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Review

The potential of lipid soluble thiamine in the treatment of cancer

  • Correction on: AIMS Biophysics 7: 452-453
  • Received: 28 November 2019 Accepted: 09 February 2020 Published: 19 February 2020
  • The resurgence of interest in cancer metabolism has linked alterations in the regulation and exploitation of metabolic pathways with an anabolic phenotype that increases biomass production for the replication of new daughter cells. To support the increase in the metabolic rate of cancer cells, a coordinated increase in the supply of nutrients, such as glucose, as well as micronutrients functioning as enzyme cofactors is required. The majority of co-enzymes are derivatives of water-soluble vitamins such as niacin, folate, pantothenic acid, pyridoxine, biotin, riboflavin and thiamine (Vitamin B1). Continuous dietary intake of these micronutrients is essential for maintaining normal health. How cancer cells adaptively regulate cellular homeostasis of cofactors and how they can regulate expression and function of metabolic enzymes in cancer is under-appreciated. Exploitation of cofactor-dependent metabolic pathways with the advent of anti-folates highlights the potential vulnerabilities and importance of vitamins in cancer biology. Vitamin supplementation products are easily accessible and patients often perceive them as safe and beneficial without full knowledge of their effects. Thus, understanding the significance of enzyme cofactors in cancer cell metabolism will provide for important dietary strategies and new molecular targets to reduce disease progression. Recent studies have demonstrated the significance of thiamine-dependent enzymes in cancer cell metabolism. Therefore, this hypothesis discusses the current knowledge in the alterations in thiamine availability, homeostasis, and exploitation of thiamine-dependent pathways by cancer cells.

    Citation: Derrick Lonsdale, Chandler Marrs. The potential of lipid soluble thiamine in the treatment of cancer[J]. AIMS Biophysics, 2020, 7(1): 17-26. doi: 10.3934/biophy.2020002

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  • The resurgence of interest in cancer metabolism has linked alterations in the regulation and exploitation of metabolic pathways with an anabolic phenotype that increases biomass production for the replication of new daughter cells. To support the increase in the metabolic rate of cancer cells, a coordinated increase in the supply of nutrients, such as glucose, as well as micronutrients functioning as enzyme cofactors is required. The majority of co-enzymes are derivatives of water-soluble vitamins such as niacin, folate, pantothenic acid, pyridoxine, biotin, riboflavin and thiamine (Vitamin B1). Continuous dietary intake of these micronutrients is essential for maintaining normal health. How cancer cells adaptively regulate cellular homeostasis of cofactors and how they can regulate expression and function of metabolic enzymes in cancer is under-appreciated. Exploitation of cofactor-dependent metabolic pathways with the advent of anti-folates highlights the potential vulnerabilities and importance of vitamins in cancer biology. Vitamin supplementation products are easily accessible and patients often perceive them as safe and beneficial without full knowledge of their effects. Thus, understanding the significance of enzyme cofactors in cancer cell metabolism will provide for important dietary strategies and new molecular targets to reduce disease progression. Recent studies have demonstrated the significance of thiamine-dependent enzymes in cancer cell metabolism. Therefore, this hypothesis discusses the current knowledge in the alterations in thiamine availability, homeostasis, and exploitation of thiamine-dependent pathways by cancer cells.



    Fractional differential equations (FDEs) appeared as an excellent mathematical tool for, modeling of many physical phenomena appearing in various branches of science and engineering, such as viscoelasticity, statistical mechanics, dynamics of particles, etc. Fractional calculus is a recently developing work in mathematics which studies derivatives and integrals of functions of fractional order [26].

    The most used fractional derivatives are the Riemann-Liouville (RL) and Caputo derivatives. These derivatives contain a non-singular derivatives but still conserves the most important peculiarity of the fractional operators [1,2,10,11,23,24]. Atangana and Baleanu described a derivative with a generalized Mittag-leffler (ML) function. This derivative is often called the Atangana-Baleanu (AB) fractional derivative. The AB-derivative in the senses of Riemman-Liouville and Caputo are denoted by ABR-derivative and ABC-derivative, respectively.

    The AB fractional derivative is a nonlocal fractional derivative with nonsingular kernel which is connected with various applications [3,5,6,8,9,13,14,15,16]. Using the advantage of the non-singular ML kernal present in the AB fractional derivatives, operators, many authors from various branches of applied mathematics have developed and studied mathematical models involving AB fractional derivatives [18,22,29,30,31,32,35,36,37].

    Mohamed et al. [25] considered a system of multi-derivatives for Caputo FDEs with an initial value problem, examined the existence and uniqueness results and obtained numerical results. Sutar et al. [32,33] considered multi-derivative FDEs involving the ABR derivative and examined existence, uniqueness and dependence results. Kucche et al. [12,19,20,21,34] enlarged the work of multi-derivative fractional differential equations involving the Caputo fractional derivative and studied the existence, uniqueness and continuous dependence of the solution.

    Inspired by the preceding work, we perceive the multi-derivative nonlinear neutral fractional integro-differential equation with AB fractional derivative of the Riemann-Liouville sense of the problem:

    dVdȷ+0Dδȷ[V(ȷ)x(ȷ,y(ȷ))]=φ(ȷ,V(ȷ),ȷ0K(ȷ,θ,V(θ))dθ,T0χ(ȷ,θ,V(θ))dθ),ȷI (1.1)
    V(0)=V0R, (1.2)

    where 0Dδȷ denotes the ABR fractional derivative of order δ(0,1), and φC(I×R×R×R,R) is a non-linear function. Let P1V(ȷ)=ȷ0K(ȷ,θ,V(θ))dθ and P2V(ȷ)=T0χ(ȷ,θ,V(θ))dθ. Now, (1.1) becomes,

    dVdȷ+0Dδȷ[V(ȷ)x(ȷ,y(ȷ))]=φ(ȷ,V(ȷ),P1V(ȷ),P2V(ȷ)),ȷI, (1.3)
    V(0)=V0R. (1.4)

    In this work, we derive a few supplemental results using the characteristics of the fractional integral operator εαδ,η,V;c+. The existence results are obtained by Krasnoselskii's fixed point theorem and the uniqueness and data dependence results are obtained by the Gronwall-Bellman inequality.

    Definition 2.1. [14] The Sobolev space Hq(X) is defined as Hq(X)={φL2(X):DβφL2(X),|β|q}. Let q[1,) and X be open, XR.

