Lactate Sensor Blood

Arxiv:2503.17049V1 [Math.Ap] 21 Mar 2025

Optimal control on a brain tumor growth model with lactate metabolism, viscoelastic effects, and tissue damage

Giulia Cavalleri

Dipartimento di matematica “F. Casorati”, Universit`a degli Studi di Pavia,

Alain Miranville

& Universit´e Le Havre Normandie, Laboratoire de Math´ematiques Appliqu´ees du Havre (LMAH), 25, rue Philippe Lebon, BP 1123, 76063 Le Havre cedex, France

Abstract

In this paper, we study an optimal control problem for a brain tumor growth model that incorporates lactate metabolism, viscoelastic effects, and tissue damage. The PDE system, introduced in [Cav+25], couples a Fisher–Kolmogorov-type equation for tumor cell density with a reaction-diffusion equation for the lactate, a quasi-static force balance governing the displacement, and a nonlinear differential inclusion for tissue damage.

The control variables, representing chemotherapy and a lactate- targeting drug, influence tumor progression and treatment response. Starting from well-posedness, regularity, and continuous dependence results proven in [Cav+25], we define a suitable cost functional and prove the existence of optimal controls.

Then, we analyze the differentiability of the control-to-state operator and establish a necessary first-order condition for treatment optimality. Key words: tumor growth models, lactate kinetics, mechanical effects, damage, optimal control, adjoint system, necessary optimality conditions.

AMS (MOS) Subject Classification: 35Q93, 49J20, 49K20, 35Q92, 92C50.

Introduction

Primary brain tumors, especially gliomas and glioblastomas, are characterized by an- giogenesis, invasive growth, and necrosis.

Despite Advances In Medical Research And

treatment strategies, the brain tumor survival rate remains low (see, e.g., [DC16]), and

1

no significant improvement has been observed recently (see, e.g., [Szo+17]). For this reason, it is crucial to explore new therapeutic approaches and to complement clinical studies with mathematical modeling. The aim of the latter is not only to predict the course of the disease, but also to optimize treatments, by adjusting the combination, timing, and dosage of therapies to achieve the best possible outcome for the patient, meaning maximum reduction of the tumor and minimum drug-related side effects. This is where the present contribution comes into play. Specifically, we are interested in a pro- tocol that combines chemotherapy with a drug targeting lactate, which seems promising from an experimental point of view (cf. [Son+08], [Tan+21], [Guy+22]). From the math- ematical perspective, only a few works go in this direction (see, e.g., [Aub+05], [Clo+09], [Gui+18], [Che+21], [Che+22]) and even fewer regard optimal control (cf. [Che+24]).

This paper seeks to study an optimal control problem starting from the model recently proposed and analyzed in [Cav+25], which describes the dynamics of a brain tumor tak- ing into account the lactate metabolism, viscoelastic effects of the tissue, and possible damage caused by surgery. The introduction of the damage variable in this context was inspired by [Cav24], by one of the authors of the present contribution, where damage was first incorporated into a tumor model. Among the huge literature regarding optimal control for tumor growth models (see, e.g., [BAD15], [Col+21], [GL16], [GLR18], [EK19], [Sig20], and the references cited therein) we would like to recall once again [Che+24], where the authors deal with a brain tumor-specific model. The resulting PDE system couples three parabolic equations, one for the tumor cell density, and the other two for the intracellular lactate concentration and the capillary lactate concentration, respec- tively. Another perspective related to the present work is the one from [GLS21], in which the authors consider a phase-field model of Cahn–Hilliard type taking into account the presence of a nutrient (such as oxygen or glucose) and mechanical effects.

