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Transcranial direct current stimulation (tDCS) is a promising treatment for major depressive disorder (MDD). This study evaluated its antidepressant and cognitive effects as a safe, effective, home-based therapy for MDD.
Methods
This double-blind, sham-controlled, randomized trial divided participants into low-intensity (1 mA, n = 47), high-intensity (2 mA, n = 49), and sham (n = 45) groups, receiving 42 daily tDCS sessions, including weekends and holidays, targeting the dorsolateral prefrontal cortex for 30 minutes. Assessments were conducted at baseline and weeks 2, 4, and 6. The primary outcome was cognitive improvement assessed by changes in total accuracy on the 2-back test from baseline to week 6. Secondary outcomes included changes in depressive symptoms (HAM-D), anxiety (HAM-A), and quality of life (QLES). Adverse events were monitored. This trial was registered with ClinicalTrials.gov (NCT04709952).
Results
In the tDCS study, of 141 participants (102 [72.3%] women; mean age 35.7 years, standard deviation 12.7), 95 completed the trial. Mean changes in the total accuracy scores from baseline to week 6 were compared across the three groups using an F-test. Linear mixed-effects models examined the interaction of group and time. Results showed no significant differences among groups in cognitive or depressive outcomes at week 6. Active groups experienced more mild adverse events compared to sham but had similar rates of severe adverse events and dropout.
Conclusions
Home-based tDCS for MDD demonstrated no evidence of effectiveness but was safe and well-tolerated. Further research is needed to address the technical limitations, evaluate broader cognitive functions, and extend durations to evaluate its therapeutic potential.
Population-wide restrictions during the COVID-19 pandemic may create barriers to mental health diagnosis. This study aims to examine changes in the number of incident cases and the incidence rates of mental health diagnoses during the COVID-19 pandemic.
Methods
By using electronic health records from France, Germany, Italy, South Korea and the UK and claims data from the US, this study conducted interrupted time-series analyses to compare the monthly incident cases and the incidence of depressive disorders, anxiety disorders, alcohol misuse or dependence, substance misuse or dependence, bipolar disorders, personality disorders and psychoses diagnoses before (January 2017 to February 2020) and after (April 2020 to the latest available date of each database [up to November 2021]) the introduction of COVID-related restrictions.
Results
A total of 629,712,954 individuals were enrolled across nine databases. Following the introduction of restrictions, an immediate decline was observed in the number of incident cases of all mental health diagnoses in the US (rate ratios (RRs) ranged from 0.005 to 0.677) and in the incidence of all conditions in France, Germany, Italy and the US (RRs ranged from 0.002 to 0.422). In the UK, significant reductions were only observed in common mental illnesses. The number of incident cases and the incidence began to return to or exceed pre-pandemic levels in most countries from mid-2020 through 2021.
Conclusions
Healthcare providers should be prepared to deliver service adaptations to mitigate burdens directly or indirectly caused by delays in the diagnosis and treatment of mental health conditions.
It is unknown how much variation in adult mental health problems is associated with differences between societal/cultural groups, over and above differences between individuals.
Methods
To test these relative contributions, a consortium of indigenous researchers collected Adult Self-Report (ASR) ratings from 16 906 18- to 59-year-olds in 28 societies that represented seven culture clusters identified in the Global Leadership and Organizational Behavioral Effectiveness study (e.g. Confucian, Anglo). The ASR is scored on 17 problem scales, plus a personal strengths scale. Hierarchical linear modeling estimated variance accounted for by individual differences (including measurement error), society, and culture cluster. Multi-level analyses of covariance tested age and gender effects.
Results
Across the 17 problem scales, the variance accounted for by individual differences ranged from 80.3% for DSM-oriented anxiety problems to 95.2% for DSM-oriented avoidant personality (mean = 90.7%); by society: 3.2% for DSM-oriented somatic problems to 8.0% for DSM-oriented anxiety problems (mean = 6.3%); and by culture cluster: 0.0% for DSM-oriented avoidant personality to 11.6% for DSM-oriented anxiety problems (mean = 3.0%). For strengths, individual differences accounted for 80.8% of variance, societal differences 10.5%, and cultural differences 8.7%. Age and gender had very small effects.
Conclusions
Overall, adults' self-ratings of mental health problems and strengths were associated much more with individual differences than societal/cultural differences, although this varied across scales. These findings support cross-cultural use of standardized measures to assess mental health problems, but urge caution in assessment of personal strengths.