    Definition 2.2. [11,17] The generalized ML function Eαδ,β(u) for complex δ,β,α with Re(δ)>0 is defined by

    Eαδ,β(u)=t=0(α)tα(δt+β)utt!,

    and the Pochhammer symbol is (α)t, where (α)0=1,(α)t=α(α+1)...(α+t1), t=1,2...., and E1δ,β(u)=Eδ,β(u),E1δ,1(u)=Eδ(u).

    Definition 2.3. [4] The ABR fractional derivative of V of order δ is

    0Dδȷ[V(ȷ)x(ȷ,y(ȷ))]=B(δ)1δddȷȷ0Eδ[δ1δ(ȷθ)δ]V(θ)dθ,

    where VH1(0,1), δ(0,1), B(δ)>0. Here, Eδ is a one parameter ML function, which shows B(0)=B(1)=1.

    Definition 2.4. [4] The ABC fractional derivative of V of order δ is

    0Dδȷ[V(ȷ)x(ȷ,y(ȷ))]=B(δ)1δȷ0Eδ[δ1δ(ȷθ)δ]V(θ)dθ,

    where VH1(0,1), δ(0,1), and B(δ)>0. Here, Eδ is a one parameter ML function, which shows B(0)=B(1)=1.

    Lemma 2.5. [4] If L{g(ȷ);b}=ˉG(b), then L{0Dδȷg(ȷ);b}=B(δ)1δbδˉG(b)bδ+δ1δ.

    Lemma 2.6. [26] L[ȷmδ+β1E(m)δ,β(±aȷδ);b]=m!bδβ(bδ±a)m+1,Em(ȷ)=dmdȷmE(ȷ).

    Definition 2.7. [17,27] The operator εαδ,η,V;c+ on class L(m,n) is

    (εαδ,η,V;c+)[V(ȷ)x(ȷ,y(ȷ))]=t0(ȷθ)α1Eαδ,η[V(ȷθ)δ]Θ(θ)dθ,ȷ[c,d],

    where δ,η,V,αC(Re(δ),Re(η)>0), and n>m.

    Lemma 2.8. [17,27] The operator εαδ,η,V;c+ is bounded on C[m,n], such that (εαδ,η,V;c+)[V(ȷ)x(ȷ,y(ȷ))]PΘ, where

    P=(nm)Re(η)t=0|(α)t||α(δt+η)|[Re(δ)t+Re(η)]|V(nm)Re(δ)|tt!.

    Here, δ,η,V,αC(Re(δ),Re(η)>0), and n>m.

    Lemma 2.9. [17,27] The operator εαδ,η,V;c+ is invertible in the space L(m,n) and φL(m,n) its left inversion is given by

    ([εαδ,η,V;c+]1)[V(ȷ)x(ȷ,y(ȷ))]=(Dη+ςc+εαδ,η,V;c+)[V(ȷ)x(ȷ,y(ȷ))],ȷ(m,n],

    where δ,η,V,αC(Re(δ),Re(η)>0), and n>m.

    Lemma 2.10. [17,27] Let δ,η,V,αC(Re(δ),Re(η)>0),n>m and suppose that the integral equation is

    ȷ0(ȷθ)α1Eαδ,η[V(ȷθ)δ]Θ(θ)dθ=φ(ȷ),ȷ(m,n],

    is solvable in the space L(m,n).Then, its unique solution Θ(ȷ) is given by

    Θ(ȷ)=(Dη+ςc+εαδ,η,V;c+)[V(ȷ)x(ȷ,y(ȷ))],ȷ(m,n].

    Lemma 2.11. [7] (Krasnoselskii's fixed point theorem) Let A be a Banach space and X be bounded, closed, convex subset of A. Let F1,F2 be maps of S into A such that F1V+F2φX V,φU. The equation F1V+F2V=V has a solution on S, and F1, F2 is a contraction and completely continuous.

    Lemma 2.12. [28] (Gronwall-Bellman inequality) Let V and φ be continuous and non-negative functions defined on I. Let V(ȷ)A+ȷaφ(θ)V(θ)dθ,ȷI; here, A is a non-negative constant.

    V(ȷ)Aexp(ȷaφ(θ)dθ),ȷI.

    In this part, we need some fixed-point-techniques-based hypotheses for the results:

    (H1) Let VC[0,T], function φ(C[0,T]×R×R×R,R) is a continuous function, and there exist +ve constants ζ1,ζ2 and ζ. φ(ȷ,V1,V2,V3)φ(ȷ,φ1,φ2,φ3)ζ1(V1φ1+V2φ2+V3φ3) for all V1,V2,V3,φ1,φ2,φ3 in Y, ζ2=maxVRf(ȷ,0,0,0), and ζ=max{ζ1,ζ2}.

    (H2) P1 is a continuous function, and there exist +ve constants C1,C2 and C. P1(ȷ,θ,V1)P1(ȷ,θ,φ1)C1(V1φ1)V1,φ1 in Y, C2=max(ȷ,θ)DP1(ȷ,θ,0), and C=max{C1,C2}.

    (H3) P2 is a continuous function and there are +ve constants D1,D2 and D. P2(ȷ,θ,V1)P2(ȷ,θ,φ1)D1(V1φ1) for all V1,φ1 in Y, D2=max(ȷ,θ)DP2(ȷ,θ,0) and D=max{D1,D2}.

    (H4) Let xc[0,I], function u(c[0,I]×R,R) is a continuous function, and there is a +ve constant k>0, such that u(ȷ,x)u(ȷ,y)kxy. Let Y=C[R,X] be the set of continuous functions on R with values in the Banach space X.

    Lemma 2.13. If (H2) and (H3) are satisfied the following estimates, P1V(ȷ)ȷ(C1V+C2),P1V(ȷ)P1φ(ȷ)CȷVφ, and P2V(ȷ)ȷ(D1V+D2),P2V(ȷ)P2φ(ȷ)DȷVφ.

    Theorem 3.1. The function φC(I×R×R×R,R) and VC(I) is a solution for the problem of Eqs (1.3) and (1.4), iff V is a solution of the fractional equation

    V(ȷ)=V0B(δ)1δȷ0Eδ[δ1δ(ȷθ)δ][V(ȷ)x(ȷ,y(ȷ))]dθ+ȷ0φ(θ,V(θ),P1V(θ),P2V(θ))dθ,ȷI. (3.1)

    Proof. (1) By using Definition 2.3 and Eq (1.3), we get

    ddȷ(V(ȷ)+B(δ)1δȷ0Eδ[δ1δ(ȷθ)δ][V(ȷ)x(ȷ,y(ȷ))]dθ)=φ(ȷ,V(ȷ),P1V(ȷ),P2V(ȷ)).

