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Least squares parameter estimation methods for material decomposition with energy discriminating detectors.

ABSTRACT: Energy resolving detectors provide more than one spectral measurement in one image acquisition. The purpose of this study is to investigate, with simulation, the ability to decompose four materials using energy discriminating detectors and least squares minimization techniques.Three least squares parameter estimation decomposition techniques were investigated for four-material breast imaging tasks in the image domain. The first technique treats the voxel as if it consisted of fractions of all the materials. The second method assumes that a voxel primarily contains one material and divides the decomposition process into segmentation and quantification tasks. The third is similar to the second method but a calibration was used. The simulated computed tomography (CT) system consisted of an 80 kVp spectrum and a CdZnTe (CZT) detector that could resolve the x-ray spectrum into five energy bins. A postmortem breast specimen was imaged with flat panel CT to provide a model for the digital phantoms. Hydroxyapatite (HA) (50, 150, 250, 350, 450, and 550 mg/ml) and iodine (4, 12, 20, 28, 36, and 44 mg/ml) contrast elements were embedded into the glandular region of the phantoms. Calibration phantoms consisted of a 30/70 glandular-to-adipose tissue ratio with embedded HA (100, 200, 300, 400, and 500 mg/ml) and iodine (5, 15, 25, 35, and 45 mg/ml). The x-ray transport process was simulated where the Beer-Lambert law, Poisson process, and CZT absorption efficiency were applied. Qualitative and quantitative evaluations of the decomposition techniques were performed and compared. The effect of breast size was also investigated.The first technique decomposed iodine adequately but failed for other materials. The second method separated the materials but was unable to quantify the materials. With the addition of a calibration, the third technique provided good separation and quantification of hydroxyapatite, iodine, glandular, and adipose tissues. Quantification with this technique was accurate with errors of 9.83% and 6.61% for HA and iodine, respectively. Calibration at one point (one breast size) showed increased errors as the mismatch in breast diameters between calibration and measurement increased. A four-point calibration successfully decomposed breast diameter spanning the entire range from 8 to 20 cm. For a 14 cm breast, errors were reduced from 5.44% to 1.75% and from 6.17% to 3.27% with the multipoint calibration for HA and iodine, respectively.The results of the simulation study showed that a CT system based on CZT detectors in conjunction with least squares minimization technique can be used to decompose four materials. The calibrated least squares parameter estimation decomposition technique performed the best, separating and accurately quantifying the concentrations of hydroxyapatite and iodine.

SUBMITTER: Le Huy Q

PROVIDER: S-EPMC3021557 | biostudies-other | 2011 Jan

REPOSITORIES: biostudies-other

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Project description:BackgroundThe importance of stochasticity in cellular processes having low number of molecules has resulted in the development of stochastic models such as chemical master equation. As in other modelling frameworks, the accompanying rate constants are important for the end-applications like analyzing system properties (e.g. robustness) or predicting the effects of genetic perturbations. Prior knowledge of kinetic constants is usually limited and the model identification routine typically includes parameter estimation from experimental data. Although the subject of parameter estimation is well-established for deterministic models, it is not yet routine for the chemical master equation. In addition, recent advances in measurement technology have made the quantification of genetic substrates possible to single molecular levels. Thus, the purpose of this work is to develop practical and effective methods for estimating kinetic model parameters in the chemical master equation and other stochastic models from single cell and cell population experimental data.ResultsThree parameter estimation methods are proposed based on the maximum likelihood and density function distance, including probability and cumulative density functions. Since stochastic models such as chemical master equations are typically solved using a Monte Carlo approach in which only a finite number of Monte Carlo realizations are computationally practical, specific considerations are given to account for the effect of finite sampling in the histogram binning of the state density functions. Applications to three practical case studies showed that while maximum likelihood method can effectively handle low replicate measurements, the density function distance methods, particularly the cumulative density function distance estimation, are more robust in estimating the parameters with consistently higher accuracy, even for systems showing multimodality.ConclusionsThe parameter estimation methodologies described in this work have provided an effective and practical approach in the estimation of kinetic parameters of stochastic systems from either sparse or dense cell population data. Nevertheless, similar to kinetic parameter estimation in other modelling frameworks, not all parameters can be estimated accurately, which is a common problem arising from the lack of complete parameter identifiability from the available data.

Project description:The objective of the study was to demonstrate that, in x-ray computed tomography (CT), more than two types of materials can be effectively separated with the use of an energy resolved photon-counting detector and classification methodology. Specifically, this applies to the case when contrast agents that contain K-absorption edges in the energy range of interest are present in the object. This separation is enabled via the use of recently developed energy resolved photon-counting detectors with multiple thresholds, which allow simultaneous measurements of the x-ray attenuation at multiple energies.To demonstrate this capability, we performed simulations and physical experiments using a six-threshold energy resolved photon-counting detector. We imaged mouse-sized cylindrical phantoms filled with several soft-tissue-like and bone-like materials and with iodine-based and gadolinium-based contrast agents. The linear attenuation coefficients were reconstructed for each material in each energy window and were visualized as scatter plots between pairs of energy windows. For comparison, a dual-kVp CT was also simulated using the same phantom materials. In this case, the linear attenuation coefficients at the lower kVp were plotted against those at the higher kVp.In both the simulations and the physical experiments, the contrast agents were easily separable from other soft-tissue-like and bone-like materials, thanks to the availability of the attenuation coefficient measurements at more than two energies provided by the energy resolved photon-counting detector. In the simulations, the amount of separation was observed to be proportional to the concentration of the contrast agents; however, this was not observed in the physical experiments due to limitations of the real detector system. We used the angle between pairs of attenuation coefficient vectors in either the 5-D space (for non-contrast-agent materials using energy resolved photon-counting acquisition) or a 2-D space (for contrast agents using energy resolved photon-counting acquisition and all materials using dual-kVp acquisition) as a measure of the degree of separation. Compared to dual-kVp techniques, an energy resolved detector provided a larger separation and the ability to separate different target materials using measurements acquired in different energy window pairs with a single x-ray exposure.We concluded that x-ray CT with an energy resolved photon-counting detector with more than two energy windows allows the separation of more than two types of materials, e.g., soft-tissue-like, bone-like, and one or more materials with K-edges in the energy range of interest. Separating material types using energy resolved photon-counting detectors has a number of advantages over dual-kVp CT in terms of the degree of separation and the number of materials that can be separated simultaneously.

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Least squares parameter estimation methods for material decomposition with energy discriminating detectors.

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