lier post-transplantation period (<3 months), when the risk and clinical conse- quences of acute rejection are higher than in stable renal patients (>6 months) [1]. Several factors such as clinical drug interactions and patient compliance can also significantly alter blood CyA concentrations and thus intensive therapeu- tic drug monitoring of CyA becomes necessary but it influences the patient’s quality of life and the cost of the care.
Models capable of predicting the future concentration and determining the optimal dosage of CyA usually aid to individualize therapy. Few studies have been done and none, to our knowledge, using machine learning or neural net- works. We propose the use of Support Vector Machines (SVM) for solving this task since they do not rely on any a priori assumption about the problem and have proven to be effective techniques in a wide range of applications [2]. To deal with the non-uniform sampling (NUS), the presence of non-stationary pro- cesses, and the high variability in the time series, we have previously clustered SVMs are state-of-the-art tools for nonlinear input-output knowledge discovery [2]. The Support Vector Regressor (SVR) is for regression and function approx- imation. Given a labelled training data set ({(x and yi ∈ R) and a nonlinear mapping to a higher dimensional space φ(·) R −→ φ(x) ∈ R , d ≤ H ), the SVR solves: where w and b define a linear regressor in the feature space, nonlinear in the input space unless φ(xi) = xi. In addition, ξi, ξ∗ and C are, respectively, positive slack variables to deal with training samples with a prediction error larger than ε and the penalization applied to these ones. The tube size ε is traded off against model complexity and slack variables via a constant ν ∈ [0, 1) which can be regarded as an upper bound on the fraction of errors and a lower bound on the fraction of Support Vectors (SV). This formulation is known as the ν-SVR [3]. The usual procedure for solving SVRs introduces the linear restrictions using Lagrange multipliers into Eq.
Kuhn-Tucker conditions and solves the Wolfe’s dual problem using quadratic programing procedures [2, 3]. We will instead use an alternative procedure that consists in solving iteratively a series of weighted least square problems [4], known as Iterative Re-Weighted Least Square (IRWLS) procedure, that is and, in order to work with reproducing kernels in Hilbert Space, we require w to be a linear combination of a subset of training samples where ei = yi − φT (xi)w − b − ε and e∗ = φT (x The column vectors y, a, a∗, β and 1, present the obvious expressions and H is known as the kernel matrix, since it is only formed by inner products of the training samples in the feature space. Consequently, neither the minimizing procedure nor the use of the regressor needs to know explicitly the form of the nonlinear mapping, φ(·), but only its kernel representation κ(·, ·). The needed transformations to obtain the IRWLS procedure from the minimization of Eq.
Fifty-seven renal allograft recipients treated in the Nephrology Service of the Hospital Universitari Dr. Peset in the city of Val` this study. Patients received a standard immunosuppressive regimen of CyA Steady state blood samples were withdrawn 12-14 hours after dose administration and measured by a specific monoclonal fluo- rescence polarization immunoassay. We collected 11 patient factors to build the models: age, gender, creatinine plasma levels, creatinine clearance, alka- line phosphatase, hematocrit, urea and bilirubin, along with dosage, CyA blood concentration and post-transplantation days. Each pattern was formed by the present and past values of the these variables in order to perform one-step- ahead prediction. We split the data into two groups: two-thirds of the patients were used to train the models and the rest for their validation using the cross- The high intersubjects variability (coefficient of variation, CV = 31%) led us to set up clusters and then building individual predictive models for each one of them. We have used the well-known K-means clustering algorithm and selected the optimal partition by evaluating the root-mean-square error (RMSE) of ded- icated models through 3-fold cross-validation experiments. Four clusters were identified with this methodology. Since the second cluster was the largest (42% of the patterns), models yielded poor results in it (RMSE>60 ng/mL) and its variability still held high values (CV=27%), we decided to perform re-clustering on it. Once again a four-clusters partition was employed for the posterior pre- diction. The clustering reduced the RMSE and revealed postoperatory days, creatinine clearance, CyA blood concentration and serum alkaline phosphatase as decisive factors. In Table 1 results are benchmarked with a multilayer per- ceptron (MLP) trained with the familiar back-propagation algorithm and the Elman recurrent neural network [5] both with and without a previous clustering.
Elman network fails with a clustering approach since NUS becomes more evident (samples from the same patient can be in different clusters) and thus contextual neurons do not deal efficiently with past time samples. The ν-SVR model outperforms the MLP since the CV[%] is subsequently reduced in each partition. Blood levels accurately predicted (%BLAP) if an error margin of 20% is fixed have reached a value of 70%, which is an excellent result considering the time series characteristics of our population. Figure 1 shows predictions in two There is, nevertheless, 14% of patients with poor predictions which can be due to errors in drug dosage administration, in recording blood sampling times or abrupt changes in each patient’s clinical condition. An additional hypothesis for this could be some kind of liver dysfunction since alkaline phosphatase has resulted in a critical clustering factor. If we discard these patients, %BLAP increases to a 88% with the ν-SVR and to a 75% with the MLP. Support Vectors are mainly placed in the early post-operatory period (68% of them in the early three months) and change of cluster according to high levels in CV[%].
In this paper we have proposed the combination of clustering and a state-of- the-art technique for knowledge gain and accuracy improvement in a complex pharmacokinetic prediction problem. By means of clustering we can identify patients’ state and their future evolution, specify confidence intervals for each cluster prediction and discover important and meaningless patient factors all the time. The power and versatility of the SVR machines allowed fast and reliable Our future work is tided to benchmark Incremental Learning using SVR and Non-Linear Mixed-Effects Modelling (NONMEM) which is a commonly used method in population pharmacokinetics.
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[2] Vladimir N. Vapnik, Statistical Learning Theory, John Wiley & Sons, New olkopf, P. L. Bartlett, A. Smola, and R. Williamson, “Shrinking the tube: a new support vector regression algorithm,” in Advances in Neural Information Processing Systems 11, M. S. Kearns, S. A. Solla, and D. A.
Cohn, Eds., Cambridge, MA, 1999, pp. 330 – 336, MIT Press.
es-Rodr´ıguez, “An IRWLS procedure for ν-SVR,” in Proceedings of the ICASSP’01, Salt Lake City, Utah. U.S.A., May 2001.
[5] S. Haykin, Neural Networks: A Comprehensive Foundation, Prentice Hall, Figure 1 CyA trough concentration predictions within two validation patients.
Table 1 Mean error (ME [ng/mL]) and root-mean-square error (RMSE [ng/mL]) of models both for training and validation.

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