    Integrating both sides of the above equation with limits 0 to ȷ, we get

    V(ȷ)+B(δ)1δȷ0Eδ[δ1δ(ȷθ)δ][V(ȷ)x(ȷ,y(ȷ))]dθV(0)=ȷ0φ(θ,V(θ),P1V(θ),P2V(θ))dθ,ȷI.

    Conversely, with differentiation on both sides of Eq (3.1) with respect to ȷ, we get

    dVdȷ+B(δ)1δddȷȷ0Eδ[δ1δ(ȷθ)δ][V(ȷ)x(ȷ,y(ȷ))]dθ=φ(ȷ,V(ȷ),P1V(ȷ),P2V(ȷ)),ȷI.

    Using Definition 2.3, we get Eq (1.3) and substitute ȷ=0 in Eq (3.1), we get Eq (1.4).

    Proof. (2) In Equation (1.3), taking the Laplace Transform on both sides, we get

    L[V(ȷ);b]+L[0Dδȷ;b][V(ȷ)x(ȷ,y(ȷ))]=L[φ(ȷ,V(ȷ),P1V(ȷ),P2V(ȷ));b].

    Now, using the Laplace Transform formula for the AB fractional derivative of the RL sense, as given in Lemma 2.5, we get

    bˉX(b)[V(ȷ)x(ȷ,y(ȷ))]V(0)+B(δ)1δbδˉX(b)bδ+δ1δ=ˉG(b),

    ˉX(b)=[V(ȷ);b] and ˉG(b)=L[φ(ȷ,V(ȷ),P1V(ȷ),P2V(ȷ));b]. Using Eq (1.4), we get

    ˉX(b)=V01bB(δ)1δbδ1ˉX(b)bδ+δ1δ[V(ȷ)x(ȷ,y(ȷ))]+1bˉG(b). (3.2)

    In Eq (3.2) applying the inverse Laplace Transform on both sides using Lemma 2.6 and the convolution theorem, we get

    L1[ˉX(b);ȷ]=V0L1[1b;ȷ]B(δ)1δ(L1[bδ1bδ+δ1δ][V(ȷ)x(ȷ,y(ȷ))]L1[ˉX(b);ȷ])+L1[ˉG(b);ȷ]L1[1b;ȷ]=V0B(δ)1δ(Eδ[δ1δȷδ][V(ȷ)x(ȷ,y(ȷ))])+φ(ȷ,V(ȷ),P1V(ȷ),P2V(ȷ))=V0B(δ)1δȷ0Eδ[δ1δ(ȷθ)δ][V(ȷ)x(ȷ,y(ȷ))]dθ+ȷ0φ(θ,V(θ),P1V(θ),P2V(θ))dθ.V(ȷ)=V0B(δ)1δȷ0Eδ[δ1δ(ȷθ)δ][V(ȷ)x(ȷ,y(ȷ))]dθ+ȷ0φ(θ,V(θ),P1V(θ),P2V(θ))dθ. (3.3)

    Theorem 3.2. Let δ(0,1). Define the operator F on C(I):

    (FV)(ȷ)=V0B(δ)1δ(ε1δ,1,δ1δ;0+)[V(ȷ)x(ȷ,y(ȷ))],VC(I). (3.4)

    (A) F is a bounded linear operator on C(I).

    (B) F satisfying the hypotheses.

    (C) F(X) is equicontinuous, and X is a bounded subset of C(I).

    (D) F is invertible, function φC(I), and the operator equation FV=φ has a unique solution in C(I).

    Proof. (A) From Definition 2.7 and Lemma 2.8, the fractional integral operator ε1δ,1,δ1δ;0+ is a bounded linear operator on C(I), such that

    ε1δ,1,δ1δ;0+[V(ȷ)x(ȷ,y(ȷ))]PV,ȷI,where
    P=Tn=0(1)nα(δn+1)(δn+1)|δ1δTδ|nn!=Tn=0(δ1δ)nTδnα(δn+2)=TEδ,2(δ1δTδ),

    and we have

    FV=|B(δ)1δ|ε1δ,1,δ1δ;0+[V(ȷ)x(ȷ,y(ȷ))]PB(δ)1δV,VC(I). (3.5)

    Thus, FV=φ is a bounded linear operator on C(I).

    (B) We consider V,φC(I). By using linear operator F and bounded operator ε1δ,1,δ1δ;0+, for any ȷI,

    |(FV)(ȷ)(Fφ)(ȷ)|=|F(Vφ)[V(ȷ)x(ȷ,y(ȷ))]|B(δ)1δ(ε1δ,1,δ1δ;0+Vφ)[V(ȷ)x(ȷ,y(ȷ))]PB(δ)1δVφ.

    Where, P=TEδ,2(δ1δTδ), then the operator F is satisfied the hypotheses with constant PB(δ)1δ.

    (C) Let U={VC(I):VR} be a bounded and closed subset of C(I), VU, and ȷ1,ȷ2I with ȷ1ȷ2.

    |(FV)(ȷ1)(FV)(ȷ2)|=|B(δ)1δ(ε1δ,1,δ1δ;0+)[V(ȷ1)u(l1,x(l))]B(δ)1δ(ε1δ,1,δ1δ;0+)[V(ȷ2)u(l2,x(l))]|B(δ)1δ|ȷ10{Eδ[δ1δ(ȷ1θ)δ]Eδ[δ1δ(ȷ2θ)δ]}[V(ȷ)x(ȷ,y(ȷ))]dθ|+B(δ)1δ|ȷ2ȷ1Eδ[δ1δ(ȷ2θ)δ][V(ȷ)x(ȷ,y(ȷ))]dθ|B(δ)1δn=0|(δ1δ)n|1α(nδ+1)ȷ10|(ȷ1θ)nδ(ȷ2θ)nδ||[V(ȷ)x(ȷ,y(ȷ))]|dθ+B(δ)1δn=0|(δ1δ)n|1α(nδ+1)ȷ2ȷ1|(ȷ2θ)nδ||[V(ȷ)x(ȷ,y(ȷ))]|dθLB(δ)1δn=0(δ1δ)n1α(nδ+1)ȷ10(ȷ2θ)nδ(ȷ1θ)nδdθ+LB(δ)1δn=0(δ1δ)n1α(nδ+1)ȷ2ȷ1(ȷ2θ)nδdθRB(δ)1δn=0(δ1δ)n1α(nδ+1){(ȷ2ȷ1)nδ+1+ȷnδ+12ȷnδ+11+(ȷ2ȷ1)nδ+1}RB(δ)1δn=0(δ1δ)n1α(nδ+2){ȷnδ+12ȷnδ+11}|(FV)(ȷ1)(FV)(ȷ2)|RB(δ)1δn=0(δ1δ)n1α(nδ+2){ȷnδ+12ȷnδ+11}. (3.6)

    Hence, if |ȷ1ȷ2|0 then |(FV)(ȷ1)(FV)(ȷ2)|0.

