Cancer Detection via Electrical Impedance
Tomography and PDE Constrained OptimalControl in Sobolev Spaces
Ugur G. Abdulla
Okinawa Institute of Science and Technology
Okinawa, Japan

ISCO 2023

1Ugur G. Abdulla

Cancer Detection via Electrical Impedance Tomography and PDE Constrain

EIT Mathematical Model

Q ∈ Rn2
Ugur G. Abdulla

Cancer Detection via Electrical Impedance Tomography and PDE Constrain

EIT Mathematical Modeln
Q ∈ Rn Electrical conductivity tensor A(x) = aij (x) ij=1 , x ∈ Q

2Ugur G. Abdulla

Cancer Detection via Electrical Impedance Tomography and PDE Constrain

EIT Mathematical Modeln
Q ∈ Rn Electrical conductivity tensor A(x) = aij (x) ij=1 , x ∈ Q(El )ml=1 ,2Ugur G. Abdulla

Cancer Detection via Electrical Impedance Tomography and PDE Constrain

EIT Mathematical Modeln
Q ∈ Rn Electrical conductivity tensor A(x) = aij (x) ij=1 , x ∈ Qmm(El )m
l=1 , Z := (Zl )l=1 ∈ R+2Ugur G. Abdulla

Cancer Detection via Electrical Impedance Tomography and PDE Constrain

EIT Mathematical Modeln
Q ∈ Rn Electrical conductivity tensor A(x) = aij (x) ij=1 , x ∈ Qmmmm
is called current pattern(El )m
l=1 , Z := (Zl )l=1 ∈ R+ I := (Il )l=1 ∈ R

2Ugur G. Abdulla

Cancer Detection via Electrical Impedance Tomography and PDE Constrain

EIT Mathematical ModelnQ ∈ Rn Electrical conductivity tensor A(x) = aij (x) ij=1 , x ∈ Qmmmm
is called current pattern(El )m
l=1 , Z := (Zl )l=1 ∈ R+ I := (Il )l=1 ∈ RmX

Il = 0l=12Ugur G. Abdulla

Cancer Detection via Electrical Impedance Tomography and PDE Constrain

EIT Mathematical Modeln
Q ∈ Rn Electrical conductivity tensor A(x) = aij (x) ij=1 , x ∈ Qmmmm
is called current pattern(El )m
l=1 , Z := (Zl )l=1 ∈ R+ I := (Il )l=1 ∈ R
m
XIl = 0l=1m
U := (Ul )m– induced constant voltage on electrodes
l=1 ∈ R

2Ugur G. Abdulla

Cancer Detection via Electrical Impedance Tomography and PDE Constrain

EIT Mathematical ModelnQ ∈ Rn Electrical conductivity tensor A(x) = aij (x) ij=1 , x ∈ Qmmm
mis called current pattern(El )ml=1 , Z := (Zl )l=1 ∈ R+ I := (Il )l=1 ∈ R
mXIl = 0l=1m
U := (Ul )m– induced constant voltage on electrodes
l=1 ∈ RmX

Ul = 0l=12Ugur G. Abdulla

Cancer Detection via Electrical Impedance Tomography and PDE Constrain

EIT Mathematical Model
Electrostatic potential u : Q → R & boundary voltages U on (El )ml=13Ugur G. Abdulla

Cancer Detection via Electrical Impedance Tomography and PDE Constrain

EIT Mathematical Model
Electrostatic potential u : Q → R & boundary voltages U on (El )ml=1−nX

aij (x)uxj x = 0,ix∈Qi,j=13
Ugur G. Abdulla

Cancer Detection via Electrical Impedance Tomography and PDE Constrain

EIT Mathematical Model
Electrostatic potential u : Q → R & boundary voltages U on (El )ml=1−nX

aij (x)uxj x = 0,ix∈Q

i,j=1

∂u(x)
= 0,
∂Nx ∈ ∂Q −m[Ell=1

3
Ugur G. Abdulla

Cancer Detection via Electrical Impedance Tomography and PDE Constrain

EIT Mathematical ModelElectrostatic potential u : Q → R & boundary voltages U on (El )ml=1−nX

aij (x)uxj x = 0,ix∈Qi,j=1

∂u(x)
= 0,
∂Nu(x) + Zlx ∈ ∂Q −∂u(x)= Ul ,∂Nm[El

l=1x ∈ El , l = 1, m

3Ugur G. Abdulla

Cancer Detection via Electrical Impedance Tomography and PDE Constrain

EIT Mathematical Model
Electrostatic potential u : Q → R & boundary voltages U on (El )ml=1−nX

aij (x)uxj x = 0,ix∈Qi,j=1∂u(x)= 0,∂Nu(x) + Zlx ∈ ∂Q −m[Ell=1∂u(x)= Ul ,∂Nx ∈ El , l = 1, m∂u(x)ds = Il ,El ∂NZl = 1, m

