N = 207 older adults (age 45–60) working full-time (>30-h/week) were randomly assigned to SLE questions framed as either ‘Live-to’ or ‘Die-by’ to evaluate competing predictions for FTP. Results indicate general support for decreasing age-change in FTP, indicated by independent-sample t-tests showing lower FTP in the ‘Die-by’ framing condition. Further general-linear model analyses were conducted to test for interaction effects of retirement planning with experimental framings on FTP and intended retirement; While retirement planning buffered FTP’s decrease, simple-effects also revealed that retirement planning increased intentions for sooner retirement, but lack of planning increased intentions for later retirement. Discussion centers on practical implications of our findings and consequences validity evidence in future empirical research of FTP in both workplace and retirement domains.Future time perspective (FTP) is defined as “perceptions of the future as being limited or open-ended” ; p. 125. On the other hand, retirement scholars regard FTP as mediating age effects for retirement planning  converge to yield static domain structures. That is, conditioning predictions on age for meaningful interpretation of relevant criteria across workplace and retirement domains, requires independence assumptions between work and retirement contexts. Independence of work – retirement domains is intuitively flawed, given the dependence of retirement from a previous worklife. Consequently, age is limitedly meaningful for the growing complexity and overlap of work and retirement contexts. This practical disconnect evidences in probabilistically opposing  age-predictions for FTP concerning the focal criteria of interest (delaying intentions/advancing preparations for retirement).Table 1).In contrast to age effects on employment and retirement, much less research attention has been given to its counterpoint, that is, subjective life expectancy SLE; . EmpiricTaken together, current empirical findings suggest that RetirPlan (accelerator) and FTP (delayer) may operate as competing mechanisms on retirement decisions Empirical support for the first mechanism, RetirPlan as-accelerator of workforce withdrawal is well-illustrated by the design of r = -0.45  work vs. non-work valuation (affect), and (2) short vs. long time horizons (cognitive), as two mechanisms linking work and retirement has not yet been evaluated in a single study. The goal of the current experiment, therefore, is to pit these competing mechanisms in an evaluative framework by manipulating SLE via attribute-framing conditions. The results are expected to clarify competing age-predictions for FTP across workplace and retirement research domains. Next, we will introduce our SLE manipulation for subsequently deriving our research hypotheses.live-to’/‘die-by’ 95-years of age? The ‘Die-by’ frame has been found to lower SLE compared to ‘Live-to’ framings.The current study evaluates the competing predictions for age-FTP with an attribute-framing manipulation of SLE and simultaneous estimation of FTP and RetirIntent. Recent findings indicate that the framing of SLE questions, as either ‘Live-to’ or ‘Die-by,’ results in an SLE difference of approximately 7–10 years . ExampleConceptually, we may re-interpret this SLE manipulation as a ‘subjective age’ manipulation. In this case, if the ‘Die-by’ frame is successfully replicated so that it lowers SLE, we may interpret this as inducing ‘older subjective age.’ Consequently, the randomization of participants and comparison on FTP enables us to infer how FTP covaries with different perceptions of one’s remaining lifetime/SLE. Restated in the context of formal hypothesis-testing, we hypothesize that the ‘Die-by’ SLE frame will reduce FTP compared to the ‘Live-to’ frame.H1: The ‘Die-by’ frame will decrease FTP compared to the ‘Live-to’ frame.Additionally, we include the variable ‘retirement intention’ in order to assess the consistency of directional effects across ‘work’ and ‘retirement’ domains. Specifically, if we accept retirement to be subsumed by overall age, then it follows that older workers (lower SLE from ‘die-by’ frame) should report closer proximity to retirement (sooner retirement). Thus, we also hypothesize that the ‘Die-by’ SLE frame will reduce retirement intentions compared to the ‘Live-to’ frame.H2: The ‘Die-by’ frame will decrease RetirIntent compared to ‘Live-to’ frame.Finally, ‘retirement planning’ is purposefully assessed before randomized treatment assignment (live-to/die-by). Therefore, because ‘retirement planning’ is domain-specific to ‘life after retirement,’ we assume that greater planning for this post-work time will reduce the negative effect of lowered SLE on FTP as a ‘global’ construct. That is, we hypothesize that RetirPlan will operate by interacting with the ‘Die-by’ frame by buffering its negative effect on FTP.H3: RetirPlan will interact with the ‘Die-by’ frame on FTP, such that high RetirPlan will reduce the negative effect.In contrast, we expect that the domain-specificity of ‘retirement planning’ and ‘retirement intention’ to conjoin, so that the hypothesized main-effect of ‘sooner retirement’ will be strengthened by more retirement planning . That is, we hypothesize that RetirPlan will operate by interacting with the ‘Die-by’ frame by strengthening its negative effect on retirement intentions.H4: RetirPlan will interact with the ‘Die-by’ frame on RetirIntent, such that high RetirPlan will strengthen the negative effect.In addition to hypothesizing SLE-manipulation effects via response processes, that is, on mean-level changes of our focal outcomes, we further explore the nomological pattern of correlates with relevant external variables, in particular, the association between RetirPlan and RetirIntent.N = 207 participated in ‘2 × 1’ between-subjects experimental design. SLE served as between-subjects factor via the framing manipulation (‘Live-to’ vs. ‘Die-by’). Inclusion criteria required that participants be presently working full-time (>30-h/week) for appropriate study of focal retirement variables. Amazon’s Mechanical Turks crowdsourcing labor market was used for participant recruitment. Participants received nominal compensation ($1.50) for completion of the online experiment-questionnaire, with an additional incentive offered via a $100 lottery drawing.A sample of Figure 2 below.This study was carried out in accordance with the recommendations of ‘international-ethical human study guidelines’ in accordance with the Declaration of Helsinki. The protocol was approved by the authors’ academic institutional review board (Georgia Institute of Technology). All subjects were given written, informed consent prior to participating in the online questionnaire. Additional educational materials on retirement preparation was provided for interested participants on completion of the questionnaire. See, Subjective life expectancy was assessed on a slider-scale with probabilities ranging from 0 to 100. Participants were instructed to use the slider to indicate their probability of living to ages 65, 75, 85, and 95-years old. Instructions varied with experimental condition, so that participants assigned to the ‘live-to’/‘die-by’ condition read, “Using the slider scale below, please indicate the likelihood you currently believe you will ‘live-to’/‘die-by’ 65 / 75 / 85 / 95-years of age. Cronbach’s internal reliability was estimated at α = 0.83.2. Participants indicated their agreement with how “true” each item described themselves using a 7-point Likert-type scale from 1 (Very untrue) to 7 (Very true). A sample item is, “(1) Many opportunities await me in the future.” Cronbach’s internal reliability was estimated at α = 0.91.Carstensen and Lang’s  original 10-item instrument was administered to assess FTPRetirIntent was assessed as a four-item composite of intended retirement age across four domains: ‘current employer,’ ‘career,’ ‘occupation,’ and ‘general workforce.’ Responses were recorded on a continuous scale. A sample item is, “I intend to retire at age 66–95.” Cronbach’s internal reliability was estimated at α = 0.89.RetirPlan was assessed with t-tests were conducted to compare means on FTP and RetireIntent. High- and low-RetirPlan groups were generated with median-splits on the RetirPlan variable in order to test interaction-hypotheses in our experimental design. In order to test for interaction-effect hypotheses, SPSS’s univariate-general linear model procedures were implemented with SLE-experimental condition (Live-to vs. Die-by) and RetPlan groups (high vs. low) entered in a full-factorial model. Type III sums-of-squares were utilized in parameter estimate calculations. Significant-interactions were further inspected with simple pairwise-comparisons of experimental conditions within each level of the data-driven moderator (low/high RetirPlan). Bonferroni’s corrections were implemented for managing Type-1 error rates from multiple-comparison tests. The observed-score general linear model was selected over latent-variable approaches3 due to its parsimony, including lower sample size requirements and our focus on competing predictions between specific relations/localized pathways .Preliminary response-quality screening, missing-data assessment, and hypotheses testing was conducted in SPSS v23. To test main-effects across our randomly allocated experimental groups, independent-sample Table 2 below. SLE-manipulation effects were successfully replicated, such that participants in the ‘Live-to’ condition reported a 10.02% greater chance of living to age 80 compared to the ‘Die-by’ condition . Notably, we observed a differential correlation between SLE-FTP that is consistent with theoretically competing age-predictions for FTP, r = 0.40(Live)/-0.30(Die). That is, the directional discrepancy may be retraced to ‘younger’ and ‘older’ populations that were the foci for early-child development and gerontological theories predating the lifespan framework’s practical integration.Summary sample characteristics are reported in M = 44.74, SD = 11.15) compared to the ‘Live-to’ condition ; t(205) = 3.44, p = 0.03, Cohen’s d = 0.48.Main effect hypotheses received mixed-support. Supporting H1, participants assigned to the ‘Die-by’ condition reported significantly lower FTP  compared to the ‘Live-to’ condition , but the effect was not significant; t(205) = 0.52, p = 0.31, Cohen’s d = 0.07. Main effect results are summarized in Figure 3 below as standardized reductions of the ‘Die-by’ frame compared to the ‘Live-to’ frame.For H2, participants assigned to the ‘Die-by’ condition also reported lower RetirIntent  = 3.01, p = 0.044. A simple-contrast estimate indicated significance in the hypothesized direction, such that the ‘Die-by’ frame’s negative effect on FTP was limited to low RetirPlan . In contrast, for high RetirPlan, the ‘Die-by’ frame had no effect on FTP . FTP’s marginal-means used for simple-contrasts are illustrated in the left-panel of Figure 4.Interaction hypotheses for RetirPlan were generally supported. Specifically, regarding H3 for RetirPlan’s buffering effect of the ‘Die-by’ frame on FTP, results indicated a significant interaction, F = 3.40, p = 0.03. A simple-contrast estimate indicated significance in the hypothesized direction, such that the ‘Die-by’ frame’s negative effect on RetirIntent was significant for high RetirPlan . For low RetirPlan, an unexpected increase in RetirIntent was also observed, although it was only marginally significant . This unexpected cross-over interaction helps to explain the unobservable main-effect for RetirIntent, which would seem