Topological asymptotic analysis for tumor identification problem

Abstract

This work is concerned with the problem of identifying the shape, size and location of a small embedded tumor from measured temperature on the skin surface. The temperature distribution in the biological tissue is governed by the Pennes model equation. The proposed approach is based on the Kohn–Vogelius formulation and the topological sensitivity analysis method. The ill-posed geometric inverse problem is reformulated as a topology optimization. The temperature field perturbation, caused by the presence of a small anomaly, is analyzed and estimated. A topological asymptotic formula, describing the variation of the considered Kohn–Vogelius type functional with respect to the presence of a small anomaly is derived.

Keywords

Asymptotic analysis topological sensitivity inverse problem Kohn–Vogelius formulation

1. Introduction

Medical infrared thermography is a non-contact, non-invasive and useful imaging technique that measures the radiation emitted from the skin surface and gives information about abnormal situations [10,11,13,15]. The temperature measurements at the skin surface is used in order to predict the existence of the tumor region as well as to study the tumor evolution after a treatment procedure [2,6].

The aim of this study is to identify the location, size and shape of a small anomaly immersed in a biological tissue from boundary measurement. The proposed approach is based on the Kohn–Vogelius formulation [3] and the topological sensitivity analysis method [1,4,9].

Let $Ω \subset R^{2}$ be a smooth and bounded domain represents the considered biological tissue. Inside the tissue Ω, we suppose the existence of an unknown small infected zone $I_{z, ε}^{*} = z^{*} + ε^{*} I^{*}$ (an embedded tumor or an other anomaly) that is characterized by its center $z^{*} \in Ω$ , its size $0 < ε^{*} ≪ 1$ and its shape $I^{*}$ with $I^{*} \subset R^{2}$ is a bounded smooth domain containing the origin. The temperatures at the skin above a tumor of melanocytes, malignant melanoma or a breast tumor, have been found to be several degrees higher than that of the surrounding area [10,11,13,15]. In the biomedical field, the infected zone is modeled as a sub-region that is characterized by a maximum temperature (peak) $θ_{m}$ . The temperature field $T_{ε}$ in the biological tissue $Ω ∖ \overline{I_{z, ε}^{*}}$ is governed by the Pennes model equation[14]. It is a well known model equation in biology simulating the heat transfer mechanism between the tissue and blood. $\begin{matrix} (1.1) & \{\begin{matrix} ρ c_{p} \frac{\partial T_{ε}}{\partial t} - div (σ \nabla T_{ε}) - ϖ_{b} ρ_{b} c_{b} (ℏ_{a} - T_{ε}) = S & in Ω ∖ \overline{I_{z, ε}^{*}} \times] 0, T [, \\ σ \nabla T_{ε} . η = θ_{N} & on Σ_{s} \times] 0, T [, \\ T_{ε} = θ_{c} & on Γ_{b} \times] 0, T [, \\ σ \nabla T^{*} . η = 0 & on Γ_{l} \times] 0, T [, \\ T_{ε} = θ_{m} & on \partial I_{z, ε}^{*} \times] 0, T [, \\ T_{ε} (\cdot, 0) = θ_{0} & in Ω ∖ \overline{I_{z, ε}^{*}}, \end{matrix} \end{matrix}$ where ρ is the density of the tissue, $c_{p}$ is the specific heat, σ is the thermal conductivity, $ϖ_{b}$ is the blood perfusion coefficient, $ρ_{b}$ is the density of the blood, $c_{b}$ the specific heat of the blood, $ℏ_{a}$ is the artery temperature, $S$ is the metabolic heat source and T is the computational time.

Here, $θ_{N}$ is an imposed heat flux on the skin surface $Σ_{s}$ with η is the outward unit normal vector, $Γ_{l}$ describes the lateral boundaries, $θ_{c}$ is a constant core temperature prescribed on $Γ_{b}$ the bottom boundary and $θ_{0}$ is the initial temperature distribution. For more details we can consult [5,7,8].

The goal of this study is to develop efficient approach for recovering the unknown parameters (size, location and size) of the infected zone $I_{z, ε}^{*}$ via the thermography concept. This diagnostic process aims to detect the unknown anomaly zone $I_{z, ε}^{*}$ from over-determined boundary data on the skin surface (the outer surface $Σ_{s}$ ) of the tissue. This clinical procedure can be modeled as a geometric inverse problem as follows:

Knowing two boundary data on the accessible surface $Σ_{s}$ : an imposed heat flux $θ_{N}$ (a Neumann condition) and a measured temperature $θ_{D}$ (a Dirichlet condition).

Determine the location, size and shape of the unknown infected zone $I_{z, ε}^{*}$ , in such a way that the temperature distribution $T_{ε}$ in the unhealthy tissue satisfies the bio-heat transfer system (1.1) with an over-specified boundary condition on the skin surface $\begin{matrix} (1.2) & \begin{matrix} σ \nabla T_{ε} . η = θ_{N} on Σ_{s} \times] 0, T [, \\ T_{ε} = θ_{D} on Σ_{s} \times] 0, T [. \end{matrix} \end{matrix}$

To analyze this geometric inverse problem, we will develop an efficient approach based on the Kohn–Vogelius method and the topological sensitivity concept. The main idea of the employed technique is illustrated in Section 2. It consists to reformulate the ill-posed inverse problem as a topology optimization problem, minimizing a Kohn–Vogelius type functional. In Section 3, we present a regularity result for the Pennes equation solution. Section 4 is concerned with a topological sensitivity analysis for the temperature field. We establish an estimate describing the temperature field behavior with respect to the anomaly size. In Section 5, we develop an asymptotic formula describing the variation of the considered Kohn–Vogelius type functional with respect to the presence of a small anomaly inside the tissue.

2. Topology optimization problem

The Kohn–Vogelius formulation consists in replacing the original over-determined boundary value problem (1.1)–(1.2) by two auxiliary well-posed problems. For each admissible anomaly zone, defined by a small domain $I_{z, ε}$ strictly included inside the biological tissue Ω, we have to solve two systems. The first system is called “Neuman problem”. It is defined using the imposed heat flux through the boundary $Σ_{s}$ $\begin{matrix} (2.1) & \{\begin{matrix} ρ c_{p} \frac{\partial T_{ε}^{N}}{\partial t} - div (σ \nabla T_{ε}^{N}) - ϖ_{b} ρ_{b} c_{b} (ℏ_{a} - T_{ε}^{N}) = S & in Ω ∖ \overline{I_{z, ε}} \times] 0, T [, \\ σ \nabla T_{ε}^{N} . η = θ_{N} & on Σ_{s} \times] 0, T [, \\ T_{ε}^{N} = θ_{c} & on Γ_{b} \times] 0, T [, \\ σ \nabla T_{ε}^{N} . η = 0 & on Γ_{l} \times] 0, T [, \\ T_{ε}^{N} = θ_{m} & on \partial I_{z, ε} \times] 0, T [, \\ T_{ε}^{N} (\cdot, 0) = θ_{0} & in Ω ∖ \overline{I_{z, ε}} . \end{matrix} \end{matrix}$ The second one is called “Dirichlet problem”. It is defined using the measured temperature on the boundary $Σ_{s}$ $\begin{matrix} (2.2) & \{\begin{matrix} ρ c_{p} \frac{\partial T_{ε}^{D}}{\partial t} - div (σ \nabla T_{ε}^{D}) - ϖ_{b} ρ_{b} c_{b} (ℏ_{a} - T_{ε}^{D}) = S & in Ω ∖ \overline{I_{z, ε}} \times] 0, T [, \\ T_{ε}^{D} = θ_{D} & on Σ_{s} \times] 0, T [, \\ T_{ε}^{D} = θ_{c} & on Γ_{b} \times] 0, T [, \\ σ \nabla T_{ε}^{D} . η = 0 & on Γ_{l} \times] 0, T [, \\ T_{ε}^{D} = θ_{m} & on \partial I_{z, ε} \times] 0, T [, \\ T_{ε}^{D} (\cdot, 0) = θ_{0} & in Ω ∖ \overline{I_{z, ε}} . \end{matrix} \end{matrix}$ We remark that if $I_{z, ε}$ coincides with the actual infected zone $I_{z, ε}^{*}$ , then the misfit between the Neumann and Dirichlet solutions vanishes. Starting from this observation, we propose in this work an identification process based on the minimization of the following Kohn–Vogelius type functional $\begin{matrix} F (Ω ∖ \overline{I_{z, ε}}) = \int_{0}^{T} \int_{Ω ∖ \overline{I_{z, ε}}} {| T_{ε}^{N} - T_{ε}^{D} |}^{2} d x d t, \end{matrix}$ where $T_{ε}^{N}$ and $T_{ε}^{D}$ are the solutions, respectively, of systems (2.1) and (2.2).