3Ugur G. Abdulla

Cancer Detection via Electrical Impedance Tomography and PDE Constrain

EIT Mathematical Model
Electrostatic potential u : Q → R & boundary voltages U on (El )ml=1−nX

aij (x)uxj x = 0,ix∈Qi,j=1

∂u(x)
= 0,
∂Nu(x) + Zlx ∈ ∂Q −m[Ell=1∂u(x)= Ul ,∂Nx ∈ El , l = 1, m

∂u(x)ds = Il ,El ∂NZl = 1, m

∂u(x) X
=aij (x)uxj ν i∂Ni,j3
Ugur G. Abdulla

Cancer Detection via Electrical Impedance Tomography and PDE Constrain

EIT Mathematical Model
Electrostatic potential u : Q → R & boundary voltages U on (El )ml=1−nX

aij (x)uxj x = 0,ix∈Qi,j=1

∂u(x)
= 0,
∂Nu(x) + Zlx ∈ ∂Q −m[Ell=1∂u(x)= Ul ,∂Nx ∈ El , l = 1, m∂u(x)ds = Il ,El ∂NZl = 1, m

∂u(x) X
=aij (x)uxj ν i∂Ni,jnXi,j=1aij (x)ξi ξj ≥ µnX

ξi2 , ∀ξ ∈ Rn ;µ > 0.

i=1
3
Ugur G. Abdulla

Cancer Detection via Electrical Impedance Tomography and PDE Constrain

Complete Electrode Model

EIT Problem: Given electrical conductivity tensor A, electrode contact
impedance vector Z, and electrode current pattern I it is required to findelectrostatic potential u and electrode voltages U :(A, Z, I) −→ (u, U )4Ugur G. Abdulla

Cancer Detection via Electrical Impedance Tomography and PDE Constrain

Complete Electrode Model

EIT Problem: Given electrical conductivity tensor A, electrode contact
impedance vector Z, and electrode current pattern I it is required to findelectrostatic potential u and electrode voltages U :(A, Z, I) −→ (u, U )I

K.-S. Cheng, D. Isaacson, J.C. Newell and D.G. Gisser, Electrodemodels for electric current computed tomography, IEEE Trans.
Biomed. Engrg., 3, 36, 1989, 918-924.

I

E. Somersalo, M. Cheney and D. Isaacson, Existence and uniquenessfor electrode models for electric current computed tomography,SIAM Journal on Applied Mathematics, 52 (1992), 1023–1040.

4Ugur G. Abdulla

Cancer Detection via Electrical Impedance Tomography and PDE Constrain

Inverse EIT Problem
Inverse EIT Problem: Given electrode contact impedance vector Z,
electrode current pattern I and boundary electrode measurement U ∗ , itis required to find electrostatic potential u and electrical conductivity
tensor A.

5Ugur G. Abdulla

Cancer Detection via Electrical Impedance Tomography and PDE Constrain

Inverse EIT Problem
Inverse EIT Problem: Given electrode contact impedance vector Z,
electrode current pattern I and boundary electrode measurement U ∗ , itis required to find electrostatic potential u and electrical conductivity
tensor A.
IA.P. Calderon, On an inverse boundary value problem, in Seminar on
Numerical Analysis and Its Applications to Continuum Physics, Soc.
Brasileira de Mathematica, Rio de Janeiro, (1980), 65–73.

5Ugur G. Abdulla

Cancer Detection via Electrical Impedance Tomography and PDE Constrain

Inverse EIT Problem
Inverse EIT Problem: Given electrode contact impedance vector Z,
electrode current pattern I and boundary electrode measurement U ∗ , itis required to find electrostatic potential u and electrical conductivity
tensor A.
I

A.P. Calderon, On an inverse boundary value problem, in Seminar on
Numerical Analysis and Its Applications to Continuum Physics, Soc.
Brasileira de Mathematica, Rio de Janeiro, (1980), 65–73.

I

U.G. Abdulla, V. Bukshtynov, S. Seif, Cancer Detection throughElectrical Impedance Tomography and Optimal Control Theory:
Theoretical and Computational Analysis, Mathematical Biosciencesand Engineering, 18, 4(2021), 4834-4859.