The model. Let Ωbe a bounded C2 domain in Rn with n = 2, 3 and let ν be its outward unit normal to ∂Ω. We are interested in the PDE system

(1.1D)

posed in the parabolic cylinder Q := Ω×(0, T), where the domain Ωrepresents the brain and the fixed time T > 0 is the medical treatment duration. In equations (1.1a) and

−Φg(Σ, Z)

for the sake of brevity. As already pointed out, a slightly modified version of (1.1) was proposed in [Cav+25]. We refer the reader to it for further modeling details and recall only the most important aspects. The variable ϕ represents the tumor cell density and

2

takes value in [0, N], where the constant N is the fixed carrying capacity of the tissue. It is ruled by the Fischer–Kolmogorov type equation (1.1a). The parabolic reaction- diffusion equation (1.1b) governs the evolution of σ, which is the intracellular lactate concentration. The quasi-static balance of forces (1.1c) for a viscoelastic tissue describes the dynamic of u, which is the small displacement field of each point with respect to a reference undeformed configuration. The local tissue damage z takes value in the interval [0, 1], where z = 0 means there is no damage while z = 1 means that the tissue is totally damaged. Its evolution is ruled by the parabolic differential inclusion (1.1d). Finally, χ1 is the concentration of a cytotoxic drug, which decreases the tumor proliferation p(σ, z) in the mass source U. Similarly, the term χ2 represents a lactate targeting drug and affects the lactate source term S. The system (1.1) is coupled with the boundary

U(0) = U0,

z(0) = z0.

(1.3)

The cost functional. Introducing the notation χ = (χ1, χ2), we define the following

(1.4)

The non-negative constants α1, . . . , α9 are weights that can not vanish all at the same time, while ϕQ, ϕΩ, σQ, σΩ, zQ are target functions. More explicitly, the term ϕQ (resp. σQ, zQ) is a desired evolution for the tumor (resp.

The Lactate, The Damage), While

ϕΩ(resp. σΩ) is a desired final configuration of the tumor (resp. concentration of the lactate). The third and eighth addends measure the size of the tumor and the magnitude of the damage at the end of the treatment. Since the presence of high mechanical stress, especially in certain areas of the brain, can compromise its functionality, we are interested in keeping it low. The sixth term serves this purpose, and γ may be, for instance, the indicator function of a subdomain of Ωwhere the stress is intended to remain low.

Finally, the last addend is a L2 regularization for the controls. We are interested in studying the minimizers of the cost functional (1.4) subject to the PDE system (1.1)– (1.3) and constrained to a suitable admissible control set. We define it as Uad = U1

U1

ad := {χ1 ∈L2(V ) ∩L∞(Q) : ∥χ1∥L2(V ) ≤Cad, χ1 ≤χ1 ≤χ1 a.e.},

Ad := {Χ2 ∈L∞(Q) : Χ2 ≤Χ2 ≤Χ2 A.E.},

where Cad > 0 is a fixed constant and χ1, χ1, χ2, χ2 ∈L∞(Q) are given threshold functions such that Uad is nonempty. The admissible control set Uad is a subset of

× L∞(Q)

equipped with its natural norm that we denote with ∥·∥U. Remark 1.1. Notice that Uad is a nonempty, closed, and convex subset of U. Moreover,

There Exists A Positive Constant R Such That

Uad ⊆UR := {χ ∈U : ∥χ∥U < R}. Aim and plan of the paper. The goal of this paper is to prove that the optimal control problem has at least one minimizer and to derive first-order necessary conditions for optimality. It is organized as follows. In Section 2, after introducing some notation and preliminary results, we enlist the hypotheses we are going to use throughout the paper. Then, we recall some known results about the state system (1.1)–(1.3) from the previous work [Cav+25] and we prove a strict separation property for the damage. In Section 3 we properly state the optimal control problem and prove that it admits at least one minimizer. In Section 4 we analyze the linearized system, and in Section 5 we use it to prove the differentiability of the control-to-state operator. Finally, in Section 6, we study the adjoint system and derive the first-order necessary conditions for optimality.

