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Research article Special Issues

Application of control theory in a delayed-infection and immune-evading oncolytic virotherapy

  • Received: 16 December 2019 Accepted: 05 February 2020 Published: 12 February 2020
  • Oncolytic virotherapy is a promising cancer treatment that harnesses the power of viruses. Through genetic engineering, these viruses are cultivated to infect and destroy cancer cells. While this therapy has shown success in a range of clinical trials, an open problem in the field is to determine more effective perturbations of these viruses. In this work, we use a controlled therapy approach to determine the optimal treatment protocol for a delayed infection from an immune-evading, coated virus. We derive a system of partial differential equations to model the interaction between a growing tumour and this coated oncolytic virus. Using this system, we show that viruses with inhibited viral clearance and infectivity are more effective than uncoated viruses. We then consider a hierarchical level of coating that degrades over time and determine a nontrivial initial distribution of coating levels needed to produce the lowest tumour volume. Interestingly, we find that a bimodal mixture of thickly coated and thinly coated virus is necessary to achieve a minimum tumour size. Throughout this article we also consider the effects of immune clearance of the virus. We show how different immune responses instigate significantly different treatment outcomes.

    Citation: Taeyong Lee, Adrianne L. Jenner, Peter S. Kim, Jeehyun Lee. Application of control theory in a delayed-infection and immune-evading oncolytic virotherapy[J]. Mathematical Biosciences and Engineering, 2020, 17(3): 2361-2383. doi: 10.3934/mbe.2020126

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  • Oncolytic virotherapy is a promising cancer treatment that harnesses the power of viruses. Through genetic engineering, these viruses are cultivated to infect and destroy cancer cells. While this therapy has shown success in a range of clinical trials, an open problem in the field is to determine more effective perturbations of these viruses. In this work, we use a controlled therapy approach to determine the optimal treatment protocol for a delayed infection from an immune-evading, coated virus. We derive a system of partial differential equations to model the interaction between a growing tumour and this coated oncolytic virus. Using this system, we show that viruses with inhibited viral clearance and infectivity are more effective than uncoated viruses. We then consider a hierarchical level of coating that degrades over time and determine a nontrivial initial distribution of coating levels needed to produce the lowest tumour volume. Interestingly, we find that a bimodal mixture of thickly coated and thinly coated virus is necessary to achieve a minimum tumour size. Throughout this article we also consider the effects of immune clearance of the virus. We show how different immune responses instigate significantly different treatment outcomes.


    Telegraph equation is introduced by Oliver Heaviside and is a linear second-order hyperbolic partial differential equations that describe the current and voltage on an electrical transmission line with distance. The model demonstrates that wave patterns can form along the line and that the electromagnetic waves can be reflected on the wire. The nonhomogeneous telegraph equation with boundary and initial conditions is given by

    utt(x,t)+2αut(x,t)+β2u(x,t)=uxx(x,t)+f(x,t),(x,t)Ω×Ω,α>β0, (1.1)

    with initial and Dirichlet boundary conditions

    u(x,0)=f0(t),ut(x,0)=f1(x),xΩ, (1.2)
    u(0,t)=g0(x),g(1,t)=g1(t),tΩ, (1.3)

    where α and β are the real constants and Ω:=[0,1]. This equation referred to as the second-order hyperbolic partial differential equation with constant coefficients, models a mixture between wave propagation and diffusion by introducing a term that accounts for effects of finite velocity to the standard heat or mass transport equation. This equation represents a damped wave motion when α>0 and β=0.