to-have been obscured. RetirIntent’s marginal means used for these calculations are illustrated in the right-panel of Figure 4.Regarding H4 for RetirPlan’s enhancing effect of the ‘Die-by’ frame on RetirIntent, results indicated a significant interaction, Table 3 below summarizes descriptive statistics for all simple-pairwise comparisons. Specifically, each row of Table 3 lists the mean differences of all 2 × 2 combinations of low/high RetirPlan for FTP and RetirIntent with Bonferroni-adjusted 95% confidence intervals [CIs] reported.Finally, r = 0.19, p = 0.04. In contrast, for the ‘Die-by’ condition, a significant negative correlation was observed, r = -0.22, p = 0.03. The latter finding supports views of RetirPlan as a mechanism that accelerates workforce withdrawal (earlier retirement).In addition to mean-level effects on focal outcomes, the patterning of external correlates was different across ‘Live-to’ and ‘Die-by’ conditions. Specifically, for the ‘Live-to’ condition, a significant positive correlation was observed between RetirPlan-RetirIntent, t-test was re-conducted, excluding the j = 3 reverse-scored FTP items. There remained a significant difference for the FTP subscore in the hypothesized direction across ‘Live-to’  and ‘Die-by’  conditions; t(205) = 3.84, p = 0.03, Cohen’s d = 0.54. The pattern of interactions also remained unchanged in terms of direction and statistical significance.Given the patterning of external correlates, we considered additional explanatory causes for our experimental findings to support substantive interpretation of RetirPlan’s moderation effects. First, we considered if the FTP main effect may be attributable to its sensitivity to ‘careless responding’ based on mix-scored items (positive- / reverse-scored). An independent-samples t-test was re-conducted separately for the low-Time and high-Time groups. The non-significant difference for RetirIntent remained for both the low-Time  and high-Time  groups. The pattern of interactions also remained unchanged in terms of direction and statistical significance. We summarize our findings and briefly elaborate their implications for future work-retirement integrative research in the section “Discussion” below.Because our RetirIntent outcome comprised no reverse-scored items, we examined if the non-significant main effect would hold across a median-split on survey duration . An independent-samples The current study identified a construct, FTP, with disproportionate external validity evidence in a shifting applied context, specifically, workers and retirees. The mono-operation bias inherent to two contrarian theoretical origins of FTP across functionally dissimilar populations (workers and retirees), thus, vesseled weak validity evidence and poor verisimilitude . We evalr = 0.40) and negative in the ‘Die-by’ condition (r = -0.30). For hypotheses, H1 was supported with a significant negative effect of the ‘Die-by’ condition on FTP. The finding generally support inveterate lifespan theories that postulate age-related decline in FTP . Furthermore, we received support for FTP’s hypothesized interaction with RetirPlan, such that participants reporting more RetirPlan were unaffected by the ‘Die-by’ framing’s negative impact on FTP.Summarizing our findings, first, we successfully replicated the SLE-framing manipulation and observed the opposing relations between SLE-FTP, which was positive in the ‘Live-to’ condition  revealed an unexpected cross-over effect, providing some indication for obscuring the hypothesized main effect. Contrast analyses supported our hypothesis for RetirPlan’s strengthening of the negative ‘Die-by’ effect. On the other hand, we also observed an unanticipated positive effect of the ‘Die-by’ condition on RetirIntent for low-RetirPlan. The cross-over effect could be interpreted as a delay of RetirIntent when shorter SLE is combined with inadequate preparation. This interpretation is consistent with recent empirical findings that indicate FTP’s dual-mechanism of prompting planning and inducing sunk-cost biases/escalations of commitment with current conditions. That is, in the same way that job characteristics may enhance occupational FTP, contracted lifespans may induce avoidance of retirement’s unknown, if ill-prepared, event space.Taken together, we hope that the current findings hold practical import for bettering the practice of career counseling and job crafting amid longer working lives. For example, practical implications of this experimental study intersect the age-integration of social institutions and domain-integration of work-life balance. To this end, results should inform ongoing areas of research on related lifespan concepts – For example, ‘active aging’ , ‘aging-later RetirIntent. Although exploratory analyses were conducted to increase substantive inference from our findings, important limitations are noteworthy for cautious interpretation. For example, from a response-modeling perspective, our focus on latent-structural relations limited our examination of potential response biases to classical approaches, such as conditioning on standard- and reverse-scored items . The current authors concur with MK designed the experiment, recruited and ran participants, and conducted primary analyses. SE provided advanced-methodological consultancy and conducted expository write-up.The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


Scientific Reports 10.1038/s41598-017-03497-z, published online 16 June 2017Correction to: The original version of this Article contained a typographical error in the spelling of the author Zoran Jakšić, which was incorrectly given as Zoran Jašić. This has now been corrected in the PDF and HTML versions of the Article.


An underestimated FDR can lead to unexpected false discoveries in the follow-up validation experiments. This issue has not been well addressed in literature, especially in the situation when the permutation procedure is necessary for p-value calculation.q-value. In practice, it is usually necessary to calculate p-value by a permutation procedure. This was also considered in our adjustment method. We used simulation data as well as experimental microarray or sequencing data to illustrate the usefulness of our method.We proposed a statistical method for the conservative adjustment of q-value, particularly in the situation that the proportion of differentially expressed genes is small or the overall differential expression signal is weak.The conservativeness of our approach has been mathematically confirmed in this study. We have demonstrated the importance of conservative adjustment of  Since the introduction of microarray technology, the concept of false discovery rate (FDR) and its related statistical methods have been well developed [q-value is a statistical method for the estimation of FDRs [Microarray and sequencing technologies have been widely used in genome-wide expression experimental for biological and medical studies –5. Aftereveloped , 7. q-va of FDRs . It has q-value is an estimation method, it is possible that it underestimates true FDRs. Then, an undesirable consequence is that many genes detected with low q-value cannot be validated in a follow-up experiment. Therefore, it is important to avoid underestimations of FDRs. However, there is a lack of statistical method to address this important issue. Furthermore, in many situations, q-values are calculated based on p-values that are evaluated based on a permutation procedure . Then, these p-values are also estimated. This increases the complexity of FDR underestimations. It is necessary to adjust the underestimation of FDRs in a comprehensive approach.Since q-value, which is one of the most frequently used procedure for estimating FDRs. We first reviewed the concepts of multiple hypothesis testing, FDR and q-value. Then, we introduced a concept of conservative adjustment. Based on the theory of rank-based quantiles, we described a non-parametric approach and we conducted simulation and application studies to illustrate its usefulness.In this study, we proposed a statistical method for the conservative adjustment of m total hypotheses, with a threshold for declaring significance, we have R hypotheses rejected (claimed positives). If we know the underlying nature of each variable, then its related hypothesis can be classified as either a true null or a false null (termed gold standard). If this information is provided, then we know the numbers U, V, W and S in Table U, V, W and S represent the numbers of true negatives, false positives, false negatives and true positives, respectively. However, in practice, the gold standard information (or the underlying nature) is usually unknown. Therefore, it is statistically interesting to evaluate U, V, W or S (or their combinations).When a large number of variables are simultaneously screened, it is usually a situation that a mixture of true and false null hypotheses is presented. . Statistically, this is a situation of multiple hypothesis testing (MHT). An illustrative summary is shown in Table V [p-value threshold), it is usually difficult to claim statistical significance. The false discovery rate (FDR) has been proposed to empirically evaluate the proportion of false positives among the claimed positives: V/R [q-value [The traditional family-wise error rate (FWER) provides a strong control on V . Since Fves: V/R . The con[q-value  is one oq-value method that is a statistical procedure for FDR evaluation. Suppose T is the test statistic and genes with test scores greater than or equal to t will be claimed significant. Let α be the p-value at t, and let f(t) and s(t) be the expected numbers of false positives and significant genes, respectively. Storey and Tibshirani [T=t could be approximated by the proportion of false positives: Storey and Tibshirani  proposedbshirani  proposedm is the total number of genes and π0 is the proportion of true null hypotheses  estimates mPr(T≥t).where p-value of t. Therefore, a permutation based p-value α. Then, the permutation version of q-value is In practice, it can be difficult to obtain the theoretical p-values, it is possible to obtain underestimated results. We have proposed a conservative adjustment of permutation p-values to address this issue [When the permutation method  is used is issue .q-value is actually a combination of three estimates: π0,α,#{T≥t}) for mPr(T≥t). must still be empirically estimated even when α can be theoretically determined. To address the underestimation of FDR , our solution is to consider a conservative adjustment of q(t).From above, notice that q-value for T=t can be defined as: α(t)=Pr(T≥t|H0) and γ(t)=Pr(T≥t).According to Eq. , the thea)% conservative adjustment of q(t) as an estimator We define the 100(1−Our solution for a0)(1−a1)(1−a2)≥1−a, then we claim that π0,α(t)>0 and If , it is difficult to calculate a lower confidence limit of π0 (the proportion of true null hypotheses) is usually conservatively estimated in practice [π0 [convest [π0. . According to our experience, it is reasonable to assume that Due to the identifiability issue, practice . Many stice [π0 . Among t[convest  is a welconvest method (or other similar methods). Then, we discuss When q-values associated with observed test scores can be evaluated, and it is difficult to accurately evaluate the q-values for unobserved test scores. Here are the mathematical details. With a test statistic T, we can obtain m observed test scores {T1,T2,…,Tm} for m variables (genes). An empirical estimate of γ(t)=Pr(T≥t) is δ(TRUE)=1 and δ(FALSE)=0. If we sort these m test scores in an increasing order: T(1)≤T(2)≤…≤Tm) can also be considered as the results from a particular permutation, they are generally included in the pool of permuted test scores. Based on r permuted test scores: α(t)=Pr(T≥t|H0) is If the theoretical distribution of rocedure  for evalr permuted test scores in an increasing order: If we sort these p-value of T=t will be t. Therefore, the permutation p-value of T=t can also be considered as a rank-based quantile estimator.Note that, if the observed test scores are not included in the pool of permuted test scores, then the permutation Since atistics , we haveatistics . Such anq-value method underestimates the true false discovery rate, we conducted a simulation study. We choose the normal distributions for simulating expression data and the Student’s t for differential expression. In this way, we could evaluate the true false discovery rate theoretically.To understand how likely the N for both groups. Then, we considered four scenarios. For the first simulation scenario, the proportion of differentially expressed genes was 10%; the expression data for differentially expressed genes were simulated from N and N for the first and the second sample groups, respectively (Δ=1). For the second simulation scenario, the proportion of differentially expressed genes was 10%; the expression data for differentially expressed genes were simulated from N and N for the first and the second sample groups, respectively (Δ=2). For the third simulation scenario, the proportion of differentially expressed genes was 20%; the expression data for differentially expressed genes were simulated from N and N for the first and the second sample groups, respectively (Δ=1). For the last simulation scenario, the proportion of differentially expressed genes was 20%; the expression data for differentially expressed genes were simulated from N and N for the first and the second sample groups, respectively (Δ=2).Ten thousand genes were simulated for two sample groups with sample size 5 for each group . For non-differentially expressed genes, the expression data were simulated from t-test was used for differential expression analysis. To evaluate p-values, we performed all possible (126) permutations for each simulated gene and pooled all 1,260,000 permuted test scores together as the empirical null distribution.  For each scenario, we conducted 100 repetitions to understand the variations in the simulation results.For all different scenarios, the Student’s q-values). In summary, when the proportion of differentially expressed genes (1−π0) becomes smaller (from 20 to 10%), it is more likely to obtain underestimated FDRs; when the signal of differential expression (Δ) becomes weaker, it is more likely to obtain underestimated FDRs. Figure π0=15%) and overall moderate differential expression signals (Δ=1.5). Each curve is a comparison between the empirical FDRs  vs. theoretical FDRs (based on one repetition of simulation). Below 0.05 theoretical FDR, some curves can be observed under the diagonal line, which indicate that these empirical FDRs  are underestimated. Furthermore, among 100 repetitions, there is a considerable portion of empirical FDRs underestimated .The simulation results are shown in Fig. ler from 0 to 10%,q-value less than 10%. However, it may be surprising that much less genes can be confirmed after the RT-PCR validation.  This hypothetical situation would be an example of under-estimation of FDR.A conservative adjustment of false discovery rate (FDR) can be useful in practice, especially before the experimental validation of genes identified from a genome-wide expression study. For example, based on a microarray or RNA-seq study, one may want to validate a few genes with N and N for the first and the second sample groups, respectively. The expression profiles of non-differentially expressed genes were simulated from N for both groups. The Student’s t-test was used for differential expression analysis. To evaluate p-values, we performed all possible (126) permutations for each simulated gene and pooled all 1,260,000 permuted test scores together as the empirical null distribution.  The convest method proposed by Langaas et al. [π0, which was used in the q-value estimation procedure [q-value. Then, we considered a conservative adjustment. We set a0=0 , a=0.05). Figure q-values.To demonstrate the above situation, we conducted a simple simulation study. 10,000 genes were simulated for two sample groups. The sample size was 5 for each group. The proportion of differentially expressed genes was 10%. The expression profiles of differentially expression genes were simulated from s et al.  was usedrocedure . In thisWe applied our method to two genome-wide expression data sets. The first one was a microarray data set and it was collected for a diabetes study. It is well-known that differential expression signals are usually weak in diabetes studies. When the sample size is not relatively large, it is usually difficult to detect true differentially expressed genes. . It is interesting to understand the adjustment effects from our method for this scenario. The second one was a RNA sequencing (RNA-seq) data set and it was collected for a prostate cancer study. It is also well-known that differential expression signals are usually strong in cancer studies. Genome-wide expression data for different types of cancer have been increasingly collected in The Cancer Genome Atlas project . The cury-value is a specific FDR and the x-value is the related number of genes with the specific FDR). Our application results can also be summarized in term of this type of curve.In practice, it is not feasible to calculate the theoretical true FDRs. The curve of estimated FDR vs. number of identified genes is widely used to for a summary of differential expression analysis. . We set a0=0 . Figure q-values (estimated FDRs) can only be as low as slightly less than 0.2. There were only several genes with q-values around 0.2. However, after our conservative adjustment, Fig. q-values are greater than 0.8. This comparison implies that most genes were very likely noise genes and the detections of differentially expressed genes by low q-values could be very unreliable. The only gene with q-value less than 0.2 is a mRNA for CD20-like precursor. However, no literature was found to support its association with diabetes diseases.For the first microarray genome-wide expression data set for a type 2 diabetes study , there wt method ), \docump-values calculation. Therefore, there was no need for an adjustment of p-values (i.e. a1=0). Then, we set a0=0 . Figure q-values (estimated FDRs) can be extremely low and lots of genes can be detected. After our conservative adjustment, Fig. q-values almost overlaps with the curve of original q-values. This comparison implies that many genes were truly differentially expressed genes and the detection of these genes by low q-values could be highly confident. After checking literature for top ranked genes, many of them have been studied to be either directly or indirectly related to prostate cancer or general cancer diseases (details not shown).For the second RNA-seq genome-wide expression data set for a prostate cancer study , there wt method ) and a2t-test calculations. We set a0=0 . Figure q-values (estimated FDRs) can only be as low as approximately 0.4 (about 20 genes). However, after our conservative adjustment, Fig. q-values are greater than 0.5. This comparison implies that more top ranked genes were likely noise genes. The only two genes with q-value less than 0.4 are mRNAs for ARNT and APCDD1. Although ARNT has been widely studied for its association with diabetes diseases, no literature was found to support the association between APCDD1 and diabetes diseases.For the third microarray genome-wide expression data set for a pancreatic-islet study , there wt method ), \documπ0 estimation. Since the estimators for π0 are usually conservative , we do q-value could be achieved. Furthermore, an independence assumption is required. It is well-known that genes work with each other during molecular and cellular processes. It would be an interesting future research topic to address the dependence among genes. Therefore, it will be our future research topics to investigate possible better upper/lower bounds for the conservative adjustment of q-value, as well as the impact of dependence on the conservative adjustment of q-value.A clear advantage of our approach is that there are rigorous mathematical theories to support it. Furthermore, no distribution assumptions are required for our conservative adjustment. However, the adjustment may be over-conservative. If we could further improve the control of upper/lower bounds , then less conservative adjustment of q-value, which is widely used to estimate false discovery rate (FDR) in practice. We provided a mathematical proof to confirm the conservativeness of our approach. We conducted simulation studies to understand how likely the q-value method would underestimate FDRs. From our simulation results, both the proportion of differentially expressed genes and the overall differential expression signal were two important factors. When both of them were relatively small/weak, it was likely to identify genes with underestimated FDRs. Our first application was based on a microarray diabetes study data set with relatively small sample size . Our third application was based on a microarray pancreatic islet study data set with relatively small sample size . The results were consistent with the conclusion from our simulation studies. Our second application was based on a RNA-seq prostate cancer study data set with relatively large sample size . According to the results, the conservatively adjusted q-values were close to the originally unadjusted q-values.In this study, we proposed a statistical method for the conservative adjustment of Mathematical Proof:X, Y and a constant c, Pr(XY≥c|X≥1)≥Pr(Y≥c) since Y≥c⇒XY≥c given X≥1 ({Y≥c} is a subset of {XY≥c} when X≥1).For any random variable q-values.R-functions for calculating conservatively adjusted π0; pvals is a list of permutation p-values based on r permutations; m is the number of permutation p-values .Notice that pi0est is a conservative estimate of