Notation And Preliminaries

Notation. In what follows, for any real Banach space X with dual space X′, we indicate its norm as ∥·∥X and the dual pairing between X′ and X as ⟨·, ·⟩X. We denote the Lebesgue and Sobolev spaces over Ωas Lp := Lp(Ω), W k,p := W k,p(Ω) and Hk := W k,2(Ω), while for the Lebesgue spaces over Γ we use Lp

H := L2

and we identify H with its dual space H′. We introduce

Where H1

0 represents the set of H1 functions with zero trace at the boundary. Addition-

4

In both cases, the natural norm induced by H2 is denoted by ∥·∥W . To simplify the no- tation, we do not always distinguish between scalar, vector, and matrix-valued spaces. For the sake of brevity, the norm of the Bochner space W k,p(0, T; X) is indicated as ∥·∥W k,p(X), omitting the time interval (0, T). With the notation C0([0, T]; X) we mean the space of continuous X-valued functions. Finally, as is customary, C represents a generic constant depending only on the problem’s data and whose value might change from line to line or even within the same line. If we want to highlight a dependency on a certain parameter, we put it as a subscript (e.g., Cτ indicates a constant that depends on τ, C0 a constant that depends on the initial data, etc.).

Useful inequalities. We will make use of classical inequalities, such as H¨older, Young, Gronwall, Poincar´e, and Poincar´e–Wirtinger. We will employ the following special cases of Gagliardo–Nirenberg inequality (see [Nir59]).

Lemma 2.1. Let Ωbe a bounded Lipschitz domain in Rd. Then, it exists a constant C

If

n = 3.

(2.2)

From this result, employing the Young inequality, it is easy to obtain an inequality that we will extensively use throughout the paper and, thus, is worth mentioning. For every δ > 0 there exists a positive constant Cδ such that, for p = 3 and p = 4, the following

(2.3)

for every v ∈V , both in dimension n = 2 and n = 3.

Hypotheses

In this section, we enlist the hypotheses under which we will work throughout the whole paper. Most of them come from [Cav+25] and are necessary to establish solutions’ well- posedness, additional regularity, and continuous dependence. However, we need stronger assumptions to prove the results related to the optimal control problem. Specifically, the higher regularity requested for the nonlinearities is because we need to handle the corresponding terms in the linearized and adjoint systems.

Where P∗, G∗Denote Positive Constants, And That

N is a positive constant.

1, K2∗, K∗

2, and S∗are given constants. (H3) The fourth-order tensors A = (aijkh), B = (bijkh) : R2 →Rn×n×n×n satisfy: A is constant, symmetric, and strictly positive definite, and

(2.9)

B ∈C0,1(R) ∩C2(R2) is bounded, symmetric, and positive definite.

Moreover, We Assume

f ∈L∞(H). In the following, being able to handle the maximal monotone operator β and its higher derivatives will be essential. To this end, the key point is proving a strict separation property for the damage variable z. This leads us to adopt a more restrictive form of the convex potential ˆβ, moving beyond the quite general hypotheses employed in the previous analysis [Cav+25].

(2.11)

bβ ∈C3(0, 1).

(2.12)

Its derivative β := bβ′ satisfies the growth conditions:

Lim

r→1−β(r) = +∞.

(2.13)

(H5) We consider a function bπ ∈C1(R)∩C3(0, 1) and we denote by π := bπ′ its derivative,

(2.14)

π is Lipschitz continuous.

(2.15)

Example 2.2. The prototypical example to keep in mind is the logarithmic potential

−C2R2

for some given and positive constants C1, C2, which clearly fulfills the hypotheses (H4) and (H5).

(H6) We Suppose That

ι ∈L∞(Q) ∩H1(0, T; H).

6

(H7) We assume that Ψ ∈W 2,∞(Ω× R × Rn×n) and that Ψ(x, ·, ·) : R × Rn×n →R is Lipschitz continuous, i.e.,

(2.17)

for a.e. x ∈Ω, for all ǫ1, ǫ2 ∈Rn×n, ϕ1, ϕ2 ∈R. In the following, for the sake of brevity, we will omit in the notation the dependence of Ψ on the point x, using Ψ(ϕ, ǫ) instead of Ψ(x, ϕ, ǫ).

Remark 2.3. We require the boundedness of Ψ(ϕ, ε(u)) to establish a separation prop- erty for the damage variable z. It is worth noting that this assumption is not as restrictive as it might initially appear. Indeed, we will prove the boundedness of ϕ, and since we are working within the framework of linear elasticity, ε(u) is expected to remain small.

