A Semi Analytic Solution of the Caputo-Fabrizio Fractional Order Mathematical Model for Cancer Treatment by Stem Cells and Chemotherapy

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A Semi Analytic Solution of the Caputo-Fabrizio Fractional Order Mathematical Model for Cancer Treatment by Stem Cells and Chemotherapy

1M. Abubakar, 2D. G. Yakubu, 3M. Abdulhameed, 4A. M. Kwami and 5Auwal Abdullahi

1,2,4Abubakar Tafawa Balewa University, (ATBU), Bauchi,

3Federal Polytechnic Bauchi, 5Federal University Kashere

Abstract:- In this paper we develop an algorithm for the semi analytic solution of the Caputo-Fabrizio fractional order mathematical model for cancer treatment by stem cell and chemotherapy using Laplace Adomian decomposition method. The recursive iterative schemes for the model were solved using the model parameters and the given initial conditions, the result obtained is in good agreement with the one presented by Alqudah (2020) validating LADMs efficiency and accuracy in finding the approximate analytic solution of the Caputo Fabrizio fractional order mathematical model for cancer treatment by stem cell and chemotherapy.

Keywords: Cancer disease, stem cells, effector cells, tumor cell, chemotherapy, Caputo-Fabrizio.

1.0 INTRODUCTION

Cancer is a disease which is non communicable but lethal, it constitutes 12% of all deaths globally where new cases and death from the disease keep rising [1]. According to [2] Cancer is responsible for 72,000 deaths in Nigeria every year, with an estimated 102,000 new cases of cancer annually, [3] added that about 116,000 new cases were recorded in only 2018 and about 41,000 died of cancer-related in Nigeria. They also added that if proper care is not taken against cancer risk factors, the cancer burden in Africa is projected to double from 1,055,172 new cancer cases in 2018 to 2,123,245 by 2040. The overall age- standardized cancer incidence rate is almost 25% higher in men than in women, with rates of 205 and 165 per 100,000, respectively. Male incidence rates vary almost five-fold across the different regions of the world, with rates ranging from 79 per 100,000 in Western Africa to 365 per 100,000 in Australia/New Zealand." (NNCCP 2018 2022).

There are more than two hundred (200) different kind of cancer that exist worldwide but the common among them include breast and cervical cancer which are common cancer in the world and the most frequent cancer among women, prostate cancer and liver cancer common in men, colorectal cancer which is common cancer in men and in women worldwide [4]. Some of the common signs and symptoms of cancer may include pyrexia, ail, tiredness, changes in skin appearance (redness, sores that won't heal, jaundice, darkening), unplanned weight loss or weight gain, lumps or tumors (mass), inconvenient swallowing, changes and difficulties with bowel or bladder function, never ceasing cough or throatiness, curtly of breath, chest pains, bleeding and discharges that cant be explained [5], [6] and [7].

Cell mutations and other factors that assist in damaging the DNA eventually led to cancer, and these factors include air pollution, smoking, heavy alcohol drinking, eating a poor diet, obesity, exposure to chemicals, and other toxins [8]. And if these triggering factors are prevented it will go a long way in minimizing the risk of effecting with cancer [9]. Different clinical procedures are used in the treatment and management of cancer include radiotherapy, chemotherapy, surgery, and immunotherapy which depend upon the patient's condition, location of the tumor, and the stage of cancer [10].

It has been for many years back biological processes are being modeled mathematically to provide means of a better understanding of the processes. [11], [12], [13], [14], [13], [15], [16], [17], [18], [16] and [19] have developed several linear, nonlinear and stochastics classical mathematical models that provided a comprehensive understanding of hematopoietic stem cell dynamic, the coexistence between the healthy and mutated cells, the rapid and uncontrolled proliferation of immature cells that invade the circulating blood which result to cancer disease.

In understanding more on the dynamics of cancer illness, and in particular to find theoretical conditions for efficient delivery of drugs under various treatments such as chemotherapy, immune therapy, radiotherapy, and to choose suitable dosages, durations and frequencies so many works have been carried out which include [20], [21], [20], [22], [23] and [24]. A new classical mathematical model of cancer treatment by stem cells and chemotherapy was developed [25] which modify the patient immune system and subsequently provide hope of a cure for the patient.