Then, the considered geometric inverse problem can be reformulated as a topology optimization problem where the unknown infected zone $I_{z, ε}^{*}$ is characterized as the solution of the following minimization problem: $\begin{array}{rcl} (P_{min}) \{\begin{matrix} Find the location “ z^{*} ”, size “ ε^{*} ” and shape “ I^{*} ” of the anomaly I_{z, ε}^{*} such that \\ F (Ω ∖ \overline{I_{z, ε}^{*}}) = {min}_{I_{z, ε} \in D_{a d}} F (Ω ∖ \overline{I_{z, ε}}), \end{matrix} \end{array}$ where $D_{a d}$ is a set of admissible domains $\begin{matrix} D_{a d} = {I_{z, ε} = z + ε I; such that \overline{I_{z, ε}} \subset Ω, 0 < ε ≪ 1 and \partial I of class C^{1}} . \end{matrix}$ To solve this optimization problem, we use the topological sensitivity analysis method. It corresponds to the study of the variation of the function to be minimized with respect to the creation of a small geometric perturbation inside the domain Ω. The adaptation of this technique to our problem leads to evaluate the Kohn–Vogelius functional variation with respect to the presence of a small infected zone $I_{z, ε} = z + ε I$ inside the biological tissue Ω.

Practically, this approach requires the development of an asymptotic expansion of the form $\begin{matrix} F (Ω ∖ \overline{I_{z, ε}}) = F (Ω) + ϕ (ε) G (z) + o (ϕ (ε)), \end{matrix}$ where

$G : Ω ⟼ R$ is a scalar function (called the topological gradient of $F$ ), represents the leading term of the variation $F (Ω ∖ \overline{I_{z, ε}}) - F (Ω)$ with respect to the insertion of a small hole inside the domain Ω.

$ϕ : R_{+} ⟼ R_{+}$ is a scalar positive function, represents the asymptotic behavior of the variation $F (Ω ∖ \overline{I_{z, ε}}) - F (Ω)$ with respect to the geometric perturbation size ε.

In order to derive the expected asymptotic formula, we need to estimate the anomaly influence on the temperature distribution. Hereafter, we start our mathematical analysis by a regularity result of the solution of a Pennes-type equation.

3. Regularity result

This section concerns a regularity result for the solution of a Pennes-type equation.

Proposition 3.1.
Let $k \in N$ and $\tilde{V} ⋐ Ω$ be a given non-empty subdomain. For $Q \in L^{2} (0, T, H^{- 1} (Ω)) \cap L^{2} (0, T, H^{k} (\tilde{V})) \cap H^{k / 2} (0, T, L^{2} (\tilde{V}))$ and $g \in L^{2} (0, T, H^{1 / 2} (\partial Ω))$ , we denote by φ the solution of the following Pennes-type equation $\begin{matrix} (3.1) & \{\begin{matrix} ρ c_{p} \frac{\partial φ}{\partial t} - div (σ \nabla φ) - ϖ_{b} ρ_{b} c_{b} (ℏ_{a} - φ) = Q & in Ω \times (0, T), \\ φ = g & on \partial Ω \times (0, T), \\ φ (\cdot, 0) = 0 & in Ω . \end{matrix} \end{matrix}$ Then, there exists a sub-domain $V_{k} ⋐ \tilde{V}$ , such that $\begin{matrix} (3.2) & φ \in L^{2} (0, T, H^{k + 2} (V_{k})) \cap H^{k / 2 + 1} (0, T, L^{2} (V_{k})) . \end{matrix}$ This regularity result is valid for the Neumann case if we replace the Dirichlet condition on $\partial Ω$ by a Neumann condition $\nabla φ . η = g$ , with $g \in L^{2} (0, T, H^{- 1 / 2} (\partial Ω))$ .
Proof.
We remark, that the compatibility conditions required to apply the standard parabolic regularity theorems are not satisfied here. To overcome this difficulty we applied an iterative process for constructing auxiliary functions verifying the required regularity conditions.
Let $V_{0} \subset R^{2}$ , such that $V_{k} ⋐ V_{0} ⋐ \tilde{V}$ . Denoting by $Υ_{0}$ an arbitrary smooth function defined in Ω and verifying $\begin{matrix} (3.3) & Υ_{0} = \{\begin{matrix} 0 & in Ω ∖ \overline{\tilde{V}}, \\ 1 & in V_{0} . \end{matrix} \end{matrix}$ Considering the function $φ_{0} = Υ_{0} φ$ we easily show that $φ_{0}$ is the solution of the following system: $\begin{matrix} (3.4) & \{\begin{matrix} ρ c_{p} \frac{\partial φ_{0}}{\partial t} - div (σ \nabla φ_{0}) - ϖ_{b} ρ_{b} c_{b} (ℏ_{a} - φ_{0}) = Q_{0} & in Ω \times (0, T), \\ φ_{0} = g & on \partial Ω \times (0, T), \\ φ_{0} (\cdot, 0) = 0 & in Ω, \end{matrix} \end{matrix}$ with $\begin{matrix} Q_{0} = Υ_{0} Q - 2 σ \nabla Υ_{0} \nabla φ - σ φ Δ Υ_{0} . \end{matrix}$ Since σ and $Υ_{0}$ are smooth in Ω, from the fact that $φ \in L^{2} (0, T; H^{1} (Ω))$ , we conclude that $Q_{0} \in L^{2} (0, T; L^{2} (Ω))$ . Using ([12], Chap. 4, Theorem 1.1), we derive that $φ_{0} \in L^{2} (0, T, H^{2} (Ω)) \cap H^{1} (0, T, L^{2} (Ω))$ . It therefore follows that $\begin{matrix} φ \in L^{2} (0, T, H^{2} (V_{0})) \cap H^{1} (0, T, L^{2} (V_{0})) . \end{matrix}$

Assume that, for each $j \in {0, \dots, k - 1}$ , there exists a domain $V_{j}$ such that $V_{k} ⋐ V_{j} ⋐ \tilde{V}$ and the solution φ satisfies $\begin{matrix} (3.5) & φ \in L^{2} (0, T, H^{j + 2} (V_{j})) \cap H^{j / 2 + 1} (0, T, L^{2} (V_{j})) . \end{matrix}$ Next, we will prove that this regularity condition remains true for the iteration $j + 1$ .

