Enhanced approaches for cluster newton method for underdetermined inverse problems

Enhanced Approaches for Cluster Newton Method for Underdetermined Inverse Problems Duong Tran Binh School of Information Science and Engineering Southeast Unversity Nanjing, China

Trang 1

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 Duc Tan

Faculty of Electrical and Electronic

Engineering Phenikaa University Hanoi, Vietnam

tantdvnu@gmail.com

Tran Quang-Huy⃰

Faculty of Physics HaNoi Pedagogical University No.2

Hanoi, Vietnam tranquanghuy@hpu2edu.vn

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

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

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

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 1 and 2 are used to subdivide period 1 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 2 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

The drug CPT-11 was initially introduced into the human body by drip method by the intravenous route Concentration

(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

[5] Since the drug inflow and outflow by pathways that alter the concentration degrees (du_i)∕dt Thus, below we can build

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

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

d

dt u = h u, t; x

(1)

Trang 2

Figure 1 Illustration of physiologically based pharmacokinetic

"ODE15s" is used to solve [7] in Matlab because we found

afterward

determining the parameters of a satisfying PBPK model

where

n: The measured noise

[5] Kinetic parameters, physiological parameters and drip

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

transmission parameters i.v have variation and are affected by

the drip transmission, so it is small

The most critical stage in the approach they propose is applicable only in the early stages in CNM Therefore, phase

1 of CNM is presented by:

1: Initial setting 1-1: Randomly select initial points {& .' (

')

*

in the box

X0

1-2: Generate randomly perturbed target values + ',')* near y* Each value of .'∗is set by:

Y*

2: For k = 0, 1, 2, …, K 1

2-1: Solve the forward problem for each point in (column vector of) X(k)

the function f 2-2: Construct a linear approximation of f

by fitting a plane to Y(k) The matrix A(k) and the shift

of an overdetermined system of linear equations:

In matrix A, the number of columns is larger than the number of rows Thus, there is an underdetermined system of

shortest scaled length as follows The vectors &4.'1(

')

*

are

2-4: Look for new points that approximates the solution

= 6 =∈ $ : 8 0 :;<> 1000

)

?

(3)

E

(4)

max

) , ,…, H 'D∗ ∗H > - (5)

min

N O ∈P QR3SR , TR ∈P QRU21 D J1 1 K 21 UV (8)

min

(10)

Trang 3

vector 4.'1 until the point ( .'1 K 4.'1) is in the domain of the

function f

abc Y = 1,2, … ,

4.'1 = 4.'1 i;< jℎ] f i;< bc

i;< bc

The relative error residual (RRE) is used to evaluate the

performance of the proposed method

Where

To deal with inverse probem, we use the output data as an

excretion profile in urine and bile and a model function of

parameters such as blood flow rate, the reaction speed of

estimated The model of the inverse problem is mentioned in:

we have a function in vector form:

where

⎩

⎪

⎨

⎪

⎧ = ; p = qsr = qs <

= t ; p = q rt = q t

s

.

s

<

u = $ ; p = q r$ = q $

s

u

s

u <

r = % ; p = q r% = q %

s

r

s

r <

= u ; p = q = q u

s

.

s

<

= u ; p = q = q u

s

<

s

t <

$ = uu ; p = q u = q uu

s

u / u

s

$ <

% = ur ; p = q r = q ur

s

r / r

s

% <

= u ; p = q = q u

s

<

15

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

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] Numerical instability of the matrix inversion is solved by

matrices dissimilar, it adds a positive constant to the diagonals

of ATA [6]

The form of Tikhonov regularization is

(16)

RŒ3Œ with σ • σ •… , σŒ• 0, σ‘ is the i-th singular value From (15), we have:

‘ “

• )

regularization parameter This parameter affects the filter as follows: i) a small z is not affected in a large σ‘ (z ≪ σ‘), i.e

= •–—

•–—k˜I•–—

•–— k˜I•–—

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

responsible for the instability of the matrix equation

Figure 2 Tikhonov filter function

In work [8], Aoki et al indicated that solving min

N O ∈P QR3SR , TR ∈P QRU21 D J1 1 K 21 UVis equivalent

1 D ∗

(14)

Trang 4

V 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

least-squares problem we need to generate

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

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 1 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

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

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%

When solving with initial CNM and the proposed

approaches after Nsum iterations, relative error residuals

(RRE) are presented in Table1 The first stage 1 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

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 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 3 After iterations 1 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 5 and 7 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

TABLE 1 PERFORMANCE COMPARISONS

Figure 3 Prediction of CPT-11 blood levels using the 500 sets of parameters

by different approaches

c

0 200 400 600 800 1000 1200 0

0.5 1 1.5 2 2.5 3 3.5

0 200 400 600 800 1000 1200 0

0.5 1 1.5 2 2.5 3 3.5

0 200 400 600 800 1000 1200 0