Thus, the existing model

dS

dt

1 S ks MS

dE E

dt

p1 ES

(S 1)

  • p2

    T M E

    (1)

    dT

    dT

    r1 bT T p3 E kT

    dt

    M T

    dM

    dt

    2 M V t

    where S t E t T t , and M t are the concentration of stem cells, effector cells, tumor cells, and chemotherapy concentration

    drug with the initial conditions S0 S

    , E0 E , T 0 T , and M 0 0 ifV

    0 . 0 t

    0 0 0 0

    Table 1: Model Parameters and their description

    PARAMETER

    DESCRIPTION

    VALUE

    S

    Stem Cells

    1

    E

    Effector Cells

    1

    T

    Tumor Cells

    1

    1

    The Decay Rate of Concentration of the Stem Cells

    -0.02825

    Rate of Produced the Effector Cells

    0.17

    The Natural Death Rate of the Effector Cells

    0.03

    b

    Carrying Capacity of the Tumor Cells

    10-9

    ks

    Fractional Stem Cells killed by Chemotherapy

    1

    p1

    Maximum Proliferation Rate of the Effector Cells

    0.1245

    r

    Tumor Growth Rate

    0.18

    p2

    Decay Rate of the Effector Cells killed by Tumor Cells and Chemotherapy

    1

    kT

    Fractional Tumor Cells killed by Chemotherapy

    0.9

    p3

    Decay Rate of the Tumor Cells killed by the Effector Cells

    0.9

    2

    Decay Rate of Chemotherapy Drug

    6.4

    V (t)

    The time dependent external influx of Chemotherapy Drug

    1

    Many areas of nowadays researches such as physics, engineering, biological sciences, finance, economics and other related areas have been observed to make a shift from classical based models that were based on integer order derivatives to fractional order derivatives. Because fractional derivatives can provide a better agreement between measured and simulated data than the classical derivatives, the fractional derivatives have a special characteristics of memory effect that depends not only upon its current state but also upon all of its historical states which does not apply to classical derivatives [26], [27] and [28]. These additional properties increase the accuracy and reliability of fractional order systems than the ordinary or classical order systems as testified by the works of [26]. [27]. [28], [29], [30], [31], [32], [33], [34], [35], [36], [37].

    2.0 FRACTIONAL DIFFERNTIAL EQUATIONS

    Fractional Calculus is one of the branches of mathematics that investigate the properties of integrals and derivatives of non integer orders which involves the notion and methods of solving the differential equations that involves fractional derivatives of the unknown function. There are three important definitions of fractional differential equations, they are the Riemann-Liouville, the Grunwald-Letnikov, and the Caputo definitions.

    ()() = lim ( )

    (1)

    (+1)

    ( ()) Grunvald Letnikov fractional derivative.

    0

    =0

    1 x

    !(+1)

    I ( ) ( f )

    (x u) 1 f (u)du

    ( ) a

    Reiman Liovelle Fractional Integral

    1 x

    a x (1 )

    c D f (x) (x t) f ' (t)dt

    a

    Caputo Fractional Derivative

    Now let [0,1] f (x) H1 (a,b) for a b such that

    M (a) x

    ( x t )

    a x 1

    CF D f (x) e

    0

    1

    f ' (t)dt

    Caputo Fabrizio Fractional Derivative

    where

    M ( ) is called Normalized function,

    H 1 (a,b)

    is a space called SOBOLEV SPACE, it contains all those functions

    { f L2 (a,b) : f ' L2 (a,b)}where

    b

    L2 (a,b) is all those function which were defined on this interval

    (a, b) such that

    f (x) 2 dt integrable function.

    a

    Thus, the existing model can be extended to Caputo Fabrizio sense as

    CF Dq S (t) q S (t) k q M (t)S (t)

    CF q

    1 s

    q q

    p1 E(t)S (t) q

    D E(t)

    E(t)

    (S 1)

    p2 T (t)

    M (t) E(t)

    q

    q

    (2)

    CF DqT (t) r q 1 bqT (t)T (t) pq E(t) k q M (t)T (t)

