Spatial dynamics of a population with stage-dependent diffusion

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Stability analysis of Crank-Nicolson and Euler schemes for time-dependent diffusion equations

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Partial reconstruction of the trajectories of a discretely observed branching diffusion with immigration and an application to inference

In this treatise we consider finite systems of branching particles where the particles move independently of each other according to d-dimensional diffusions. Particles are killed at a position dependent rate, leaving at their death position a random number of descendants according to a position dependent reproduction law. In addition particles immigrate at constant rate (one immigrant per immigration time). A process with above properties is called a branching diffusion withimmigration (BDI). In the first part we present the model in detail and discuss the properties of the BDI under our basic assumptions. In the second part we consider the problem of reconstruction of the trajectory of a BDI from discrete observations. We observe positions of the particles at discrete times; in particular we assume that we have no information about the pedigree of the particles. A natural question arises if we want to apply statistical procedures on the discrete observations: How can we find couples of particle positions which belong to the same particle? We give an easy to implement 'reconstruction scheme' which allows us to redraw or 'reconstruct' parts of the trajectory of the BDI with high accuracy. Moreover asymptotically the whole path can be reconstructed. Further we present simulations which show that our partial reconstruction rule is tractable in practice. In the third part we study how the partial reconstruction rule fits into statistical applications. As an extensive example we present a nonparametric estimator for the diffusion coefficient of a BDI where the particles move according to one-dimensional diffusions. This estimator is based on the Nadaraya-Watson estimator for the diffusion coefficient of one-dimensional diffusions and it uses the partial reconstruction rule developed in the second part above. We are able to prove a rate of convergence of this estimator and finally we present simulations which show that the estimator works well even if we leave our set of assumptions.

Diffusion of laser polarized gases in MRI

The use of Magnetic Resonance Imaging (MRI) as a diagnostic tool is increasingly employing functional contrast agents to study or contrast entire mechanisms. Contrast agents in MRI can be classified in two categories. One type of contrast agents alters the NMR signal of the protons in its surrounding, e.g. lowers the T1 relaxation time. The other type enhances the Nuclear Magnetic Resonance (NMR) signal of specific nuclei. For hyperpolarized gases the NMR signal is improved up to several orders of magnitude. However, gases have a high diffusivity which strongly influences the NMR signal strength, hence the resolution and appearance of the images. The most interesting question in spatially resolved experiments is of course the achievable resolution and contrast by controlling the diffusivity of the gas. The influence of such diffusive processes scales with the diffusion coefficient, the strength of the magnetic field gradients and the timings used in the experiment. Diffusion may not only limit the MRI resolution, but also distort the line shape of MR images for samples, which contain boundaries or diffusion barriers within the sampled space. In addition, due to the large polarization in gaseous 3He and 129Xe, spin diffusion (different from particle diffusion) could play a role in MRI experiments. It is demonstrated that for low temperatures some corrections to the NMR measured diffusion coefficient have to be done, which depend on quantum exchange effects for indistinguishable particles. Physically, if these effects can not change the spin current, they can do it indirectly by modifying the velocity distribution of the different spin states separately, so that the subsequent collisions between atoms and therefore the diffusion coefficient can eventually be affected. A detailed study of the hyperpolarized gas diffusion coefficient is presented, demonstrating the absence of spin diffusion (different from particle diffusion) influence in MRI at clinical conditions. A novel procedure is proposed to control the diffusion coefficient of gases in MRI by admixture of inert buffer gases. The experimental measured diffusion agrees with theoretical simulations. Therefore, the molecular mass and concentration enter as additional parameters into the equations that describe structural contrast. This allows for setting a structural threshold up to which structures contribute to the image. For MRI of the lung this allows for images of very small structural elements (alveoli) only, or in the other extreme, all airways can be displayed with minimal signal loss due to diffusion.