I

U.G. Abdulla, S.Seif, Discretization and Convergence of the EIT
Optimal Control Problem in Sobolev Spaces with Dominating Mixed
Smoothness, Contemporary Mathematics, Volume 784, 2023.

5Ugur G. Abdulla

Cancer Detection via Electrical Impedance Tomography and PDE Constrain

Optimal Control Problem

6Ugur G. Abdulla

Cancer Detection via Electrical Impedance Tomography and PDE Constrain

Optimal Control Problem
J (v) =m ZX
l=12Ul − u(x)ds − Il + β|U − U ∗ |2 → minZlEl

6Ugur G. Abdulla

Cancer Detection via Electrical Impedance Tomography and PDE Constrain

Optimal Control Problem
J (v) =m ZXl=1

2Ul − u(x)ds − Il + β|U − U ∗ |2 → minZlElm
\X
VR = v = (A, U ) ∈ L∞ (Q; Mn×n ) H  (Q; Mn×n ) × Rm
Ul = 0,l=1
T

2nkAkL∞ + kAkH  + |U | ≤ R, ξ Aξ ≥ µ|ξ| , ∀ξ ∈ R , µ > 0

6Ugur G. Abdulla

Cancer Detection via Electrical Impedance Tomography and PDE Constrain

Optimal Control Problem
J (v) =m ZX

2Ul − u(x)ds − Il + β|U − U ∗ |2 → minZlEll=1m
\X
VR = v = (A, U ) ∈ L∞ (Q; Mn×n ) H  (Q; Mn×n ) × Rm
Ul = 0,l=1T2nkAkL∞ + kAkH  + |U | ≤ R, ξ Aξ ≥ µ|ξ| , ∀ξ ∈ R , µ > 0−nX

aij (x)uxj xi = 0,x∈Q

i,j=1

6
Ugur G. Abdulla

Cancer Detection via Electrical Impedance Tomography and PDE Constrain

Optimal Control Problem
J (v) =m ZX

2Ul − u(x)ds − Il + β|U − U ∗ |2 → minZlEll=1m
\X
VR = v = (A, U ) ∈ L∞ (Q; Mn×n ) H  (Q; Mn×n ) × Rm
Ul = 0,l=1Tn

2kAkL∞ + kAkH  + |U | ≤ R, ξ Aξ ≥ µ|ξ| , ∀ξ ∈ R , µ > 0−nX

aij (x)uxj xi = 0,x∈Q

i,j=1

∂u(x)
= 0,
∂Nx ∈ ∂Q −m[Ell=1

6Ugur G. Abdulla

Cancer Detection via Electrical Impedance Tomography and PDE Constrain

Optimal Control Problem
J (v) =m ZX

2Ul − u(x)ds − Il + β|U − U ∗ |2 → minZlEll=1m
\X
VR = v = (A, U ) ∈ L∞ (Q; Mn×n ) H  (Q; Mn×n ) × Rm
Ul = 0,l=1Tn

2kAkL∞ + kAkH  + |U | ≤ R, ξ Aξ ≥ µ|ξ| , ∀ξ ∈ R , µ > 0−nX

aij (x)uxj xi = 0,x∈Qi,j=1

∂u(x)
= 0,
∂Nu(x) + Zlx ∈ ∂Q −∂u(x)= Ul ,∂Nm[Ell=1x ∈ El , l = 1, m
6Ugur G. Abdulla

Cancer Detection via Electrical Impedance Tomography and PDE Constrain

Optimal Control Problem with Increased Input DataU 1 = U, I 1 = I, U j = (Uj , ..., Um , U1 , ..., Uj−1 ), j = 2, ..., m

7
Ugur G. Abdulla

Cancer Detection via Electrical Impedance Tomography and PDE Constrain

Optimal Control Problem with Increased Input Data
U 1 = U, I 1 = I, U j = (Uj , ..., Um , U1 , ..., Uj−1 ), j = 2, ..., mK(v) =m Xm ZX
j=1 l=1

2
Ulj − uj (x)ds − Ilj + β|U − U ∗ |2 → min (0.1)
ZlEluj = uj (·; A, U j ) ∈ H 1 (Q), j = 1, ..., m

7
Ugur G. AbdullaCancer Detection via Electrical Impedance Tomography and PDE Constrain

Optimal Control Problem with Increased Input Data
U 1 = U, I 1 = I, U j = (Uj , ..., Um , U1 , ..., Uj−1 ), j = 2, ..., mK(v) =m Xm ZX
j=1 l=1