    (FV) is equicontinuous on I.

    (D) By Lemmas 2.9 and 2.10, φC(I), and we get

    (ε1δ,1,δ1δ;0+)1[V(ȷ)x(ȷ,y(ȷ))]=(D1+β0+ε1δ,β,δ1δ;0+)[V(ȷ)x(ȷ,y(ȷ))],ȷ(m,n). (3.7)

    By Eqs (3.4) and (3.5), we have

    (F1)[V(ȷ)x(ȷ,y(ȷ))]=(B(δ)1δε1δ,1,δ1δ;0+)1[V(ȷ)x(ȷ,y(ȷ))]=1δB(δ)(D1+β0+ε1δ,β,δ1δ;0+)[V(ȷ)x(ȷ,y(ȷ))],ȷ(m,n),

    where βC with Re(β)>0. This shows F is invertible on C(I) and

    (FV)[V(ȷ)x(ȷ,y(ȷ))]=[V(ȷ)x(ȷ,y(ȷ))],ȷI,

    has the unique solution,

    V(ȷ)=(F1[V(ȷ)x(ȷ,y(ȷ))])=1δB(δ)(D1+β0+ε1δ,β,δ1δ;0+)[V(ȷ)x(ȷ)),y(ȷ))],ȷ(m,n). (3.8)

    Theorem 4.1. Let φC(I×R×R×R,R). Then, the ABR derivative 0Dδȷ[V(ȷ)x(ȷ,y(ȷ))]=φ(ȷ,V(ȷ),P1V(ȷ),P2V(ȷ)),ȷI, is solvable in C(I), and the solution in C(I) is

    V(ȷ)=1δB(δ)(D1+β0+ε1δ,β,δ1δ;0+)[V(ȷ)x(ȷ,y(ȷ))],ȷI, (4.1)

    where βC,Re(β)>0, and ˆφ(ȷ)=ȷ0φ(θ,V(θ),P1V(θ),P2V(θ))dθ,ȷI.

    Proof. The corresponding fractional equation of the ABR derivative

    0Dδȷ[V(ȷ)x(ȷ,y(ȷ))]=φ(ȷ,V(ȷ),P1V(ȷ),P2V(ȷ)),ȷI,

    is given by

    B(δ)1δȷ0Eδ[δ1δ(ȷθ)δ][V(ȷ)x(ȷ,y(ȷ))]dθ=ȷ0φ(θ,V(θ),P1V(θ),P2V(θ))dθ,ȷI.

    Using operator F of Eq (3.4), we get

    (FV)(s)=ȷ0φ(θ,V(θ),P1V(θ),P2V(θ))dθ=ˆφ(ȷ),ȷI. (4.2)

    Equations (3.7) and (4.2) are solvable, and we get

    V(ȷ)=1δB(δ)(D1+β0+ε1δ,β,δ1δ;0+)[V(ȷ)x(ȷ,y(ȷ))],ȷI;βC,Re(β)>0. (4.3)

    Theorem 4.2. Let φC(I×R×R×R,R) satisfy (H1)(H3) with L=supȷIω(ȷ), where ω(ȷ)=ζ(1+Cȷ+DT), if L=min{1,12T}. Then problem of (1.3) and (1.4) has a solution in C(I) provided

    2B(δ)TEδ,2(δ1δ)Tδ1δ1. (4.4)

    Proof. Define

    R=V0+NφT1LTB(δ)TEδ,2(δ1δ)Tδ1δ,

    where Nφ=supȷIφ(ȷ,0,0,0). Let U={VC(I):VR}. Consider F1:XA and F2:XA given as

    (F1V)(ȷ)=V0+ȷ0φ(θ,V(θ),P1V(θ),P2V(θ))dθ,ȷI,(F2V)(ȷ)=(F)[V(ȷ)x(ȷ,y(ȷ))],ȷI.

    Let V=F1V+F2V,VC(I) is the fractional Eq (3.1) to the problems (1.3) and (1.4).

    Hence, the operators F1 and F2 satisfy the Krasnoselskii's fixed point theorem.

    Step (ⅰ) F1 is a contraction.

    By (H1)(H3) on φ, V,φC(I) and ȷI,

    |F1V(ȷ)F2φ(ȷ)|ω(ȷ)|V(ȷ)φ(ȷ)|RVφ. (4.5)

    This gives, F1VF2φRTVφ,V,φC(I).

    Step (ⅱ) F2 is completely continuous. By using Theorem 3.3 and Ascoli-Arzela theorem, F2=F is completely continuous.

    Step (ⅲ) F1V+F2φU, for any V,φU, using Theorem 3.3, we obtain

    (F1V+F2φ)(ȷ)(F1V)(ȷ)+(F2φ)(ȷ)V0+ȷ0φ(θ,V(θ),P1V(θ),P2V(θ))dθ+ε1δ,1,δ1δ;0+φV0+ȷ0φ(θ,V(θ),P1V(θ),P2V(θ))dθ+B(δ)1δTEδ,2(δ1δTδ)φV0+ȷ0φ(θ,V(θ),P1V(θ),P2V(θ))φ(θ,0,0,0)dθ+ȷ0φ(θ,0,0,0)dθ+B(δ)1δTEδ,2(δ1δTδ)LV0+ȷ0ζ(V+CȷV+DTV)dθ+Nφȷ0dθ+B(δ)1δTEδ,2(δ1δTδ)LV0+ζ(1+Cȷ+DT)ȷ0Vdθ+Nφȷ0dθ+B(δ)1δTEδ,2(δ1δTδ)LV0+ω(ȷ)Rȷ0dθ+Nφȷ0dθ+B(δ)1δTEδ,2(δ1δTδ)LV0+LRT+NφT+B(δ)1δTEδ,2(δ1δTδ)L. (4.6)

    By definition of R, we get

    V0+NφT=L(1RT+B(δ)TEδ,2(δ1δTδ)1δ). (4.7)

    Using the Eq (4.5) in (4.7), we get condition of Eq (4.4).

