Analytical Models of Exoplanetary Atmospheres. II. Radiative Transfer via the Two-Stream Approximation

We present a comprehensive analytical study of radiative transfer using the method of moments and include the effects of non-isotropic scattering in the coherent limit. Within this unified formalism, we derive the governing equations and solutions describing two-stream radiative transfer (which approximates the passage of radiation as a pair of outgoing and incoming fluxes), flux-limited diffusion (which describes radiative transfer in the deep interior) and solutions for the temperature-pressure profiles. Generally, the problem is mathematically under-determined unless a set of closures (Eddington coefficients) is specified. We demonstrate that the hemispheric (or hemi-isotropic) closure naturally derives from the radiative transfer equation if energy conservation is obeyed, while the Eddington closure produces spurious enhancements of both reflected light and thermal emission. We concoct recipes for implementing two-stream radiative transfer in stand-alone numerical calculations and general circulation models. We use our two-stream solutions to construct toy models of the runaway greenhouse effect. We present a new solution for temperature-pressure profiles with a non-constant optical opacity and elucidate the effects of non-isotropic scattering in the optical and infrared. We derive generalized expressions for the spherical and Bond albedos and the photon deposition depth. We demonstrate that the value of the optical depth corresponding to the photosphere is not always 2/3 (Milne's solution) and depends on a combination of stellar irradiation, internal heat and the properties of scattering both in optical and infrared. Finally, we derive generalized expressions for the total, net, outgoing and incoming fluxes in the convective regime.

Therapy response with diffusion MRI: an update

The efficiency of an oncological treatment regimen is often assessed by morphological criteria such as tumour size evaluated by cross-sectional imaging, or by laboratory measurements of plasma biomarkers. Because these types of measures typically allow for assessment of treatment response several weeks or even months after the start of therapy, earlier response assessment that provides insight into tumour function is needed. This is particularly urgent for the evaluation of newer targeted therapies and for fractionated therapies that are delivered over a period of weeks to allow for a change of treatment in non-responding patients. Diffusion-weighted MRI (DW-MRI) is a non-invasive imaging tool that does not involve radiation or contrast media, and is sensitive to tissue microstructure and function on a cellular level. DW-MRI parameters have shown sensitivity to treatment response in a growing number of tumour types and organ sites, with additional potential as predictive parameters for treatment outcome. A brief overview of DW-MRI principles is provided here, followed by a review of recent literature in which DW-MRI has been used to monitor and predict tumour response to various therapeutic regimens.

A Parallel Solver for the Time-Periodic Navier–Stokes Equations

We investigate parallel algorithms for the solution of the Navier–Stokes equations in space-time. For periodic solutions, the discretized problem can be written as a large non-linear system of equations. This system of equations is solved by a Newton iteration. The Newton correction is computed using a preconditioned GMRES solver. The parallel performance of the algorithm is illustrated.

Apparent diffusion coefficient in the aging mouse brain: a magnetic resonance imaging study

Novel magnetic resonance imaging sequences have and still continue to play an increasing role in neuroimaging and neuroscience. Among these techniques, diffusion-weighted imaging (DWI) has revolutionized the diagnosis and management of diseases such as stroke, neoplastic disease and inflammation. However, the effects of aging on diffusion are yet to be determined. To establish reference values for future experimental mouse studies we tested the hypothesis that absolute apparent diffusion coefficients (ADC) of the normal brain change with age. A total of 41 healthy mice were examined by T2-weighted imaging and DWI. For each animal ADC frequency histograms (i) of the whole brain were calculated on a voxel-by-voxel basis and region-of-interest (ROI) measurements (ii) performed and related to the animals' age. The mean entire brain ADC of mice <3 months was 0.715(+/-0.016) x 10(-3) mm2/s, no significant difference to mice aged 4 to 5 months (0.736(+/-0.040) x 10(-3) mm2/s) or animals older than 9 months 0.736(+/-0.020) x 10(-3) mm2/s. Mean whole brain ADCs showed a trend towards lower values with aging but both methods (i + ii) did not reveal a significant correlation with age. ROI measurements in predefined areas: 0.723(+/-0.057) x 10(-3) mm2/s in the parietal lobe, 0.659(+/-0.037) x 10(-3) mm2/s in the striatum and 0.679(+/-0.056) x 10(-3) mm2/s in the temporal lobe. With advancing age, we observed minimal diffusion changes in the whole mouse brain as well as in three ROIs by determination of ADCs. According to our data ADCs remain nearly constant during the aging process of the brain with a small but statistically non-significant trend towards a decreased diffusion in older animals.

