Enhanced approaches for cluster newton method for underdetermined inverse problems

2021 8th NAFOSTED Conference on Information and Computer Science (NICS) Enhanced Approaches for Cluster Newton Method for Underdetermined Inverse Problems Duong Tran Binh School of Information Science and Engineering Southeast Unversity Nanjing, China duongdtvt@gmail.com Nguyen Thi Thu Faculty of Electronic Engineering, Hanoi University of Industry thunt@haui.edu.vn Uyen Nguyen Duc Radio The Voice of Vietnam Broadcasting College No.1 Hanam, Vietnam uyenvov@gmail.com Tran Quang-Huy⃰ Faculty of Physics HaNoi Pedagogical University No.2 Hanoi, Vietnam tranquanghuy@hpu2edu.vn Tran Duc Tan Faculty of Electrical and Electronic Engineering Phenikaa University Hanoi, Vietnam tantdvnu@gmail.com Furthermore, we also proposed an efficient iteration procedure for CN method for inverse parameter identification in pharmacokinetics Reducing the noise level of y* is necessary when the cluster of points is close to the contribution line X* after each iteration for numerical stability to be appropriate The two sub-stages and are used to subdivide period The significant perturbation of y* is used for the first number of iterations in substage 1, followed by the small perturbation of y* for the remaining iterations in substage The more stable moving point cluster and the number of iterations and computation time saved are the results proving this approach when used Compared to the method of using regular Tikhonov, this method is slightly better Abstract— Along with many solutions for determining the inverse parameter in pharmacokinetics, with this work, we propose two improved approaches to the original cluster Newton method Applying Tikhonov regularization for hyperplane fitting in the CN method is the first method, and the efficient iterative process for the CN method is the next When using these proposed approaches, it has been demonstrated that numerical experiments of both approaches can bring benefits such as saving iterations, reduced computation time, and clustering of points They also move more stably and asymptotically with the diversity of solutions Keywords— Pharmacokinetics (PK), Cluster Newton method (CNM), Tikhonov regularization I INTRODUCTION The number of equations is smaller than variables in undetermined inverse pharmacokinetics frequently occurs It is because the complex mechanisms of the human body are often not explained by the data we collect By simulating complex activities through mathematical modeling, we gain valuable insight into in vivo pharmacokinetics The undetermined inverse problem with the ability to find multiple solutions simultaneously was proposed by Aoki et al [1] recently evolved into a new algorithm The method that has proven to be more reliable, robust, and efficient than the Levenberg-Marquardt method is the Newton cluster To improve the original Newton Cluster method as [2], [3], [4] several approaches are proposed II PHYSIOLOGICALLY BASED PHARMACOKINETIC (PBPK) The drug CPT-11 was initially introduced into the human body by drip method by the intravenous route Concentration ,…, and its metabolites patterns of drugs CPT-11, (SN-38, SN-38G, NPC, and APC) in each part of the body (blood, fat, GI, liver, and NET) established by the PBPK model developed by Arikuma [5] 1, …, 55, representing the chemical compound inflow and outflow of each chemical compound in each interconnected section Consequently, the system of first-order differential equations of concentrations can be constructed denoted in Figure were modeled in The roads denoted , … , [5] Since the drug inflow and outflow by pathways that alter the concentration degrees (du_i)∕dt Thus, below we can build : an ODE system to model the concentration In the original CN method in step 2.2, we still need to improve the stability However, using the backslash operator is good enough to fit a hyperplane due to (i) Lack of stability properties of the solution because parameter recognition is a presumptive inverse problem In particular, the measurement noise and modeling error can be significantly amplified By adopting the normative approach, these problems can be overcome More concretely, the problem of minimizing the data mismatch and the regularization term can be formulated as stable identification parameters; (ii) A quadratic filter based on the contribution of the individual values of the matrix operator that acts as the Tikhonov regularization The effective removal of singularity values is lower than the values of the usual parameters, leading to an unstable matrix equation To solve the overdetermined system for fitting the hyperplane in Step 2.2 in the Newton initial cluster method, we propose using the Tikhonov regularization d (1) u = h u, t; x dt Drip, glycemic, metabolic, and excretory routes are the four pathways by which