Titanium (Ti) has been used for long in dentistry and medicine for implant purpose. During the years, not only the commercially pure Ti but also some alloys such as binary and tertiary Ti alloys were used. The aim of this review is to describe and compare the current literature on binary Ti alloys, including Ti–Zr, Ti–In, Ti–Ag, Ti–Cu, Ti–Au, Ti–Pd, Ti–Nb, Ti–Mn, Ti–Mo, Ti–Cr, Ti–Co, Ti–Sn, Ti–Ge and Ti–Ga, in particular to mechanical, chemical and biological parameters related to implant application. Literature was searched using the PubMed and Web of Science databases, as well as google without limiting the year, but with principle key terms such as ‘ Ti alloy’, ‘binary Ti ’, ‘Ti-X’ , ‘dental implant’ and ‘medical implant’. Only laboratory studies that intentionally for implant or biomedical applications were included. According to available literatures, we might conclude that most of the binary Ti alloys with alloying <20% elements of Zr, In, Ag, Cu, Au, Pd, Nb, Mn, Cr, Mo, Sn and Co have high potential as implant materials, due to good mechanical performance without compromising the biocompatibility and biological behaviour compare to cp-Ti. Titanium (Ti) is a transition metal and element with the atomic number of 22. Ti has a lustrous finishing and characterized with silver colour, low density and high strength. It has a high ability to resist corrosion in various media such as sea water, aqua regia and chlorine . Ti is a−1/2. For β-Ti, beta phase was observed at room temperature after quenching, or sometimes even upon air cooling. Thus, it is ready for cold working (forming), and could be solution-treated, quenched and aged to give higher strength together with low ductility. However, its fatigue performance is poor. The tensile strength is about 1220–1450 MPa and fracture toughness of >50 MPa m−1/2.Ti is a dimorphic metal with two phase, α and β phase. α-Ti is hexagonal close-packed (hcp) crystal lattice, and β-Ti is body-centered cubic (bcc) lattice. During the processing, Ti exists as α-Ti when the temperature is lower than 883 °C. When Ti is heated exceeding 883 °C, the atoms in the hcp crystal lattice packed closer with each other and become β-Ti . α-Ti isIndeed, the commercially pure Ti (cp-Ti) is divided into four Grades from 1 to 4, according to the purity and the processing oxygen content . These dIn order to replace a missing tooth, a lot of materials, such as cobalt–chromium  and stainless steel, had been attempted to make an implant. The development of materials science and technology improved the materials for implant application. Nowadays, Ti becomes the most popular implant material due to its advantages.x) as film, is very stable and could separate the bulk Ti material from its surrounding. Thus, Ti has a high ability to resist the corrosion. The TiOx layer is typically around 3–10 nm thick that stably stayed onto the Ti surface, and the oxide film on the surface − + H3O+−) which incur negative charge on the surface, and the acidic type [Ti-OH2]+ incur positive charge for 2]+. Thus, surface treatment methods such as the acid etching could not only roughen the surface that provide a harbouring site for negatively charged osteoblast [2]+, which is hydrophilic and documented for biological activity enhancement [2]+ on Ti surface could be a good strategy for Ti-osteoblast bonding without the addition of bi-positively charged growth factors/proteins +, a storage of acidic medium is recommended which might be a tactic in the commercial product ‘SLActive’  that use an acidified saline  to preserve effectively the hydrophilicity and nanostructure [However, during the storage and the equilibrium nature of the oxide, the nano-oxide layer will become thicker due to the time and quality of exposed storage atmosphere . In suchtructure .In general, titanium is a good choice for intraosseous applications not only due to the biocompatibility, but also mechanically titanium could be processed and machined in a rapid manner such that the shapes and sizes could be easily controlled. Nevertheless, one of the disadvantages of the titanium could be the aesthetic problem since Ti is grey in colour, such that the dark colour would be seen through the thin mucosa if the soft tissue situation is not optimal. Ti also proceeded with some other drawbacks, e.g. low deformability and wear resistance, and high reactivity with the surrounding impurities (such as oxygen and nitrogen) at elevated temperatures , 27. MorOne of the tactics—alloying Ti with a variety of elements—might become viable to enhance some of these properties, such as increase the corrosion resistance, lower the modulus of elasticity, and improve the machinability. This is because the Ti-alloys properties are related to their respective phases/crystalline structures, such that by adding some alloying elements could stabilize certain phases. In addition, some metal substrates could combine with this oxide layer in order to prevent absorption and disintegration of coating , 31. ThuCommonly, alloy could broadly defined as ‘a mixture of a metal and another element(s) which has metallic bonding character’. In titanium, a lot of attempts, such as using Silver (Ag), Aluminium (Al), Copper (Cu), Iron (Fe), Vanadium (V) and Zinc (Zn), have been done in order to improve the material.In fact, as aforementioned, the titanium has three forms: α, β and α-β. With the addition of the alloying elements, the phase compositions could be adjusted and thus change the bulk Ti-alloy properties. For example, Al acts as α-phase stabilizers which could improve the strength and lower the weight of the alloy. V is β-phase stabilizer that could improve the ductility and formability. Thus, by adding of the Al and V, the temperature for α-phase transform to β-phase has also been changed to a range, i.e. both α and β phases exist in the temperature range. The most commonly used Ti-alloy in dental implant , 32 is T2O3 ceramic ball heads. However, Ti-6Al–4 V alloy has the disadvantage of low wear resistance [Ti–6Al–4V  is one of the most commonly used tertiary titanium alloy, that is usable as a biomedical implant. As the symbol implies, Ti–6Al-4V has the composition of 6 wt% aluminium and 4 wt% vanadium. In particular, Ti–6Al–4V alloy has a higher strength that could be used in various applications, such as for anchorage stems of femoral components, together with Co–Cr–Mo alloy or Alsistance , high elsistance  , is a neutral element when dissolved in Ti. Zirconium belongs to Group 4 (according to new IUPAC name) in the periodic table, which is the same as titanium and hafnium, have similar chemical structure and properties. Thus, they have been recognized as non-toxic and non-allergic. Zirconium is a transition metal with atomic number 40 and atomic weight of 91.22 amu. Being as a greyish-white lustrous metal, Zirconium has extremely high melting (1857 °C) and boiling (4409 °C) points. Zirconium has a great resistance to corrosion, which is similar to Titanium, and is therefore highly biocompatible  since th2), has started to use in dentistry, due to its biocompatibility and possesses the capacity to osseointegrate [et al. [et al. [Start from 1990s, the oxide form of zirconium, Zirconia  is lowered by increasing the amount of Zr  regardless α-Ti or β-Ti. Thus, the casting process could be easily facilitated. In addition, from the knowledge of casting titanium , reducinet al. [Recently, due to their good corrosion resistance and biocompatibility, binary Ti–Zr alloys have been developed for dental applications  which clet al.  demonstret al. , which iet al. [et al. [Kobayashi et al.  investig [et al.  has claiFor a long time, Indium (In) has been used in Pd- and Ag-based porcelain-fused-metal (PFM) applications. In fact, during the firing process of porcelain, the indium oxides film would formed on the metal surface, which could be served as a ‘bonding agent’ between metal and porcelain , 75. In Several studies on experimental Ti-In alloys has shown that Ti-In alloys were biocompatible. Further, with the addition of indium to Ti could improve clinical performance in terms of mechanical properties, corrosion resistance, and biocompatibility  of dentaet al. [For binary Ti-In alloys, Wang  found thet al.  has showet al. [et al. [Han et al.  has furt [et al.  that the [et al.  showed aet al. [et al. [in vitro cytotoxicity test which showed Ti–Ag alloys and cp-Ti alloys seem to be cytocompatible with each other. They recommend that, for dental application, Ti–5Ag and Ti–20Ag might be more suitable according to the surface passive film and cytotoxicity viewpoints.Oh et al.  reported [et al.  confirme2Cu was found. Thus, Ti–Cu alloys that have a nearly eutectoid composition was expected to have higher strength and less ductility than cp-Ti.Copper has been used for long time as dental casting alloys. Study  has showStudies , 81 reveMoreover, in terms of grindability, some Ti–Cu and Ti–Ag alloys had shown a better outcome than cp-Ti , 82. At per se belongs to Group 11 in the periodic table, same as silver and copper. Gold, similar to silver and copper, stabilizes β-phase of titanium to a lower temperature according to the binary equilibrium phase diagrams.Gold (Au), has been mainly used for dental prostheses in dentistry for a long time, particularly for casting, since gold has good corrosion resistance, suitable melting point, and could achieve appropriate mechanical properties by alloying . Gold pe3Au forms. Therefore, gold, as an alloying element for titanium, might be expected to positively affect on the mechanical properties and grindability of titanium, similar to silver and copper.According to the phase diagram, similar to the Ti-Cu and Ti-Ag alloys, the Ti-Au alloy has the eutectoid point with titanium. At the gold concentration of 15.3% the intermetallic compound Tiet al. [Hwang et al. [et al. [et al. [ca. 1 at% Ag, Au, Pd and Pt) vs. commercial Ti-6Al-7Nb in the fluoridated artificial saliva. Takahashi et al. [Moser et al.  reported [et al.  utilized [et al.  utilizedi et al. , 88 alsoet al. [et al. [Lee et al.  studied  [et al. , they exet al. [Manganese (Mn), is one of the trace elements that could be found in human body which has lower toxicity than Al and V. Mn can be added to tri-calcium phosphate by doping and become a bioceramic which showed a good cell compatibility . In a reet al.  has inveet al. [−, the Ti–20Cr alloy had a greater corrosion resistant than cp-Ti. In addition, according to phase diagram, alloying high Cr content (46%) to Ti could reduce the liquidus temperature from the high melting point of cp-Ti (1670 °C) to a minimum (1410 °C), similar to Ti–Zr as previously discussed.Chromium (Cr), is well-known in dentistry due to the Co–Cr application that could control the anodic activity of the alloy, and could passivate Ti . Takemotet al.  reportedet al. [Hsu et al.  reportedet al. [et al. also studied the fractography of the specimen, and fractographs of SEM showed that the Ti–10Cr alloy has: (i) coarse cleavage facets in the fractured surface; and (ii) some terrace-type morphology. In addition, the Ti–20Cr has shown to be ductile, such that it has even 460% more elastic recovery capability than cp-Ti, whereas other composition of Ti–Cr alloys did not exhibit such properties. As shown in their unetched optical micrographs, large number of slip bands was observed on the surfaces of the Ti–20Cr alloy. That said, slippage dislocations are the reasons of the Ti–20Cr alloy deformation. Therefore, if Ti–Cr alloys are needed to be use for prosthetic dental applications, many properties such as mechanical properties and deformation behaviour should be further studied.Ho et al.  investiget al. [According to Bania  and Ho eet al. , a minimet al. , and eveet al. , due to et al. . Indeed,et al.  at low cV) of the Ti–Sn alloys, e.g. 30 wt% Sn showed a high hardness value of 357 HV. To illustrate the CAD/CAM processability, grindability of the alloys was also studied [Ti–Sn has been found nontoxic and nonallergic . Thus, T studied . In factCobalt (Co), is a widely accepted biocompatible element  might haet al. [et al. [6Ge5 and subsequently TiGe2 would be preferentially nucleated and crystalized [Lin et al.  has stud [et al.  has repo [et al. , for thestalized . For Ti–stalized  i.e. coret al. and Song et al. [Titanium and its alloys has the properties of attracting cells, including osteoblasts and bacteria, due to various properties such as surface charge , existeng et al. , 108 evag et al.  and biomg et al.  could beg et al.  and Ti–6g et al.  have beeBinary Ti alloys, in particular to Ti––Zr, Ti–In, Ti–Ag, Ti–Cu, Ti–Au, Ti–Pd, Ti–Nb, Ti–Mn, Ti–Cr, Ti–Mo, Ti–Sn and Ti–Co, with the alloying components <20% has a high potential as implant materials due to good mechanical performance without compromising the biological behaviour compare to cp-Ti. Further investigation on the inherent anti-biofilm properties could be a future topic for these alloys.Conflict of interest statement. None declared.


