Mass distribution campaigns of insecticide-treated nets for malaria prevention are usually accompanied by intensive behaviour change communication (BCC) to encourage hanging and use of nets. However, data on the effectiveness of these communication efforts are scarce. In preparation for the next round of mass campaigns in Nigeria, a secondary analysis of existing data from post-campaign surveys was undertaken to investigate the influence of BCC on net hanging and use.p < 0.2. To quantify the effects from BCC, a treatment effect model was used with an inverse-probability weight regression adjustment.Surveys were undertaken between 2009 and 2012 in ten states in Nigeria using standardized questionnaires. Two-stage cluster sampling was used to select households in each study site. Outcomes were defined as the effects of BCC message exposure and recall on knowledge, attitudes, perception as well as intentions and actual use. From the univariable analysis, potential confounders and explanatory variables were identified and key effects explored in multivariable linear or logistic regression models; terms in the models were kept if they had a marginal significance with p < 0.001). All BCC outcomes showed a significant increase in net use, which was strongest for the confidence to take action regarding nets with an overall effect of 17 %-point increase of net use comparing poor and excellent confidence levels. Intention to use every night increased net use by 15 %-points and discussing net use in the family by 8 % points. All these effects were statistically significant (p < 0.001).More than half of the respondents  had heard a message about net use or hanging during or after the distribution campaign, with media cited as the most common source of information. Attitude towards net use was positively linked to the number of messages recalled and was overall better in the northern study sites. The number of messages recalled was also the strongest predictor of knowledge  contains supplementary material, which is available to authorized users. Over the past 10 years, the population in sub-Saharan Africa with access to insecticide-treated nets (ITNs) has increased significantly from around 3 % in 2004 to 49 % in 2014 . StudiesOne challenge when implementing universal coverage net distribution is ensuring that ITNs/LLINs are used consistently ; informaQualitative studies have suggested (within the limitations inherent in such studies) multiple reasons for not using an ITN, unrelated to access and availability , 18 and Nigeria is the most populous country in Africa and 100 % of the population lives in an area of high transmission for malaria ; in 2013In preparation of the new LLIN distribution exercise, efforts were undertaken to identify shortcomings of the previous distribution campaign in order to optimize procedures for implementation and follow-up. This included a literature review on aspects of net and ITN use in Nigeria, which identified a difference in net culture between different parts of the country but also a lack of understanding of the determinants of net hanging and use and how these can best be influenced by BCC activities. As one of the outcomes of the review, two immediate operational research activities were suggested to inform the implementation of the campaign roll-out: (i) a mapping of BCC activities by all partners and stakeholders in the recent past with respect to messages, channels, content, intensity, and quality of messaging; and, (ii) a secondary analysis of existing data from post-campaign surveys that had been undertaken by different projects to assess the campaign outcomes and which included detailed information on net hanging and use as well as information of BCC exposure and message recall. Examples of IEC/BCC materials on LLIN use that were used by stakeholders included television and radio jingles and documentaries, audio songs, radio drama and magazines, billboards, brochures and toolkits, calendar, community stage drama reports, promotional gift items , posters, leaflets, stickers and fliers, IPC flipcharts and job aid cards. This paper reports on the findings of the secondary data analysis.The objectives for the secondary data analysis were: (i) to explore the relationship between channels of LLIN-related BCC messages, their recall, and subsequent attitudes and behaviours; (ii) to make recommendations linked to these findings on strategies to intensify utilization of bed nets.The surveys were undertaken in between 2009 and 2012 in ten states in Nigeria in the context of two projects: the Support to the National Malaria Programme (SuNMaP) funded by UKaid, and the NetWorks project funded by the US President’s Malaria Initiative (PMI). All surveys were planned and implemented by Malaria Consortium with fieldwork contracted out to research and marketing services (RMS). Data sets were kindly provided by Malaria Consortium.The geographic location of the ten sampled states and the corresponding rainfall pattern is shown in Additional file A standardized questionnaire was used throughout with only minor modifications between states. The basic design was taken from the Malaria Indicator Survey (MIS) with additional modules added with respect to the registration and distribution process from the campaigns including: the hanging of nets and any difficulties encountered, retention of campaign nets and fate of previously owned nets, exposure to net related messages with source of information and the key messages recalled, as well as a section exploring attitudes, beliefs and knowledge using a set of Likert score questions, i.e., by asking respondents to choose on a scale of agreement. In November 2011 these questions were expanded for the remaining four surveys to include a broader spectrum of knowledge, attitudes and practices (KAP) statements. For the net module questions of reasons for non-use of nets were added.Sample size calculations were based on the two domains of the first survey, Kano State, and aimed at detecting at least a 10 % point difference between the domains if the outcome estimate in one domain is 50 % and assuming an alpha error of 0.05, beta-error of 0.2, design effect of 1.75 and non-response rate of 5 %. This resulted in a sample of 510 households per domain  divided into 30 clusters per domain  with 17 households per cluster. This 60 × 17 design of a total of 1020 households per survey was applied to all other states with the exception of Cross River which had 75 clusters and a target sample of 1275 households as two areas were over-sampled for another project. The actual achieved samples for households, individuals and nets after data cleaning is given in Table Data cleaning and preparation was done by Malaria Consortium analysts from data files which had been double-entered and validated against the paper forms collected in the field. An asset-based wealth index using principle component analysis was calculated and wealth quintiles created. The data sets were screened for the variables of interest and those not needed or available in all data sets were dropped. A new variable was created to identify the state so that data sets could be appended and a unique identifier be maintained for each observation, i.e., household identification (ID) plus state/survey etc. Explanatory variables for timing of the survey with respect to rains  no ITN; (ii) fewer