drug flow rates in [nmol/min] are quantitatively described Their typical values and the 60 parameters associated with these paths, including , … , are listed in [5] along with Tables B1-B5 Based on data from clinical observations, estimating the parameters of this model is defined as an inverse problem XXX-X-XXXX-XXXX-X/XX/$XX.00 ©20XX IEEE 978-1-6654-1001-4/21/$31.00 ©2021 IEEE 2021 8th NAFOSTED Conference on Information and Computer Science (NICS) III ORIGINAL CLUSTER NEWTON METHOD (CNM) The most critical stage in the approach they propose is applicable only in the early stages in CNM Therefore, phase of CNM is presented by: 1: Initial setting 1-1: Randomly select initial points {& X0 .' * ( ') in the box They are stored in a matrix X(0) whose size is 60×l, each column corresponds to a point ' in + ∗ * ' ,') 1-2: Generate randomly perturbed near y* Each value of '∗ is set by: ) , ,…, Figure Illustration of physiologically based pharmacokinetic "ODE15s" is used to solve [7] in Matlab because we found ,…, can be that this is a first-order ODE system estimated since u (x1, …, x60; t) can be obtained depending on the time t and the parameters x1, …, x60 immediately afterward where : ∗ ⊂ = +n → =@ ∈ ) : :;< > 1000 : max C ) , ,…, ? O N O ∈P QR3SR , TR ∈PQR (3) U 1CG FE ` = J K2 UV 4:4 ] K` 2∗ U2 I1 J K D J1 1 K2 UV (7) (8) by fitting a plane to Y(k) The matrix A(k) and the shift can be estimated using the least-squares solution constant of an overdetermined system of linear equations: (4) , ,…, are variables described in the Arikuma's model [5] Kinetic parameters, physiological parameters and drip infusion parameters i.v have variation of ± 50% ( = ⋯ = = 0.5), ± 30% = ⋯ = $ = 0, respectively), ±5% = 0.05) Since the values of the inter-partial ( %= variation are smaller than ±50% [9], the variability of the kinetic parameters was chosen The altered physiological parameters were selected according to [10] The drip [ O ∈P SR3\ (5) 2-2: Construct a linear approximation of f n: The measured noise = H>- where each column of Y corresponds to the solution of the function f : a mapping function $ ∗ 2-1: Solve the forward problem for each point in (column vector of) X(k) (2) ∈ ∗ 2: For k = 0, 1, 2, …, K1 : The measured data =6 D values where - = 0.1 , thus, the target accuracy of stage is /10% The vectors '∗ are put in the jth column of the matrix Y* near a box is a problem of Finding the points in ⊂ determining the parameters of a satisfying PBPK model ∗ ∗ ' max H target where =J 1 is a 103l matrix whose columns are all 4.' 2-3: Compute the update vector for all columns of X(k) using a linear approximation It means to find 4.' satisfies: ∗ ' W ' K 4.'1 X K Y = 1,2, … , (9) In matrix A, the number of columns is larger than the number of rows Thus, there is an underdetermined system of linear 4.' that satisfies Equation (12) Among all the solutions of Equation (12), we choose the vector 4.' with the (10) * shortest scaled length as follows The vectors &4.' ( ') are shown as a matrix S(k) are the minimum norm solution: transmission parameters i.v have variation and are affected by the drip transmission, so it is small where = , ,…, 2-4: Look for new points that approximates the solution manifold X* by updating X(k) We can shrink the length of the 2021 8th NAFOSTED Conference on Information and Computer Science (NICS) vector 4.' until the point ( function f abc Y = 1,2, … , dℎ] f i;< bc ' K 4.' ∉X i;< bc K 4.' ) is in the domain of the ' The inverse problem here means determining the coefficients of a system of ordinary differential equations using the metabolism model of the anticancer drug CPT-11 ,…, depends on the and its metabolites parameters ,…, A system that maps the inputs ,…, to the output , … , is what we need to estimate The parameters in the human body are unknown; we can only measure the outputs The PBPK model is used to find many mutable biological states of the patient [15-17] 4.' = 4.' i;< jℎ] f = 1k K` Numerical instability of the matrix inversion is solved by Tikhonov’s to the w];]w]xf‖DJ ‖ problem To make the matrices dissimilar, it adds a positive constant to the diagonals of ATA [6] The form of Tikhonov regularization is The relative error residual (RRE) is used to evaluate the performance of the proposed method c' = ) , ,…, Where = ' D ' max š ∗ ∗ š (11) ↔ ' IV THE FIRST PROPOSED APPROACH • == To deal with inverse probem, we use the output data as an excretion profile in urine and bile and a model function of → , ∈ Moreover, model PBPK : parameters such as blood flow rate, the reaction speed of need to be enzyme, tissue volume, and so on ∈ estimated The model of the inverse problem is mentioned in: where ,…, where ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ,…, ,…, = = = u = r t $ % t = = = = = = u u uu ur u $ % u ›s ;p ,…, ,…, ;p = q ;p = q ;p = q ;p = q ;p = q ;p = q ;p = q s s s ;p = q ;p = q s s s s s s s ;p = q u r r rt r$ r% ,…, u ,…, =q =q =q =q =q =q =q s s s s =q s s t s $ s =q =q s s u u uu ur u % u u r ;p ›s u •— r / / / / / r u r ) •— – •— – k˜ •— – = of 1/σ‘ , since (14) ƒs “ σ‘ 19 (18) are Tikhonov filters, and z is the •— – •— – k˜ I •— – ˜ ≪ Figure shows the Tikhonov filter function It filters out the small singular components while retaining the large components Therefore, Tikhonov regularization behaves as a second-order filter on the contribution of the singular values of the matrix operator The result of this filtering operation is to effectively remove the singular values of a lower order than z, which are the ones responsible for the instability of the matrix equation < u = ’ – = •—k˜ I •–— = 1; ii) a small z can reduce the magnification – ›s (17) regularization parameter This parameter affects the filter as follows: i) a small z is not affected in a large σ‘ (z ≪ σ‘ ), i.e (13) we have a function in vector form: ‡ , =‡ =‡ (16) Assuming that J€3• is real and full rank (w ≫ ;), J = ƒΣ… s where U,V are orthogonal matrices, U = ‡u , … , uˆ ∈ Rˆ3ˆ , V = ‡v , … , vŒ ∈ RŒ3Œ and Σ = diag σ , … , σŒ ∈ RŒ3Œ with σ • σ • … , σŒ • 0, σ‘ is the i-th singular value From (15), we have: (12) = = w];]w]xf ‖ D J ‖ z K ‖ ‖ s = J J K z| _ A~ 18 w];]w]xfa < < < < t $ % < 15 < < Figure Tikhonov filter function < In work [8], Aoki et al indicated that solving O U2 D J 1 K UV is equivalent N O ∈PQR3SR, TR ∈PQR < to w];]w]xf‖ D J ‖ Therefore, Tikhonov regularization in the proposed method is ‖ D J ‖ K z‖ ‖ 2021 8th NAFOSTED Conference on Information and Computer Science (NICS) V With the same number of iterations of 10, the approach yields a significant reduction in computation time The calculation time of the proposed method is 416.989932 seconds, while the time of the original CNM is 461.882349 seconds; that is, the computation time is reduced by 9.76% after ten iterations The proposed approaches only need five iterations compared to the original CNM, which required seven iterations It can be explained as follows: Tikhonov filter function can filter out small single components; therefore, these values are not involved in computation, while the initial CNM takes longer to still calculating these values THE SECOND PROPOSED APPROACH An efficient iterative procedure for the CN method has been proposed in this subsection In the original CNM, it can be seen that after each iteration, the points cluster approaches the root manifold X* To ensure the correctness of least squares problem (current disturbance is 10%), it is necessary to generate randomly shuffled target values of y* After each iteration, the point cluster of points will move towards the solution manifold X*, but the degree of perturbation does not change The values of the elements in the point cluster are much different from those in the root manifold when the point cluster is far from the X* multiplicity Therefore, a certain level of perturbation is large enough to make a significant difference between the elements in the points cluster and the solution manifold to ensure the correctness of the leastsquares problem we need to generate The results of the CPT-11 blood levels prediction using the 500 sets of parameters found by the original CNM and the proposed approaches are shown in Fig After iterations and 3, we can see no difference between the original CNM and the proposed approaches by observation We notice an evident difference between the initial CNM and the proposed approaches after iterations and The cluster of points in the proposed approaches moves more steadily towards the manifold solution Compared with the proposed method, some samples are still scattered away from the cluster's center in the original CNM Moreover, it can also be noticed that the nature of the Tikhonov regularity offers a more stable and reasonable solution The values of the elements in the point cluster are not much different from those in the root manifold when the point cluster is close to the root manifold X* Therefore, to ensure the correctness of the least-squares problem, we only need to create a smaller perturbation level that produces a significant difference between the elements in the point cluster and the multiplicity of solutions TABLE PERFORMANCE COMPARISONS Therefore, for numerical stability, after each iteration, when the cluster of points is close to the X* solution path, reducing the noise level of y* is necessary and appropriate Two Sub-Stages are established in stage For some iterations, the first sub-stage is solved using a large perturbation degree of y*, while sub-period two the times The remaining iteration is solved using a small perturbation level of y* Using this method, the numerical results show that we can also save the number of iterations and computation time because the cluster of