By Jin Keun Seo, Eung Je Woo (auth.), Habib Ammari (eds.)
This quantity supplies an advent to a desirable study quarter to utilized mathematicians. it really is dedicated to supplying the exposition of promising analytical and numerical concepts for fixing not easy biomedical imaging difficulties, which set off the research of fascinating matters in a variety of branches of mathematics.
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Extra info for Mathematical Modeling in Biomedical Imaging I: Electrical and Ultrasound Tomographies, Anomaly Detection, and Brain Imaging
To distinguish them, we denote τ (r) := σ(r,ω2 ) (r,ω2 ) (r) := (r, ω1 ). In the multiσ(r,ω1 ) , κ(r) := (r,ω1 ) , σ(r) := σ(r, ω1 ), frequency TAS model, we use a weighted frequency-diﬀerence of Neumann data gω2 − αgω1 instead of a time-diﬀerence gω − gω∗ where α is chosen as σ+iω2 κ α := τ σ+iω |Γ . 31) (3σc )2 (2σc +σn )2 g0 (ξ1 , ξ2 ), σ= σn in ΩL \ D and σc in D = n c in ΩL \ D in D. 22 shows an example of the real and imaginary parts of the difference image gω2 − αgω1 on the probe region Γ .
In order to distinguish between them, we set σ(r, ω) = σb (ω) σa (ω) if r ∈ Ω \ D if r ∈ D b (ω) if r ∈ Ω \ D if r ∈ D. and (r, ω) = a (ω) Complex conductivities σa +iω a and σb +iω b are constants at each frequency ω but they are changing with the frequency ω. 002 1 10 100 1k 10k Frequency [Hz] 100k 1M Fig. 17 (a) σ and (b) ω spectra. (c) Magniﬁed ω spectra of non-biological materials only 1 MFEIT and MREIT 27 σ and ω spectra of seven diﬀerent materials. Conductivities of non-biological materials including saline, agar, polyacrylamide (PAA), animal hide gelatin (AHG) and TX151 can be adjusted by changing the amount of NaCl.
Noting that EIT suﬀers from the ill-posed nature of the corresponding inverse problem, we introduce MREIT which utilizes internal information on the induced magnetic ﬁeld in addition to the boundary voltage subject to an injection current to produce threedimensional images of conductivity and current density distributions. Since 2000, imaging techniques in MREIT have been advanced rapidly and now are at the stage of in vivo animal experiments. In both EIT and MREIT, we inject currents through electrodes placed on the surface of a subject.