Compressed sensing (CS) has been applied to accelerate magnetic resonance imaging (MRI) for many years. Due to the lack of translation invariance of the wavelet basis, undersampled MRI reconstruction based on discrete wavelet transform may result in serious artifacts. In this paper, we propose a CS-based reconstruction scheme, which combines complex double-density dual-tree discrete wavelet transform (CDDDT-DWT) with fast iterative shrinkage/soft thresholding algorithm (FISTA) to efficiently reduce such visual artifacts. The CDDDT-DWT has the characteristics of shift invariance, high degree, and a good directional selectivity. In addition, FISTA has an excellent convergence rate, and the design of FISTA is simple. Compared with conventional CS-based reconstruction methods, the experimental results demonstrate that this novel approach achieves higher peak signal-to-noise ratio (PSNR), larger signal-to-noise ratio (SNR), better structural similarity index (SSIM), and lower relative error. MRI prok-space sampling [k-space sampling pattern to shorten the sampling time, increase the imaging speed, and reduce the motion artifacts. However, aliasing artifacts may occur in these sampling methods. The high-frequency part contains less image information than the low-frequency part. Hence, by using undersampling patterns, the information about details is lost in the reconstructed images. Besides, the substantial aliasing artifacts appear incoherent. In case that the sampling ratio is extremely low, it is almost impossible to remove the significant aliasing artifacts from real signals [k-space data. The discrete wavelet transform (DWT) is widely applied in CS-MRI, but it is sensitive to shift, lacks information about phase, and has poor directionality [ℓ1-norm regularization. These approaches, such as conjugate gradient [O(1/k2), where k is the number of iterations.In recent years, a variety of techniques have been proposed to enhance the quality of MRI, which can be roughly classified into three categories : incohersampling , spiralssampling , radial sampling , and Gausampling ) takes a signals , 8, 9. Fionality . Waveletionality . The staionality . Similarionality , 19. Thegradient , two-stegradient , 22, andgradient , have begradient  can be eTo enhance the image reconstruction quality and reduce the reconstruction artifacts, in this paper, we propose a novel reconstruction scheme, which combines CDDDT-DWT with FISTA. Although dual-tree complex wavelet transform has also been exploited in the literature , its dirThe remainder of this paper is organized as follows. φh(t) and φg(t) and four distinct wavelets φh,i(t) and φg,i(t), where φh,1(t) is an offset from φh,2(t) by one-half and φg,1(t) is an offset from φg,2(t) by one-half. One pair of wavelets φh,i(t) and φg,i(t) form an approximation of the Hilbert transform pair [φg,1(t) ≈ ℋ{φh,1(t)}, φg,2(t) ≈ ℋ{φh,2(t)}.The CDDDT-DWT is an overcompleted discrete wavelet transform that combines double-density DWT  with duaorm pair ; namely,L0L0, H0H0) and 16 high-frequency subbands  to describe the details of the recovered image.Two-dimensional (2D) double-density dual-tree DWT includes 2D real double-density dual-tree DWT and 2D complex double-density dual-tree DWT. The former is constructed from two oversampled 2D double-density DWT in parallel, which is redundant by a factor of two. Lp and Hp make up the filter banks of the first-level decomposition, where Lp represents a scale filter and Hp depicts eight wavelet filters, while Lw and Hg denote the filter bank structures of the second-level decomposition. Each level generates four low-frequency subbands  and 32 high-frequency subbands  through 2D CDDDT-DWT transform. Similar to other wavelet transforms, the redundant transform is achieved by recursively applying low-frequency subbands to complete the decomposition of each level. For each pair of subbands, CDDDT-DWT takes their summation and difference to produce the 32 oriented wavelets, describing a total of 16 main directions. Besides, each main direction contains two distinct wavelet representations, which indicate the real part of a complex-valued 2D wavelet function and the imaginary part, respectively [The latter is formed by utilizing four oversampled 2D double-density DWT in parallel to the input image. The filter bank structure of this transform can be obtained by extending the one illustrated in ectively .x denotes the fully sampled image, y is the k-space data acquired from a MR scanner, and ξ (ξ > 0) is a parameter appropriately chosen based on the noise level, which controls the difference between the object image and the reconstructed one. Fu is the undersampled Fourier transform in MRI. ϕ(·) is called the regularization function in the transform domain, which is generally nonsmooth. This optimization problem can be potentially solved by total variation- (TV-) based approaches [τ is a positive regularization parameter. To solve (ℓ0 “norm” (‖x‖0 = |{i : ≠0}|) is chosen as the regularization function. ϕ(x) = ‖x‖0 especially provides the simplest way to enforce the sparsity:x can be sparsely represented in this selected domain Φ. Here, Φi  is the 16 high-frequency subbands of CDDDT-DWT, which serves as a new sparse basis. However, the solution of  = (1/2)‖Fux − y‖22 represents the quadratic term, which is a convex function with Lipschitz continuous gradient, and g(x) = τ‖Φix‖1, i = 1,2,…, 16 is a nonsmooth convex regularizer.Since  convex,  can be c convex,  to effec convex, , f(x) = zk, we can get the gradient of f(x) at zk byFuT(Fuzk − y) denotes the gradient of f(x) at the given point zk, which is a specific combination of the previous estimate values xk, xk−1. The original FISTA based on wavelet transform has been well studied in the literature with a backtracking step size or a constant step size ρ, both of which can provide an improved global convergence rate of O(1/k2) [The FISTA algorithm is applied to solve the optimization problem of . For a g O(1/k2) . For simxg, we can getwh is the new wavelet coefficients, which can be adjusted by the proximal forward-backward iterative scheme [ℓ1-norm is nonsmooth, it is separable and CDDDT-DWT has the characteristics of tight frame. It is known that the shrinkage thresholding function with threshold T is utilized to obtain the modified wavelet coefficients wh′:Applying the sparsity transform CDDDT-DWT to a local optimal image e scheme  to catchxk is updated byThe recovered image i−1,  i = 1,2,…, 16) is applied by the synthesis filter bank structure, constituted by inverse order of the analysis filter bank [β is a threshold relaxation factor to adjust T, which optimizes T and reduces the calculation time. When the stop condition T ≤ ϵ is satisfied, we obtain the optimal solution of  for solving  is depick-space data of the axial brain are acquired by a 3T GE MR750 scanner using fast spin echo sequence . The spine MR data are a fully sampled k-space data obtained by a 3T GE MR750 system with FRFSE sequence . For the sake of brevity, the size of all testing images is scaled to 256 × 256. All experiments are performed using MATLAB 2014b on a desktop computer with a 3.2 GHz Intel core i5-4460 CPU.To evaluate the performance of the proposed reconstruction algorithm, we implement the complex double-density dual-tree wavelet and conventional wavelet using the software in , 29. Thek-space pattern and radial undersampling pattern are used to undersample the fully sampled k-space raw data. For most of MR images, the k-space signal with a large magnitude is generally localized in the central part. k-space and less high-frequency signals in the peripheral region of k-space. The radial undersampling pattern displayed in Gaussian random In this work, FISTA based on three different sparsity transforms is utilized to solve the optimization problem of . These tk-space data. In the axial brain experiment, we set the optimal parameters ϵ = 0.001, ρ = 1, β = 0.9, and T = 0.0095, while parameters in the spine experiment are set as ϵ = 0.0001, ρ = 1, β = 0.9, and T = 0.01. For different testing datasets, the tuning parameter τ is set to different values and the total number of iterations in all the cases is set to 120. Additionally, in both simulation and experiments on in vivo data, we add Gaussian white noise to simulate a realistic environment. For simplicity, the same standard derivation σ = 0.01 is used.We first conduct the experiment on the Shepp-Logan phantom image shown in x denotes images reconstructed from fully sampled data, M and N are the number of rows and columns in the input image, respectively. MAX means the maximum possible pixel value in the input image data, and The peak signal-to-noise ratio (PSNR), signal-to-noise ratio (SNR), structural similarity (SSIM) index, and relative error (Rel.Err) are used to evaluate the FISTA-CDDDT recovery performance. The PSNR is calculated using the following equation:The SNR is defined asp and q are the various local windows from the same local window in the two different images to be compared. μp and μq are the mean of p and q, respectively, while θp and θq represent their variance. θpq is the covariance of p and q. c1, c2, and c3 are three variables used to increase the stability of the results. Both SSIM index and SNR have the same criterion as PSNR; that is, the reconstruction quality is directly proportional to the value of the metrics.The definition of the SSIM index is given byThe Rel.Err is defined asNote that, for all the figures in this part, various approaches are labeled by different colors below the images. The green dotted lines mean FISTA-DWT, the pink dotted lines denote FISTA-CDT, and the blue lines represent the proposed FISTA-CDDDT.ℓ1-norm based CS techniques may result in poor performance with substantial artifacts. Consequently, CS methods cannot guarantee the quality of the reconstructed image at low sampling ratios.For further analysis, the PSNRs of the reconstructed images using different methods are plotted at various sampling ratios. It is clear from k-space data. The k-space data from each coil are reconstructed separately, and the final image is generated by the sum-of-square method [Figures e method .Note that the PSNRs of reconstructed axial brain images using radial sampling mask by FISTA-DWT, FISTA-CDT, and FISTA-CDDDT are 33.99 dB, 37.58 dB, and 38.87 dB, respectively. The magnified images are shown in Figures σ = 0.01. These results further demonstrate that the proposed FISTA-CDDDT is superior to the other two methods, because it exhibits the highest SNR, best SSIM index, and lowest Rel.Err.Figures Considering the superiority of CDDDT-DWT in preserving edges and maintaining higher directional selectivity, the proposed reconstruction approach combines the CDDDT-DWT with FISTA to produce better recovery results with a faster convergence rate. Although the ISTA and TwIST can be integrated with CDDDT-DWT as well, both of them were designed for simple regularization problems. Besides, they have some drawbacks that cannot be ignored. ISTA based on the operator-splitting strategy is a promising method, which has been successfully used in signal recovery. However, it belongs to the first-order algorithm that converges quite slow. As a variant of ISTA, TwIST is also an iterative thresholding algorithm, which is not guaranteed to converge globally. In contrast, FISTA has a faster convergence rate and better reconstruction accuracy, as proved in .l1-norm regularization is studied in this work. Furthermore, FISTA-CDDDT first uses proximal forward-back optimization to approximate the linearized function f(x), then applies shrinkage thresholding function to solve the minimization problem due to the separable characteristics of l1-norm, and finally adopts the specific linear combination of xk and xk−1 to smartly select the search points. Besides, the threshold relaxation technique can further reduce the computational cost. Consequently, our method can gain a more accurate solution with a dramatically improved complexity of O(1/k2).It is worth noting that Zhu et al.  designedIn this paper, we develop a new image reconstruction method for CS-MRI based on complex double-density dual-tree wavelet transform. The filter bank structure of the CDDDT-DWT is explored. This novel approach has been applied to Shepp-Logan phantom and axial brain and spine image reconstruction and compared with two popular methods, namely, FISTA-DWT and FISTA-CDT. The reconstructed results demonstrate that our scheme improves the PSNR and SNR as well as SSIM index and reduces the reconstructed artifacts significantly. In both simulation and experiments on in vivo data, we use the FISTA as the reconstruction algorithm. However, it can only solve the unconstrained minimization problems. An algorithm that can solve both unconstrained and constrained convex optimization problems will be studied in the future work.