    In this paper, we employ the multi-wavelet Galerkin method to solve nonhomogeneous telegraph equation (1.1) with initial (1.2) and boundary conditions (1.3). The Alpert's multi-wavelet bases are infinitely differentiable while having small compact support. In other words, these bases have combined the advantages of both finite difference and spectral bases. The first application of Alpert's multi-wavelet bases to the solution of PDEs is the adaptive solution of nonlinear time-dependent PDEs [1]. In this approach, the multiresolution representation of the derivative operator introduced, and then an adaptive solver developed for both linear and nonlinear PDEs. Further, multi-wavelet methods have been developed for PDEs such as conservation laws [10,16,23]. For other similar studies to related PDEs we refer to [4,5,6,7]

    The required conditions for the existence of a unique solution to a nonhomogeneous telegraph equation with initial and boundary conditions an integral boundary condition via Galerkin's method investigated in [3,14]. The reproducing kernel Hilbert space method is utilised to solve this equation [3]. Lakestani et al. [19] employed a numerical solution based on the Galerkin and collocation method to solve this equation appropriately. In [18], the authors proposed the differential quadrature algorithm to obtain an approximate solution of the two-dimensional telegraph equation. A fast and simple method based on the Chebyshev wavelets method is proposed by Heydari et al., [15]. In this paper, the matrices of integration and differentiation are applied to reduce complexity. Mittal et al. [20] used cubic B-spline collocation method, whereas Dehghan and Shorki [11] proposed an algorithm based on thin plates spline radial basis functions using collocation points for solving this equation. A high accuracy method for the long-time evolution of the acoustic wave equation is introduced in [21]. Authors of [8] presented dual reciprocity boundary integral equation method. Due to the importance of this equation, many numerical methods have been proposed to solve the telegraph equation such as Quardatic B-spline collocation method [12], collocation method based on Chebyshev cardinal function [9], Lagrange interpolation and modified cubic B-spline differential quadrature methods [17], a hybrid meshless method [26], generalized finite difference method [25].

    The paper is structured as follows. A brief introduction of the Alpert's multi-wavelets is provided in Section 2. In Section 3, the wavelet Galerkin method is used to approximate the solution of the problem, and the convergence analysis is investigated. Some numerical experiments are solved to illustrate the efficiency and accuracy of the proposed method in Section 4. finally conclusions are included in Section 5.

    Assume that Ω:=[0,1]=bBXJ,b is the finite discretizations of Ω, where XJ,b:=[xb,xb+1],bB:={0,,2J1} with JZ+{0}, are determined by the point xb:=b/(2J). On this discretization, appling the dilation D2j and the translation Tb operators to primal scaling functions {ϕ00,0,,ϕr10,0}, one can introduce the subspaces

    VrJ:=Span{ϕkj,b:=D2jTbϕk,bBj,kR}L2(Ω),r0,

    of scaling functions. Here R={0,1,,r1} and the primal scaling functions are the Lagrange polynomials of degree less than r that introduced in [1].

    Every function pL2(Ω) can be represented in the form

    pPrJ(p)=bBJkRpkJ,bϕkJ,b, (2.1)

    where .,. denotes the L2-inner product

    f,g=10|fg|dx,

    and PrJ is the orthogonal projection that maps L2(Ω) onto the subspace VrJ. To find the coefficients pkJ,b that are determined by p,ϕkJ,b=XJ,bf(x)ϕkJ,b(x)dx, we shall compute these integrals. We apply the r-point Gauss-Legendre quadrature by a suitable choice of the weights ωk and nodes τk for kR to avoid these integrals [1,24], via

    pkJ,b2J/2ωk2p(2J(τk+12+b)),kR, bBJ, (2.2)

    Convergence analysis of the projection PrJ(p) is investigated for the r-times continuously differentiable function pCr(Ω).

    PrJ(p)p2Jr24rr!supx[0,1]|p(r)(x)|. (2.3)

    For the full proof of this approximation and further details, we refer the readers to [2]. Thus we can conclude that PrJ(p) converges to p with rate of convergence O(2Jr).

    Let ΦrJ be the vector function ΦrJ:=[Φr,J,0,,Φr,J,2J1]T and consists of vectors Φr,J,b:=[ϕ0J,b,,ϕr1J,b]. The vector function ΦrJ includes the scaling functions and called multi-scaling function. Furthermore, by definition of vector P that includes entries Pbr+k+1:=pkJ,b, we can rewrite Eq (2.2) as follows

    PrJ(p)=PTΦrJ, (2.4)

    where P is an N-dimensional vector (N:=r2J). The building blocks of these bases construction can be applied to approximate a higher-dimensional function. To this end, one can introduce the two-dimensional subspace Vr,2J:=VrJ×VrJL2(Ω)2 that is spanned by

    {ϕkJ,bϕkJ,b:b,bBJ,k,kR}.