Consequently, under hypothesis (2.17), the boundedness of Ψ(ϕ, ε(u)) would follow ask- ing that Ψ(0, 0) ∈L∞(Ω). However, as we are unable to prove the boundedness of ε(u) mathematically, it becomes necessary to explicitly impose Ψ the boundedness of as an additional condition. The boundedness of the higher derivatives is required to handle the linearized coefficients in the linearized and adjoint systems.

(H8) Regarding the boundary conditions, we suppose that

(2.18)

where M0 is a fixed positive constant. (H9) Regarding the initial conditions, we assume that

0 < Ess Inf(Z0),

ess sup(z0) < 1.

(2.22)

Remark 2.4. Since the domain of such ˆβ is the physically meaningful interval [0, 1], the initial datum z0 must be chosen such that its values lie within this interval. Recalling again that our goal is to prove a separation property for the damage, we request that z0 stays bounded away from 0 and 1.

Finally, regarding the cost functional J , we make the following assumptions. (H10) The coefficients α1, . . . , α9 are nonnegative constants that can not vanish all at the same time.

Φq, Σq, Zq ∈L2(Q),

ϕΩ, σΩ∈H.

7

(H12) The coefficient γ : Ω× R →[0, +∞) is a Carath´eodory function such that

(2.24)

for a.e. x ∈Ω. Moreover, it exists a constant Cγ > 0 such that

(2.25)

for a.e. x ∈Ωand for all ϕ ∈[0, N]. Even if γ also depends on the point x ∈Ω, in the following, we will employ the shorter notation γ(ϕ) instead of γ(x, ϕ). For the same reason, the partial derivative of γ with respect to the variable ϕ will be denoted by γ′(ϕ).

Previous Results

Definition 2.5 (State system). We say that the quadruplet (ϕ, σ, u, z) is a weak solu-

(2.26D)

a.e. in (0, T), for every η ∈V and θ ∈V0. Since χ is fixed and regular enough, well-posedness and continuous dependence can be proved as in [Cav+25]. Note that z has values in (0, 1) because z belongs to the domain of β. The precise statement is contained in the theorem below.

8

Theorem 2.6. The following statements hold. (i) For every χ ∈U, the state system admits a unique solution in the sense of Defini- tion 2.5, with the following additional regularity

U ∈W 1,∞(0, T; W0),

z ∈H1(0, T; V ) ∩L∞(0, T; W). (ii) For every χ ∈UR and its associated solution to the state system (ϕ, σ, u, z), the

(2.27)

for a positive constant CR that depends on R, the initial data, and the assigned functions, but not on χ. (iii) For every couple of controls χ, χ ∈UR with associated solutions to the state system given respectively by (ϕ, σ, u, z) and (ϕ, σ, u, z), the following continuous depen-

Dence Inequality Is Satisfied

∥ϕ −ϕ∥L∞(H)∩L2(V ) + ∥σ −σ∥L∞(H)∩L2(V ) + ∥u −u∥H1(V0)

+∥Z −Z∥L∞(H)∩L2(V ) ≤Cr∥Χ −Χ∥L2(Q)

for a certain positive constant CR that depends on R, the initial data, and the assigned functions, but not on χ, χ. Remark 2.7. From (ii) and (iii) in Theorem 2.6, employing standard interpolation results, we are able to prove the following continuity estimates which we will need later

On. Precisely, For Every Χ, Χ In Ur, We Have:

∥ϕ −ϕ∥L∞(L4) + ∥ε(u) −ε(u)∥L∞(L4) + ∥z −z∥L∞(L4) ≤CR∥χ −χ∥

(2.28)

Applying Gagliardo–Nirenberg interpolation inequality from Lemma 2.1, estimates (ii)

L2(Q).

