Abstract
In this paper, the effect of inhibition of monocarboxylate transporters on intracellular and capillary lactate concentrations is investigated using an optimal control problem. A control term representing the concentration of the inhibitor is used in an ODE model that models lactate kinetics between the cell and the capillary. Finally, some numerical simulations were performed to confirm the efficiency of the control term for the problem.
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Appendices
Appendix A
Proof of Theorem 2.2
Proof
-
The solution is nonnegative. System (1.4)–(1.6) can be viewed as
$$\begin{aligned} \begin{array}{lll} u^{\prime }(t)=f(t,u(t),v(t)) \\ v^{\prime }(t) =g(t,u(t),v(t)). \end{array} \end{aligned}$$
Using Assumption (E), for \(u=0,\, v\ge 0\), then
For \(u\ge 0,\, v=0\) and using Assumption (E), see that F is a positive function and \(L >0\). Thus,
Hence, the system is quasipositive and so the solution
-
Existence of solution.
System (1.4)–(1.6) can be rewritten in the form
where \(X(t):=(u(t),v(t))\) and \(F(t,X(t)):= (f(t,u,v),g(t,u,v))\). We know that for nonnegative \(u_1\) and \(u_2\)
and for nonnegative \(v_1\) and \(v_2\)
As well, using Assumption (E), we have
and
We deduce that F is globally Lipschitz with respect to u and v, so System (1.4)–(1.6) admits a unique solution \((u,v)\in C^1([0,T],\mathbb {R}_{+})^2\).
-
Continuous dependence on control.
Let \(w_1, w_2\) be two controls in \(\mathcal {W}_{ad}\) and \((u_1,v_1)\), \((u_2,v_2)\) be their corresponding solutions of System (1.4)–(1.6) with same initial data. Set \(u=u_1-u_2\), \(v = v_1 - v_2\), and \(w= w_1 -w_2\), then we have
and
Step one. Multiply Equation (A.1) by u in \(\mathbb {R_{+}}\), we find
Thanks to Cauchy–Schwarz inequality, we get
Now using Assumption (E) and Young’s inequality, we have
On the other hand, multiplying Equation (A.2) by v in \(\mathbb {R}_+\), using Assumption (E) and Young’s inequality, we get
Combining Equation (A.3) and Equation (A.4), then integrating over [0, t], where \(t \in [0,T]\), we get
So, owing to Gronwall’s inequality, we obtain
On the other hand, since
then, using Equation (A.5), we find
Similarly, we have
Step two. Similarly to step one, multiplying Equation (A.1) and Equation (A.2) by \(u^{\prime }\) and \(v^{\prime }\), respectively, we get
Let \(\delta >0\). Applying Young’s inequality, we obtain
where \(c >0\) is a constant independent of \(\delta \). Choosing \(\delta<<1\) so that \(1 -c \delta =\dfrac{1}{2}\), we obtain
By similar way, we obtain
Further, integrating (A.8) and (A.9) over \([0,t], \, t\in [0,T]\), and using (A.5), we get
and
and hence,
\(\square \)
Proof of Theorem 4.1
Proof
Let \((U_1,V_1)\), \((U_2,V_2)\) be two solutions of System (4.1)–(4.3), which can be written in the form
where \(X =(U,V)\) and \(H(t,U,V)=(F(t,U,V),G(t,U,V))\), we know that
which yields that H is Lipchitz with respect to X, and therefore, System (4.1)–(4.3) admits a unique solution \((U,V)\in C^1([0,T],\mathbb {R}_+)^2\) (see [8]). \(\square \)
Proof of Theorem 5.1
Proof
The remainders \(\rho _h\) and \(\theta _h\) satisfy
and
Setting \(f(u) =\dfrac{k}{k +u}\) and \(g(v) =\dfrac{k^{\prime }}{k^{\prime }+v}\), Taylor with integral remainder gives
and
However, the remainders
are bounded, so that
Thus, \(\rho \) satisfies
Equivalently, we have
and \(\theta \) satisfies
which is equivalent to
We need to prove some estimates.
Estimate 1. Multiplying (A.10) by \(\rho _h\) and (A.11) by \(\theta _h\), we find
and
Consequently, after using Assumptions (E) and (F), and Cauchy–Schwarz inequality, we get
and
Combining (A.12) and (A.13), integrating over [0, t], and using Holder’s inequality, we get
Consequently, using Young’s inequality, Assumption (F), Equations (A.5), (A.6), and (A.7), we have
Estimate 2. Multiplying (A.10) by \(\rho ^{\prime }_h\), and (A.11) by \(\theta ^{\prime }_h\), we find
and
Then, using Cauchy–Schwarz inequality, we get
and
Combining (A.15) and (A.16), and integrating over [0, t], in addition to that, using Assumption (F) and Young’s inequality, we have
Further, using Equations (A.14), (A.5), (A.6), and (A.7), we infer
and hence the result. \(\square \)
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Alsayed, H., Fakih, H., Miranville, A. et al. On an Optimal Control Problem Describing Lactate Transport Inhibition. J Optim Theory Appl 198, 1049–1076 (2023). https://doi.org/10.1007/s10957-023-02271-8
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DOI: https://doi.org/10.1007/s10957-023-02271-8
Keywords
- Monocarboxylate transporters
- Glioma treatment
- Lactate dehydrogenase transport
- First-order necessary optimality conditions