Abnormal reward functioning is central to anhedonia and amotivation symptoms of schizophrenia (SCZ). Reward processing encompasses a series of psychological components. This systematic review and meta-analysis examined the brain dysfunction related to reward processing of individuals with SCZ spectrum disorders and risks, covering multiple reward components.
Methods
After a systematic literature search, 37 neuroimaging studies were identified and divided into four groups based on their target psychology components (i.e. reward anticipation, reward consumption, reward learning, effort computation). Whole-brain Seed-based d Mapping (SDM) meta-analyses were conducted for all included studies and each component.
Results
The meta-analysis for all reward-related studies revealed reduced functional activation across the SCZ spectrum in the striatum, orbital frontal cortex, cingulate cortex, and cerebellar areas. Meanwhile, distinct abnormal patterns were found for reward anticipation (decreased activation of the cingulate cortex and striatum), reward consumption (decreased activation of cerebellum IV/V areas, insula and inferior frontal gyri), and reward learning processing (decreased activation of the striatum, thalamus, cerebellar Crus I, cingulate cortex, orbitofrontal cortex, and parietal and occipital areas). Lastly, our qualitative review suggested that decreased activation of the ventral striatum and anterior cingulate cortex was also involved in effort computation.
Conclusions
These results provide deep insights on the component-based neuro-psychopathological mechanisms for anhedonia and amotivation symptoms of the SCZ spectrum.
Glutamatergic dysfunction has been implicated in sensory integration deficits in schizophrenia, yet how glutamatergic function contributes to behavioural impairments and neural activities of sensory integration remains unknown.
Methods
Fifty schizophrenia patients and 43 healthy controls completed behavioural assessments for sensory integration and underwent magnetic resonance spectroscopy (MRS) for measuring the anterior cingulate cortex (ACC) glutamate levels. The correlation between glutamate levels and behavioural sensory integration deficits was examined in each group. A subsample of 20 pairs of patients and controls further completed an audiovisual sensory integration functional magnetic resonance imaging (fMRI) task. Blood Oxygenation Level Dependent (BOLD) activation and task-dependent functional connectivity (FC) were assessed based on fMRI data. Full factorial analyses were performed to examine the Group-by-Glutamate Level interaction effects on fMRI measurements (group differences in correlation between glutamate levels and fMRI measurements) and the correlation between glutamate levels and fMRI measurements within each group.
Results
We found that schizophrenia patients exhibited impaired sensory integration which was positively correlated with ACC glutamate levels. Multimodal analyses showed significantly Group-by-Glutamate Level interaction effects on BOLD activation as well as task-dependent FC in a ‘cortico-subcortical-cortical’ network (including medial frontal gyrus, precuneus, ACC, middle cingulate gyrus, thalamus and caudate) with positive correlations in patients and negative in controls.
Conclusions
Our findings indicate that ACC glutamate influences neural activities in a large-scale network during sensory integration, but the effects have opposite directionality between schizophrenia patients and healthy people. This implicates the crucial role of glutamatergic system in sensory integration processing in schizophrenia.
The coronavirus disease 2019 pandemic caused substantial changes to healthcare delivery and antibiotic prescribing beginning in March 2020. To assess pandemic impact on Clostridioides difficile infection (CDI) rates, we described patients and trends in facility-level incidence, testing rates, and percent positivity during 2019–2020 in a large cohort of US hospitals.
Methods:
We estimated and compared rates of community-onset CDI (CO-CDI) per 10,000 discharges, hospital-onset CDI (HO-CDI) per 10,000 patient days, and C. difficile testing rates per 10,000 discharges in 2019 and 2020. We calculated percent positivity as the number of inpatients diagnosed with CDI over the total number of discharges with a test for C. difficile. We used an interrupted time series (ITS) design with negative binomial and logistic regression models to describe level and trend changes in rates and percent positivity before and after March 2020.
Results:
In pairwise comparisons, overall CO-CDI rates decreased from 20.0 to 15.8 between 2019 and 2020 (P < .0001). HO-CDI rates did not change. Using ITS, we detected decreasing monthly trends in CO-CDI (−1% per month, P = .0036) and HO-CDI incidence (−1% per month, P < .0001) during the baseline period, prior to the COVID-19 pandemic declaration. We detected no change in monthly trends for CO-CDI or HO-CDI incidence or percent positivity after March 2020 compared with the baseline period.