The same can be performed for ϕ and z. Similarly, we have

9

In fact, applying Gagliardo–Nirenberg inequality and the embedding W 1,4֒ →L∞(Ω),

V ,

where the last inequality follows from the energy estimate in (ii). Elevating to the power

= Cr∥Φ −Φ∥L1(V ) ≤Cr∥Φ −Φ∥L2(V ) ≤Cr∥Χ −Χ∥L2(Q)

thanks to the continuous estimate (iii). The same holds for the damage. Finally, the

Following Inequality Is Satisfied

∥ϕ −ϕ∥L4(L3) + ∥z −z∥L4(L3) ≤CR∥χ −χ∥L2(Q).

L2(Q),

and the same for z.

A Strict Separation Property For The Damage

Differently from [Cav+25], we impose more restrictive assumptions on the potential ˆβ, as well as boundedness for ι and Ψ, which enable us to establish a separation property for the damage z.

Proposition 2.8. There exist 0 < r∗≤r∗< 1 which may depend on the data of the problem and on R such that, for every χ ∈UR with the associated solution to the state

(2.31)

a.e. in Q. Proof. We claim that there exist 0 < r∗≤r∗< 1 such that

2. Ess Sup(Z0) ≤R∗,

3. β(r) + π(r) + ∥ι∥L∞+ ∥Ψ∥L∞≤0 for all r ∈(0, r∗),

10

4. β(r) + π(r) −∥ι∥L∞−∥Ψ∥L∞≥0 for all r ∈(r∗, 1). We can find such r∗, r∗satisfying conditions 1 and 2 because of hypothesis (H9) ensuring that z0 remains bounded away from 0 and 1. Moreover, regarding points 3 and 4, we recall that, from the specific choice we made for the potential ˆβ, it holds

Β(R) →+∞If R →1−

where, from hypothesis (H5), π ∈C0(R) so is bounded over [0, 1] and, from hypotheses (H6) and (H7), ι and Ψ are bounded. Let χ be an arbitrary control in UR and S(χ) = (ϕ, σ, u, z). We test the equation (2.26d) with (z −r∗)+, obtaining:

|(Z −R∗)+|2 Dx ,

where we used the property 4 in order to obtain the inequality. Integrating in time over

|(Z0 −R∗)+|2 Dx = 0

because, accordingly to property 2, z0 is smaller or equal to r∗almost everywhere, thus (z0 −r∗)+ = 0. This proves that (z −r∗)+ = 0 and, equivalently, that z is smaller or equal to r∗almost everywhere in Q. Proceeding in the same way, we test equation (2.26d) with −(z −r∗)−. Employing inequality 3 and then integrating in time and using 1, we obtain that (z −r∗)−is equal to 0. Thus, we have that z is bigger or equal to r∗ almost everywhere in Q.

Remark 2.9. Let us highlight the fact that thanks to the strict separation property we have just proved, from now on we will treat β as a regular potential. Moreover, we

∥Β(Z)∥L∞+ ∥Β′(Z)∥L∞+ ∥Β′′(Z)∥L∞≤Cr,

since β ∈C2(0, 1) by hypothesis (H4).

The Optimal Control Problem

In view of the previous Theorem 2.6, we introduce the so-called control-to-state operator or solution operator, which maps every control χ to the unique solution to the associated

11

state problem. More precisely, we introduce the state-space

I

.

S : U →V,

χ 7→(ϕ, σ, u, z).

(3.1)

Notice that S is well-defined over U and Lipschitz continuous over UR. We introduce

The Reduced Cost Functional As

J(χ) := J (S(χ), χ).

(3.2)

Then, the optimal control problem can be stated as

(3.3)

which means that we search a minimizer for the functional J subject to the PDE system (1.1)–(1.3) and constrained to Uad. Theorem 3.1. There exists at least one minimizer χ∗∈Uad to the optimal control problem (3.3).