    3 T

    2

    2

    CF Dq M (t) q M (t) V t

      1. ADOMIAN DECOMPOSITION METHOD

        Among the several semi analytic methods for the analytical approximation of both linear and nonlinear problem is Adomian Decomposition Method (ADM). A great mathematician George Adomian from America developed the method to serve as a semi analytical method for solving numerous mathematical problems of partial and ordinary differential equation in linear and nonlinear system [38]. Now with extension to fractional order derivatives [39]. ADM provides efficient algorithm for the approximate analytical solutions [38]. ADM warrants the solution to both nonlinear initial value problems and boundary value problems without unphysical restrictive assumptions such as required by linearization, perturbation, adhoc assumptions, guessing the initial terms or a set of basic functions and so forth ( [40], [39], [41]). Hence ADM solves nonlinear operator equation for analytic non linearity by providing an easily computable, rapidly verifiable and rapidly convergent sequence of analytic approximate functions [42].

        Many analytical approximations have been carried out using ADM to provide an approximate semi analytic solution of real-life problems, among which are application of Adomian Decomposition method on Mathematical model of malaria [41], solution of typhoid fever model by Adomian decomposition method [43].

      2. Adomian Decomposition Method in solving system ODEs

    A system of ordinary differential equations of the first order as defined by [42], can be considered as

    y ' f x, y ,…y

    1 1 1 n

    y

    y

    f

    f

    '

    '

    2 2 x, y1 ,…yn

    .

    .

    .

    y

    y

    f

    f

    '

    '

    n n x, y1 ,…yn

    (3)

    where each equation represents the first derivative of one of the unknown functions as a mapping depending on the independent

    variable x , and n unknown functions

    f1,… fn .

    [42] has considered the solution of (3) by presenting the system (3) in the ith equation as

    x

    x

    Lyi f1 x, y1,…, yn

    i 1,2,…,n

    (4)

    where L is the linear operator

    d with inverse

    L1

    0 ()dx . Applying the inverse operator on (2) the following canonical

    dx

    dx

    form are suitable for applying ADM.

    x

    yi yi (0) 0 fi (x, y1 ,…,yn )dx

    i 1,2,…,n

    (5)

    As usual in ADM the solution of (3) is considered to be as the sum of a series

    yi fij

    j0

    And the integrand in (3) as the sum of the following series

    fi (x, y1 ,…,yn ) Aij ( fi0 ,…, fij )

    j0

    (6)

    (7)

    where

    Aij ( fi0 , fi1,…, fin ) are called Adomian polynomials.

    Substituting (6) and (7) into (5) we get

    x

    fij

    j0

    yi (0) 0 Aij ( fi0 ,…, fij )

    j0

    (8)

    x

    From which

    yi (0) 0 Aij ( fi0 ,…, f

    j0

    fi,0 yi (0)

    x

    x

    fi,n1 0 Aij ( fi,0 ,…, fi, j )dx

    n 0,1,2,…

    (9)

    4.0 CAPUTO FABRIZIO FRACTIONAL DERIVATIVE BY LAPLACE ADOMIAN DECOMPOSITION METHOD Fractional order mathematical models are also solved using the Laplace Adomian Decomposition techniques. This has

    been proved efficient through researches as [43] who carried out numerical solution of integro differential equations of fractional order by Laplace decomposition method. [39], on modelling of epidemic childhood diseases with Caputo Fabrizio derivative by using the Laplace Adomian Decomposition method.

    Taking Laplace of both side of the Caputo Fabrizio fractional derivative as

    cf

    1 x

    x t

    1

    '

    L 0 Dx f x L e f t dt

    (10)

    Since

    1 0

    1 is constant, we shift it outside

    1

    1 x

    x t

    1

    '

    0

    0

    1 L e f

    t dt dt

    (11)

    x

    Since e

    0

    xt

    1

    f ' t is a convolution of two function, therefore it is written as

    1

    1

    Lcf D f x 1 Le t f ' t

    (12)

    0 x 1

    By the property of convolution of two functions

    1

    1

    1 Le t Lf ' t

    (13)

    1

    Using the properties of Laplace transform

    1 1 SLf t f 0

    (14)

    1 s

    1

    Simplifying (12) we get

    1 1

    SF S F 0

    (15)

    1 S 1

    Hence the Laplace transform of Caputo Fabrizio fractional derivative is given as

    Lcf D f x SFS F0

    (16)

    0 x S 1 S

    Applying (16) the Laplace transform of the Caputo Fabrizio on both side of (2)