Predicting and monitoring cancer treatment response with diffusion-weighted MRI

An imaging biomarker that would provide for an early quantitative metric of clinical treatment response in cancer patients would provide for a paradigm shift in cancer care. Currently, nonimage based clinical outcome metrics include morphology, clinical, and laboratory parameters, however, these are obtained relatively late following treatment. Diffusion-weighted MRI (DW-MRI) holds promise for use as a cancer treatment response biomarker as it is sensitive to macromolecular and microstructural changes which can occur at the cellular level earlier than anatomical changes during therapy. Studies have shown that successful treatment of many tumor types can be detected using DW-MRI as an early increase in the apparent diffusion coefficient (ADC) values. Additionally, low pretreatment ADC values of various tumors are often predictive of better outcome. These capabilities, once validated, could provide for an important opportunity to individualize therapy thereby minimizing unnecessary systemic toxicity associated with ineffective therapies with the additional advantage of improving overall patient health care and associated costs. In this report, we provide a brief technical overview of DW-MRI acquisition protocols, quantitative image analysis approaches and review studies which have implemented DW-MRI for the purpose of early prediction of cancer treatment response.

Diffusion-weighted MR imaging of the placenta in fetuses with placental insufficiency

PURPOSE: To evaluate diffusion-weighted magnetic resonance (MR) imaging of the human placenta in fetuses with and fetuses without intrauterine growth restriction (IUGR) who were suspected of having placental insufficiency. MATERIALS AND METHODS: The study was approved by the local ethics committee, and written informed consent was obtained. The authors retrospectively evaluated 1.5-T fetal MR images from 102 singleton pregnancies (mean gestation ± standard deviation, 29 weeks ± 5; range, 21-41 weeks). Morphologic and diffusion-weighted MR imaging were performed. A region of interest analysis of the apparent diffusion coefficient (ADC) of the placenta was independently performed by two observers who were blinded to clinical data and outcome. Placental insufficiency was diagnosed if flattening of the growth curve was detected at obstetric ultrasonography (US), if the birth weight was in the 10th percentile or less, or if fetal weight estimated with US was below the 10th percentile. Abnormal findings at Doppler US of the umbilical artery and histopathologic examination of specimens from the placenta were recorded. The ADCs in fetuses with placental insufficiency were compared with those in fetuses of the same gestational age without placental insufficiency and tested for normal distribution. The t tests and Pearson correlation coefficients were used to compare these results at 5% levels of significance. RESULTS: Thirty-three of the 102 pregnancies were ultimately categorized as having an insufficient placenta. MR imaging depicted morphologic changes (eg, infarction or bleeding) in 27 fetuses. Placental dysfunction was suspected in 33 fetuses at diffusion-weighted imaging (mean ADC, 146.4 sec/mm(2) ± 10.63 for fetuses with placental insufficiency vs 177.1 sec/mm(2) ± 18.90 for fetuses without placental insufficiency; P < .01, with one false-positive case). The use of diffusion-weighted imaging in addition to US increased sensitivity for the detection of placental insufficiency from 73% to 100%, increased accuracy from 91% to 99%, and preserved specificity at 99%. CONCLUSION: Placental dysfunction associated with growth restriction is associated with restricted diffusion and reduced ADC. A decreased ADC used as an early marker of placental damage might be indicative of pregnancy complications such as IUGR.

Diffusion-weighted MR imaging of native and transplanted kidneys

Applications of diffusion-weighted (DW) magnetic resonance (MR) imaging outside the brain have gained increasing importance in recent years. Owing to technical improvements in MR imaging units and faster sequences, the need for noninvasive imaging without contrast medium administration, mainly in patients with renal insufficiency, can be met successfully by applying this technique. DW MR imaging is quantified by the apparent diffusion coefficient (ADC), which provides information on diffusion and perfusion simultaneously. By using a biexponential fitting process of the DW MR imaging data, these two entities can be separated, because this type of fitting process can serve as an estimate of both the perfusion fraction and the true diffusion coefficient. DW MR imaging can be applied for functional evaluation of the kidneys in patients with acute or chronic renal failure. Impairment of renal function is accompanied by a decreased ADC. Acute ureteral obstruction leads to perfusion and diffusion changes in the affected kidney, and renal artery stenosis results in a decreased ADC. In patients with pyelonephritis, diffuse or focal changes in signal intensity are seen on the high-b-value images, with increased signal intensity corresponding to low signal intensity on the ADC map. The feasibility and reproducibility of DW MR imaging in patients with transplanted kidneys have already been demonstrated, and initial results seem to be promising for the assessment of allograft deterioration. Overall, performance of renal DW MR imaging, presuming that measurements are of high quality, will further boost this modality, particularly for early detection of diffuse renal conditions, as well as more accurate characterization of focal renal lesions.