B-matrix. The proposed error bounds are better than some existing results. The advantages of the results obtained are illustrated by numerical examples.In this paper, new error bounds for the linear complementarity problem are obtained when the involved matrix is a weakly chained diagonally dominant  LCP) is to find a vector LCP has various applications in the free boundary problems for journal bearing, the contact problem, and the Nash equilibrium point of a bimatrix game [A linear complementarity problem  matrix if A is diagonally dominant, i.e.,A of the form A matrix wcdd) B-matrix if it can be written in the form wcdd matrix whose diagonal entries are all positive.A matrix et al. [M is a B-matrix: Let B-matrix with the formGarcía-Esnaola et al.  gave theet al. [B-matrix with the form To improve the bound in , Li et aet al.  presentementclass2pt{minimM is a weakly chained diagonally dominant (wcdd) B-matrix, Li et al. [wcddB-matrix with the form Recently, when i et al.  gave a bmentclass2pt{minimM is a B-matrix since a B-matrix is a weakly chained diagonally dominant B-matrix [This bound in  holds whB-matrix .Now, some notation is given, which will be used in the sequel. Let M is a wcddB-matrix. Numerical examples are given to verify the corresponding results in Section The rest of this paper is organized as follows: In Section M is a wcddB-matrix. Firstly, several lemmas, which will be used later, are given.In this section, some new upper bounds for Letbe a wcddB-matrix with the formwhereis defined as , an, an8), aesult in  holds frB-matrix is also a wcddB-matrix, then by Theorem Since a Letbe aB-matrix with the formwhereis defined as (Thenwhereis defined as in Theorem mentclass2pt{minimmentclass2pt{minimWe next give a comparison of the bounds in  and 5)  5) as foLetbe a wcddB-matrix with the formwhereis defined as , an, anLet\ in Theorem (ii)n is very large, one needs more computations to obtain these upper bounds by (When ounds by  than by ounds by .In this section, we present numerical examples to illustrate the advantages of our derived results.B-matrices in [Consider the family of rices in :\documeentclass1pt{minimaentclass1pt{minimawcddB-matrix in [Consider the atrix in :\documeM is a weakly chained diagonally dominant B-matrix, which improve some existing results. A numerical example shows that the given bounds are efficient.In this paper, we present some new upper bounds for


Nature Communications 10.1038/s41467-017-02269-7; published online 28 February 2018Correction to: The original PDF version of this Article had an incorrect volume number of ‘8’; it should have been ‘9’. This has been corrected in the PDF version of the Article. The HTML version was correct from the time of publication.


The paradigm of the rubber hand illusion was applied to a shadow to determine whether the body-shadow is a good candidate for the alternative belonging to our body. Three kinds of shadows, a physical hand, a hand-shaped cloth, and a rectangle cloth, were tested for this purpose. The questionnaire results showed that both anatomical similarity and visuo-proprioception correlation were effective in enhancing illusory ownership of the shadow. According to the proprioceptive drift measurement, whether the shadow purely originated from the physical body was a critical factor in yielding the significantly positive drift. Thus, results demonstrated that the shadow can distort illusory ownership with the rubber hand illusion paradigm, but the proprioception was clearly distorted only when the body-shadow was purely applied. This implies the presence of special cognitive processing to discriminate the self-body shadow from the others. There is a well-known and old game for children called “ka-ge-fu-mi” in Japan wherein a chaser (tagger) tries to step on the shadow of the others (resulting in switching of roles in the game). In this imaginary world, touching the body-shadow is identical to touching the physical body. Similar beliefs, such that an attack to the body-shadow is equivalent to physical body, have been found in various parts of the world, suggesting a special involvement with the body-shadows in cognitive processing.Although seeing a shadow of an object is found to have a positive effect on recognizing the object accurately in many studies e.g.,  or even Therefore, it is probable that the body-shadow is processed as a substitution or extension of the physical body casting that shadow, but it is so difficult to assume that the participants in the related experiments felt their shadow just as a part of the body amid the relevant perceptual distortion. In other words, the related studies have demonstrated that the body-shadow can affect the cognitive processing in a level of body-schema but not in a level of body image . This seAn obvious feeling we naturally have for each part of the self-body is called body ownership. Rubber hand illusion (RHI) can provide body ownership to a fake hand based on spatio-temporal correlation between visual and tactile stimulation . AnotherIn this research, four shadow environments were designed for illusion induction to examine the effect of body-shadow in RHI . Specifishadow type , hand movements , and statement type (illusion vs. control). The results  = 20.40, p < .001, and statement type, F = 51.84, p < .001, in addition to a significant interaction among the three factors, F = 14.03, p < .001. Hand movement did not yield any significant effect, F = 0.00, p = .98, n.s. A follow-up test with simple effects analysis revealed that the static trial was accompanied by a significant predominance of the average rating for the ownership illusion statements over that of the ownership control statements for the three shadows, Hand + : F = 52.65, p < .001; Handshape−: F = 21.41, p < .001; Rectangle + : F = 4.88, p < .03; Rectangle − : F =2.83, p = .10, n.s., whereas such a significant trend was only shown in the active shadows in the active trial, Hand+: F = 104.05, p < .001; Handshape−: F = 0.62, p = .43, n.s.; Rectangle+: F = 35.93, p < .001; Rectangle−: F = 0.04, p = .84, n.s. In addition, multiple comparisons using Ryan’s method (p < .001 for Hand+ vs. Rectangle+ and Handshape− vs. Rectangle−). Finally, the active trial provided significantly higher ratings for the illusion ownership statements as compared with the static trial for the active shadows, F = 11.80, p < .001 for Hand+; F = 27.55, p < .001 for Rectangle+), whereas a significant trend toward the opposite direction was found for the static shadows, F = 28.24, p < .001 for Handshape−; F = 7.41, p < .008 for Rectangle−.First, ownership statements for both static and active trials were analyzed using a three-way ANOVA with within-subject factors: s method  revealedshadow type and statement type and revealed a significant main effect of shadow type, F = 20.4, p < .001, and statement type, F = 51.84, p < .001, in addition to significant interaction, F = 14.37, p < .001. Simple effects analysis revealed that the active trial was accompanied by a significant predominance of the average rating for the agency statements over that of the agency control statements only for active shadows, F = 19.58, p < 0.001 for Hand + ; F = 28.87, p < .001 for Rectangle+.Next, agency statements for the active trial were analyzed using a two-way ANOVA with within-subject factors shadow type, hand movement, and sight type (eyes-open vs. eyes-closed). The results  = 5.56, p < .003, and sight type, F = 6.95, p < .019, whereas hand movement did not yield a significant effect, F = 0.57, p = .46, n.s. In addition, the multiple comparisons (Ryan’s method) under the eyes-open measurement demonstrated that the drift for Hand+ was significantly higher than that of the three other shadows . Finally, one-tailed Welch’s t-test comparison with zeroes showed that Hand+ had a significant positive drift in both the static and active trials with the eyes-open measurement but not with the eyes-closed measurement, t(15) = 4.01, p < .002 for static trial; t(15) = 3.55, p < .003 for active trial, whereas the other shadows did not have any significant effect.Finally, we analyzed proprioception change. Prior to the analysis, the proprioceptive drift of each participant was calculated for each sight condition (eyes-open and eyes-closed) using a common pre-illusion proprioception of the right hand (baseline), which was pooled and averaged per sight condition to restrain dispersion. The obtained proprioceptive drifts (post-illusion proprioceptive height minus pre-illusion proprioceptive height) were analyzed using a three-way ANOVA with within-subject factors The questionnaire analysis revealed that both the anatomic similarity to the body-part and the visuo-proprioception correlation were effective in enhancing illusory ownership. This demonstrates that the shadow could distort illusory ownership with the RHI paradigm. In contrast, these effects on the proprioceptive drift were uncertain. In fact, there was no significant difference in the drift between the static and active trials, regardless of the type of the shadow shape. In addition, the hand-shaped cloth’s shadow (Handshape−) did not yield a significantly greater drift than the rectangle cloth’s shadows in any situations. Thus, the proprioceptive drift was less sensitive to both the anatomic similarity and the visuo-proprioception correlation as compared with the illusory ownership. These points support the recent view that the proprioceptive distortion does not have a causal link with a feeling of limb ownership .As is unusual for the study of RHI, we tested the proprioceptive drift measurement not only in the eyes-closed situation but also with eyes open toward the shadow (eyes-open measurement) to probe the effect of seeing the shadow in detail. Contrary to the former reports with the other artificial hands, all the shadows, including the body-shadow, did not yield the significant positive drift in the eyes-closed measurement, even for the active trial. Although we do not have a convincing explanation for this disagreement, it is invalid to suppose that this is consistently reproducible considering that we found the opposite trend for the body-shadow in some preliminary experiments. With the eyes-open measurement, the physical hand’s shadow (Hand+) was the only shadow that provided with the significant positive drift; notably, its drift was also significantly greater than for the hand-shaped shadow. Thus, the proprioceptive drift did not occur merely because the subjects see the hand-shaped shadow; the shadow should be the innocent body-shadow. Here, it is also remarkable that even the static trial yielded the significant positive drift toward the body-shadow. As the static trial did not involve the physical hand movement, the results indicate that just seeing the body-shadow results in visuo-proprioceptive recalibration. In the study of RHI, a few studies have reported that changes in hand proprioception can occur even without visuo-tactile correlation e.g., . ConsideIn total, 16 students  participated in the experiment. All participants were recruited from the