    SLS S 0 L q S (t) k q M (t)S (t)

    S q1 S

    SLT T 0

    1 s

    q q

    q

    q

    p1 E(t)S (t) q

    S q1 S

    L

    E(t)

    (S 1)

    p2 T (t)

    M (t) E(t)

    (18)

    SLE E0 Lr q 1 bqT (t)T (t) pq E(t) k q M (t)T (t)

    S q1 S 3 T

    SLM M 0 L q M (t) V t

    S q1 S 2

    By using the boundary conditions and taking inverse Laplace transform of (18)

    S t S 0 L1 S q1S L q S (t) k q M (t)S (t)

    s 1 s

    Et E0 L1 S q1S L q q E(t) pq E(t)S (t) pq T (t) M (t)E(t)

    s 1 2

    (19)

    s

    s

    3

    3

    T

    T

    T t T 0 L1 S q1S Lr q 1 bqT (t)T (t) pq E(t) k q M (t)T (t)

    M t M 0 L1 S q1S L q M (t) V t

    s 2

    Assume that the solutions S , E , T and M are in the form of infinite series

    S(t) Sn (t) , E(t) En (t) , T (t) Tn (t) and M (t) Mn (t)

    n0

    n0

    n0

    n0

    S t S 0 L1 S q1S L q S (t) k q M (t)S (t)

    s 1 s

    Et E0 L1 S q1S L q q E(t) pq E(t)S (t) pqT (t)E(t) pq M (t)E(t)

    s 1 2 2

    (20)

    s

    s

    3

    3

    T

    T

    T t T 0 L1 S q1S Lr q 1 bqT (t)T (t) pq E(t)T (t) k q M (t)T (t)

    M t M 0 L1 S q1S L q M (t) V t

    s 2

    By the Adomian polynomials, the nonlinear terms T t Et , M t Et , Et S t , M t S t and M t T t as follows

    An t T (t)E(t) , Bn t M t Et , Cn t Et St , Dn t M t St and Gn t M tT t

    n0

    n0

    n0

    n0

    (21)

    n0

    where

    An (t) , Bn (t) , Cn (t) , Dn (t) and Gn (t) are Adomian polynomials and defines as

    1 d n n

    n k

    An (t) n 1 dn Tk (t)

    Ek (t)

    (22)

    1 d n

    k 0

    n

    k 0

    n

    k

    0

    Bn (t) n 1 dn M k (t)

    Ek (t)

    (23)

    1 d n

    k 0

    n

    k 0

    n

    k

    0

    Cn (t) n 1 dn Ek (t)

    Sk (t)

    (24)

    1 d n

    k 0

    n

    k 0

    n

    k

    0

    Dn (t) n 1 dn M k (t)

    Sk (t)

    (25)

    1 d n

    k 0

    n

    k 0

    n

    k

    0

    Gn (t) n 1 dn M k (t) Tk (t)

    (26)

    k 0

    k 0

    0

    Now by applying (19-24) into (18) and taking the Laplace transform of both side we have

    L S

    t S0 S q1S L q S

    t k q

    D t

    n

    n0

    s s

    1 n

    n0

    s n

    n0

    L

    E t E0 S q1S L q q

    E t pq C

    t pq

    A t pq

    B t

    n

    n0

    s s

    n n0

    1. n n0

    2. n n0

    2 n

    n0

    L T

    t T 0 S q1S Lrq 1 bqT

    (t) T

    t ) pq

    A t k q

    G t

    s

    s

    n s

    n0

    n n n0

    3 n

    n0

    T n n0

    L M

    t M 0 S q1S L q M

    t V t

    n

    n0

    s s

    2 n

    n0

    (27)

    From (27) we get

    L(S ) L(S ) .L(S

    ) … S0 S q1S q L(S ) L(S ) .L(S

    ) … k q L(D ) L(D ) .L(D ) …

    0 1 2

    s s 1 0 1 2

    s 0 1 2

    On equating we have obtain

    S0

    L(S0 ) s

    L(S ) S q1 S q L(S ) k q L(D )

    1 s 1 0 s 0

    .L(S ) S q1 S q L(S ) k q L(D )

    2 s 1 1 s 1

    .

    .

    .