Conclusions:
While there was a slight downward trajectory in CDI trends prior to March 2020, no significant change in CDI trends occurred during the COVID-19 pandemic despite changes in infection control practices, antibiotic use, and healthcare delivery.
While there has been much research on welfare exit and entry into employment, less research has looked at return to government assistance. Applying survival analysis on data from a national government assistance programme in Singapore, we found two important factors of welfare return to which activation programmes need to pay greater attention. First, return was more likely if former beneficiaries accumulated a higher number of types of arrears rather than higher dollar values of arrears. This new finding contributes to the emerging literature on bandwidth tax, and suggests the importance of designing programmes that relieve mental accounting due to debt and poverty. Second, return was more likely if respondents had an infant or toddler child. This points to the importance of a range of support policies including affordable and accessible childcare, exemption from work requirement in receipt of welfare, and family leave for low-wage workers.
We present an overview of the Middle Ages Galaxy Properties with Integral Field Spectroscopy (MAGPI) survey, a Large Program on the European Southern Observatory Very Large Telescope. MAGPI is designed to study the physical drivers of galaxy transformation at a lookback time of 3–4 Gyr, during which the dynamical, morphological, and chemical properties of galaxies are predicted to evolve significantly. The survey uses new medium-deep adaptive optics aided Multi-Unit Spectroscopic Explorer (MUSE) observations of fields selected from the Galaxy and Mass Assembly (GAMA) survey, providing a wealth of publicly available ancillary multi-wavelength data. With these data, MAGPI will map the kinematic and chemical properties of stars and ionised gas for a sample of 60 massive (
${>}7 \times 10^{10} {\mathrm{M}}_\odot$
) central galaxies at
$0.25 < z <0.35$
in a representative range of environments (isolated, groups and clusters). The spatial resolution delivered by MUSE with Ground Layer Adaptive Optics (
$0.6-0.8$
arcsec FWHM) will facilitate a direct comparison with Integral Field Spectroscopy surveys of the nearby Universe, such as SAMI and MaNGA, and at higher redshifts using adaptive optics, for example, SINS. In addition to the primary (central) galaxy sample, MAGPI will deliver resolved and unresolved spectra for as many as 150 satellite galaxies at
$0.25 < z <0.35$
, as well as hundreds of emission-line sources at
$z < 6$
. This paper outlines the science goals, survey design, and observing strategy of MAGPI. We also present a first look at the MAGPI data, and the theoretical framework to which MAGPI data will be compared using the current generation of cosmological hydrodynamical simulations including EAGLE, Magneticum, HORIZON-AGN, and Illustris-TNG. Our results show that cosmological hydrodynamical simulations make discrepant predictions in the spatially resolved properties of galaxies at
$z\approx 0.3$
. MAGPI observations will place new constraints and allow for tangible improvements in galaxy formation theory.
Previously reported associations between hospital-level antibiotic use and hospital-onset Clostridioides difficile infection (HO-CDI) were reexamined using 2012–2018 data from a new cohort of US acute-care hospitals. This analysis revealed significant positive associations between total, third-generation, and fourth-generation cephalosporin, fluoroquinolone, carbapenem, and piperacillin-tazobactam use and HO-CDI rates, confirming previous findings.
Brain-derived neurotrophic factor (BDNF) gene may be involved in the pathogenesis of schizophrenia by virtue of its effects on neurotransmitter systems that are dysregulated in psychiatric disorder. The common functional polymorphism Val66Met (or rs6265) within the BDNF gene has been reported to be associated with age of onset in schizophrenia. We investigated the relationship between BDNF polymorphisms rs6265 and rs11030101 and early-onset schizophrenia in the Chinese population.
Subjects and methods
The tag single nucleotide polymorphisms (tag SNPs) rs11030101 and rs6265 in the BDNF gene were genotyped in 360 early-onset schizophrenics and 399 controls subjects. Single nucleotide polymorphism association and haplotype analysis were performed.
Results
There were significant differences in allele and genotype frequencies between patient and normal control subjects for rs11030101 (χ2 = 5.130407, df = 1, p = 0.023553; χ2 = 6.121, df = 2, p = 0.047). No statistically significant differences were found in allele or genotype between patient and normal control subjects for rs6265. Stratification of the study by gender of the samples yielded significant evidence for an association with the polymorphisms rs11030101 in female population (genotype-wise: χ2 = 7.758, df = 2, p = 0.021).