2
Ulj − uj (x)ds − Ilj + β|U − U ∗ |2 → min (0.1)
ZlEl

uj = uj (·; A, U j ) ∈ H 1 (Q), j = 1, ..., m−nXaij (x)ujxj x = 0,ix∈Qi,j=17Ugur G. Abdulla

Cancer Detection via Electrical Impedance Tomography and PDE Constrain

Optimal Control Problem with Increased Input Data
U 1 = U, I 1 = I, U j = (Uj , ..., Um , U1 , ..., Uj−1 ), j = 2, ..., mK(v) =m Xm ZX
j=1 l=1

2
Ulj − uj (x)ds − Ilj + β|U − U ∗ |2 → min (0.1)
Zl
Eluj = uj (·; A, U j ) ∈ H 1 (Q), j = 1, ..., m−nXaij (x)ujxj x = 0,ix∈Qi,j=1∂uj (x)= 0,∂Nx ∈ ∂Q −m[Ell=17Ugur G. Abdulla

Cancer Detection via Electrical Impedance Tomography and PDE Constrain

Optimal Control Problem with Increased Input Data
U 1 = U, I 1 = I, U j = (Uj , ..., Um , U1 , ..., Uj−1 ), j = 2, ..., mK(v) =m Xm Z
X
j=1 l=1

2
Ulj − uj (x)ds − Ilj + β|U − U ∗ |2 → min (0.1)
Zl
Eluj = uj (·; A, U j ) ∈ H 1 (Q), j = 1, ..., m−nXaij (x)ujxj x = 0,ix∈Qi,j=1∂uj (x)= 0,∂Nuj (x) + Zlx ∈ ∂Q −m[Ell=1∂uj (x)= Ulj ,∂Nx ∈ El , l = 1, m
7Ugur G. Abdulla

Cancer Detection via Electrical Impedance Tomography and PDE Constrain

Theorem on Fréchet Differentiability
The functional K(v) is differentiable on VR in the senseof Fréchet and00the Fréchet gradient K0 (σ, U ) = KA
(A, U ), KU(A, U ) ∈ L0 × Rm ism
Xψxj p ujxq

ni
Ulj − ujds − IljZlElZ0KA(A, U ) = −p,q=1

j=10KU(A, U ) =m X
m hZ
X2j=1 l=1,δl,θkj − wθkj (s) mdsZl
k=1El

where ψ j (·), j = 1, ..., m, be a solution of the adjoined PDE problemwith u(·), U and I replaced with uj (·), U j , I j respectively, and(k − j + 1,if j ≤ k,
θkj =m + k − j + 1,
if j > k.

8Ugur G. Abdulla

Cancer Detection via Electrical Impedance Tomography and PDE Constrain

Gradient Method in a Banach Space+1N NãN(x) = aNij (x) + αN ψxi uxj , i, j = 1, ..., n,ij

9Ugur G. Abdulla

Cancer Detection via Electrical Impedance Tomography and PDE Constrain

Gradient Method in a Banach Space+1N NãN(x) = aNij (x) + αN ψxi uxj , i, j = 1, ..., n,ij

ŨkN +1 = UkN − αNm ZhX2

UlN − uN (s)ds − IlZlEl
l=1i+2β(UkN − Uk∗ ) , k = 1, ..., m.

Z

1(δlk − wkN (s))dsZlEl

9Ugur G. Abdulla

Cancer Detection via Electrical Impedance Tomography and PDE Constrain

Gradient Method in a Banach Space+1N NãN(x) = aNij (x) + αN ψxi uxj , i, j = 1, ..., n,ij

ŨkN +1 = UkN − αNm ZhX2

UlN − uN (s)ds − IlZlEl
l=1i+2β(UkN − Uk∗ ) , k = 1, ..., m.

µ,N +1+1aij (x) = ãN(x),ijR,Z1(δlk − wkN (s))dsZlEl+1if ãN(x) ≤ µ,ij+1if µ ≤ ãN(x) ≤ R,ijN +1if ãij (x) > R.
m

UkN +1 = ŨkN +1 −1 X N +1Ũkk = 1, ..., mm
k=1

9Ugur G. Abdulla

Cancer Detection via Electrical Impedance Tomography and PDE Constrain

Discretization via Finite Differences

Jh ([v]h ) =m  X
X
l=1

Γlα

A(Êlh )2Ul − uα− Il + β|U − U ∗ |2 → minZl

10Ugur G. Abdulla

Cancer Detection via Electrical Impedance Tomography and PDE Constrain

Discretization via Finite Differences

Jh ([v]h ) =m  X
X
l=1

FhR :=ΓlαA(Êlh )[v]h = ([σ]h , U )2Ul − uα− Il + β|U − U ∗ |2 → minZlm
X

Ul = 0, k[σ]h k2H̃ 1 (Q ) + |U |2Rm ≤ R2 ,hl=1σα ≥ σ0 > 0, ∀α ∈ A(Qh )10Ugur G. Abdulla