    Thus by this assumption, to derive an approximation of the function pL2(Ω)2 by the projection operator PrJ, we have

    pPrJ(p)=bBjr1k=0r1k=0bBjPrb+(k+1),rb+(k+1)ϕkJ,b(x)ϕkJ,b(y)=ΦrJT(x)PΦrJ(y), (2.5)

    where components of the square matrix P of order N are obtained by

    Prb+(k+1),rb+(k+1)2Jωk2ωk2p(2J(ˆτk+b),2J(ˆτk+b)), (2.6)

    where ˆτk=(τk+1)/2. Consider the 2r-th partial derivatives of f:Ω2R are continuous. Utilizing this assumption, the error of this approximation can be bounded as follows

    PrJppMmax21rJ4rr!(2+21Jr4rr!), (2.7)

    where Mmax is a constant.

    By reviewing the spaces VrJ, it is obvious these bases are nested. Hence there exist complement spaces WrJ such that

    VrJ+1=VrJWrJ,JZ{0}, (2.8)

    where denotes orthogonal sums. These subspaces are spanned by the multi-wavelet basis

    WrJ=Span{ψkJ,b:=D2JTbψk:bBJ,kR}.

    According to (2.8), the space VJ may be inductively decomposed to VrJ=Vr0(J1j=0Wrj). This called multi-scale decomposition and spanned by the multi-wavelet bases and single-scale bases. This leads us to introduce the multi-scale projection operator MrJ. Assume that the projection operator Qrj the maps L2(Ω) onto Wrj. Thus we obtain

    pMrJ(p)=(Pr0+J1j=0Qrj)(p), (2.9)

    and consequently, any function pL2(Ω) can be approximated as a linear combination of multi-wavelet bases

    pMrJ(p)=r1k=0pk0,0ϕk0,0+J1j=0bBjkR˜pkj,bψkj,b, (2.10)

    where

    pk0,0:=p,ϕk0,0,˜pkj,b:=p,ψkj,b. (2.11)

    Note that, we can compute the coefficients pk0,0 by using (2.2). But multi-wavelet coefficients from zero up to higher-level J1 in many cases must be evaluated numerically. To avoid this problem, we use multi-wavelet transform matrix TJ, introduced in [22,24]. This matrix connects multi-wavelet bases and multi-scaling functions, via,

    ΨrJ=TJΦrJ, (2.12)

    where ΨrJ:=[Φr,0,b,Ψr,0,b,Ψr,1,b,,Ψr,J1,b]T is a vector with the same dimension ΦrJ (here Ψr,j,b:=[ψ0j,b,,ψr1j,b]). This representation helps to rewrite Eq (2.10) as to form

    pMrJ(p)=˜PJTΨrJ, (2.13)

    where we have the N-dimensional vector ˜PJ whose entries are pk0,0 and ˜pkj,b and is given by employing the multi-wavelet transform matrix TJ as ˜PJ=TJPJ. Note that according to the properties of TJ we have T1J=TTJ.

    The multi-wavelet coefficients (details) become small when the underlying function is smooth (locally) with increasing refinement levels. If the multi-wavelet bases have Nrψ vanishing moment, then details decay at the rate of 2JNrψ [16]. Because vanishing moment of Alpert's multi-wavelet is equal to r, one can obtain ˜pkJ,bO(2Jr) consequently. This allows us to truncate the full wavelet transforms while preserving most of the necessary data. Thus we can set to zero all details that satisfy a certain constraint ε using thresholding operator Cε

    Cε(˜PJ)=ˉPJ, (2.14)

    and the elements of ˉPJ are determined by

    ˉpkj,b:={˜pkj,b,(j,b,k)Dε,0,else,bBj, j=0,,J1,k=0,,r1, (2.15)

    where Dε:={(j,b,k):|˜pkj,b|>ε}. Now we can bound the approximation error after thresholding via

    PrJpPrJ,DεpL2(Ω)Cthrε, (2.16)

    where PrJ,Dε(p) is the projection operator after thresholding with the threshold ε and Cthr>0 is constant independent of J,ε.