Conclusions
Our study indicates that the BDNF play major roles in the susceptibility to early-onset and female schizophrenia in the Chinese population.
The purpose of this study was to investigated the prevalence child depression in primary schools.
Methods
3685 students from Grade 3 to Grade 5 were selected from 7 primary schools of Pudong district in Shanghai by random and cluster sampling. The study design consisted of a screening stage in which the Center for Epidemiological Studies Depression Scale for Children(CES-DC) were used, and a clinical interview stage in which the K-SADS-present state version (K-SADS) and DSM-IV were used. The diagnoses of depressive disorder were made according the DSM-IV criteria.
Results
The prevalence of children depression was 1.60% (95%CI = 1.19%∼2.00%). The prevalence rate of male(2.08%) was significant higher than that of female (1.09%)(X2=5.40, P = 0.02). The rate of depressive disorder increased with age from 0.57% (8 years old) to 2.47% (12 years old). The prevalence of depression was no significant difference between ages from 8 to 12 years old (X2 = 4.49, P = 0.34).
Conclusion
The prevalence rate of children depression in Shanghai is low. The prevalence of depression among boys is much higher than that of girls.It shows the prevalence of depression is no significant difference between ages from 8 to 12 years old.
Accumulated studies indicate that schizophrenia patients are prone to present the type 2 diabetes symptoms, but the potential mechanisms behind their association remain unknown. Here we explored the pathogenetic association between SCZ and T2D based on pathway analysis and protein-protein interaction. To explore the pathway crosstalk, we constructed a pathway-based network including all of those significant pathways. Our results revealed that some pathways are shared by both SCZ and T2D diseases through a number of susceptibility genes. With 382 unique susceptibility proteins for SCZ and T2D, we further built a protein-protein interaction network by extracting their nearest interacting neighbours. Among 2,104 retrieved proteins, 364 of them were found simultaneously interacted with susceptibility proteins of both SCZ and T2D, and proposed as new candidate risk factors for both diseases. Moreover, some proteins were hub proteins with high connectivity and interacted with multiple proteins involved in both diseases.Some of these hub proteins are the components of our identified enriched pathways, including calcium signaling, g-secretase mediated ErbB4 signaling, adipocytokine signaling, insulin signaling, AKT signaling and type II diabetes mellitus pathways. Through the integration of multiple lines of information, we proposed that those signaling pathways, such as AKT signaling, that contain susceptibility genes for both diseases, could be the key pathways to bridge SCZ and T2D. AKT could be one of the important shared components and may play a pivotal role to link both of the pathogenetic processes. Our study provides the general pathway-based view of pathogenetic association between two diseases.
rs10761482 in ANK3 gene showed a significant association with schizophrenia in a genome-wide association study (GWAS). Another marker rs10994336 in ANK3 with the risk of bipolar disorder (BD) which might have more genetic overlap with schizophrenia, had been reported in two meta-analyses of GWAS. In this study, we investigated the association between ANK3 polymorphisms and the susceptibility of schizophrenia in Chinese Han population.
Methods
Population-based (schizophrenia patients = 516 and controls = 400) and family based (trios of early onset schizophrenia= 81) study was performed through genotyping the most promising makers rs10761482, rs10994336, and two missenses rs3808942 and rs3808943 near promoter of ANK3. Particularly, we conducted an association analysis for the combined case-control and family samples.
Results
Our population-based study replicated the association between rs10761482 (P = 0.0268 with C allele) and schizophrenia, and detected a novel association with rs10994336 (P = 4.0 × 10−4 with T allele). Haplotype analysis revealed the higher frequencies of C-T, and T-C (rs10761482–10994336) in the cases than controls (P = 0.0032 and P = 0.0012, respectively). In the family study, the C allele of rs10761482 (P = 0.0940) and T allele of rs10994336 (P = 0.0832) were slightly over-transmitted, and T-C was significantly associated with schizophrenia (P = 0.0304). The results from the combined samples analysis were consistent with independent analysis. rs10761482, rs10994336, C-T, and T-C were significantly associated with schizophrenia (P = 3.3 × 10−6∼3.9 × 10−5), whilst rs3808942 and rs3808943 did not reach normal significance.