Firstly, one can easily take a sub-domain $V_{j + 1}$ such that $V_{k} ⋐ V_{j + 1} ⋐ V_{j}$ . Let $Υ_{j + 1}$ be a smooth function verifying $\begin{matrix} (3.6) & Υ_{j + 1} = \{\begin{matrix} 0 & in Ω ∖ \overline{V_{j}}, \\ 1 & in V_{j + 1} . \end{matrix} \end{matrix}$ Then the function $\begin{matrix} φ_{j + 1} = Υ_{j + 1} φ . \end{matrix}$ is the solution of the next system: $\begin{matrix} (3.7) & \{\begin{matrix} ρ c_{p} \frac{\partial φ_{j + 1}}{\partial t} - div (ν \nabla φ_{j + 1}) - ϖ_{b} ρ_{b} c_{b} (ℏ_{a} - θ_{j + 1}) = Q_{j + 1} & in Ω \times (0, T), \\ φ_{j + 1} = g & on \partial Ω \times (0, T), \\ φ_{j + 1} (\cdot, 0) = 0 & in Ω, \end{matrix} \end{matrix}$ where $\begin{matrix} Q_{j + 1} = Υ_{j + 1} Q - 2 σ \nabla Υ_{j + 1} \nabla φ - σ φ Δ Υ_{j + 1} . \end{matrix}$ Applying ([12], Chap. 4, Proposition 2.3), it follows that $Q_{j + 1} \in L^{2} (0, T, H^{j + 1} (V_{j})) \cap H^{(j + 1) / 2} (0, T, L^{2} (V_{j}))$ . Thanks to ([12], Chap. 4, Theorem 5.3), we conclude that $\begin{matrix} φ_{j + 1} \in L^{2} (0, T, H^{j + 3} (V_{p})) \cap H^{(j + 3) / 2} (0, T, L^{2} (V_{j})) . \end{matrix}$ Then, we obtain $\begin{matrix} φ \in L^{2} (0, T, H^{j + 3} (V_{j + 1})) \cap H^{(j + 3) / 2} (0, T, L^{2} (V_{j + 1})) . \end{matrix}$

Finally, the regularity result (3.2) can be derived by repeating this procedure up to the iteration k. □

4. Sensitivity analysis

In this section, we discuss the temperature field sensitivity with respect to the presence of a small anomaly. We will examine the behavior of the temperature variation $Θ_{ε}^{β} = T_{ε}^{β} - T_{0}^{β}$ (where $β = N$ or D) with respect to the anomaly size ε. Here $T_{0}^{β}$ represents the temperature field in the safe tissue Ω (i.e. in the absence of any anomaly). Then, $Θ_{ε}^{β}$ satisfies the system $\begin{matrix} (4.1) & \{\begin{matrix} ρ c_{p} \frac{\partial Θ_{ε}^{β}}{\partial t} - div (σ \nabla Θ_{ε}^{β}) + ϖ_{b} ρ_{b} c_{b} Θ_{ε}^{β} = 0 & in Ω \times] 0, T [, \\ Θ_{ε}^{β} = 0 & on Γ_{b} \times] 0, T [, \\ σ \nabla Θ_{ε}^{β} . η = 0 & on Γ_{l} \times] 0, T [, \\ Θ_{ε}^{β} = - T_{0}^{β} & on \partial I_{z, ε} \times] 0, T [, \\ Θ_{0}^{β} (\cdot, 0) = 0 & in Ω, \end{matrix} \end{matrix}$ and the boundary condition $\begin{matrix} (4.2) & \{\begin{matrix} σ \nabla Θ_{ε}^{β} . η = 0 & on Σ_{s} \times] 0, T [, if β = N, \\ Θ_{ε}^{β} = 0 & on Σ_{s} \times] 0, T [, if β = D . \end{matrix} \end{matrix}$ We will prove in this section that the leading term of the temperature variation $T_{ε}^{β} - T_{0}^{β}$ is governed by the quantity $\begin{matrix} H^{β} (x, t) = 2 π T_{0}^{β} (z, t) E (x - z), (x, t) \in Ω ∖ \overline{I_{z, ε}} \times] 0, T [, \end{matrix}$ where E is the Green’s function for the two dimensional Laplace operator $\begin{matrix} E (y) = - \frac{1}{2 π} log (| y |), \forall y \in R^{2} . \end{matrix}$ The following theorem gives an estimate of the temperature perturbation caused by the presence of a small anomaly $I_{z, ε}$ inside the tissue Ω.

Theorem 4.1.
Let $I_{z, ε}$ be a small anomaly created inside the tissue Ω. Then, there exist a constant $c > 0$ and a small parameter $ε_{0} > 0$ , such that the temperature field variation verifies the estimate $\begin{array}{l} {‖ T_{ε}^{β} - T_{0}^{β} - \frac{1}{log ε} H^{β} ‖}_{L^{\infty} (0, T; L^{2} (Ω ∖ \overline{I_{z, ε}}))} + {‖ T_{ε}^{β} - T_{0}^{β} - \frac{1}{log ε} H^{β} ‖}_{L^{2} (0, T; H^{1} (Ω ∖ \overline{I_{z, ε}}))} ⩽ \frac{c}{| log ε |} \end{array}$ for each $ε \in] 0, ε_{0} [$ .

To establish this result, we need some preliminary results.
4.1. Preliminary lemmas

Here, we state two preliminary estimates which will be used in our proof. The first one is devoted to an energy estimate for a Pennes type equation. It is summarized in Lemma 4.2. The second one is concerned with some useful estimates related to the Laplace fundamental solution E (presented in Lemma 4.3).

Lemma 4.2.
Let $q \in] 1, + \infty [$ , $Q \in L^{2} (0, T; H^{1} (Ω))$ and $φ \in H^{1} (0, T; H^{1} (Ω))$ be the solution of the following Pennes type equation $\begin{matrix} \{\begin{matrix} ρ c_{p} \frac{\partial φ}{\partial t} - div (σ (x) \nabla φ) + ϖ_{b} ρ_{b} c_{b} φ = Q & in Ω \times (0, T), \\ φ = 0 & on Γ_{1} \times (0, T), \\ σ \nabla φ . η = 0 & on Γ_{2} \times (0, T), \\ φ (\cdot, 0) = 0 & in Ω, \end{matrix} \end{matrix}$ where $Γ_{1}$ and $Γ_{2}$ are two parts of the boundary $\partial Ω$ , such that $Γ_{1} \cap Γ_{2} = \emptyset$ and $Γ_{1} \cup Γ_{2} = \partial Ω$ .

Then, there exists a constant $c > 0$ such that $\begin{array}{l} ‖ φ ‖_{L^{\infty} (0, T; L^{2} (Ω))} + ‖ φ ‖_{L^{2} (0, T; H^{1} (Ω))} & ⩽ c ‖ Q ‖_{L^{2} (0, T; L^{q} (Ω))} . \end{array}$
Proof.
Due to Green’s formula, from the previous system, one can obtain $\begin{array}{l} ρ c_{p} \int_{Ω} \frac{\partial φ}{\partial t} (x, t) φ (x, t) d x + \int_{Ω} σ (x) {| \nabla φ (x, t) |}^{2} d x + ϖ_{b} ρ_{b} c_{b} \int_{Ω} {| φ (x, t) |}^{2} d x \\ = \int_{Ω} Q (x, t) φ (x, t) d x . \end{array}$ Let $t_{0} \in [0, T]$ , since $\frac{\partial φ}{\partial t} (x, t) φ (x, t) = \frac{1}{2} \frac{d}{d t} {(φ (x, t))}^{2}$ and $φ (\cdot, 0) = 0$ in Ω, it follows $\begin{array}{l} \frac{ρ c_{p}}{2} \int_{Ω} {| φ (\cdot, t_{0}) |}^{2} d x + \int_{0}^{t_{0}} \int_{Ω} σ (x) {| \nabla φ (x, t) |}^{2} d x d t + ϖ_{b} ρ_{b} c_{b} \int_{0}^{t_{0}} \int_{Ω} {| φ (x, t) |}^{2} d x \\ = \int_{0}^{t_{0}} \int_{Ω} Q (x, t) φ (x, t) d x d t \\ (4.3) & ⩽ \int_{0}^{T} \int_{Ω} | Q (x, t) φ (x, t) | d x d t . \end{array}$ Let $r \in N^{}$ such that $\frac{1}{q} + \frac{1}{r} = 1$ . Thanks to Cauchy-Schwartz inequality, we obtain $\begin{array}{l} \int_{0}^{T} \int_{Ω} | Q (x, t) φ (x, t) | d x d t & ⩽ ‖ Q ‖_{L^{2} (0, T; L^{q} (Ω))} ‖ φ ‖_{L^{2} (0, T; L^{r} (Ω))} . \end{array}$ From the fact $q \in] 1, + \infty [$ , it follows $r \in] 1, + \infty [$ , which implies that $H^{1} (Ω) ↪ L^{r} (Ω)$ . Then, there exits a constant $c_{1} > 0$ , independent of ε, such that $\begin{array}{l} \int_{0}^{t_{0}} \int_{Ω} σ (x) {| \nabla φ (x, t) |}^{2} d x d t + ϖ_{b} ρ_{b} c_{b} \int_{0}^{t_{0}} \int_{Ω} {| φ (x, t) |}^{2} d x \\ (4.4) & ⩽ c_{1} ‖ Q ‖_{L^{2} (0, T; L^{q} (Ω))} ‖ φ ‖_{L^{2} (0, T; H^{1} (Ω))} \\ (4.5) & \frac{ρ c_{p}}{2} \int_{Ω} {| φ (\cdot, t_{0}) |}^{2} d x ⩽ c_{1} ‖ Q ‖_{L^{2} (0, T; L^{q} (Ω))} ‖ φ ‖_{L^{2} (0, T; H^{1} (Ω))} . \end{array}$
Estimate of* $‖ φ ‖_{L^{2} (0, T; H^{1} (Ω))}$ : Let $ν_{min} = min {σ_{min}, ϖ_{b} ρ_{b} c_{b}}$ .