Image registration for triggered and non-triggered DTI of the human kidney: Reduced variability of diffusion parameter estimation

BACKGROUND: To investigate if non-rigid image-registration reduces motion artifacts in triggered and non-triggered diffusion tensor imaging (DTI) of native kidneys. A secondary aim was to determine, if improvements through registration allow for omitting respiratory-triggering. METHODS: Twenty volunteers underwent coronal DTI of the kidneys with nine b-values (10-700 s/mm2 ) at 3 Tesla. Image-registration was performed using a multimodal nonrigid registration algorithm. Data processing yielded the apparent diffusion coefficient (ADC), the contribution of perfusion (FP ), and the fractional anisotropy (FA). For comparison of the data stability, the root mean square error (RMSE) of the fitting and the standard deviations within the regions of interest (SDROI ) were evaluated. RESULTS: RMSEs decreased significantly after registration for triggered and also for non-triggered scans (P < 0.05). SDROI for ADC, FA, and FP were significantly lower after registration in both medulla and cortex of triggered scans (P < 0.01). Similarly the SDROI of FA and FP decreased significantly in non-triggered scans after registration (P < 0.05). RMSEs were significantly lower in triggered than in non-triggered scans, both with and without registration (P < 0.05). CONCLUSION: Respiratory motion correction by registration of individual echo-planar images leads to clearly reduced signal variations in renal DTI for both triggered and particularly non-triggered scans. Secondarily, the results suggest that respiratory-triggering still seems advantageous.J. Magn. Reson. Imaging 2014. (c) 2014 Wiley Periodicals, Inc.

Contribution of the apparent diffusion coefficient in perilesional edema for the assessment of brain tumors

OBJECTIVES: Diffusion-weighted MRI is sensitive to molecular motion and has been applied to the diagnosis of stroke. Our intention was to investigate its usefulness in patients with brain tumor and, in particular, in the perilesional edema. METHODS: We performed MRI of the brain, including diffusion-weighted imaging and mapping of the apparent diffusion coefficient (ADC), in 16 patients with brain tumors (glioblastomas, low-grade gliomas and metastases). ADC values were determined by the use of regions of interest positioned in areas of high signal intensities as seen on T2-weighted images and ADC maps. Measurements were taken in the tumor itself, in the area of perilesional edema and in the healthy contralateral brain. RESULTS: ADC mapping showed higher values of peritumoral edema in patients with glioblastoma (1.75 x 10(-3)mm(2)/s) and metastatic lesions (1.61 x 10(-3)mm(2)/s) compared with those who had low-grade glioma (1.40 x10(-3)mm(2)/s). The higher ADC values in the peritumoral zone were associated with lower ADC values in the tumor itself. CONCLUSIONS: The higher ADC values in the more malignant tumors probably reflect vasogenic edema, thereby allowing their differentiation from other lesions.

Effect of vascular targeting agent in rat tumor model: dynamic contrast-enhanced versus diffusion-weighted MR imaging