Conclusions
Our data strongly support ANK3 gene is a schizophrenia susceptibility gene, and also provide further evidence for the shared susceptibility loci between schizophrenia and BD.
rs10761482 in ANK3 gene showed a significant association with schizophrenia in a genome-wide association study (GWAS). Another marker rs10994336 in ANK3 with the risk of bipolar disorder (BD) which might have more genetic overlap with schizophrenia, had been reported in two meta-analyses of GWAS.
Objective
In this study, we investigated the association between ANK3 polymorphisms and the susceptibility of schizophrenia in Chinese Han population.
Methods
Population-based (schizophrenia patients = 516 and controls = 400) and family based (trios of early onset schizophrenia= 81) study was performed through genotyping the most promising makers rs10761482, rs10994336, and two missenses rs3808942 and rs3808943 near promoter of ANK3. Particularly, we conducted an association analysis for the combined case-control and family samples.
Results
Our population-based study replicated the association between rs10761482 (P = 0.0268 with C allele) and schizophrenia, and detected a novel association with rs10994336 (P = 4.0 × 10−4 with T allele). Haplotype analysis revealed the higher frequencies of C-T, and T-C (rs10761482–10994336) in the cases than controls (P = 0.0032 and P = 0.0012, respectively). In the family study, the C allele of rs10761482 (P = 0.0940) and T allele of rs10994336 (P = 0.0832) were slightly over-transmitted, and T-C was significantly associated with schizophrenia (P = 0.0304). The results from the combined samples analysis were consistent with independent analysis. rs10761482, rs10994336, C-T, and T-C were significantly associated with schizophrenia (P = 3.3 × 10−6∼3.9 × 10−5), whilst rs3808942 and rs3808943 did not reach normal significance.
Conclusions
Our data strongly support ANK3 gene is a schizophrenia susceptibility gene, and also provide further evidence for the shared susceptibility loci between schizophrenia and BD.
Schizophrenia is a chronic psychiatry disorder with high heritability. Schizophrenic patients with early age at onset trend to have more genetic component and thus may be an attractive subpopulation for genetic studies. Brain-derived neurotrophimc factor (BDNF) is considered as candidate gene for schizophrenia. A single nucleotide polymorphism (BDNF Val66Met) was reported to be associated with schizophrenia, although discrepancy remains. The aim of this study was to evaluate the association between BDNF Val66Met polymorphism and schizophrenia using an early onset sample in Chinese Han population. Our sample consisted of 353 schizophrenic patients with onset before age 18 and 394 healthy age and sex matched controls. All subjects were ethnically homogenous Han Chinese origin. No significant differences of genotype or allele distribution were identified between the patients and controls. However, the Met allele was significantly associated with an earlier age at onset in male schizophrenic patients (Kaplan-Meier log-rank test P = 0.005), but not in females (P = 0.289). The BDNF Val66Met polymorphism has an important effect on the age at onset of schizophrenia in a gender-specific manner, and this may provided a significant genetic clue for the etiology of schizophrenia. Therefore, further studies are required to uncover the exact role of BDNF in the development of schizophrenia.
We investigated the relationship between tyrosine hydroxylase (TH) polymorphisms rs11042978, rs2070762 and rs6356 and early-onset schizophrenia in the Chinese Han population.
Subjects and methods
The tag single nucleotide polymorphisms (tag SNPs) rs11042978, rs2070762 and rs6356 in the TH gene were genotyped in 315 early-onset schizophrenics (188 male patients,127 female patients)and 391 controls subjects (219 males,172 females). Single nucleotide polymorphism association and haplotype analysis were performed.
Results
There were significant differences in allele and genotype frequencies between patients and normal control subjects for rs11042978 allele (χ2 = 4.47, df = 1, P = 0.034) and genotype (χ2 = 6.35, df = 2, P = 0.042). No statistically significant differences were found in allele or genotype between patients and normal control subjects for rs2070762 and rs6356. The haplotype analysis revealed that there were significant differences between patients and normal control subjects for haplotypes GAC (χ2 = 6.35, P = 0.012).
Conclusions
Our study indicates that the TH gene may play major roles in the susceptibility to early-onset schizophrenia in the Chinese population.
The present study compared the expression profile and made the classification with the leukocytes by using whole-genome cRNA microarrays among patients with SSD, major depressive disorder (MDD) and healthy controls.