    Let us consider the generalized telegraph equations (TE) on the region Ω×Ω governed by the partial differential equation

    2ut2(x,t)+2αux(x,t)+β2u(x,t)=2ux2(x,t)+f(x,t),(x,t)Ω×Ω,α>β0, (3.1)

    with initial and Dirichlet boundary conditions

    u(x,0)=f0(t),ut(x,0)=f1(x),xΩ, (3.2)
    u(0,t)=g0(x),g(1,t)=g1(t),tΩ. (3.3)

    In order to derive the multi-wavelet Galerkin method for solving TE (3.1), we assume that the approximate solution u can be expanded by the Alpert's multi-wavelet bases ΨrJ, i.e.,

    u(x,t)uJ(x,t)=ΨrJT(x)UΨrJ(t). (3.4)

    Taking the first and second derivative with respect to x and t from both sides of the Eq (3.4), one can get

    ut(x,t)ΨrJT(x)UDψΨrJ(t),utt(x,t)ΨrJT(x)UD2ψΨrJ(t),ux(x,t)ΨrJT(x)DTψUΨrJ(t),uxx(x,t)ΨrJT(x)D2ψTUΨrJ(t), (3.5)

    where the matrix Dψ is used to represents the derivative of multi-wavelet defined by [10,23].

    Inserting (3.4) into (3.1) and employing (3.5) we obtain the residual via

    rrJ(x,t)=ΨrJT(x)(UD2ψ+2αUDψ+β2UD2ψTUψ˜FT)ΨrJ(t), (3.6)

    where the vector ˜F is obtained the same way as the direction in (2.5) i.e.,

    fΨrJT(x)˜FΨrJ(t),

    with ˜F=TJFT1J. The Galerkin method requires rrJ to satisfy rrJ,ψkj,bψkj,b=0. Multiplying (3.6) by ΨrJ(x) from left and ΨrJT(t) from right and integrating, we end up with

    Λ:=UD2ψ+2αUDψ+β2UD2ψTU˜FT=0, (3.7)

    where we employ orthonormality of multi-wavelet bases and the local support of these bases.

    Equation (3.7) gives (N2)2 independent equations

    Λi,j=0,i=3:N, j=2:N1.

    We obtain 4N4 other equations from boundary conditions (3.1) and (3.2) via equations (3.3) and (3.5),

    UΨrJ(0)=˜F0,UDψΨrJ(0)=˜F1,ΨrJT(0)U=˜GT0,ΨrJT(1)U=˜GT1.

    The problem becomes a system of linear equations with N2 equations and N2 unknowns,

    AU=Υ, (3.8)

    where U and Υ are the vectorization of U and ˜F, respectively. It should be noted here that since most of the elements of the matrix A are zero, We use appropriate methods such as the generalized minimal residual method (GMRES). After solving this system the approximate solution is implicitly represented by (3.4).

    Convergence analysis

    To investigate the convergence analysis of the multi-wavelet Galerkin method, we put

    2uJt2(x,t)+2αuJx(x,t)+β2uJ(x,t)=2uJx2(x,t)+fJ(x,t), (3.9)

    subtracting this equation from (3.1), we obtain

    e(x,t):=2eJt2(x,t)+2αeJx(x,t)+β2eJ(x,t)2eJx2(x,t)(f(x,t)fJ(x,t)), (3.10)

    where eJ:=uuJ. Taking L2-norm from both sides and using the triangle inequality yields

    e22eJt2(x,t)+2αeJx(x,t)+β2eJ(x,t)2+2eJx2(x,t)2+f(x,t)fJ(x,t)2. (3.11)