Faculty of Design and Architecture, Nagoya City University, and signed informed consent prior to their participation. All participants received a book of tokens as compensation (1000 Yen). All experiments were conducted in accordance with the Declaration of Helsinki. The study protocols were approved by the ethics committee of the Nagoya City University.We set up three acrylic transparent plates vertically at a suitable interval, where the participant could see a shadow from any of the three kinds of objects . They could be viewed on the screen of the top plate via a light source installed on the floor, with the palm of the right hand on the middle plate, 12 cm below the top plate. The shadow on the top screen approximately magnified the objects below by 5% to 10%. Four shadow environments were designed for illusion induction: first, an active hand shadow (Hand+) where the top plate shows the right hand’s shadow in a straightforward manner; second, a static, hand-shaped shadow (Handshape−) showing the shadow of the hand-shaped cloth (on the bottom panel) that remained stable regardless of the hand movement; third, an active rectangle shadow (Rectangle+) showing the shadow of a rectangle cloth (clinging to the right hand) that moved in sync with the hand movement; and fourth, a static rectangle shadow (Rectangle−) showing the shadow of the rectangle cloth (on the bottom panel) that remained stable regardless of the hand movement .hand movement —is independent from the classification of static or active shadows. Thus, the illusion induction was accompanied by the shadow’s movement only when the active trial was applied for the active shadows (Hand+ and Rectangle+); the static trial did not involve the shadow’s movement for not only the static shadows but also the active shadows. Subsequent to this illusion induction, they were asked again to localize the right hand with the sight condition that was applied at the pre-illusion stage. Notably, the eyes-open measurement was done without the participants taking their eyes off the shadow to measure proprioceptive height during the prior illusion induction as accurately as possible. Quickly after they answered four- or six-item questions (see lower right of


Viruses are key players in ocean ecology and biogeochemistry, not only because of their functional roles but also partially due to their sheer abundance Fuhrman, . BecauseV, mL−1) and microbial cells  are approximately proportional. A rule-of-thumb “virus-to-microbe” ratio of 10 has been used  regression on log-transformed V and M data, to our knowledge none have tested the linear model V = aM+b. Unlike V = αM, the linear model V = aM+b has the same number of parameters as the nonlinear model V = αMβ, is consistent with a decreasing V/M as M increases, and is not a special case of the nonlinear model. We interpret V = aM+b to consider V as the sum of two pools—a pool of viruses whose abundance is proportional to microbial abundance (aM), and a background pool of other viruses and virus-like particles whose abundance is unrelated to microbial abundance (b). Because this model is simple and its parameters are comparatively straightforward to interpret and possibly predict, its ability to explain observations should be evaluated before concluding that V and M are nonlinearly related.While several studies  and subsurface  samples, totaling 27 datasets. We repeated their OLS regression of log10-transformed V and M data . We also fit the model V = aM+b after log-transformation:We first reanalyzed the datasets considered in Wigington et al. . They anV = b as M → 0 and to V = aM as M → ∞. We analyzed log-transformed data to be consistent with Wigington et al.  we found n = 18) we found Knowles et al.  and Paria et al.  (n = 221ls Table .V-M relationship. Maranger and Bird (n = 149), and Danovaro et al.  represent burst size, virus-microbe interactions, viral decay, and microbial growth respectively , novel experiments with model systems, or other types of data that improve upon the indiscriminant measure of virus-like particles may help answer this question (Baran et al., b. For now, we contest that a linear model remains plausible.These analyses highlight the need for alternative methods to quantify viruses in the environment relative to their hosts. Measurements from marine environments with extremely low microbial densities (because the two models diverge for low values of BC conceived the research, performed the analyses, and wrote the paper. MC, CF, and MF assisted in the writing process and revised the manuscript.The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


In this paper, we first present a new lattice-based PKE scheme on SIS, proving that it achieves CPA-security under DBi-ISIS assumption. Compared to some lattice-based schemes, ours has some advantages and is quite efficient as well as great simplicity. Similarly, we give a lattice-based PKE with multiple bits which is CPA secure under DBi-ISIS assumption. We hope that our contributions help to pave the way for the development of lattice-based PKEs in the future work. Nowadays, with the development of technologies, such as cloud computing and quantum information technology, quantum computing power becomes stronger and stronger. As a result, traditional cryptosystems, e.g number-theoretic cryptosystems, could be almost broken by quantum computers. To handle this case, lattice has come up as a powerful technique to resist quantum computers and been gradually used to construct cryptography primitives which can be against quantum attack. Maybe lattice-based cryptography will be a replacement to number-theoretic cryptography in cryptography field.the shortest independent vectors problem SIVP (one variant of SVP)   factors, because they remain secure even against quantum computers.Over the last decades, lattice has emerged as a very attractive foundation for cryptography. The appeal of lattice-based primitives stems from the fact that their security can often be based on worst-case hard problems such as shortest vector problem (SVP)  was one of the most fundamental primitives in cryptography. The first security notion for PKE was indistinguishability of encryptions under chosen-plaintext attacks (indistinguishability against chosen-plaintext attacks) (IND-CPA or CPA) Stinson , also knIn 2008, Gentry et al.  was at lIn 2014, Wang et al. . We takeThis paper is organized as follows. “Notations Assume that n is the the main security parameter in this paper. Bold lower-case letters denote vectors in the column form, e.g. x. Bold capital letters denote matrix, e.g. A, and the transposition of A is X uniformly at random is denoted by Here mainly describe some definitions and properties on lattice Regev , 2009.k linearly independent column vectors n-dimensional lattice n dimensional real space Bx is the usual matrix-vector multiplication.Regev  Given k Regev  For inteThe most well-known basic hard computational lattice problems are SVP  minimizing the quantity (SIVP) (Micciancio The approximation variants of these problems: B of a full-rank (SVP (Decision Version): nd Regev .n-dimensional lattice basis B. The goal is to output a set of n linearly independent lattice vectors (The hard-on-average problem first proposed by Ajtai  was the A) Regev .q, a uniformly random   = n. Let A, and A, such that for S and Given security parameters (DBi-ISIS problem)   if for all PPT adversaries negl such that:. The challenger gives pk as well as oracle access to sk private.Queries phase 1m where message m must be in the plaintext space associated with pk. The challenger responds with Encrypt  . Then computex secretly  and output ciphertext C.Decrypt follows by the form of We utilize the “Game hopping” Dent  to proveIfDBi-ISISproblem is hard for a PPT algorithmthen the lattice-based PKE scheme in “section has indistinguishable encryptions under CPA. Namely, our PKE shcheme is CPA secure underDBi-ISISassumption.Suppose there exists an adversary GenKey is the same as in Let m from Although the receiver cannot calculate the plaintext m being encrypted. Virtually, when m. Obviously, the first element m. Taken together, the ciphertext in m and hence contains no information about m. Thus we getNow we discuss that the ciphertext in Setup The adversary m.)Queries phase 1 The adversary m, where m is an alleged plaintext.Challenge The adversary b, constructs the challenge ciphertext C is a valid encryption of C is independent of b in the We note that Queries phase 2 Same as phase 1. But the adversary is not allowed to query the Guess Then b. If C is completely random. Because C is a valid ciphertext. DBi-ISIS game, DBi-ISIS problem is hard, there must exist a negligible function negl(n) such thatS has advantage at most negl(n) in solving DBi-ISIS problem). It follows that our PKE in “has indistinguishable encryptions under CPA. By Definition 14, we haveWe see that if Pub.size) and computation complexity (e.g Enc.comp.) of PKEs. Pub.size means the size of the public key, others are so; Comput.Comp. means the computation complexity and is estimated by the number of the multiplications in n is the security parameter, q is a polynomial function of n.For the comparison to be meaningful, we consider the latticed-based PKEs. Table mentclass2pt{minimpriv.size in our system is nearly the same as that of Regev .X secretly.Then computeDecrypt Upon receiving M.Correctness If the system is run honestly, the message M is obtained correctly.M by computingTo show the correctness of our scheme, Theorem 16 indicates that our extended PKE of matrix form in “If DBi-ISIS problem is hard for a PPT algorithmthen the lattice-based PKE in “section is security against CPA under DBi-ISISassumption.The proof of Theorem 16 is similar to that of Theorem 15, omit it here.In this paper, we present a simple PKE scheme that achieves CPA security under the DBi-ISIS assumption. We build it on previous works of Wang et al.  and RegeIn addition, we extend the lattice-based PKE in “Note that our PKE schemes in both “


Immunoglobulin genes are generated during differentiation of B lymphocytes by joininggene segments. A mouse pre-B cell contains a functional immunoglobulin heavy-chain gene,but no light-chain gene. Although there is only one heavy-chain locus, there are two lightchainloci: κ and λ.It has been reported that κ loci in the germ-line configuration are never(in man) or very rarely (in the mouse) present in cells with functionally rearranged λ-chaingenes. Two explanations have been proposed to explain this: (a) the ordered rearrangementtheory, which postulates that light-chain gene rearrangement in the pre-B cell is firstattempted at the κ locus, and that only upon failure to produce a functional κ chain is therean attempt to rearrange the λ locus; and (b) the stochastic theory, which postulates thatrearrangement at the λ locus proceeds at a rate that is intrinsically much slower than that atthe κ locus. We show here that λ-chain genes are generated whether or not the κ locus haslost its germ-line arrangement, a result that is compatible only with the stochastic theory.