By taking $t_{0} = T$ in (4.4), we deduce that there exists $c_{2} > 0$ such that $\begin{array}{l} (4.6) & ‖ φ ‖_{L^{2} (0, T; H^{1} (Ω))} & ⩽ c_{2} ‖ Q ‖_{L^{2} (0, T; L^{q} (Ω))} . \end{array}$ Here, we choose $c_{2} = \frac{c_{1}}{ν_{min}}$ .

Estimate of $‖ φ ‖_{L^{\infty} (0, T; L^{2} (Ω))}$ : By combining (4.5) and (4.6), we obtain $\begin{array}{l} \int_{Ω} {| φ (\cdot, t_{0}) |}^{2} d x & ⩽ \frac{2 c_{1} c_{2}}{ρ c_{p}} ‖ Q ‖_{L^{2} (0, T; L^{q} (Ω))}^{2}, \forall t_{0} \in [0, T] . \end{array}$ Then, there exists a constant $c_{3} > 0$ such that $\begin{array}{l} (4.7) & ‖ φ ‖_{L^{\infty} (0, T; L^{2} (Ω))} & ⩽ c_{3} ‖ Q ‖_{L^{2} (0, T; L^{q} (Ω))} . \end{array}$

Here, we take $c_{3} = \sqrt{\frac{2}{ρ c_{p} ν_{min}}} c_{1}$

The desired estimate: From (4.6) and (4.7), we conclude that there exists a constant $c = max (c_{2}, c_{3})$ such that $\begin{array}{l} (4.8) & ‖ φ ‖_{L^{\infty} (0, T; L^{2} (Ω))} + ‖ φ ‖_{L^{2} (0, T; H^{1} (Ω))} ⩽ c ‖ Q ‖_{L^{2} (0, T; L^{q} (Ω))} . \end{array}$
□

In the following lemma, we establish some estimates related to the Laplace operator Green’s function E. Lemma 4.3.
In the two dimensional case, the Laplace operator Green’s function E admits the following properties. There exists a constant $c > 0$ , independent of ε, such that $\begin{array}{l} (4.9) & {‖ E (x - z) ‖}_{H^{1 / 2} (\partial Ω)} ⩽ c, \\ (4.10) & {‖ E (y) ‖}_{H^{1 / 2} (\partial I)} ⩽ c, \\ (4.11) & {‖ E (x - z) ‖}_{L^{2} (Ω ∖ \overline{I_{z, ε}})} ⩽ c, \\ (4.12) & {‖ \nabla E (x - z) ‖}_{L^{q} (Ω ∖ \overline{I_{z, ε}})} ⩽ c, \forall q \in] 1, 2 [, \\ (4.13) & {‖ \nabla E (x - z) ‖}_{L^{2} (Ω ∖ \overline{I_{z, ε}})} ⩽ c \sqrt{| log ε |} . \end{array}$
Proof.

Let $δ_{0} > 0$ such that $\overline{B (z, δ_{0})} \subset Ω$ . Since $z \notin Ω_{z, δ_{0}} : = Ω ∖ \overline{B (z, δ_{0})}$ , the function $x \mapsto E (x - z)$ belongs to $C^{\infty} (\overline{Ω_{z, δ_{0}}})$ and the term $‖ E (x - z) ‖_{H^{1 / 2} (\partial Ω)}$ is uniformly bounded, i.e. there exists $c > 0$ such that $\begin{matrix} {‖ E (x - z) ‖}_{H^{1 / 2} (\partial Ω)} ⩽ {‖ E (x - z) ‖}_{H^{1} (Ω_{z, δ_{0}})} ⩽ c . \end{matrix}$

Due to the fact that $I$ is an open domain containing the origin, then there exists $δ_{1} > 0$ such that $\overline{B (0, δ_{1})} \subset I$ . It follows that the function $y \mapsto E (y)$ belongs to $C^{\infty} (\overline{I_{0, δ_{1}}})$ where $I_{0, δ_{1}} : = I ∖ \overline{B (0, δ_{1})}$ . Consequently, $\begin{array}{rcl} ‖ E ‖_{H^{1 / 2} (\partial I)} & : = & inf {‖ υ ‖_{H^{1} (I_{0, δ_{1}})}, υ = E on \partial I} \\ ⩽ & ‖ E ‖_{L^{2} (I_{0, δ_{1}})} + ‖ \nabla E ‖_{L^{2} (I_{0, δ_{1}})} = O (1) . \end{array}$

The estimate of the term $‖ E (x - z) ‖_{L^{2} (Ω ∖ \overline{I_{z, ε}})}$ , is deduced from the fact that the domain Ω is bounded in such a way that there exists $δ_{2} > 0$ such that $\overline{Ω} \subset B (z, δ_{2})$ , $\forall z \in Ω$ . We have $\begin{matrix} (4.14) & Ω ∖ \overline{I_{z, ε}} - z : = {x - z, x \in Ω ∖ \overline{I_{z, ε}}} \subset C (0, δ_{1} ε, δ_{2}) : = {y \in R^{2}, δ_{1} ε < | y | < δ_{2}} . \end{matrix}$ Since $0 \notin \overline{C (0, δ_{1} ε, δ_{2})}$ , the function $x \mapsto E (x - z)$ is smooth in $C (0, δ_{1} ε, δ_{2})$ . Employing the polar coordinates, one can derive $\begin{array}{l} {‖ E (x - z) ‖}_{L^{2} (Ω ∖ \overline{I_{z, ε}})} & ⩽ \frac{1}{\sqrt{2 π}} {(\int_{δ_{1} ε}^{δ_{2}} r {(log r)}^{2} d r)}^{\frac{1}{2}} . \end{array}$ Exploiting the fact that the function $r \mapsto r {(log r)}^{2}$ is integrable near the origin, we conclude that there exists $c > 0$ , independent of ε, such that $\begin{array}{l} {‖ E (x - z) ‖}_{L^{2} (Ω ∖ \overline{I_{z, ε}})} & ⩽ c . \end{array}$