Methods
Gene expression profiling was conducted in peripheral blood leucocytes from drug-free first-episode subjects with SSD, MDD, and matched controls (8 subjects in each group) using global mRNA expression arrays. Support vector machines (SVMs) were utilized for training and testing on candidate signature expression profiles from signature selection step.
Results
We identified SSD and MDD gene signatures from blood-based gene expression profile and build a SSD- MDD disorder model with higher predictive power. Firstly, we identified 63 differentially expressed SSD signatures in contrast to control (P <= 5.0E-4) and 30 differentially expressed MDD signatures in contrast to control, respectively. Then, 123 gene signatures were identified with significantly differential expression level between SSD and MDD. Secondly, in order to conduct priority selection for biomarkers for SSD and MDD together, we selected top gene signatures from each group of pair-wise comparison results, and merged the signatures together to generate better profiles used for clearly classify SSD and MDD sets in the same time. In details, we tried different combination of signatures from the three pair-wise compartmental results and finally determined 48 gene expression signatures with 100% accuracy.
Conclusion
Blood cell-derived RNA may have significant value for performing diagnostic functions and identifying disease biomarkers in SSD and MDD. These 48 gene model could classify SSD, MDD, and healthy controls.
To study the relationship between insulin-like growth factor 1 receptor (IGF1R)and subsyndromal symptomatic depression (SSD).
Methods:
In this case-control study, real-time quantitative reverse transcriptase polymerase chain reaction (RT-qPCR) with TaqMan MGB was used to analyzing the differences of IGF1R gene mRNA expression in peripheral leukocytes between subsyndromal symptomatic depression group(n = 47) and healthy controls(n = 52). At the same time Hamilton Depression Rating Scale -17(HAMD17) were assessed.
Results:
IGF1R gene mRNA expression was 0.21 ± 0.11 in SSD group, 0.56 ± 0.37 in healthy group, and there was significant difference between both groups on IGF1R expression(z = 39.54, P < 0.001). the expression levels of IGF1R in SSD patients was not correlated with Hamilton score(r = −0.292, p = 0.275).
Conclusion:
This study suggested that the decreased expression of IGF1R were related with the pathophysiology of SSD.
The implementation of advanced multi-level modulation schemes such as quadrature phase-shift keying (QPSK) in contrast to the conventional on–off keying is crucial to further boost the terahertz (THz) communications speed. Thereby, carrier phase noise reduction in the THz range is one of the key goals that need to be urgently achieved. In this paper, the photonic-based THz sources and the phase noise problem are briefly summarized. Then, a low phase-noise photonic source based on the stimulated Brillouin scattering (SBS) optical fiber cavity is first applied for a 300-GHz-band QPSK wireless communication link. The highest data rate at forward-error-correction limited condition was 15 Gbaud utilizing the SBS-based photonic source with a small transmit power of ~ −36 dBm. Its transmission characteristics are evaluated and compared with the conventional optical frequency comb generator (OFCG)-based source at 5 Gbaud. The proposed SBS-based photonic source has been proven to offer better performances than the OFCG-based source with respect to the phase noise, optical carrier to noise ratio, and bit error rate in communications.
The present study deals with a theoretical investigation of a close-contact melting (CCM) process involving a vertical cylinder on a horizontal isothermal surface, where the liquid phase is a non-Newtonian viscoplastic fluid that behaves according to the Bingham model. Accordingly, a new approach is formulated based on the thin layer approximation and different quasi-steady process assumptions. By analytical derivation, an algebraic equation that relates the molten layer thickness and the solid bulk height is developed. The problem is then solved numerically, coupled with another equation for the melting rate. The new model shows that as the yield stress increases the melting rate decreases and the molten layer thickness increases. It is found that under certain conditions, the model can be reduced to a form that allows an analytical solution. The approximate model predicts an exponential dependence of both the melt fraction and the molten layer thickness. Comparison between the numerical and analytical solutions shows that the analytical approximation provides an excellent estimation for sufficiently large values of the yield stress. Dimensional analysis, which is supported by the analytical model results, reveals the dimensionless groups that govern the problem. For the general case, the melt fraction is a function of two dimensionless groups. For the analytical approximation, it is shown that the melt fraction is governed by a single dimensionless group and that the molten layer thickness is governed by two dimensionless groups.