We present a numerical spectroscopic study of visible and infrared laser radiation in a biological tissue. We derive a solution of a general two-dimensional time dependent radiative transfer equation in a tissue-like medium. The used model is suitable for many situations especially when the external source is time-dependent or continuous. We use a control volume-discrete ordinate method associated with an implicit three-level second-order time differencing scheme. We consider a very thin rectangular biological-tissue-like medium submitted to a visible or a near infrared light sources. The RTE is solved for a set of different wavelength source. All sources are assumed to be monochromatic and collimated. The energetic fluence rate is computed at a set of detector points on the boundaries. According to the source type, we investigate either the steady-state or transient response of the medium. The used model is validated in the case of a heterogeneous tissue-like medium using referencing experimental results from the literature. Also, the developed model is used to study changes on transmitted light in a rat-liver tissue-like medium. Optical properties depend on the source wavelength and they are taken from the literature. In particular, light-transmission in the medium is studied for continuous wave and for short pulse. Diffuse light imaging and spectroscopy aims to investigate tissue physiology in subsurface. There is a spectral window existing within tissues in the near-infrared. The absorption of hemoglobin and water is small in the near-infrared, but elastic scattering from organelles and other microscopic interfaces is large. In tissue optics, any measuring is devoted for either spectroscopy or imaging. Spectroscopy is useful for measurement of time-dependent variations in the absorption and scattering of large tissue volumes. Many spectroscopic methods involving laser interactions in scattering media were employed to study different types of biological tissue –3. PhotoSPN equation provides accurate solutions for a time-independent RTE. Boulanger and Charette [The purpose of this paper is to produce a computational Laser spectroscopic study in a biological tissue. We aim to examine light-transmission through a biological tissue for different wavelength sources. Our concern is with modeling light propagation using the radiative transfer theory. We derive a model that unifies the basic applications of diffuse optical tomography which differ essentially by the time behavior of the external source. In particular, we will work with two choices for the external source, continuous wave and short-pulse. The simplest and easiest source to use is the continuous wave (CW). In this case, the source amplitude is constant, and the transmitted amplitude is measured as a function of source-detector separation or wavelength. The second type is a short-time-pulse. In this scheme, a short pulse is launched into the medium, and the temporal point spread function of the transmitted pulse is measured. First, we must say a few words about another way to model propagation of visible and near-infrared light in biological media which is the diffusion equation –11. It iCharette  have useCharette , 22 wherCharette , 24. SucCharette  investigIn this work, we derive a control volume discrete-ordinate method associated with an implicit three-level second-order time differencing scheme to solve a two-dimensional time-dependent radiative transfer equation in a general biological tissue medium. We investigate many cases concerning the properties of the medium and the characteristics of the source. We will prove that the used model suits for all these applications. First, the proposed model is tested in the case of a heterogeneous tissue-like phantom illuminated by a continuous wave source. Experimental results are taken from . NumericLight propagation in a human biological tissue is modeled by using a general time-dependent RTE :Iλ  is the monochromatic directional energetic radiance at the spatial position-vector r = . When used in a dimensional form, the radiance Iλ  is often expressed in units of W·cm−2sr−1. The photon direction is described using a unitary direction vector  Ω;μa(r)  and  μs(r) are the absorption and the scattering coefficients, respectively. They are both given in units of cm−1. c is the speed of light in vacuum. n is the medium's refractive index. Also we make use of reduced scattering coefficient:Sλ  is an injected monochromatic radiance at the medium boundary; The source term p  describes the probability that during a scattering event a photon with direction  Ω′ is scattered into the direction  Ω. In this paper, we use a commonly-applied phase function in tissue optics. It is the Henyey-Greenstein phase function [The phase function function , 28: g is the anisotropy factor and θ is the angle between Ω and  Ω′. The integral of radiance over all angles Ω at one point r yields the fluence rate: λ will be omitted in all the rest of this paper.The radiance in the boundary is the sum of the external source contribution and the partly reflected radiance due to the refractive index mismatch. This givesrb is a vector position on the boundary and n is the unit outer normal vector. The reflectivity R is given for each direction using a Fresnel's relationship:φref  and φtra are the reflection and the transmission angle respectively. In the air, the transmission angles, is calculated through the following Snell's law: nm depends on the tissue and n0 is taken as unity. The critical angle is determined by the following equation:d will be called detected fluence rate and we will take AP = 45° in all our investigations.V. The spatial calculus domain is divided into a set of I × J elementary uniform volumes ∆V with unitary depth. This yields a calculus domain subdivided in I ∗ J rectangular cells. Each -cell is Δx∆y-sized. So, we can write. In this paper, we use a control volume discrete-ordinate method , 30 assoξm = cos(Ωm · xn) and ηm = cos(Ωm · yn) with nx  and  ny being two normal vectors to the -plane and the -plane, respectively. wm is a weighting factor depending on the chosen quadrature formula. In this work, we use a constant weight quadrature. After angular discretisation, the phase function term becomes W and S and they are unknown on the faces E and N of the -cell and also in the centre P. Therefore, we need two complementary relations to eliminate IE,m and IN,m.These relations can be obtained by using interpolation formula:α is an interpolation parameter. Using the above formulae, -cell, we can obtain results over the cells  and  by using the boundary conditions and the following relations:ξm > 0  and  ηm > 0), then we get the following equation: Ii,j,mn+1Ii,j,mn = ISi,j,mn = S.n ≥ 1,To solve , we use Ii,j,mn)updatedk+1 becomes a linear combination of the iterated value k+1 and the previously computed value k asechnique . So the ρ being a relaxation parameter whose value is usually between 1 and 2. In most of our calculus, we have used: ρ = 1.1. The solution is obtained when the relative discrepancy value, ϵ = |k+1 − k/k|, is smaller than a tolerance value tol. In that case, we take k as a solution. It will be noted −8 and for initial condition, we take x = Δy and a uniform time step Δt = 10−13s. Also it is important to mention that when a continuous source is injected, another iterating process is implemented to attempt the steady-state solution. This is obtained by iterating the solution Ωm)  is divided into four subsets depending on the sign of ξm and ηm. By the use of direction cosine sign, all cases can be given through the following general equation:It is worth noting that, in all our simulations, we take a tolerance value: tol = 10In all our investigations, the power laser-source is assumed to be equivalent to a forward collimated radiance injected at a pinpoint on the bottom side of the boundary. We use two types of source:−2sr−1.a continuous wave source with a uniform equivalent intensity value of 20 mWcm−2sr−1,a short-Laser-pulse source, applied only at the first instant of calculus. It is a 100fs-pulse having an equivalent radiance value of 200 mWcmIn our presented results on the boundary, we use either calculated detected fluence rate or normalized detected fluence rate:ϕd  is the computed detected fluence rate at a detector point d, D is the number of the detector points on the concerned side of the boundary, and wd is a weighting factor from the generalized trapezoidal integration rule [nΔt, as:ion rule . Also, dx < 1/(μa + μs′). In these calculations, we have used Δx = Δy = 0.05 cm.To test our model, we have considered a case of 678 nm-light punctual continuous wave source. Our Model is tested by comparing obtained results with phantom experimental data reported in . Figure α = 0.5) shows the closest results to the experimental measurements. In the rest of investigations, the interpolation parameter value will be putted 0.5. Also In this investigation, we consider a rat liver tissue-like medium as it is shown in  In this investigation, we study the same medium as in the just precedent paragraph but we use a short-pulsed source. In all cases, it is a 100fs-laser pulse injected in the medium through a point on the middle of the bottom side of the boundary.This paper investigated situations concerning biological tissue interaction with Laser radiation by using radiative transfer theory. More precisely, a computational radiative transfer model is used to establish a numerical Laser spectroscopic study in a rat-liver-tissue-like medium. The response of the medium to both time-dependent and time-independent sources is explored. Obtained Results are tested by using experimental measurements from the literature and by using Monte Carlo simulations. The maximal transmission from the tissue is observed in the red and the near infrared part of radiation spectrum. Also multiple scattering is prevailing in 488 nm while more and more absorption is observed in 2100 nm.