Proof. The reduced cost functional is proper and non-negative, so infχ∈Uad J(χ) is finite and non-negative. Let {χn}n∈N ⊆Uad be a minimizing sequence for J, meaning that

Lim

n→+∞J(χn). We denote the corresponding solution to the state system as (ϕn, σn, un, zn) = S(χn). Since the sequence {χn}n∈N ∈Uad, it is uniformly bounded in U and, consequentially, there exists a χ∗∈U such that, along a subsequence that we do not relabel,

Weakly-∗

in L∞(Q). Notice that Uad is convex and closed in the space L2(0, T; V ) × L2(Q) thus, it is sequen- tially weakly closed. This justifies the fact that the limit χ∗belongs to Uad. Moreover,

12

by the uniform boundedness of the solution sequence from (ii) in Theorem 2.6, applying Banach–Alaouglu (see, e.g., [Br´e11]) and Aubin–Lions Theorems (see [Sim86, Section 8, Corollary 4]), there exists a (ϕ∗, σ∗, u∗, z∗) ∈VS such that, along a further subsequence,

In L2(0, T; V ),

a.e.

In L2(0, T; H),

a.e.

In L2(0, T; V ),

a.e. in Q. The first step of the proof consists of proving that S(χ∗) = (ϕ∗, σ∗, u∗, z∗). To do so, thanks to the convergences above, we can pass to the limit in the PDE system satisfied by (ϕn, σn, un, zn) and χn, from which (ϕ∗, σ∗, u∗, z∗) is the unique solution to the state system associated with χ∗. The second step is showing that

Inf

χ∈Uad J(χ) = J(χ∗). To this end, we write the reduced cost functional as the sum of the following terms

Γ(Φ)Ε(U)∥2

L2(Q) + α8∥z(T)∥L1(Ω). By the weak convergences above and weak lower semicontinuity of the L2-norm, we

Lim Inf

n→+∞J1(χn) ≥J1(χ∗).

(3.4)

Notice that, by uniform boundedness of the sequence from (ii) in Theorem 2.6, and Aubin–Lions Theorem (see [Sim86, Section 8, Corollary 4]), we have the strong conver-

Strongly

in C0([0, T]; H).

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Again from (ii) in Theorem 2.6, {σ∗(T)}n∈N is uniformly bounded in H. Thus, extracting

(3.8)

where we are able to identify the limit with σ∗(T) because of convergence (3.6). Since

(3.9)

for every η ∈L2(0, T; H) thanks to Dominated Convergence Theorem. Moreover, from

Weakly

in L2(0, T; H).

Weakly

in L2(0, T; H).

(3.11)

By the weak lower semicontinuity of the L2-norm and the weak convergences (3.8), (3.11), and by the strong continuity of the L2-norm and the strong convergences (3.5),

N→+∞J2(Χn) ≥J2(Χ∗)

from which the thesis follows. We aim to establish first-order optimality conditions for the optimal control problem (3.3). To do so, the standard procedure is to prove the Fr´echet differentiability of the control-to-state operator. With this purpose, we linearize the state system.

The Linearized State System

We consider a fixed control χ ∈Uad with the associated state given by (ϕ, σ, u, z) = S(χ). For every small h = (h1, h2) ∈U, we consider the perturbed variables

(4.1)

and the corresponding state system. Linearizing it near (ϕ, σ, u, z), χ and approximating the nonlinearities with their first order Taylors’expantions, we have that (ξ, ρ, ω, ζ), h

14

where, for the sake of better readability, we introduced the following notation:

D2 = −Ψ,Ε(Φ, Ε(U)),

d3 = −β′(z) −π′(z). Remark 4.1. Notice that, since all the assigned functions are Lipschitz continuous and bounded, and because of the regularity we have already proved for (ϕ, σ, u, z) in

A1, . . . , A4, B1, . . . , B4, D1, D2 ∈L∞(Q),

and they are uniformly bounded by a constant that depends on R. Moreover,

C1, C2 ∈L∞(0, T; Lp)

for any p ∈[1, 6] and their norm is uniformly bounded by a certain CR, because

|C1| + |C2| ≤C|Ε(U)|,

and u ∈W 1,∞(0, T; W)֒ →W 1,∞(0, T; W 1,p). Regarding the last term, thanks to the regularity of β in its domain and to the separation property we proved in Proposition 2.8,

D3 ∈L∞(Q),

and its norm is bounded by a constant that depends on R.

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