A MICROFLUIDIC METHOD TO MEASURE DIFFUSION IN HYDROGELS

A novel microfluidic method is proposed for studying diffusion of small molecules in a hydrogel. Microfluidic devices were prepared with semi-permeable microchannels defined by crosslinked poly(ethylene glycol) (PEG). Uptake of dye molecules from aqueous solutions flowing through the microchannels was observedoptically and diffusion of the dye into the hydrogel was quantified. To complement the diffusion measurements from the microfluidic studies, nuclear magnetic resonance(NMR) characterization of the diffusion of dye in the PEG hydrogels was performed. The diffusion of small molecules in a hydrogel is relevant to applications such asdrug delivery and modeling transport for tissue-engineering applications. The diffusion of small molecules in a hydrogel is dependent on the extent of crosslinking within the gel, gel structure, and interactions between the diffusive species and the hydrogel network. These effects were studied in a model environment (semi-infinite slab) at the hydrogelfluid boundary in a microfluidic device. The microfluidic devices containing PEG microchannels were fabricated using photolithography. The unsteady diffusion of small molecules (dyes) within the microfluidic device was monitored and recorded using a digital microscope. The information was analyzed with techniques drawn from digital microscopy and image analysis to obtain concentration profiles with time. Using a diffusion model to fit this concentration vs. position data, a diffusion coefficient was obtained. This diffusion coefficient was compared to those from complementary NMR analysis. A pulsed field gradient (PFG) method was used to investigate and quantify small molecule diffusion in gradient (PFG) method was used to investigate and quantify small molecule diffusion in hydrogels. There is good agreement between the diffusion coefficients obtained from the microfluidic methods and those found from the NMR studies. The microfluidic approachused in this research enables the study of diffusion at length scales that approach those of vasculature, facilitating models for studying drug elution from hydrogels in blood-contacting applications.

Quantifying The Diffusion And Erosion Of Degradable Hydrogels Using A Microfluidic Technique

Hydrogels are composed of cross-linked networks of hydrophilic polymers that are biocompatible due to their high water content. Mass transfer through hydrogels has been suggested as an effective method of drug delivery, specifically in degradable polymers to minimize lasting effects within the body. Diffusion of small molecules in poly (ethylene glycol) diacrylate (PEG-DA) and dextran methacrylate (dex-MA) hydrogels was characterized in a microfluidic device and by complementary techniques. Microfluidic devices were prepared by crosslinking a formulation of hydrogel and photo-initiator, with and without visible dye, using photolithography to define a central microchannel. Channel sizes within the devices were approximately 600 ¿m to simulate vessels within the body. The microfluidic technique allows for both image and effluent analyses. To visualize the diffusive behavior within the dextran hydrogel, methylene blue and sulforhodamine 101 dyes were used in both elution and uptake experiments. Three analysis techniques for measuring diffusion coefficients were used to quantify the diffusion of solute in the hydrogel, including optical microscopy, characterization of device effluent, and NMR analyses. The optical microscopy technique analyzes images of the dye diffusion captured by a stereomicroscope to generate dye concentration v. position profiles. The data was fit to a diffusion model to determine diffusion coefficients and the dye release profile. In a typical elution experiment, aqueous solution is pumped through the microchannel and dye diffuses out of the hydrogel and into the aqueous phase. During elution, images are taken at regular time intervals and the effluent was collected. Analysis of the device effluent was performed using ultraviolet-visible (UV/Vis) spectroscopy to determine the effluent dye concentration and thus a short-time diffusion coefficient. Nuclear magnetic resonance (NMR) was used to determine a free diffusion coefficient of molecules in hydrogel without the effect of a concentration gradient. Diffusion coefficients for methylene blue and sulforhodamine 101 dyes in dex-MA hydrogel calculated using the three analysis methods all agree well. It was determined that utilizing a combination of the three techniques offers greater insight into molecular diffusion in hydrogels than employing each technique individually. The use of the same microfluidic devices used to measure diffusion is explored in the use of studying the degradation of dex-MA hydrogels. By combining what is known about the degradation rate in regards to the effect of pH and crosslinking and the ability to use a dye solution in contrast to establish the hydrogel boundaries could be a novel approach to studying hydrogel degradation.