    (F1V+F2φ)(ȷ)L(2B(δ)TEδ,2(δ1δ)Tδ1δ),ȷI. (4.8)

    (F1V+F2φ)(ȷ)L,ȷI. This gives, F1V+F2φU, V,φX.

    From Steps (ⅰ)–(ⅲ), all the conditions of Lemma 2.11 follow.

    Theorem 4.3. By Theorem 4.2, the Eqs (1.3) and (1.4) have a unique solution in C(I).

    Proof. (1) The problems (1.3) and (1.4) have an operator equation form as:

    (ε1δ,1,δ1δ;0+)[V(ȷ)x(ȷ,y(ȷ))]=ˆφ(ȷ),ȷI, (4.9)

    where,

    ˆφ(ȷ)=1δB(δ)(V0V(ȷ)+ȷ0φ(θ,V(θ),P1V(θ),P2V(θ))dθ),ȷI.

    By Theorem 4.2, Eq (4.7) is solvable in C(I), by Lemma 2.10 we get a unique solution of Eqs (1.3) and (1.4),

    V(ȷ)=(D1+β0+ε1δ,β,δ1δ;0+)[V(ȷ)x(ȷ,y(ȷ))],VC(I).

    Proof. (2) Let V,φ be solutions of Eqs (1.3) and (1.4). By fractional integral operators and (H1)(H3), we find, for any ȷI,

    |V(ȷ)φ(ȷ)||B(δ)1δ(ε1δ,1,δ1δ;0+(Vφ))[V(ȷ)x(ȷ,y(ȷ))]|+ȷ0|φ(θ,V(θ),P1V(θ),P2V(θ))φ(θ,φ(θ),P1φ(θ),P2φ(θ))|dθ|B(δ)1δȷ0Eδ[δ1δ(ȷθ)δ](V(θ)φ(θ))dθ|+ȷ0ζ(|V(θ)φ(θ)|+C|V(θ)φ(θ)|+D|V(θ)φ(θ)|)dθB(δ)1δȷ0Eδ(|δ1δTδ|)|V(θ)φ(θ)|dθ+ȷ0ζ(1+C+D)|V(θ)φ(θ)|dθB(δ)1δȷ0Eδ(δ1δTδ)|V(θ)φ(θ)|dθ+ȷ0[V(ȷ)x(ȷ,y(ȷ))]|V(θ)φ(θ)|dθȷ0[B(δ)1δEδ(δ1δTδ)+[V(ȷ)x(ȷ,y(ȷ))]]|V(θ)φ(θ)|dθ|V(ȷ)φ(ȷ)|ȷ0[B(δ)1δEδ(δ1δTδ)+[V(ȷ)x(ȷ,y(ȷ))]]|V(θ)φ(θ)|dθ. (4.10)

    Theorem 5.1. By Theorem 4.2, if V(ȷ) is a solution of Eqs (1.3) and (1.4), then

    |V(ȷ)|{|V0|+NφT}exp(ȷ0[B(δ)1δEδ(δ1δTδ)+[V(ȷ)x(ȷ,y(ȷ))]]dθ),ȷI, (5.1)

    where, Nφ=supȷI|φ(ȷ,0,0,0)|.

    Proof. If V(ȷ) is a solution of Eqs (1.3) and (1.4), for all ȷI,

    |V(ȷ)||V0|B(δ)1δȷ0Eδ(|δ1δ(ȷθ)δ|)|V(θ)|dθ+ȷ0|φ(θ,V(θ),P1V(θ),P2V(θ))|dθ
    |V0|B(δ)1δȷ0Eδ(δ1δ(ȷθ)δ)|V(θ)|dθ+ȷ0|φ(θ,V(θ),P1V(θ),P2V(θ))φ(θ,0,0,0)|dθ+ȷ0|φ(θ,0,0,0)|dθ|V0|B(δ)1δȷ0Eδ(δ1δTδ)|V(θ)|dθ+ȷ0ζ(|V(θ)|+C|V(θ)|+D|V(θ)|)dθ+Nφȷ|V0|B(δ)1δȷ0Eδ(δ1δTδ)|V(θ)|dθ+ȷ0ζ(1+C+D)|V(ȷ)|dθ+NφT|V0|B(δ)1δȷ0Eδ(δ1δTδ)|V(θ)|dθ+ȷ0[V(ȷ)x(ȷ,y(ȷ))]|V(θ)|dθ+NφT{|V0|+NφT}+ȷ0[B(δ)1δEδ(δ1δTδ)+[V(ȷ)x(ȷ,y(ȷ))]]|V(θ)|dθ.

    By Lemma 2.12, we get

    |V(ȷ)|{|V0|+NφT}exp(ȷ0[B(δ)1δEδ(δ1δTδ)+[V(ȷ)x(ȷ,y(ȷ))]]dθ),ȷI. (5.2)

    We discuss data dependence results for the problem

    dφdȷ+0Dδȷ[V(ȷ)x(ȷ,y(ȷ))]=˜φ(ȷ,φ(ȷ),P1φ(ȷ),P2φ(ȷ)),ȷI, (6.1)
    φ(0)=φ0R. (6.2)

    Theorem 6.1. Equation (4.2) holds, and ξk>0, where k=1,2 are real numbers such that,

    |V0φ0|ξ1,|φ(ȷ,V(ȷ),P1V(ȷ),P2V(ȷ))˜φ(ȷ,φ(ȷ),P1φ(ȷ),P2φ(ȷ))|ξ2,ȷI. (6.3)

    φ(ȷ) is a solution of ABR fractional derivative Eqs (6.1) and (6.2), and V(ȷ) is a solution of Eqs (1.3) and (1.4).

