The function of Green for the bioheat equation of Pennes in an axisymmetric unbounded domain
Abstract
The function of Green associated to a linear partial differential operator P(D) in a domain Ω acting at point x0 of the domain, is a distribution G(x, x0) such that
P(D)G(x, x0) = δ(x − x0),
where δ is the Dirac’s delta distribution. The property P(D)G(x, x0) = δ(x − x0) of a Green’s function can be exploited to solve differential equations of the form P(D)u = f, because
∫Ω P(D)G(x, x0) f(x0) dx0 = f(x) = P(D)u,
Hence
P(D)u = P(D) ( ∫Ω G(x, x0) f(x0) dx0 ),
which implies that u = G(x, x0) f(x0) dx0. Not every operator P(D) admits a Green’s function. And the Green’s function, if it exists, is not unique, but adding boundary conditions it will be unique. In regular Sturm–Liouville problems, there is a standard way to obtain the corresponding Green’s function, and after that, as the domain is bounded, to incorporate the initial and boundary conditions using also the Green’s function. But the method doesn’t work if the domain is not bounded, because the justification is based on the use of the Green’s Theorem. In this paper we find the Green’s function for the Penne’s bioheat equation, see [1], in an unbounded domain consisting in the space ℝ3 with an infinite cylindrical hole. This type of problems appears in radiofrequency (RF) ablation with needle-like electrodes, which is widely used for medical techniques such as tumor ablation or cardiac ablation to cure arrhythmias. We recall that theoretical modeling is a rapid and inexpensive way of studying different aspects of the RF process.
Keywords
Download Options
Introduction
In the theory of heat conduction in perfused biological tissues, the so-called Pennes’ bioheat equation, that is
ρc ∂T/∂t = ∇·(k∇T) + ωbcb(Tb − T) + Q
plays a central role. In (1), T(x) denotes the temperature at every point x of a biological tissue lying in a domain Ω ⊂ ℝ³ in the interior of the tissue. The thermal conductivity of the tissue is represented by k, and ρ and c are the density and specific heat of the tissue. ωb and cb denote density and specific heat perfusion coefficient of the blood, while Tb represents the arterial blood temperature and Q is a heat source.
We consider the following infinite spatial domain:
Ω = {(x, y, z) ∈ ℝ³ : x² + y² ≥ r0² }
with source
S = S(x, y, z, t)
bounded in Ω, and initial and boundary conditions only dependent on r = √(x² + y²) and the temporal variable t. This is the geometry used for problems related to radiofrequency ablation with needle-like electrodes. In this case, we may suppose that the temperature T is radially symmetric and independent of the time variable z, so that T = T(r, t).
Switching to cylindrical coordinates, Pennes’ bioheat equation (1) becomes:
ρc ∂T/∂t = k ( ∂²T/∂r² + (1/r) ∂T/∂r ) + ωbcb(Tb − T) + S(r, t)
The initial and boundary conditions we will consider in this paper are the following:
limr→∞ T(r, t) = Tb, t > 0
T(r, 0) = T0(r), r ≥ r0
−k ∂T/∂r (r0, t) = φ(t), t > 0
Where φ(t) can be interpreted as an infinitesimal temperature in the boundary of the hole. For example, RF ablation with internally cooled-like electrodes is widely used for medical techniques due to its capability to decrease tissue damage close to the electrode surface. In this case, φ(t) can be given as
φ(t) = −k (Te(t) − T(r0, t))
where Te(t) is the temperature of the electrode, and k is the thermal conductivity of the tissue.
Then (2) becomes
ρc ∂T/∂t = k ( ∂²T/∂r² + (1/r) ∂T/∂r ) − ρbcb(T − Tb) + S(r, t)
limr→∞ T(r, t) = Tb
T(r, 0) = T0(r)
−k ∂T/∂r (r0, t) = φ(t)
Conclusion
∂/∂ρ [ (−ρ) ∂V/∂ρ ] + ∂V/∂ξ + ρβV = ρS1(ρ, ξ)
V(ρ, 0) = g1(ρ), ∀ρ > 1
limρ→∞ V(ρ, ξ) = h1(ξ), ∀ξ > 0
V(1, ξ) = f1(ξ), ∀ξ > 0
Define
H(ρ, ξ) = V(ρ, ξ) − h1(ξ)
Then H(ρ, ξ) satisfies the equation:
∂/∂ρ [ (−ρ) ∂H/∂ρ ] + ∂H/∂ξ + ρβH = ρ( S1(ρ, ξ) + h′1(ξ) + βh1(ξ) )
H(ρ, 0) = g1(ρ) − h1(0), ∀ρ > 1
limρ→∞ H(ρ, ξ) = 0, ∀ξ > 0
H(1, ξ) = f1(ξ) − h1(ξ), ∀ξ > 0
Then
V(ρ, ξ) = ∫1∞ ∫0ξ G(ρ, ρ0, ξ, ξ0) S1(ρ0, ξ0) dξ0 dρ0
+ ∫1∞ ∫0ξ G(ρ, ρ0, ξ, ξ0) [ g1(ρ0) − h1(0) ] δ(ξ0) dξ0 dρ0
− L−1 { K0(ρ√(s + β)) / K0(√(s + β)) } L−1 { K0(ρ√(s + β)) / K0(√(s + β)) } ⋆ ( h1(ξ) − f1(ξ) )
= ∫1∞ ∫0ξ G(ρ, ρ0, ξ, ξ0) S1(ρ0, ξ0) dξ0 dρ0
+ ∫1∞ G(ρ, ρ0, ξ) g1(ρ0) dρ0
+ ∫1∞ ∫0ξ G(ρ, ρ0, ξ, ξ0) ( h′1(ξ0) + βh1(ξ0) − h1(0) ) δ(ξ0) dξ0 dρ0
− L−1 { K0(ρ√(s + β)) / K0(√(s + β)) } L−1 { K0(ρ√(s + β)) / K0(√(s + β)) } ⋆ ( h1(ξ) − f1(ξ) )