Genitourinary applications of diffusion-weighted MR imaging in the pelvis

Diffusion-weighted (DW) magnetic resonance (MR) imaging has a large number of potential clinical applications in the female and male pelvis and can easily be added to any routine MR protocol. In the female pelvis, DW imaging allows improvement of staging in endometrial and cervical cancer, especially in locally advanced disease and in patients in whom contrast medium administration should be avoided. It can also be helpful in characterizing complex adnexal masses and in depicting recurrent tumor after treatment of various gynecologic malignancies. DW imaging shows promising results in monitoring treatment response in patients undergoing radiation therapy of cervical cancer. An increase in apparent diffusion coefficient (ADC) values of responders precedes changes in size and may therefore allow early assessment of treatment success. In the male pelvis, the detection of prostate cancer in the peripheral zone is relatively easier than in the central gland based on the underlying ADC values, whereas overlapping values reported in the central gland still need further research. DW imaging might also be applied in the noninvasive evaluation of bladder cancer to differentiate between superficial and muscle-invasive tumors. Initial promising results have been reported in differentiating benign from malignant pelvic lymph nodes based on the ADC values; however, larger-scale studies will be needed to allow the detection of lymph node metastases in an individual patient. Prerequisites for successfully performing DW imaging of the female and male pelvis are standardization of the DW imaging technique, including the choice of b values, administration of an antiperistaltic drug, and comparison of DW findings with those of morphologic MR imaging.

Functional evaluation of transplanted kidneys with diffusion-weighted and BOLD MR imaging: initial experience

PURPOSE: To prospectively evaluate feasibility and reproducibility of diffusion-weighted (DW) and blood oxygenation level-dependent (BOLD) magnetic resonance (MR) imaging in patients with renal allografts, as compared with these features in healthy volunteers with native kidneys. MATERIALS AND METHODS: The local ethics committee approved the study protocol; patients provided written informed consent. Fifteen patients with a renal allograft and in stable condition (nine men, six women; age range, 20-67 years) and 15 age- and sex-matched healthy volunteers underwent DW and BOLD MR imaging. Seven patients with renal allografts were examined twice to assess reproducibility of results. DW MR imaging yielded a total apparent diffusion coefficient including diffusion and microperfusion (ADC(tot)), as well as an ADC reflecting predominantly pure diffusion (ADC(D)) and the perfusion fraction. R2* of BOLD MR imaging enabled the estimation of renal oxygenation. Statistical analysis was performed, and analysis of variance was used for repeated measurements. Coefficients of variation between and within subjects were calculated to assess reproducibility. RESULTS: In patients, ADC(tot), ADC(D), and perfusion fraction were similar in the cortex and medulla. In volunteers, values in the medulla were similar to those in the cortex and medulla of patients; however, values in the cortex were higher than those in the medulla (P < .05). Medullary R2* was higher than cortical R2* in patients (12.9 sec(-1) +/- 2.1 [standard deviation] vs 11.0 sec(-1) +/- 0.6, P < .007) and volunteers (15.3 sec(-1) +/- 1.1 vs 11.5 sec(-1) +/- 0.5, P < .0001). However, medullary R2* was lower in patients than in volunteers (P < .004). Increased medullary R2* was paralleled by decreased diffusion in patients with allografts. A low coefficient of variation in the cortex and medulla within subjects was obtained for ADC(tot), ADC(D), and R2* (<5.2%), while coefficient of variation within subjects was higher for perfusion fraction (medulla, 15.1%; cortex, 8.6%). Diffusion and perfusion indexes correlated significantly with serum creatinine concentrations. CONCLUSION: DW and BOLD MR imaging are feasible and reproducible in patients with renal allografts.