moving points is more stable Compared to Tikhonov's method of regularization use, this approach is slightly better VI Methods Total time (sec) RRE after each iteration (1-10) The original CNM 3.1452 0.8131 0.5434 0.2516 0.1401 0.1324 0.1200 0.1155 0.1124 0.1115 349 The first approach 3.1452 0.8214 0.5170 0.1990 0.1430 0.1182 0.1126 0.1132 0.1123 0.1133 932 The second approach 3.1452 0.8054 0.5270 0.1871 0.1268 0.1154 0.1111 0.1064 0.1049 0.1036 260 The original CNM The first approach 461.882 416.989 417.197 The second approach CPT-11 in blood NUMERICAL EXPERIMENTS AND RESULTS Number of samples Nsamp=500, Number of iterations Niter=10, Accuracy of function evaluation δ_ODE = 10-3, regularization parameters λ=10^(-4), 10% perturbation level are parameters of numerical test of the first proposed approach 3.5 3.5 3.5 3 2.5 2.5 2.5 2 1.5 1.5 1.5 1 0.5 0.5 0.5 0 200 400 600 800 1000 0 1200 200 1st iteration 400 600 800 1000 0 1200 200 1st iteration 400 3.5 3.5 3.5 3 2.5 2.5 2.5 2 1.5 1.5 1.5 1 0.5 0.5 0.5 0 200 400 600 800 1000 1200 CPT-11 in blood CPT-11 in blood Total number of iterations Niter=10, number of iterations of the first Substage N1-iter=2, number of iterations of the second Substage N1-iter=8 where the number of samples Nsamp=500 are the numerical parameters test of the method second proposed approach The degree of perturbation in the second stage is 6%, the accuracy of function evaluation δ_ODE = 10-3, the degree of perturbation for the first stage is 10% 600 1st iteration 800 1000 1200 0 c 200 3rd iteration 400 600 800 1000 1200 0 200 3rd iteration 400 600 800 1000 1200 1000 1200 1000 1200 3rd iteration CPT-11 in blood 3.5 3 2.5 2.5 2.5 2 1.5 1.5 1.5 1 0.5 0.5 0.5 0 When solving with initial CNM and the proposed approaches after Nsum iterations, relative error residuals (RRE) are presented in Table1 The first stage can remark that the smallest RRE that the algorithm can achieve is about 0.11, that is 11%, denoted RREmin Initially, it was evident that CNM needs seven iterations to get RREmin, compared to the proposed approach, which required only five iterations Two iterations are saving at this stage If we need to find many possible solutions thus, we can skip a large number of samples Therefore, the saving of iterations is significant because the complexity is reduced 3.5 3.5 200 400 600 800 1000 1200 0 200 5th iteration 400 600 800 1000 1200 0 CPT-11 in blood 3.5 3 2.5 2.5 2.5 2 1.5 1.5 1.5 1 0.5 0.5 0.5 400 600 800 7th iteration 1000 1200 0 200 400 600 600 800 CPT-11 in blood CPT-11 in blood 3.5 200 400 5th iteration 3.5 0 200 5th iteration 800 7th iteration 1000 1200 0 200 400 600 800 7th iteration Figure Prediction of CPT-11 blood levels using the 500 sets of parameters by different approaches 2021 8th NAFOSTED Conference on Information and Computer Science (NICS) VII CONCLUSIONS The proposed methods are proved to be reliable, robust, and efficient They can be used in the field of undetermined inverse pharmacokinetics In future works, we will optimize the regularization parameter in the Newton cluster method The experiment data will be applied to confirm the performance of these proposed methods We, in this work, have successfully proposed two approaches in improving the accuracy to determine the inverse parameter in pharmacokinetics based on the original cluster Newton method The proposed method is experimentally shown to have the advantages mentioned in summary However, it is still necessary to develop a more efficient method and, at the same time, generate the perturbation of y* to determine the optimal percentage and factors 10 11 REFERENCES 12 Aoki Y, Hayami K, De Sterck H, Konagaya A Cluster Newton method for sampling multiple solutions of underdetermined inverse problems: Application to a parameter identification problem in pharmacokinetics SIAM Journal on Scientific Computing 2014;36(1): B14–B44 Van Nguyen T., Huy T.Q., Nguyen V.D., Thu N.T., Tan T.D (2020) An Improved Approach for Cluster Newton Method in Parameter Identification for Pharmacokinetics In: Solanki V., Hoang M., Lu Z., Pattnaik P (eds) Intelligent Computing in Engineering Advances in Intelligent Systems and Computing, vol 1125 Springer, Singapore, ISBN 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"Cluster Newton Method for Sampling Multiple Solutions of an Underdetermined Inverse Problem: Parameter Identification for Pharmacokinetics." SIAM Journal on Scientific Computing (accepted for. .. Institute of Informatics, Tokyo, 2011, at http://www nii ac jp/TechReports//11-002E html.) (2011) Gaudreau, Philippe, et al "Improvements to the cluster Newton method for underdetermined inverse problems."... proposed methods We, in this work, have successfully proposed two approaches in improving the accuracy to determine the inverse parameter in pharmacokinetics based on the original cluster Newton method