    (28)

    L(S

    ) S q1 S q L(S

    ) k q L(D )

    n1

    s 1 n s

    n

    q

  • q LE

    LE

    …. pq LC

    LC

    LE

    LE . … E 0

    S q1S s 0 0

    1 0 0

    0 1 s

    s

    pq LA

    LA

    … pq LB

    LB

    LE0

    E0

    s

    2 0 0

    2 0 0

    (29)

    S q1S q

  • q

    q

    q

    q

    L E1 .

    s s

    L E0

    p1 L C0

    p2 L A0

    p2 L B0

    S q1S q

  • q

    q

    q

    q

    L E2

    .

    .

    .

    s s

    L E1

    p1 L C1

    p2 L A1

    p2 L B1

    (30)

    S q1S q

  • q

    q

    q

    q

    q q

    q q

    L En1 s s

    L En

    p1 L Cn

    p2 L An

    p2 L Bn

    T 0 S q1S r LT LT …) (b LN LN …

    0 1 0 1

    0 1 0 1

    L T0 L T1

    s s

    pq (LA

    LA ) … pq k q LG

    LG …

    3 0 1

    3 T 0

    1

    (32)

    LT0

    T 0

    s

    LT S q1S r q LT bq LN pq (LA k q LG

    1 s 0 0 3 0 T 0

    LT S q1S r q LT ) (bq LN pq (LA k q LG

    2 s 1

    .

    .

    .

    1 3 1 T 1

    (31)

    LT

    S q1S r q LT

    ) (bq LN

    pq (LA

    k q LG

    n1 s n

    .

    n 3 n T n

    LM

    LM LM

    … M 0 S q1S q LM LM LM

    … V t

    (32)

    0

    LM 0

    1 2

    M 0

    s

    s s 2 0 1 2

    LM

    S q1S q LM

    V t

    1 s 2 0

    s

    LM

    S q1S q LM

    V t

    2 s 2 1

    .

    .

    .

    s

    (33)

    LM

    S q1S q LM

    V t

    n1

    s 2

    n1

    s

    Summary of the Laplace inverse of the system of equation and the iterative schemes are as follows

    S0 S0 , E0 E0 , T0 T0 , M0 M 0 = 1 (34)

    1

    1

    1

    1

    0

    0

    S q S

  • k q M

0 S0

1 q(t 1)

(35)

s

s

E q q M pq E S pqT E

1 q(t 1)

(36)

1 0 1 0 0 2 0 0

T r q bqT T pq (T E k q M T )(1 q(t 1))

(37)

2

2

1 0 0

3 0 0

T 0 0

2

2

1

1

0

0

M q M

qV (t)(1 q(t 1))

(38)

  1. q q S

    • k q M S k q M S

    • k q M S

      1 q(t 1)

      2 1 1 0

      s 0 0

      s 0 1

      s 1 0

      q q S

    • k q M S

      k q M S

    • k q M S

      1 q(t 1)

      (39)

      S 1 1 0

      s 0 0

      s 0 1

      s 1 0

      1 q(t 1)

      2 ( q M qV (t))1 q(t 1)S

      2 0 2 0

      E q q M pq C pq A 1 q(t 1)

      2 1 1 1 2 1

      E q q q M qV (t)1 q(t 1) pq E S

      • E S

        pq T E

      • T E

        1 q(t 1)

        2 2 0 2

        1 0 1 1 0

        2 0 1 1 0

        1

        1

        1

        1

        0

        0

        0

        0

        s

        s

        0

        0

        0

        0

        0

        0

        q q q M qV (t)1 q(t 1) pq q E S

        k q E M S 1 q(t 1)

        2 0 2

        0

        0

        S

        q q M

      • pq E S

    • pqT E

      1 q(t 1)

      E2

      2

      0

      2

      0

      0 1 0 0

      0

      1

      0

      1

      2 0 0

      0

      2

      0

      2

      0

      0

      0

      0

      1 q(t 1)

      pq T q q M

      • pq E S pqT E

        1 q(t 1)

        0

        0

        E r q bqT T

    • pq T E k q M

      T 1 q(t 1)

      0 0 0

      3 0 0

      t 0 0

      (40)

      T r q bqT T pq A k q G 1 q(t 1)