The estimates (4.12) and (4.13) can be derived with an explicit calculus. From (4.14), we obtain for all $q \in] 1, 2]$ $\begin{array}{rcl} {‖ \nabla E (x - z) ‖}_{L^{q} (Ω ∖ \overline{I_{z, ε}})} & ⩽ & ‖ \nabla E ‖_{L^{q} (C (0, δ_{1} ε, δ_{2}))} \\ = & {(\frac{1}{{(2 π)}^{q}} \int_{C (0, δ_{1} ε, δ_{2})} {| \frac{1}{y} |}^{q} d x)}^{1 / q} \\ (4.15) & = & {({(2 π)}^{1 - q} \int_{δ_{1} ε}^{δ_{2}} r^{1 - q} d r)}^{1 / q} . \end{array}$ Then, when $q \in] 1, 2 [$ , we have $\begin{matrix} {‖ \nabla E (x - z) ‖}_{L^{q} (Ω ∖ \overline{I_{z, ε}})} ⩽ {(\frac{{(2 π)}^{1 - q}}{2 - q} [{(δ_{2})}^{2 - q} - {(δ_{1} ε)}^{2 - q}])}^{1 / q} = O (1) . \end{matrix}$ In the case $q = 2$ , $‖ \nabla E (x - z) ‖_{L^{2} (Ω ∖ \overline{I_{z, ε}})}$ admits the following behavior $\begin{matrix} {‖ \nabla E (x - z) ‖}_{L^{2} (Ω ∖ \overline{I_{z, ε}})} ⩽ \frac{1}{\sqrt{2 π}} {(log (δ_{2}) - log (δ_{1} ε))}^{1 / 2} = O (\sqrt{| log ε |}) . \end{matrix}$
□

4.2. Proof of Theorem 4.1

Denoting by $R_{ε}^{β}$ the quantity to be estimated $\begin{array}{l} R_{ε}^{β} (x, t) = T_{ε}^{β} (x, t) - T_{0}^{β} (x, t) - \frac{1}{log ε} H^{β} (x, t), \\ (4.16) & (x, t) \in Ω ∖ \overline{I_{z, ε}} \times] 0, T [, for β = N, D . \end{array}$ The Neumann ( $β = N$ ) and Dirichlet ( $β = D$ ) cases can be treated by the same arguments. In this proof, we only develop the Neumann case. We can easily verify that $R_{ε}^{N}$ is the solution of the following system: $\begin{matrix} \{\begin{matrix} ρ c_{p} \frac{\partial R_{ε}^{N}}{\partial t} - div (σ \nabla R_{ε}^{N}) + ϖ_{b} ρ_{b} c_{b} R_{ε}^{N} = Q^{N} & in Ω ∖ \overline{I_{z, ε}} \times] 0, T [, \\ σ \nabla R_{ε}^{N} . η = - \frac{2 π σ}{log ε} T_{0}^{N} (z, t) \nabla E (x - z) . η & on Σ_{s} \times] 0, T [, \\ R_{ε}^{N} = - \frac{2 π}{log ε} T_{0}^{N} (z, t) E (x - z) & on Γ_{b} \times] 0, T [, \\ σ \nabla R_{ε}^{N} . η = 0 & on Γ_{l} \times] 0, T [, \\ R_{ε}^{N} = - T_{0}^{N} (x, t) - \frac{2 π}{log ε} T_{0}^{N} (z, t) E (x - z) & on \partial I_{z, ε} \times] 0, T [, \\ R_{ε}^{N} (\cdot, 0) = 0 & in Ω ∖ \overline{I_{z, ε}} . \end{matrix} \end{matrix}$ with $Q^{N}$ is the associated source term, given by $\begin{matrix} Q^{N} (x, t) = \frac{2 π}{log ε} (T_{0}^{N} (z, t) \nabla σ . \nabla E (x - z) - ρ c_{p} \frac{\partial T_{0}^{N}}{\partial t} (z, t) E (x - z) + ϖ_{b} ρ_{b} c_{b} T_{0}^{N} (z, t) E (x - z)) \end{matrix}$ In order to establish the expected estimate, we decompose the function $R_{ε}^{N}$ in three parts $\begin{matrix} R_{ε}^{N} = R_{ε}^{1, N} + R_{ε}^{2, N} + R_{ε}^{3, N} \end{matrix}$ and we estimate each term separately.

Estimate of the first term: $R_{ε}^{1, N}$ satisfies $\begin{matrix} \{\begin{matrix} ρ c_{p} \frac{\partial R_{ε}^{1, N}}{\partial t} - div (σ \nabla R_{ε}^{1, N}) + ϖ_{b} ρ_{b} c_{b} R_{ε}^{1, N} = \frac{2 π}{log ε} T_{0}^{N} (z, t) \nabla σ . \nabla E (x - z) & in Ω ∖ \overline{I_{z, ε}} \times] 0, T [, \\ σ \nabla R_{ε}^{1, N} . η = 0 & on Σ_{s} \times] 0, T [, \\ R_{ε}^{1, N} = 0 & on Γ_{b} \times] 0, T [, \\ σ \nabla R_{ε}^{1, N} . η = 0 & on Γ_{l} \times] 0, T [, \\ R_{ε}^{1, N} = 0 & on \partial I_{z, ε} \times] 0, T [, \\ R_{ε}^{1, N} (\cdot, 0) = 0 & in Ω ∖ \overline{I_{z, ε}} . \end{matrix} \end{matrix}$ Applying Lemma 4.2 with $q \in] 1, 2 [$ , it follows $\begin{array}{l} {‖ R_{ε}^{1, N} ‖}_{L^{\infty} (0, T; L^{2} (Ω ∖ \overline{I_{z, ε}}))} + {‖ R_{ε}^{1, N} ‖}_{L^{2} (0, T; H^{1} (Ω ∖ \overline{I_{z, ε}}))} \\ ⩽ \frac{c}{| log ε |} {‖ T_{0}^{N} (z, t) ‖}_{L^{\infty} (0, T)} {‖ \nabla σ . \nabla E (x - z) ‖}_{L^{q} (Ω ∖ \overline{I_{z, ε}})} . \end{array}$ Exploiting Lemma 4.3 and the smoothness of $T_{0}^{N}$ and σ, we obtain $\begin{array}{l} {‖ R_{ε}^{1, N} ‖}_{L^{\infty} (0, T; L^{2} (Ω ∖ \overline{I_{z, ε}}))} + {‖ R_{ε}^{1, N} ‖}_{L^{2} (0, T; H^{1} (Ω ∖ \overline{I_{z, ε}}))} \\ ⩽ \frac{c}{| log ε |} {‖ \nabla E (x - z) ‖}_{L^{q} (Ω ∖ \overline{I_{z, ε}})} \\ = O (\frac{1}{| log ε |}) . \end{array}$