    Proof. Let V,φ are the solution of Eqs (1.3) and (1.4), (6.1) and (6.2) respectively. We find for any

    |V(ȷ)φ(ȷ)||V0φ0|+B(δ)1δȷ0Eδ(|δ1δ(ȷθ)δ|)|V(θ)φ(θ)|dθ+ȷ0|φ(θ,V(θ),P1V(θ),P2V(θ))˜φ(s,φ(θ),P1φ(θ),P2φ(θ))|dθ|V0φ0|+B(δ)1δȷ0Eδ(|δ1δ(ȷθ)δ|)|V(θ)φ(s)|dθ+ȷ0|φ(θ,V(θ),P1V(θ),P2V(θ))φ(θ,φ(θ),P1φ(θ),P2φ(θ))|dθ+ȷ0|φ(θ,V(θ),P1V(θ),P2V(θ))˜φ(θ,φ(θ),P1φ(θ),P2φ(θ))|dθξ1+B(δ)1δȷ0Eδ(δ1δTδ)|V(θ)φ(θ)|dθ+ȷ0ζ(|V(θ)φ(θ)|+C|V(θ)φ(θ)|+D|V(θ)φ(θ)|)dθ+ξ2ȷ0dθξ1+B(δ)1δȷ0Eδ(δ1δTδ)|V(θ)φ(θ)|dθ+ȷ0ζ(1+C+D)|V(θ)φ(θ)|dθ+ξ2ȷξ1+B(δ)1δȷ0Eδ(δ1δTδ)|V(θ)φ(θ)|dθ+ȷ0[V(ȷ)x(ȷ,y(ȷ))]|V(θ)φ(θ)|dθ+ξ2T|V(ȷ)φ(ȷ)|ξ1+ξ2T+ȷ0[B(δ)1δEδ(δ1δTδ)+[V(ȷ)x(ȷ,y(ȷ))]]|V(ȷ)φ(θ)|dθ.

    By Lemma 2.12, we get

    |V(ȷ)φ(ȷ)|(ξ1+ξ2T)exp(ȷ0[B(δ)1δEδ(δ1δTδ)+[V(ȷ)x(ȷ,y(ȷ))]]dθ),ȷI. (6.4)

    Let any λ,λ0R and

    dVdȷ+0Dδȷ[V(ȷ)x(ȷ,y(ȷ))]=Θ(ȷ,V(ȷ),P1V(ȷ),P2V(ȷ),λ),ȷI, (7.1)
    V(0)=V0R. (7.2)
    dVdȷ+0Dδȷ[V(ȷ)x(ȷ,y(ȷ)]=Θ(ȷ,V(ȷ),P1V(ȷ),P2V(ȷ),λ0),ȷI, (7.3)
    V(0)=V0R. (7.4)

    Theorem 7.1. Let the function Θ satisfy Theorem 4.2. Suppose there exists ω,uC(I,R+) such that,

    |Θ(ȷ,V,P1V,P2V,λ)Θ(ȷ,φ,P1φ,P2φ,λ)|ω(ȷ)|Vφ|,|Θ(ȷ,V,P1V,P2V,λ)Θ(ȷ,V,P1V,P2V,λ0)|u(ȷ)|λλ0|.

    If V1,V2 are the solutions of Eqs (7.1) and (7.3), then

    |V1(ȷ)V2(ȷ)|PT|λλ0|exp(ȷ0[B(δ)1δEδ(δ1δTδ)+[V(ȷ)x(ȷ,y(ȷ))]]dθ),ȷI, (7.5)

    where P=supȷIu(ȷ).

    Proof. Let, for any ȷI,

    |V1(ȷ)V2(ȷ)|B(δ)1δ|ȷ0Eδ[δ1δ(ȷθ)δ](V2(θ)V1(θ)dθ)|+ȷ0|Θ(θ,V1(θ),P1V1(θ),P2V1(θ),λ)Θ(θ,V2(θ),P1V2(θ),P2V2(θ),λ0)|dθB(δ)1δȷ0Eδ(|δ1δ(ȷθ)δ|)|V1(θ)V2(θ)|dθ+ȷ0|Θ(θ,V1(θ),P1V1(θ),P2V1(θ),λ)Θ(θ,V2(θ),P1V2(θ),P2V2(θ),λ)|dθ+ȷ0|Θ(θ,V2(θ),P1V2(θ),P2V2(θ),λ)Θ(θ,V2(θ),P1V2(θ),P2V2(θ),λ0)|dθB(δ)1δȷ0Eδ(δ1δ(ȷθ)δ)|V1(θ)V2(θ)|dθ+ȷ0ζ(|V1(θ)V2(θ)|+C|V1(θ)V2(θ)|+D|V1(θ)V2(θ)|)dθ+ȷ0u(θ)|λλ0|dθB(δ)1δȷ0Eδ(δ1δTδ)|V1(θ)V2(θ)|dθ+ȷ0ζ(1+C+D)|V1(θ)V2(θ)|dθ+Pȷ|λλ0|B(δ)1δȷ0Eδ(δ1δTδ)|V1(θ)V2(θ)|dθ+ȷ0[V(ȷ)x(ȷ,y(ȷ))]|V1(θ)V2(θ)|dθ+PT|λλ0|ȷ0[B(δ)1δEδ(δ1δTδ)+[V(ȷ)x(ȷ,y(ȷ))]]|V1(θ)V2(θ)|dθ+PT|λλ0|.

    By Lemma 2.12,

    |V1(ȷ)V2(ȷ)|PT|λλ0|exp(ȷ0[B(δ)1δEδ(δ1δTδ)+[V(ȷ)x(ȷ,y(ȷ))]]dθ),ȷI. (7.6)

    Consider a nonlinear ABR fractional derivative with neutral integro-differential equations of the form:

    dVdȷ+0D12ȷ[V(ȷ)x(ȷ,y(ȷ))]=φ(ȷ,V(ȷ),P1V(ȷ),P2V(ȷ)),ȷI=[0,2], (8.1)
    V(0)=1R. (8.2)

    φ:(I×R×R×R)R is a continuous nonlinear function such that,

    φ(ȷ,V(ȷ),P1V(ȷ),P2V(ȷ))=|V(ȷ)|+13+M(ȷ)+N(ȷ),ȷI,

    and

    M(ȷ)=B(12){ȷE12,2(ȷ12)+E12(ȷ12)ȷ1},N(ȷ)=B(12){E12,2(ȷ12)+ȷE12(ȷ12)1}.

    We observe that for all V,φR and ȷI,

    |φ(ȷ,V,P1V,P2V)φ(ȷ,φ,P1φ,P2φ)|=|(|V(ȷ)|+13+M(ȷ)+N(ȷ))(|φ(ȷ)|+13+M(ȷ)+N(ȷ))|13|Vφ|. (8.3)

    The function φ satisfies (H1)(H4) with constant 13. From Theorem 4.2, we have δ=12 and T = 2 which is substitute in Eq (4.2), and we get

    B(12)<18E12,2(212). (8.4)

    If the function B(δ) satisfies Eq (8.4), then Eqs (8.1) and (8.2) have a unique solution.