      2 1 1 3 1 T 1

      r q bq r q bqT T pq T E

      <> k q M T 1 q(t 1)r q bqT T

      0 0 3 0 0 T 0 0

      0 0

      T

      q q M

      • pq E S pqT E

        pq T E k M T 1 q(t 1) pq 0

        0 1 0 0 2 0 0

3 0 0

T 0 0

3 1 q(t 1)

1 q(t 1)

2 E r q bqT T

  • pq T E

    k q M T 1 q(t 1)

    0 0 0 3 0 0

    T 0 0

    r q bqT T

    pq T E k q M T 1 q(t 1)

    k q M

    0

    0

    0 0 3 0 0

    T 0 0

    T

    q M

    q 1 q(t 1)T

    2 0 2 0

    (41)

    2

    2

    2

    2

    1

    1

    2

    2

    M q M qV (t)1 q(t 1)

    2

    2

    2

    2

    0

    2

    2

    2

    0

    2

    (42)

    M 2 M V (t) 1 q(t 1)

    q q q

    2

    2

    0

    2

    2

    2

    0

    2

    qV (t) 1 q(t 1)

    In a similar passion we obtain the recursive iterative formula for Caputo Fabrizio fractional order mathematical model for cancer treatment by stem cell and chemotherapy as

    S 1 q(t 1) q S k q D

    n1

    1 n s n

    n

    n

    n

    n

    z n

    z n

    E 1 q(t 1) q q E

    • pq C

    1

    1

  • pq A

  • pq B

z

n

z

n

n1

n

n

3

n

T

n

n

n

3

n

T

n

(43)

2

2

n

n

Tn1

1 q(t 1) r q bqT T pq A k q G

Mn1

1 q(t 1) q M

V (t)

4.0 RESULTS

Concentration of Cells

Concentration of Cells

In order to validate our semi analytic solution of the Caputo Fabrizio Fractional order mathematical model by the derived recursive iterative scheme through Laplace Adomian Decomposition method we consider the value of q = 1 corresponding to the classical model of Alqudah (2020) which is shown in figure 1a. where the present result presented in figure 1b is in good agreement with the result presented by Algudah (2020).

1.2

1

0.8

0.6

0.4

0.2

0

S(t)

E(t)

T(t)

M(t)

1.2

1

0.8

0.6

0.4

0.2

0

S(t)

E(t)

T(t)

M(t)

1 2 3 4 5 6 7 8 9 10 11

Time(t)

1 2 3 4 5 6 7 8 9 10 11

Time(t)

Fig. 1a: Numerical solution of the classical by Alqudah (2020) Fig. 1b: Semi Analytic Solution by the LADM

We now give the numerical simulation of the individual compartment with varying values of the fractional order; the result is presented below.

Concentration of Stem

Cells

Concentration of Stem

Cells

1.2

1

0.8

0.6

q = 1

q = 0.85

1.2

Concentration of Effectors

Cells

Concentration of Effectors

Cells

1

0.8

0.6

q = 1

q = 0.85

0.4 q = 0.90

0.2 q = 0.95

0.4 q = 0.90

0.2 q = 0.95

0

1 3 5 7 9 11

Time(t)

0

q = 0.98

1 3 5 7 9 11

Time(t)

q = 0.98

Figure 2: Concentration of Stem Cells with varying values of q Figure 3: Concentration of Effector Cells with varying values of q

Concentration of Tumor

Cells

Concentration of Tumor

Cells

1.2

1

0.8

0.6

q = 1

q = 0.85

1.2

Concentration of

Chemotheraphy

Concentration of

Chemotheraphy

1

0.8

0.6

q = 1

q = 0.85

0.4 q = 0.90

0.2 q = 0.95

0

0.4 q = 0.90

0.2 q = 0.95

0

1 3 5 7 9 11

Time(t)

q = 0.98

1 3 5 7 9 11

Time(t)

q = 0.98

Figure 4: Concentration of Tumor Cells Figure 5: Concentration of Chemotherapy Drug

5.0 CONCLUSION

In this paper we develop a recursive iterative formula for the solution of Caputo-Fabrizio fractional order mathematical model for cancer treatment by stem cell and chemotherapy. The result obtained by this method is in good agreement with classical solution of Alqudah (2020) as shown in figure 1a and 1b. Furthermore, numerical simulations were carried out at different value of q for each of the individual cells were presented in both tables and figures.

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