Estimate of the second term: $R_{ε}^{2, N}$ satisfies $\begin{matrix} \{\begin{matrix} ρ c_{p} \frac{\partial R_{ε}^{2, N}}{\partial t} - div (σ \nabla R_{ε}^{2, N}) + ϖ_{b} ρ_{b} c_{b} R_{ε}^{2, N} \\ = \frac{2 π}{log ε} [ϖ_{b} ρ_{b} c_{b} T_{0}^{N} (z, t) E (x - z) - ρ c_{p} \frac{\partial T_{0}^{N}}{\partial t} (z, t) E (x - z)] & in Ω ∖ \overline{I_{z, ε}} \times] 0, T [, \\ σ \nabla R_{ε}^{2, N} . η = - \frac{2 π σ}{log ε} T_{0}^{N} (z, t) \nabla E (x - z) . η & on Σ_{s} \times] 0, T [, \\ R_{ε}^{2, N} = - \frac{2 π}{log ε} T_{0}^{N} (z, t) E (x - z) & on Γ_{b} \times] 0, T [, \\ σ \nabla R_{ε}^{2, N} . η = 0 & on Γ_{l} \times] 0, T [, \\ R_{ε}^{2, N} = 0 & on \partial I_{z, ε} \times] 0, T [, \\ R_{ε}^{2, N} (\cdot, 0) = 0 & in Ω ∖ \overline{I_{z, ε}} . \end{matrix} \end{matrix}$ Using the fact that $div E (x - z) = 0$ in $Ω ∖ \overline{I_{z, ε}}$ , we derive $\begin{array}{l} {‖ R_{ε}^{2, N} ‖}_{L^{\infty} (0, T; L^{2} (Ω ∖ \overline{I_{z, ε}}))} + {‖ R_{ε}^{2, N} ‖}_{L^{2} (0, T; H^{1} (Ω ∖ \overline{I_{z, ε}}))} \\ ⩽ \frac{c}{| log ε |} ({‖ E (x - z) ‖}_{L^{2} (Ω ∖ \overline{I_{z, ε}})} + {‖ E (x - z) ‖}_{H^{1} (Ω_{r})}) \end{array}$ where $r > 0$ is chosen such that $\overline{I_{z, ε}} \subset B (z, r)$ and $Ω_{r} = Ω ∖ \overline{B (z, r)})$ . Since $‖ E (x - z) ‖_{L^{2} (Ω ∖ \overline{I_{z, ε}})}$ and $‖ E (x - z) ‖_{H^{1} (Ω_{r})}$ are uniformly bounded, then we have $\begin{array}{rcl} {‖ R_{ε}^{2, N} ‖}_{L^{\infty} (0, T; L^{2} (Ω ∖ \overline{I_{z, ε}}))} + {‖ R_{ε}^{2, N} ‖}_{L^{2} (0, T; H^{1} (Ω ∖ \overline{I_{z, ε}}))} ⩽ \frac{c}{| log ε |} . \end{array}$

Estimate of the third term: $R_{ε}^{3, N}$ satisfies $\begin{matrix} \{\begin{matrix} ρ c_{p} \frac{\partial R_{ε}^{3, N}}{\partial t} - div (σ \nabla R_{ε}^{3, N}) + ϖ_{b} ρ_{b} c_{b} R_{ε}^{3, N} = 0 & in Ω ∖ \overline{I_{z, ε}} \times] 0, T [, \\ σ \nabla R_{ε}^{3, N} . η = 0 & on Σ_{s} \times] 0, T [, \\ R_{ε}^{3, N} = 0 & on Γ_{b} \times] 0, T [, \\ σ \nabla R_{ε}^{1, N} . η = 0 & on Γ_{l} \times] 0, T [, \\ R_{ε}^{3, N} = - T_{0}^{N} (x, t) - \frac{2 π}{log ε} T_{0}^{N} (z, t) E (x - z) & on \partial I_{z, ε} \times] 0, T [, \\ R_{ε}^{3, N} (\cdot, 0) = 0 & in Ω ∖ \overline{I_{z, ε}} . \end{matrix} \end{matrix}$ From the identity $E (x - z) = E ((x - z) / ε) - \frac{1}{2 π} log ε$ , we obtain $\begin{array}{l} {‖ R_{ε}^{3, N} ‖}_{L^{\infty} (0, T; L^{2} (Ω ∖ \overline{I_{z, ε}}))} + {‖ R_{ε}^{3, N} ‖}_{L^{2} (0, T; H^{1} (Ω ∖ \overline{I_{z, ε}}))} \\ ⩽ c {{‖ T_{0}^{N} (x, t) - T_{0}^{N} (z, t) ‖}_{L^{2} (0, T; H^{1 / 2} (\partial I_{z, ε})} \\ + \frac{2 π}{| log ε |} {‖ T_{0}^{N} (z, t) ‖}_{L^{2} (0, T)} {‖ E ((x - z) / ε) ‖}_{H^{1} (Ω ∖ \overline{I_{z, ε}})}} . \end{array}$ Since $T_{0}^{N}$ is sufficiently smooth in $Ω \times (0, T)$ and $‖ E ((x - z) / ε) ‖_{H^{1} (Ω ∖ \overline{I_{z, ε}})}$ is uniformly bounded with respect to ε, it follows $\begin{array}{rcl} {‖ R_{ε}^{3, N} ‖}_{L^{\infty} (0, T; L^{2} (Ω ∖ \overline{I_{z, ε}}))} + {‖ R_{ε}^{3, N} ‖}_{L^{2} (0, T; H^{1} (Ω ∖ \overline{I_{z, ε}}))} ⩽ \frac{c}{| log ε |} . \end{array}$

5. Asymptotic analysis

This section is concerned with the development of an asymptotic formula describing the variation of the function $F$ with respect to the presence of a small anomaly $I_{z, ε}$ inside the tissue Ω.

Exploiting the definition of $F$ and replacing the temperature field $T^{β}$ by $T^{β} - θ_{m}$ (where $β = N$ or D) in (2.1) and (2.2), the quantity $F (Ω ∖ \overline{I_{z, ε}})$ can be rewritten as $\begin{matrix} F (Ω ∖ \overline{I_{z, ε}}) = \int_{0}^{T} \int_{Ω ∖ \overline{I_{z, ε}}} {| T_{ε}^{N} - T_{ε}^{D} |}^{2} d x d t, \end{matrix}$ where $T_{ε}^{N}$ and $T_{ε}^{D}$ are respectively the solutions to the following systems with an homogeneous condition on the anomaly boundary $\begin{array}{l} (5.1) & \{\begin{matrix} ρ c_{p} \frac{\partial T_{ε}^{N}}{\partial t} - div (σ \nabla T_{ε}^{N}) + ϖ_{b} ρ_{b} c_{b} T_{ε}^{N} = ℜ & in Ω ∖ \overline{I_{z, ε}} \times] 0, T [, \\ σ \nabla T_{ε}^{N} . η = θ_{N} & on Σ_{s} \times] 0, T [, \\ T_{ε}^{N} = θ_{c} - θ_{m} & on Γ_{b} \times] 0, T [, \\ σ \nabla T_{ε}^{N} . η = 0 & on Γ_{l} \times] 0, T [, \\ T_{ε}^{N} = 0 & on \partial I_{z, ε} \times] 0, T [, \\ T_{ε}^{N} (\cdot, 0) = θ_{0} - θ_{m} & in Ω ∖ \overline{I_{z, ε}}, \end{matrix} \\ (5.2) & \{\begin{matrix} ρ c_{p} \frac{\partial T_{ε}^{D}}{\partial t} - div (σ \nabla T_{ε}^{D}) + ϖ_{b} ρ_{b} c_{b} T_{ε}^{D} = ℜ & in Ω ∖ \overline{I_{z, ε}} \times] 0, T [, \\ T_{ε}^{D} = θ_{D} - θ_{m} & on Σ_{s} \times] 0, T [, \\ T_{ε}^{D} = θ_{c} - θ_{m} & on Γ_{b} \times] 0, T [, \\ σ \nabla T_{ε}^{D} . η = 0 & on Γ_{l} \times] 0, T [, \\ T_{ε}^{D} = 0 & on \partial I_{z, ε} \times] 0, T [, \\ T_{ε}^{D} (\cdot, 0) = θ_{0} - θ_{m} & in Ω ∖ \overline{I_{z, ε}}, \end{matrix} \end{array}$ with $ℜ = S + ϖ_{b} ρ_{b} c_{b} (ℏ_{a} + θ_{m})$ .