PURPOSE: To compare dynamic contrast material-enhanced magnetic resonance (MR) imaging and diffusion-weighted MR imaging for noninvasive evaluation of early and late effects of a vascular targeting agent in a rat tumor model. MATERIALS AND METHODS: The study protocol was approved by the local ethics committee for animal care and use. Thirteen rats with one rhabdomyosarcoma in each flank (26 tumors) underwent dynamic contrast-enhanced imaging and diffusion-weighted echo-planar imaging in a 1.5-T MR unit before intraperitoneal injection of combretastatin A4 phosphate and at early (1 and 6 hours) and later (2 and 9 days) follow-up examinations after the injection. Histopathologic examination was performed at each time point. The apparent diffusion coefficient (ADC) of each tumor was calculated separately on the basis of diffusion-weighted images obtained with low b gradient values (ADC(low); b = 0, 50, and 100 sec/mm(2)) and high b gradient values (ADC(high); b = 500, 750, and 1000 sec/mm(2)). The difference between ADC(low) and ADC(high) was used as a surrogate measure of tissue perfusion (ADC(low) - ADC(high) = ADC(perf)). From the dynamic contrast-enhanced MR images, the volume transfer constant k and the initial slope of the contrast enhancement-time curve were calculated. For statistical analyses, a paired two-tailed Student t test and linear regression analysis were used. RESULTS: Early after administration of combretastatin, all perfusion-related parameters (k, initial slope, and ADC(perf)) decreased significantly (P < .001); at 9 days after combretastatin administration, they increased significantly (P < .001). Changes in ADC(perf) were correlated with changes in k (R(2) = 0.46, P < .001) and the initial slope (R(2) = 0.67, P < .001). CONCLUSION: Both dynamic contrast-enhanced MR imaging and diffusion-weighted MR imaging allow monitoring of perfusion changes induced by vascular targeting agents in tumors. Diffusion-weighted imaging provides additional information about intratumoral cell viability versus necrosis after administration of combretastatin.

Diffusion-weighted imaging of the parotid gland: Influence of the choice of b-values on the apparent diffusion coefficient value

PURPOSE: To determine how the ADC value of parotid glands is influenced by the choice of b-values. MATERIALS AND METHODS: In eight healthy volunteers, diffusion-weighted echo-planar imaging (DW-EPI) was performed on a 1.5 T system, with b-values (in seconds/mm2) of 0, 50, 100, 150, 200, 250, 300, 500, 750, and 1000. ADC values were calculated by two alternative methods (exponential vs. logarithmic fit) from five different sets of b-values: (A) all b-values; (B) b=0, 50, and 100; (C) b=0 and 750; (D) b=0, 500, and 1000; and (E) b=500, 750, and 1000. RESULTS: The mean ADC values for the different settings were (in 10(-3) mm2/second, exponential fit): (A) 0.732+/-0.019, (B) 2.074+/-0.084, (C) 0.947+/-0.020, (D) 0.890+/-0.023, and (E) 0.581+/-0.021. ADC values were significantly (P <0.001) different for all pairwise comparisons of settings (A-E) of b-values, except for A vs. D (P=0.172) and C vs. D (P=0.380). The ADC(B) was significantly higher than ADC(C) or ADC(D), which was significantly higher than ADC(E). ADC values from exponential vs. logarithmic fit (P=0.542), as well as left vs. right parotid gland (P=0.962), were indistinguishable. CONCLUSION: The ADC values calculated from low b-value settings were significantly higher than those calculated from high b-value settings. These results suggest that not only true diffusion but also perfusion and saliva flow may contribute to the ADC.

Living Renal Allograft Transplantation: Diffusion-weighted MR Imaging in Longitudinal Follow-up of the Donated and the Remaining Kidney.