    V(ȷ)=ȷ3+1,ȷ[0,2]. (8.5)

    In this research article, we explored multi-derivative nonlinear neutral fractional integro-differential equations involving the ABR fractional derivative. The elementary results of the existence, uniqueness and dependence solution on various data are based on the Prabhakar fractional integral operator εαδ,η,V;c+ involving a generalized ML function. The existence results are obtained by Krasnoselskii's fixed point theorem, and the uniqueness and data dependence results are obtained by the Gronwall-Bellman inequality with continuous functions.

    The research on Existence and data dependence results for neutral fractional order integro-differential equations by Khon Kaen University has received funding support from the National Science, Research and Innovation Fund.

    The authors declare no conflict of interest.



    Conflict of interest



    Clinical studies of TTFD have been performed by Dr. Lonsdale since 1973 under independent investigator license IND 11019. Dr. Marrs declares no conflicts of interest.

    [1] Elliott RL, Jiang XP, Head JF (2012) Mitochondria organelle transplantation: introduction of normal epithelial mitochondria into human cancer cells inhibits proliferation and increases drug sensitivity. Breast Cancer Res Tr 136: 347-354. doi: 10.1007/s10549-012-2283-2
    [2] Jonus HC, Byrnes CC, Kim J, et al. (2020) Thiamine mimetics sulbutiamine and benfotiamine as a nutraceutical approach to anticancer therapy. Biomed Pharmacother 121: 109648. doi: 10.1016/j.biopha.2019.109648
    [3] Anderson MW, Maronpot RR, Reynolds SH (1988) Role of genes in chemical carcinogenesis: extrapolation from rodents to humans. IARC Sci Publ 89: 477-485.
    [4] Torry DS, Cooper GM (1991) Proto-oncogenes in development and cancer. Am J Reprod Immunol 25: 129-132. doi: 10.1111/j.1600-0897.1991.tb01080.x
    [5] Schwartz L, Supuran CT, Alfarouk KO (2017) The Warburg effect and the hallmarks of cancer. Anti-cancer Agent Me 17: 164-70. doi: 10.2174/1871520616666161031143301
    [6] Kaipparettu BA, Ma Y, Park JH, et al. (2019) Correction: Crosstalk from non-cancerous mitochondria can inhibit tumor properties of metastatic cell suppressing oncogenic pathways. PLoS One 14: e0221671. doi: 10.1371/journal.pone.0221671
    [7] Babaian A, Mager DL (2016) Androgynous regional viral promoter exaptation in human cancer. Mob DNA 7: 24. doi: 10.1186/s13100-016-0080-x
    [8] Cancarini I, Krogh V, Agnoli C, et al. (2015) Micronutrients involved in one-carbon metabolism and risk of breast cancer subtypes. PLoS One 10: e0138318. doi: 10.1371/journal.pone.0138318
    [9] Wu Y, Sarkissyan M, Vadgama JV (2015) Epigenetics in breast and prostate cancer. Methods Mol Biol 1238: 425-466. doi: 10.1007/978-1-4939-1804-1_23
    [10] Fabbri M, Calore F, Paone A, et al. (2013) Epigenetic regulation of miRNAs in cancer. Adv Exp Med Biol 754: 137-148. doi: 10.1007/978-1-4419-9967-2_6
    [11] Dawson MA, Kouzarides T (2012) Cancer epigenetics: from mechanism to therapy. Cell 150: 12-27. doi: 10.1016/j.cell.2012.06.013
    [12] Mahmoud AM, Ali MM (2019) Methyl donor micronutrients that modify DNA methylation and cancer outcome. Nutrients 11: 608. doi: 10.3390/nu11030608
    [13] Sechi GP, Sechi E, Fois C, et al. (2007) Wernicke's encephalopathy: new clinical settings and recent advances in diagnosis and management. Lancet Neurol 6: 442-455. doi: 10.1016/S1474-4422(07)70104-7
    [14] Sweet RL, Zastre JA (2013) HIF1-α-mediated gene expression induced by vitamin B. Int J Vitam Nutr Res 83: 188-197. doi: 10.1024/0300-9831/a000159
    [15] Lonsdale D, Marrs C (2017)  Thiamine deficiency disease, dysautonomia and high calorie malnutrition London: Academic Press.
    [16] Zastre JA, Sweet RL, Hanberry BS, et al. (2013) Linking vitamin B1 with cancer cell metabolism. Cancer Metab 1: 16. doi: 10.1186/2049-3002-1-16
    [17] Isenberg-Grzeda E, Shen MJ, Alici Y, et al. (2017) High rates of thiamine deficiency among inpatients with cancer referred for psychiatric consultation: results of a single site prevalence study. Psycho-oncology 26: 1384-1389. doi: 10.1002/pon.4155
    [18] Onishi H, Ishida M, Uchida N, et al. (2018) The rate and treatment outcome of thiamine deficiency in cancer patients diagnosed with delirium: A preliminary study. J Clin Oncol 36: 205-205. doi: 10.1200/JCO.2018.36.34_suppl.205
    [19] Choi EY, Gomes WA, Haigentz M, et al. (2016) Association between malignancy and non-alcoholic Wernicke's encephalopathy: a case report and literature review. Neuro-Oncology Pract 3: 196-207. doi: 10.1093/nop/npv036
    [20] Antunez E, Estruch R, Cardenal C, et al. (1998) Usefulness of CT and MR imaging in the diagnosis of acute Wernicke's encephalopathy. Am J Roentgenol 171: 1131-1137. doi: 10.2214/ajr.171.4.9763009
    [21] Seligmann H, Levi R, Konijn AM, et al. (2001) Thiamine deficiency in patients with B-chronic lymphocytic leukemia: a pilot study. Postgrad Med J 77: 582-585. doi: 10.1136/pmj.77.911.582
    [22] Gangat N, Phelps A, Lasho TL, et al. (2019) A prospective evaluation of vitamin B1 (thiamine) level in myeloproliferative neoplasms: clinical correlations and impact of JAK2 inhibitor therapy. Blood Cancer J 9: 1-4. doi: 10.1038/s41408-018-0167-3
    [23] Sechi GP, Sechi E, Fois C, et al. (2016) Advances in clinical determinants and neurological manifestations of B vitamin deficiency in adults. Nutr Rev 74: 281-300. doi: 10.1093/nutrit/nuv107