In order to evaluate the variation $F (Ω ∖ \overline{I_{z, ε}}) - F (Ω)$ , we introduce two adjoint states $Ψ_{0}^{N}$ and $Ψ_{0}^{D}$ , respectively solutions of the following auxiliary problems: $\begin{array}{l} (A^{N}) \{\begin{matrix} - ρ c_{p} \frac{\partial Ψ_{0}^{N}}{\partial t} - div (σ \nabla Ψ_{0}^{N}) + ϖ_{b} ρ_{b} c_{b} Ψ_{0}^{N} = - 2 (T_{0}^{N} - T_{0}^{D}) & in Ω \times] 0, T [, \\ σ \nabla Ψ_{0}^{N} . η = 0 & on Σ_{s} \times] 0, T [, \\ Ψ_{0}^{N} = 0 & on Γ_{b} \times] 0, T [, \\ σ \nabla Ψ_{0}^{N} . η = 0 & on Γ_{l} \times] 0, T [, \\ Ψ_{0}^{N} (\cdot, T) = 0 & in Ω, \end{matrix} \\ (A^{D}) \{\begin{matrix} - ρ c_{p} \frac{\partial Ψ_{0}^{D}}{\partial t} - div (σ \nabla Ψ_{0}^{D}) + ϖ_{b} ρ_{b} c_{b} Ψ_{0}^{D} = - 2 (T_{0}^{D} - T_{0}^{N}) & in Ω \times] 0, T [, \\ Ψ_{0}^{D} = 0 & on Σ_{s} \times] 0, T [, \\ Ψ_{0}^{D} = 0 & on Γ_{b} \times] 0, T [, \\ σ \nabla Ψ_{0}^{D} . η = 0 & on Γ_{l} \times] 0, T [, \\ Ψ_{0}^{D} (\cdot, T) = 0 & in Ω . \end{matrix} \end{array}$

The following theorem summarizes the obtained asymptotic formula.

Theorem 5.1.

The variation of the function $F$ , with respect to the presence of a small internal anomaly $I_{z, ε} = z + ε I$ inside the tissue domain Ω, satisfies $\begin{array}{l} F (Ω ∖ {\overline{I}}_{z, ε}) - F (Ω) & = - \frac{1}{log ε} G (z) + o (\frac{- 1}{| log ε |}), \end{array}$ where $G$ is the topological gradient, defined as $\begin{matrix} G (x) = 2 π \int_{0}^{T} [T_{0}^{N} (z, t) Ψ_{0}^{N} (z, t) + T_{0}^{D} (z, t) Ψ_{0}^{D} (z, t)] d t, \forall x \in Ω . \end{matrix}$

Proof.

The variation of the function $F$ can be written as follows $\begin{array}{l} F (Ω ∖ \overline{I_{z, ε}}) - F (Ω) \\ = \int_{0}^{T} \int_{Ω ∖ \overline{I_{z, ε}}} {| T_{ε}^{N} - T_{ε}^{D} |}^{2} - {| T_{0}^{N} - T_{0}^{D} |}^{2} d x d t - \int_{0}^{T} \int_{I_{z, ε}} {| T_{0}^{N} - T_{0}^{D} |}^{2} d x d t \\ = \int_{0}^{T} \int_{Ω ∖ \overline{I_{z, ε}}} {| (T_{ε}^{N} - T_{0}^{N}) - (T_{ε}^{D} - T_{0}^{D}) |}^{2} d x d t + \int_{0}^{T} \int_{I_{z, ε}} {| T_{0}^{N} - T_{0}^{D} |}^{2} d x d t \\ - 2 \int_{0}^{T} \int_{Ω} (T_{0}^{N} - T_{0}^{D}) (T_{ε}^{N} - T_{0}^{N}) d x d t \\ (5.3) & - 2 \int_{0}^{T} \int_{Ω} (T_{0}^{D} - T_{0}^{N}) (T_{ε}^{D} - T_{0}^{D}) d x d t \end{array}$ In the two last integral terms, $T_{ε}^{N}$ and $T_{ε}^{D}$ are extended by zero in $I_{z, ε} \times] 0, T [$ . We will estimate the previous integral terms separately.

Estimate of the first integral term: Using the identity ${(a - b)}^{2} ⩽ 2 (a^{2} + b^{2})$ , we deduce that the first integral term in the previous equality satisfies $\begin{array}{l} \int_{0}^{T} \int_{Ω ∖ \overline{I_{z, ε}}} {| (T_{ε}^{N} - T_{0}^{N}) - (T_{ε}^{D} - T_{0}^{D}) |}^{2} d x d t \\ ⩽ 2 {‖ T_{ε}^{N} - T_{0}^{N} ‖}_{L^{2} (0, T; L^{2} (Ω ∖ \overline{I_{z, ε}}))}^{2} + 2 {‖ T_{ε}^{D} - T_{0}^{D} ‖}_{L^{2} (0, T; L^{2} (Ω ∖ \overline{I_{z, ε}}))}^{2} . \end{array}$ By the decomposition (4.16), we have $\begin{array}{l} {‖ T_{ε}^{β} - T_{0}^{β} ‖}_{L^{2} (0, T; L^{2} (Ω ∖ \overline{I_{z, ε}}))}^{2} \\ ⩽ 2 {‖ R_{ε}^{β} ‖}_{L^{2} (0, T; L^{2} (Ω ∖ \overline{I_{z, ε}}))}^{2} + \frac{2}{| log ε |^{2}} {‖ H^{β} ‖}_{L^{2} (0, T; L^{2} (Ω ∖ \overline{I_{z, ε}}))}^{2} \\ ⩽ 2 {‖ R_{ε}^{β} ‖}_{L^{2} (0, T; L^{2} (Ω ∖ \overline{I_{z, ε}}))}^{2} + \frac{8 π^{2}}{| log ε |^{2}} {‖ T_{0}^{β} (z, \cdot) ‖}_{L^{2} (0, T)}^{2} {‖ E (x - z) ‖}_{L^{2} (Ω ∖ \overline{I_{z, ε}})}^{2} . \end{array}$ Thanks to Theorem 4.1 and the fact that $‖ E (x - z) ‖_{L^{2} (Ω ∖ \overline{I_{z, ε}})}$ is uniformly bounded, we obtain $\begin{array}{rcl} {‖ T_{ε}^{β} - T_{0}^{β} ‖}_{L^{2} (0, T; L^{2} (Ω ∖ \overline{I_{z, ε}}))}^{2} = o (\frac{1}{| log ε |}), β = N or D . \end{array}$ Then, the first integral term admits the estimate $\begin{array}{rcl} \int_{0}^{T} \int_{Ω ∖ \overline{I_{z, ε}}} {| (T_{ε}^{N} - T_{0}^{N}) - (T_{ε}^{D} - T_{0}^{D}) |}^{2} d x d t = o (\frac{1}{| log ε |}) . \end{array}$

Estimate of the second integral term: The estimate of the second integral term is deduced from the smoothness of the functions $T_{0}^{N}$ and $T_{0}^{D}$ near the point z. Then, $\begin{array}{rcl} \int_{0}^{T} \int_{I_{z, ε}} {| T_{0}^{N} - T_{0}^{D} |}^{2} d x d t = O (| I_{z, ε} |) = o (\frac{1}{| log ε |}), \end{array}$ where $| I_{z, ε} |$ is the Lebesgue measure of the domain $I_{z, ε}$ (i.e. $| I_{z, ε} | = O (ε^{2})$ ).