Cancer Detection via Electrical Impedance Tomography and PDE Constrain

Discretization via Finite Differences

Jh ([v]h ) =m  X
X
l=1FhR :=ΓlαA(Êlh )[v]h = ([σ]h , U )2Ul − uα− Il + β|U − U ∗ |2 → minZlm
X

Ul = 0, k[σ]h k2H̃ 1 (Q ) + |U |2Rm ≤ R2 ,hl=1σα ≥ σ0 > 0, ∀α ∈ A(Qh )hnX
A(Q+h)

σαnXuαxi ηαxi +i=1mX
1
l=1=ZlXmXUl X
l=1Zl

Γlα uα ηα + Jh ([u]h , [η]h )A(Êlh )Γlα ηαA(Êlh )10Ugur G. Abdulla

Cancer Detection via Electrical Impedance Tomography and PDE Constrain

Convergence of the Method of Finite Differences
The sequence of discrete optimal control problems Eh approximates theoptimal control problem E with respect to functional, i.e.
lim Jh∗ = J∗h→011Ugur G. Abdulla

Cancer Detection via Electrical Impedance Tomography and PDE Constrain

Convergence of the Method of Finite Differences
The sequence of discrete optimal control problems Eh approximates theoptimal control problem E with respect to functional, i.e.
lim Jh∗ = J∗h→0Interpolation of the discrete control minimizers areIis precompact in Tikhonov topology of H̃ 1 (Q) × Rm formed withthe product of the weak topology of H̃ 1 (Q) and Euclidean topologyof Rm ;I

is precompact in Tikhonov topology of C 0,µ (Q) × Rm , 0 < µ < 21
formed with the product of the strong topology of Hölder space
C 0,µ (Q) and Euclidean topology of Rm ;and all the corresponding limit points v∗ = (σ∗ , U∗ ) are optimal controls.

11Ugur G. Abdulla

Cancer Detection via Electrical Impedance Tomography and PDE Constrain

Convergence of the Method of Finite Differences
The sequence of discrete optimal control problems Eh approximates theoptimal control problem E with respect to functional, i.e.
lim Jh∗ = J∗h→0Interpolation of the discrete control minimizers areI

is precompact in Tikhonov topology of H̃ 1 (Q) × Rm formed withthe product of the weak topology of H̃ 1 (Q) and Euclidean topologyof Rm ;I

is precompact in Tikhonov topology of C 0,µ (Q) × Rm , 0 < µ < 21
formed with the product of the strong topology of Hölder space
C 0,µ (Q) and Euclidean topology of Rm ;and all the corresponding limit points v∗ = (σ∗ , U∗ ) are optimal controls.
I

Multilinear interpolations of the discrete state vectors [u([v]h0 , )]h0
converge to the optimal state vector u = u(x; v∗ ) weakly in H 1 (Q),strongly in L2 (Q), and almost everywhere on Q.
11Ugur G. Abdulla

Cancer Detection via Electrical Impedance Tomography and PDE Constrain

Numerical Results

E6E5E4

E70.5
E3

E8

I6

I3

I8

0.35
E19I4

0.4

E2EI5

I7

0.45

0.3

I

2II

910.25EE

1016

E110.2
0.15

E15
E12

E130.1

E14I

I

16

10I15

I11
I12

I13

I14

12Ugur G. Abdulla

Cancer Detection via Electrical Impedance Tomography and PDE Constrain

Cancer Detection though Electrical ImpedanceTomography (EIT) and Optimal Control Theory
0.5

540.45

30.4
20.35

Ul

100.3

−10.25−20.2−30.15−4−502468101214

160.1

l13Ugur G. Abdulla

Cancer Detection via Electrical Impedance Tomography and PDE Constrain

Cancer Detection though Electrical Impedance
Tomography (EIT) and Optimal Control Theory
0.5

540.45

30.4
20.35

Ul

1
00.3

−10.25−20.2−30.15−4−502468101214160.1

l0.5

540.45

30.4
20.35

Ul

100.3−10.25−20.2−30.15−4−502468101214160.1

l13Ugur G. Abdulla

Cancer Detection via Electrical Impedance Tomography and PDE Constrain