    [24] Sechi GP, Batzu L, Agrò L, et al. (2016) Cancer-related Wernicke-Korsakoff syndrome. Lancet Oncol 17: e221-e222. doi: 10.1016/S1470-2045(16)30109-7
    [25] Seyfried TN (2015) Cancer as a mitochondrial metabolic disease. Front Cell Dev Biol 3: 43. doi: 10.3389/fcell.2015.00043
    [26] Boros LG (2000) Population thiamine status and varying cancer rates between western, Asian and African countries. Anticancer Res 20: 2245-2248.
    [27] Fiolet T, Srour B, Sellem L, et al. (2018) Consumption of ultra-processed foods and cancer risk: results from NutriNet-Santé prospective cohort. BMJ 360: k322. doi: 10.1136/bmj.k322
    [28] Berner LA, Keast DR, Bailey RL, et al. (2014) Fortified foods are major contributors to nutrient intakes in diets of US children and adolescents. J Acad Nutr Diet 114: 1009-1022. doi: 10.1016/j.jand.2013.10.012
    [29] Via M (2012) The malnutrition of obesity: micronutrient deficiencies that promote diabetes. ISRN Endocrinol 2012: 103472.
    [30] Li Q, Shu Y (2014) Role of solute carriers in response to anticancer drugs. Mol Cell Ther 2: 15. doi: 10.1186/2052-8426-2-15
    [31] Liu S, Huang H, Lu X, et al. (2003) Down-regulation of thiamine transporter THTR2 gene expression in breast cancer and its association with resistance to apoptosis11RPG-00-031-01-CDD from the American Cancer Society (JAM). Mol Cancer Res 1: 665-673.
    [32] Sweet R, Paul A, Zastre J (2010) Hypoxia induced upregulation and function of the thiamine transporter, SLC19A3 in a breast cancer cell line. Cancer Biol Ther 10: 1101-1111. doi: 10.4161/cbt.10.11.13444
    [33] Akanji MA, Rotimi D, Adeyemi OS (2019) Hypoxia-inducible factors as an alternative source of treatment strategy for cancer. Oxid Med Cell Longev Available from: https://doi.org/10.1155/2019/8547846.
    [34] Zera K, Sweet R, Zastre J (2016) Role of HIF-1α in the hypoxia inducible expression of the thiamine transporter, SLC19A3. Gene 595: 212-220. doi: 10.1016/j.gene.2016.10.013
    [35] Dhir S, Tarasenko M, Napoli E, et al. (2019) Neurological, psychiatric, and biochemical aspects of thiamine deficiency in children and adults. Front Psychiatry 10: 207. doi: 10.3389/fpsyt.2019.00207
    [36] Bentz S, Cee A, Endlicher E, et al. (2013) Hypoxia induces the expression of transketolase-like 1 in human colorectal cancer. Digestion 88: 182-192. doi: 10.1159/000355015
    [37] Briston T, Yang J, Ashcroft M (2011) HIF-1α localization with mitochondria: a new role for an old favorite? Cell Cycle 10: 4170-4171. doi: 10.4161/cc.10.23.18565
    [38] Stacpoole PW (2017) Therapeutic targeting of the pyruvate dehydrogenase complex/pyruvate dehydrogenase kinase (PDC/PDK) axis in cancer. JNCI-J Natl Cancer I 109.
    [39] Hanberry BS, Berger R, Zastre JA (2014) High-dose vitamin B1 reduces proliferation in cancer cell lines analagous to dichloracetate. Cancer Chemoth Pharm 73: 586-594. doi: 10.1007/s00280-014-2386-z
    [40] Liu X, Montissol S, Uber A, et al. (2018) The effects of thiamine on breast cancer cells. Molecules 23: 1464. doi: 10.3390/molecules23061464
    [41] Sutendra G, Kinnaird A, Dromparis P, et al. (2014) A nuclear pyruvate dehydrogenase complex is important for the generation of acetyl-CoA and histone acetylation. Cell 158: 84-97. doi: 10.1016/j.cell.2014.04.046
    [42] Inouye K, Katsura E (1965) Etiology and pathology of beriberi. Thiamine and beriberi Tokyo: Igaku Shoin Ltd, 1-28.
    [43] Fujita T, Suzuoki Z (1973) Enzymatic studies on the metabolism of the tetrahydrofurfuryl mercaptan moiety of thiamine tetrahydrofurfuryl disulfide I. Microsomal S-transmethylase. J Biochem 74: 717-722. doi: 10.1093/oxfordjournals.jbchem.a130296
    [44] Fujita T, Suzuoki Z, Kozuka S, et al. (1973) Enzymatic studies on the metabolism of the tetrahydrofurfuryl mercaptan moiety of thiamine tetrahydrofurfuryl disulfide. II. Sulfide and sulfoxide oxygenases in microsomes. J Biochem 74: 723-732. doi: 10.1093/oxfordjournals.jbchem.a130297
    [45] Fujita T, Suzuoki Z (1973) Enzymatic studies on the metabolism of tetrahydrofurfuryl disulfide. III. Oxidative cleavage of the tetrahydrofuran moiety. J Biochem 74: 733-738. doi: 10.1093/oxfordjournals.jbchem.a130298
    [46] Volvert ML, Seyen S, Piette M, et al. (2008) Benfotiamine, a synthetic S-acyl thiamine derivative has different mechanisms of action and a different pharmacological profile than lipid-soluble thiamine disulfide derivatives. BMC Pharmacol 8: 10. doi: 10.1186/1471-2210-8-10
    [47] Lonsdale D (2004) Thiamine tetrahydrofurfuryl disulfide: a little-known therapeutic agent. Med Sci Monitor 10: RA199-RA203.
    [48] Isenberg-Grzeda E, Shen MJ, Alici Y, et al. (2017) High rates of thiamine deficiency among inpatients with cancer referred for psychiatric consultation: results of a single site prevalence study. Psycho oncology 26: 1384-1389. doi: 10.1002/pon.4155
    [49] Selye H (1946) The general adaptation syndrome and the diseases of adaptation. J Clin Endocr 6: 117-230. doi: 10.1210/jcem-6-2-117
    [50] Skelton FR (1950) Some specific and non-specific effects of thiamin deficiency in the rat. Proc Soc Exp Biol Med 73: 516-519. doi: 10.3181/00379727-73-17729
    [51] Zbinden G (1962) Therapeutic use of vitamin B1 in diseases other than beriberi. Therapeutic use of vitamin B1 and diseases of the beriberi. Ann NY Acad Sci 98: 550-561. doi: 10.1111/j.1749-6632.1962.tb30576.x
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