Estimate of the third integral term: The weak formulation of the Neumann adjoint problem $(A^{N})$ implies $\begin{array}{l} - ρ c_{p} \int_{0}^{T} \int_{Ω} \frac{\partial Ψ_{0}^{N}}{\partial t} (T_{ε}^{N} - T_{0}^{N}) d x d t + \int_{0}^{T} \int_{Ω} σ \nabla Ψ_{0}^{N} \nabla (T_{ε}^{N} - T_{0}^{N}) d x d t \\ + ϖ_{b} ρ_{b} c_{b} \int_{0}^{T} \int_{Ω} Ψ_{0}^{N} (T_{ε}^{N} - T_{0}^{N}) d x d t \\ = - 2 \int_{0}^{T} \int_{Ω} (T_{0}^{N} - T_{0}^{D}) (T_{ε}^{N} - T_{0}^{N}) d x d t . \end{array}$ Since $Ψ_{0}^{N} (\cdot, T) = (T_{ε}^{N} - T_{0}^{N}) (\cdot, 0) = 0$ in Ω, an integration by parts gives $\begin{array}{l} - ρ c_{p} \int_{0}^{T} \int_{Ω} \frac{\partial Ψ_{0}^{N}}{\partial t} (T_{ε}^{N} - T_{0}^{N}) d x d t = ρ c_{p} \int_{0}^{T} \int_{Ω} \frac{\partial (T_{ε}^{N} - T_{0}^{N})}{\partial t} Ψ_{0}^{N} d x d t . \end{array}$ Applying Green formula to the problem satisfied by the variation $T_{ε}^{N} - T_{0}^{N}$ in $Ω ∖ \overline{I_{z, ε}}$ , we conclude that $\begin{array}{l} ρ c_{p} \int_{0}^{T} \int_{Ω ∖ \overline{I_{z, ε}}} \frac{\partial (T_{ε}^{N} - T_{0}^{N})}{\partial t} Ψ_{0}^{N} d x d t + \int_{0}^{T} \int_{Ω ∖ \overline{I_{z, ε}}} σ \nabla (T_{ε}^{N} - T_{0}^{N}) \nabla Ψ_{0}^{N} d x d t \\ + ϖ_{b} ρ_{b} c_{b} \int_{0}^{T} \int_{Ω ∖ \overline{I_{z, ε}}} (T_{ε}^{N} - T_{0}^{N}) Ψ_{0}^{N} d x d t = \int_{0}^{T} \int_{\partial I_{z, ε}} σ \nabla (T_{ε}^{N} - T_{0}^{N}) . η Ψ_{0}^{N} d s d t . \end{array}$ Combining the last equality, we deduce $\begin{array}{l} - 2 \int_{0}^{T} \int_{Ω} (T_{0}^{N} - T_{0}^{D}) (T_{ε}^{N} - T_{0}^{N}) d x d t \\ = \int_{0}^{T} \int_{\partial I_{z, ε}} σ \nabla (T_{ε}^{N} - T_{0}^{N}) . η Ψ_{0}^{N} d s d t + ρ c_{p} \int_{0}^{T} \int_{I_{z, ε}} \frac{\partial Ψ_{0}^{N}}{\partial t} T_{0}^{N} d x d t \\ (5.4) & - \int_{0}^{T} \int_{I_{z, ε}} σ \nabla Ψ_{0}^{N} \nabla T_{0}^{N} d x d t - ϖ_{b} ρ_{b} c_{b} \int_{0}^{T} \int_{I_{z, ε}} Ψ_{0}^{N} T_{0}^{N} d x d t . \end{array}$ By the smoothness of the functions $T_{0}^{N}$ and $Ψ_{0}^{N}$ , one can check $\begin{array}{l} ρ c_{p} \int_{0}^{T} \int_{I_{z, ε}} \frac{\partial Ψ_{0}^{N}}{\partial t} T_{0}^{N} d x d t - \int_{0}^{T} \int_{I_{z, ε}} σ \nabla Ψ_{0}^{N} \nabla T_{0}^{N} d x d t - ϖ_{b} ρ_{b} c_{b} \int_{0}^{T} \int_{I_{z, ε}} Ψ_{0}^{N} T_{0}^{N} d x d t \\ (5.5) & = o (\frac{1}{| log ε |}) . \end{array}$ Using the decomposition (4.16), it follows $\begin{array}{l} \int_{0}^{T} \int_{\partial I_{z, ε}} σ \nabla (T_{ε}^{N} - T_{0}^{N}) . η Ψ_{0}^{N} d s d t \\ (5.6) & = \int_{0}^{T} \int_{\partial I_{z, ε}} σ \nabla R_{ε}^{N} . η Ψ_{0}^{N} d s d t + \frac{1}{log ε} \int_{0}^{T} \int_{\partial I_{z, ε}} σ \nabla H^{N} . η Ψ_{0}^{N} d s d t . \end{array}$ Trace theorem implies $\begin{array}{rcl} | \int_{0}^{T} \int_{\partial I_{z, ε}} σ \nabla R_{ε}^{N} . η Ψ_{0}^{N} d s d t | ⩽ {‖ σ \nabla R_{ε}^{N} . η ‖}_{L^{2} (0, T; H^{- 1 / 2} (\partial I_{z, ε}))} {‖ Ψ_{0}^{N} ‖}_{L^{2} (0, T; H^{1} (I_{z, ε}))} \end{array}$ Due to Theorem 4.1 and the fact that $‖ Ψ_{0}^{N} ‖_{L^{2} (0, T; H^{1} (I_{z, ε}))} = o (ε)$ , we obtain $\begin{array}{rcl} (5.7) & \int_{0}^{T} \int_{\partial I_{z, ε}} σ \nabla R_{ε}^{N} . η Ψ_{0}^{N} d s d t = o (\frac{1}{| log ε |}) . \end{array}$ By regularity argument, the functions σ and $Ψ_{0}^{N}$ can be approximated as $σ (x) = σ (z) + O (ε)$ and $Ψ_{0}^{N} (x, t) = Ψ_{0}^{N} (z, t) + O (ε)$ in the neighborhood of $I_{z, ε}$ . Then, from the definition of $H^{N}$ (i.e. $H^{N} (x, t) = 2 π T_{0}^{N} (z, t) E (x - z)$ ), we derive that the last integral term in (5.6) satisfies $\begin{array}{l} \frac{1}{log ε} \int_{0}^{T} \int_{\partial I_{z, ε}} σ \nabla H^{N} . η Ψ_{0}^{N} d s d t \\ = \frac{2 π}{log ε} σ (z) \int_{0}^{T} T_{0}^{N} (z, t) Ψ_{0}^{N} (z, t) d t \int_{\partial I_{z, ε}} \nabla E (x - z) . η d x + o (\frac{1}{| log ε |}) . \end{array}$ From the fact that $- Δ E (x - z) = δ_{z}$ in $I_{z, ε}$ it follows $\int_{\partial I_{z, ε}} \nabla E (x - z) . η d x = - 1$ , which implies $\begin{array}{l} \frac{1}{log ε} \int_{0}^{T} \int_{\partial I_{z, ε}} σ \nabla H^{N} . η Ψ_{0}^{N} d s d t \\ (5.8) & = - \frac{2 π}{log ε} σ (z) \int_{0}^{T} T_{0}^{N} (z, t) d t Ψ_{0}^{N} (z, t) + o (\frac{1}{| log ε |}) . \end{array}$ Finally, using the established estimates (5.4)–(5.8), it follows that the third integral term verifies $\begin{array}{l} - 2 \int_{0}^{T} \int_{Ω} (T_{0}^{N} - T_{0}^{D}) (T_{ε}^{N} - T_{0}^{N}) d x d t \\ = - \frac{2 π}{log ε} σ (z) \int_{0}^{T} T_{0}^{N} (z, t) Ψ_{0}^{N} (z, t) d t + o (\frac{1}{| log ε |}) . \end{array}$

Estimate of the fourth integral term: The last integral term can be estimated by the same technique developed in the previous paragraph (for the third term). Then $\begin{array}{l} - 2 \int_{0}^{T} \int_{Ω} (T_{0}^{N} - T_{0}^{D}) (T_{ε}^{D} - T_{0}^{D}) d x d t \\ = - \frac{2 π}{log ε} σ (z) \int_{0}^{T} T_{0}^{D} (z, t) Ψ_{0}^{D} (z, t) d t + o (\frac{1}{| log ε |}), \end{array}$ which achieves the proof. □

Footnotes

Acknowledgements

Researchers Supporting Project number (RSP-2020/153), King Saud University, Riyadh, Saudi Arabia.

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