    Now suppose that

    eJ(x,t):=ΨrJT(x)EΨrJ(t),

    where E is the (N×N) matrix and thus, one can write

    e2ED2ψ+2αEDψ+β2E2+D2ψE2+FFJ2,E2D2ψ+2αDψ+β2IN+D2ψ2+FFJ2,

    where we utilize the orthonormality of multi-wavelet bases. According to the previous section, for any function p, when multi-wavelet bases hava high vanishing moments and the function p is smooth, p,ΨrJ decays fast in J. By means of vanishing moments of Alpert's multi-wavelets and the matrix norms inequalities, we get

    e2NE2D2ψ+2αDψ+β2IN+D2ψ+FFJ2

    Obviously, using (2.7), we can find

    e22η21rJ4rr!(2+21Jr4rr!), (3.12)

    where η:=κMN with κ:=D2ψ+2αDψ+β2IN+D2ψ and M is a constant. Consequently, e20 when J.

    To show the efficiency and accuracy of will employ the proposed method to obtain the approximate solution of the following examples. All of the computations have been done by Maple and MATLAB simultaneously.

    Example 4.1. Assume the telegraph equation (3.1) with initial and boundary conditions (3.2) and (3.3). Let

    f0(x)=tan(x2),f1(x)=12(1+tan2(x2)),g0(t)=tan(t2),g1(t)=tan(1+t2),

    and function

    f(x,t)=α(1+tan2(x+t2))+β2tan(x+t2).

    The exact solution of this equation is u(x,t)=tan(x+t2) [8,20].

    In Table 1, with fixed α=10 and β=5, choosing different refinement levels J and multiplicity r guarantees our convergence investigation. In Table 2, results are also compared with other methods [12,20] in terms of L,L2 errors at different times. In [12,20], The space and time discretized by the rate of 0.001 while for the proposed method this is 0.25. In view of this and the results, our method is better much more than them. Taking r=7 and J=2, the approximate solution and L errors are shown in Figure 1.

    Table 1.  Effects of parameters r and J on L, L2 errors for Example 1.
    r J t=0.2 t=0.4 t=0.6 t=0.8 t=1.0
    4 2 L 5.96e5 9.54e5 9.54e5 5.96e5 1.21e5
    L2 1.84e6 2.39e6 4.34e5 7.12e5 5.46e5
    3 L 2.77e5 4.43e5 4.43e5 2.77e5 1.23e5
    L2 9.56e7 1.13e6 2.05e5 3.30e5 2.52e5
    5 2 L 2.98e5 2.55e5 2.54e5 3.01e5 1.3e5
    L2 6.85e7 1.17e6 1.35e6 1.63e6 2.50e5
    3 L 2.09e5 1.48e5 1.48e5 2.09e5 1.23e5
    L2 4.59e7 7.80e7 9.22e7 1.12e6 1.66e5

     | Show Table
    DownLoad: CSV
    Table 2.  Comparison of the L,L2 errors using presented method and others taking α=10 and β=5 for Example 1.
    presented method Mittal et al. [20] Dosti et al. [12]
    t L Ł2 L Ł2 L Ł2
    0.2 1.14e8 1.94e8 3.61e4 2.18e4 2.77e4 3.32e8
    0.4 2.96e8 3.98e8 1.04e4 5.66e4 7.18e4 2.31e7
    0.6 5.36e7 5.94e7 2.60e3 1.15e4 1.38e3 8.21e7
    0.8 1.13e6 5.88e7 7.63e3 2.61e3 3.09e3 3.24e6
    1.0 5.82e7 8.02e7 4.66e2 1.04e2 1.34e2 3.28e5

     | Show Table
    DownLoad: CSV
    Figure 1.  Plot of the approximate solution (left) and L errors (right) taking r=7 and J=2 for Example 1.

    Example 4.2. In this example, we consider u(x,t)=(xx2)t2et is the analytical solution of the telegraph equation (1.1) with initial and boundary conditions

    f0(x)=f1(x)=0,g0(t)=g1(t)=0,

    and

    f(x,t)=(22t+t2)(xx2)et+2t2et.

    L2,L errors are reported in Table 3 taking r=5 ad J=3. Results have been compared with the results of [11,20]. These results indicate that the proposed method solves this equation better than them. Time and space steps in these papers have been reported Δt=0.001 and h=0.01 while they are equal to 0.125 in our simulation. The graph of numerical solution uJ and L error are shown in Figure 2 and the exact and approximate solution at different values of the time t and space x are plotted in Figure 3.