Purpose To determine whether diffusion-weighted (DW) magnetic resonance (MR) imaging in living renal allograft donation allows monitoring of potential changes in the nontransplanted remaining kidney of the donor because of unilateral nephrectomy and changes in the transplanted kidney before and after transplantation in donor and recipient, respectively, and whether DW MR parameters are correlated in the same kidney before and after transplantation. Materials and Methods The study protocol was approved by the local ethics committee; written informed consent was obtained. Thirteen healthy kidney donors and their corresponding recipients prospectively underwent DW MR imaging (multiple b values) in donors before donation and in donors and recipients at day 8 and months 3 and 12 after donation. Total apparent diffusion coefficient (ADCT) values were determined; contribution of microcirculation was quantified in perfusion fraction (FP). Longitudinal changes of diffusion parameters were compared (repeated-measures one-way analysis of variance with post hoc pairwise comparisons). Correlations were tested (linear regression). Results ADCT values in nontransplanted kidney of donors increased from a preexplantation value of (188 ± 9 [standard deviation]) to (202 ± 11) × 10(-5) mm(2)/sec in medulla and from (199 ± 11) to (210 ± 13) × 10(-5) mm(2)/sec in cortex 1 week after donation (P < .004). Medullary, but not cortical, ADCT values stayed increased up to 1 year. ADCT values in allografts in recipients were stable. Compared with values obtained before transplantation in donors, the corticomedullary difference was reduced in allografts (P < .03). Cortical ADCT values correlated with estimated glomerular filtration rate in recipients (R = 0.56, P < .001) but not donors. Cortical ADCT values in the same kidney before transplantation in donors correlated with those in recipients on day 8 after transplantation (R = 0.77, P = .006). FP did not show significant changes. Conclusion DW MR imaging depicts early adaptations in the remaining nontransplanted kidney of donors after nephrectomy. All diffusion parameters remained constant in allograft recipients after transplantation. This method has potential monitoring utility, although assessment of clinical relevance is needed. © RSNA, 2013 Online supplemental material is available for this article.

An estimate of heavy quark momentum diffusion coefficient in gluon plasma

We calculate the momentum diffusion coefficient for heavy quarks in SU(3) gluon plasma at temperatures 1-2 times the deconfinement temperature. The momentum diffusion coefficient is extracted from a Monte Carlo calculation of the correlation function of color electric fields, in the leading order of expansion in heavy quark mass. Systematics of the calculation are examined, and compared with perturbtion theory and other estimates.

Nonperturbative estimate of the heavy quark momentum diffusion coefficient

We estimate the momentum diffusion coefficient of a heavy quark within a pure SU(3) plasma at a temperature of about 1.5Tc. Large-scale Monte Carlo simulations on a series of lattices extending up to 1923×48 permit us to carry out a continuum extrapolation of the so-called color-electric imaginary-time correlator. The extrapolated correlator is analyzed with the help of theoretically motivated models for the corresponding spectral function. Evidence for a nonzero transport coefficient is found and, incorporating systematic uncertainties reflecting model assumptions, we obtain κ=(1.8–3.4)T3. This implies that the “drag coefficient,” characterizing the time scale at which heavy quarks adjust to hydrodynamic flow, is η−1D=(1.8–3.4)(Tc/T)2(M/1.5  GeV)  fm/c, where M is the heavy quark kinetic mass. The results apply to bottom and, with somewhat larger systematic uncertainties, to charm quarks.

The effect of dissolved magnesium on diffusion creep in calcite

We experimentally tested a series of synthetic calcite marbles with varying amounts of dissolved magnesium in a standard triaxial deformation machine at 300 MPa confining pressure, temperatures between 700 and 850°C, stresses between 2 and 100 MPa, and strain rates between 10−7 and 10−3 s−1. The samples were fabricated by hot isostatic pressing of a mixture of calcite and dolomite at 850°C and 300 MPa. The fabrication protocol resulted in a homogeneous, fine-grained high-magnesian calcite aggregate with minimal porosity and with magnesium contents between 0.07 and 0.17 mol% MgCO3. At stresses below 40 MPa the samples deformed with linear viscosity that depended inversely on grain size to the 3.26±0.51 power, suggesting that the mechanisms of deformation were some combination of grain boundary diffusion and grain boundary sliding. Because small grain sizes tended to occur in the high-magnesium calcite, the strength also appeared to vary inversely with magnesium content. However, the strength at constant grain size does not depend on the amount of dissolved magnesium, and thus, the impurity effect seems to be indirect. At stresses higher than 40 MPa, the aggregates become non-linearly viscous, a regime we interpret to be dislocation creep. The transition between the two regimes depends on grain size, as expected. The activation energy for diffusion creep is 200±30 kJ/mol and is quite similar to previous measurements in natural and synthetic marbles deformed at similar conditions with no added magnesium.