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Notes: Abstract A spatially and temporally discrete numerical approximation scheme is developed for the identification of a class of semilinear parabolic systems with unknown boundary parameters. The identification problem is formulated as a least squares fit to data subject to an equivalent representation for the dynamics in the form of an abstract evolution equation. Finite-dimensional difference equation state approximations are constructed using a cubic spline-based, Galerkin method and the Padé rational function approximations to the exponential. A sequence of approximating identification problems result, the solutions of which are shown to exist and, in a certain sense, approximate solutions to the original identification problem. Numerical results for two examples, one involving the modeling of biological mixing in deep sea sediment cores, and the other, the estimation of transport parameters for indoor mixing, are discussed. In both examples, the identification is based upon actual experimental data.

Notes: Abstract Purpose To compare chemotherapy treatment monitoring in astrocytoma by 201thallium single photon emission computed tomography (SPECT) and photon magnetic resonance spectroscopy (1H-MRS) with magnetic resonance imaging (MRI), and to evaluate the influence of morphological tumor changes on cerebral 201thallium uptake and metabolic changes in 1H-MRS. Materials and methods Six patients with highly malignant astrocytomas were followed with quantitative 201thallium SPECT, MRI, and 1H-MRS during chemotherapy. Maximum follow-up included six examinations per patient by either method during 18 months. Criteria were set for: (1) regression (≥ 25% tumor reduction), (2) status quo (〈 25% reduction and 〈 25% increase), and (3) progression of disease (≥ 25% tumor increase). Results were compared with the clinical state of disease. Changes of tumor volume, contrast enhancement, necrosis, hemorrhage and edema on MRI were compared to changes in 201thallium uptake volumes and 1H-MRS metabolite ratios. Results Six patients were followed with a total of twenty-four examinations with 201thallium SPECT, MRI and 1H-MRS, respectively, between February 1997 and October 1998. Five patients developed clinical progression of disease, 4 out of 5 cases showed SPECT progression, 4 out of 5 cases MRI progression, and 1 out of 2 interpretable cases 1H-MRS progression at final assessment before clinical deterioration. During the phase of clinically stable disease; (A) the criterion for regression or status quo was met in 10 out of 13 assessments with SPECT, 11 out of 13 with MRI, and 8 out of 9 interpretable 1H-MRS; (B) the criterion for progression was met in 3 out of 13 with SPECT, 2 out of 13 with MRI, and 1 out of 9 interpretable 1H-MRS. The accuracy of SPECT, MRI, and 1H-MRS in identifying changes of tumor burden concordant with patients' clinical course was 78%, 83%, and 82%, respectively. SPECT regression was associated with MRI decrease of tumor size, contrast enhancement, edema and hemorrhage. SPECT progression was associated with MRI increase of the same parameters and the increase of necrosis. 1H-MRS regression was associated with decrease of edema. 1H-MRS progression was associated with increase of tumor size, hemorrhage, and increase or decrease of contrast enhancement. Conclusions Both 201thallium SPECT and 1H-MRS evaluation showed sensitivity for detection of astrocytoma progression. We did not find a higher accuracy of SPECT or MRS than of MRI in astrocytoma chemotherapy monitoring. Treatment induced MRI changes were associated with 201thallium uptake variations. 1H-MRS was difficult to apply for astrocytoma treatment monitoring. Improvements regarding size of measurement area such as multivoxel MRS and fat suppression pulses appeared desirable, and also the use of functional techniques with superior resolution such as dual isotope SPECT. However, our results suggest that 201thallium SPECT and 1H-MRS can provide additional information to MRI for chemotherapy efficacy evaluation in selected cases.