    Table 3.  Comparison of the L,L2 errors using presented method and others taking α=8 and β=4 for Example 2.
    presented method Mittal et al. [20] Dehghan et al. [11]
    L Ł2 L Ł2 L Ł2
    t=1 2.81e017 1.31e06 5.92e5 4.55e5 1.85e5 1.44e4

     | Show Table
    DownLoad: CSV
    Figure 2.  Plots of approximate solution (left) and absolute value errors (right) with α=8 and β=4 taking r=5 and J=2 for Example 2.
    Figure 3.  Comparison of numerical and exact solution of Example 2 with α=8 and β=4, taking r=5 and J=2, at different values of the space x and time level t.

    Example 4.3. Consider the Eq (1.1) with the right hand side function

    f(x,t)=2αsin(t)sin(x)+β2cos(t)sin(x),

    and the initial and boundary conditions

    f0(x)=sin(x),f1(x)=0,g0(t)=0g1(t)=cos(t)sin(1),

    The exact solution of this problem is given in [8,20], as

    u(x,t)=cos(t)sin(x).

    The effect of multiplicity parameter r is show in Figure 4. Table 4, shows a comparison among the L,L2 errors for the proposed method and other methods [12,20]. Given this table, we can find the proposed method very flexible and better than others. The effects of the refinement level J and multiplicity parameter r on L,L2 errors are given in Table 5.

    Figure 4.  Effect of multiplicity parameter r on L and percentage of sparsity taking J=2 for Example 3.
    Table 4.  Comparison of the L,L2 errors using presented method and others taking α=6 and β=2 for Example 3.
    presented method
    h=0.25,Δt=0.25
    Mittal et al. [20]
    h=0.005,Δt=0.001
    Dosti et al. [12]
    h=0.005,Δt=0.001
    t L Ł2 L Ł2 L Ł2
    0.2 5.70e10 8.58e12 6.83e5 3.43e5 2.42e5 1.70e10
    0.4 3.83e10 1.34e11 1.49e4 8.58e5 7.93e5 2.67e9
    0.6 3.84e10 1.67e10 2.24e4 1.34e4 1.21e4 6.78e9
    0.8 5.74e10 3.81e10 2.90e4 1.75e4 1.49e4 1.07e8
    1.0 1.46e10 2.81e10 3.44e4 2.09e4 1.65e4 1.34e8

     | Show Table
    DownLoad: CSV
    Table 5.  Effects of parameters r and J on L, L2 errors for Example 3.
    r J t=0.2 t=0.4 t=0.6 t=0.8 t=1.0
    5 2 L 5.96e5 9.54e5 9.54e5 5.96e5 1.21e5
    L2 1.84e6 2.39e6 4.34e5 7.12e5 5.46e5
    3 L 2.77e5 4.43e5 4.43e5 2.77e5 1.23e5
    L2 9.56e7 1.13e6 2.05e5 3.30e5 2.52e5
    6 2 L 2.98e5 2.55e5 2.54e5 3.01e5 1.3e5
    L2 6.85e7 1.17e6 1.35e6 1.63e6 2.50e5
    3 L 2.09e5 1.48e5 1.48e5 2.09e5 1.23e5
    L2 4.59e7 7.80e7 9.22e7 1.12e6 1.66e5

     | Show Table
    DownLoad: CSV

    In this study, the multi-wavelet Galerkin method was used to obtain an approximate solution of the telegraph equation. This method reduces the problem to a sparse system of linear equations, and then this system is solved by the GMRES method. The convergence analysis was investigated and some numerical tests were guaranteed it. Numerical experiments were shown the ability and flexibility of the proposed method in comparison to other methods.

    This project was supported by Researchers Supporting Project number (RSP-2020/210), King Saud University, Riyadh, Saudi Arabia.

    The writers state that they have no known personal relationships or competing financial interests that could have appeared to affect the work reported in this work.



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