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.

Noninvasive assessment of acute ureteral obstruction with diffusion-weighted MR imaging: a prospective study

PURPOSE: To prospectively assess the potential of noninvasive diffusion-weighted magnetic resonance (MR) imaging to depict changes in microperfusion and diffusion in patients with acute unilateral ureteral obstruction. MATERIALS AND METHODS: The local ethics committee approved the study protocol. Informed consent was obtained. Diffusion-weighted MR imaging was performed in 21 patients (two women, 19 men; mean age, 43 years +/- 10 [standard deviation]) with acute unilateral ureteral obstruction due to a calculus diagnosed at unenhanced computed tomography. A control group (one woman, 15 men; mean age, 44 years +/- 12) underwent the same MR protocol. Standard processing yielded an apparent diffusion coefficient (ADC) ADCT; the separation of microperfusion and diffusion contributions yielded the perfusion fraction FP and the pure diffusion coefficient ADCD. ADCT, ADCD, and FP were compared between obstructed and contralateral unobstructed kidneys and with control values. For statistical analysis, nonparametric rank tests were used. A P value of less than .05 was considered significant. RESULTS: No significant differences were observed between the ADCT of the medulla or cortex of the obstructed and unobstructed kidneys. Compared with control kidneys, only medullary ADCT was slightly increased in the obstructed kidney (P < .04). However, the ADCD in the medulla of the obstructed and unobstructed kidneys was significantly higher than that in control subjects (201 x 10(-5) mm2/sec +/- 16 and 199 x 10(-5) mm2/sec +/- 20 vs 189 x 10(-5) mm2/sec +/- 12; P < .008 and P < .03, respectively). FP of the cortex of the obstructed kidney was significantly lower than that in the unobstructed kidney (20.2% +/- 4.8 vs 24.0% +/- 5.8; P < .002); FP of the medulla was slightly lower in the obstructed kidney than in the unobstructed kidney (18.3% +/- 5.9 vs 20.7% +/- 6.4; P = .05). CONCLUSION: Diffusion-weighted MR imaging allows noninvasive detection of changes in renal perfusion and diffusion during acute unilateral ureteral obstruction, as exemplified in patients with a ureteral calculus.

Dynamic contrast-enhanced and diffusion-weighted MRI for early detection of tumoral changes in single-dose and fractionated radiotherapy: evaluation in a rat rhabdomyosarcoma model

We aimed to examine different intratumoral changes after single-dose and fractionated radiotherapy, using diffusion-weighted (DW) and dynamic contrast-enhanced (DCE) MRI in a rat rhabdomyosarcoma model. Four WAG/Rij rats with rhabdomyosarcomas in the flanks received single-dose radiotherapy of 8 Gy, and four others underwent fractionated radiotherapy (five times 3 Gy). In rats receiving single-dose radiotherapy, a significant perfusion decrease was found in the first 2 days post-treatment, with slow recuperation afterwards. No substantial diffusion changes could be seen; tumor growth delay was 12 days. The rats undergoing fractionated radiotherapy showed a similar perfusion decrease early after the treatment. However, a very strong increase in apparent diffusion coefficient occurred in the first 10 days; growth delay was 18 days. DW-MRI and DCE-MRI can be used to show early tumoral changes induced by radiotherapy. Single-dose and fractionated radiotherapy induce an immediate perfusion effect, while the latter induces more intratumoral necrosis.

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.