than one ITN/three persons; (iii) one ITN/three persons; (iv) one ITN/two persons; (v) one ITN or more/persons. The last two categories were considered as enough nets for all household members. A similar variable was created for net ownership in general with the following three categories: (i) households with no nets at all; (ii) those with any nets but no net from the campaign; and, (iii) households with at least one net from the campaigns. Exposure to any messages on nets (hanging or use) in the 6 months preceding the survey was recorded in the questionnaires together with all the sources of information. These were then grouped into five major groups, namely campaign , media , mediators , health system  and social network (friends and family).BCC impact is generally not easy to measure as the effects are complex and not caused by a single event of exposure to a message, but rather by repeated exposures from multiple sources modified by various internal and external factors. Therefore, three steps were used to define BCC exposure-dependent outcomes as a composite measure of the BCC effects, which could then directly be linked to net hanging and use in univariable and multivariable analysis: (i) exposure to messages about net hanging and use was explored with respect to coverage (any exposure) and intensity (number of information sources)To quantify the treatment effects from BCC, i.e., the impact of BCC exposure on ITN use, a treatment effect model was used with an inverse-probability weight regression adjustment (IPWRA in Stata). Treatment effect models use predicted exposure to an intervention (here BCC) as a variable in a simultaneous equation that predicts the outcome behaviour of interest , therebyStatistical analysis was done with STATA 13.1 . All cross tabulations and testing of means in the initial univariable analyses were done applying the sampling weights from the original surveys and adjusting 95 % confidence intervals (CI) for the design effect of the cluster sampling design by using the appropriate survey commands (SVY). In addition the survey was defined as second level of cluster in the analysis. All models also applied the survey analysis settings.p < 0.2. However, due to the characteristics of the data set comprising ten surveys from different years and states within Nigeria, the variables year of survey, season, time since campaign, and region were maintained in all models irrespective of significance. These variables were used instead of the state itself in order to allow the separate analysis of specific aspects, such as seasonality, which would have otherwise been absorbed in the state variable thereby confounding the results. Fit of the models was assessed by the coefficient of determination (R-squared) and by likelihood-ratio tests.From the univariable analysis, potential confounders and explanatory variables were identified and then the key effects explored in multivariable linear or logistic regression models. In general, terms in the models were kept if they had a marginal significance with More than half of the respondents  reported having heard any message about net hanging and use during or after the campaign, ranging from 45.9 % in Kano to 70.5 % in Enugu. The rate of recall increased with time since the campaign from 37.9 % in the first 3 months, to 56.8 % 3–9 months afterwards, and to 63.8 % after 9–16 months afterwards. Recall of messages was seen to be dependent on the ownership of nets  were the most commonly cited information sources in both groups , followed by information channels directly linked to the campaign (campaign leaflets and information from the mobilization team) with 19.7 and 33.7 %. This was followed by health systems  with 18.6 and 31.9 %, and mediators  with 15.7 and 26.8 %. Social networks (family or friends) were the least cited source with 14.8 and 25.4 %. A detailed breakdown by sub-category is shown in Additional file p < 0.001), whilst owning a radio (independent of socio-economic status) was positively associated . Among households that could recall two or more messages, only small households without children were significantly less likely to have received any message .The determinants of information exposure were explored in a multivariable logistic regression model Table  and strap < 0.001), slightly favouring wealthier households  and was strongly associated with the first year of campaigns . Individuals who recalled campaign BCC were significantly more likely to own a campaign net . Interestingly, recall of campaign BCC had significantly lower odds in the north . The health system channel was strongly associated with the campaign: those exposed to this information channel had significantly higher odds of owning a campaign net  and had a similarly strong increasing trend after the campaign (p < 0.001). Households with a better educated head of household were most likely to recall messages from this source , whilst small households with no young children  and households with older heads of household  were less likely to pick up messages from the health system. Communication through mediators, also associated with the campaign, showed a strong increase with time since the campaign  but otherwise reached exactly those who were less favoured by the health system, namely older household heads  and those with lower education . In addition, this channel was more associated with the northern campaigns  and appeared to be more emphasized in the early campaigns . Media-related BCC was positively associated with radio ownership . Media-related BCC also favoured wealthier  and better-educated families  and was marginally favouring disadvantaged small families without young children . Finally, BCC through social networks was linked to urban , wealthier  and younger households  with young children . Communication through this channel appears to be more emphasized in the late campaigns .Considering that the overall result of information exposure is the sum of the effects of each of the five communication channels, each of these channels was examined in a separate regression model to determine the complementarity of their profiles (whether specific channels are better suited to reach particular sections of the target population) and detailed results are shown in Table p < 0.0001) with a high coefficient of determination (R-squared 0.70) meaning that 70 % of the variation of messages recalled in the data set can be explained by the number of information sources alone. However, only close to half (44.4 %) of those with any information and 25.9 % of all respondents could recall more than one source of information and 14.4 and 8.4 %, respectively, three or more information sources. Subsequently, only 30.9 %  of respondents recalled two or more messages, and this proportion was somewhat higher among those who had received any nets from the campaign reaching 44 %.Figure Finally, the relationship between specific messages and communication channels was explored using the variables previously identified as determinates of exposure and results are presented in Table BCC outcomes were defined as the effects of message exposure and recall on knowledge, attitudes, perception as well as intentions to use nets. Measurements of KAP and belief were measured using the Likert score questions  or five  messages. Confidence to take action was significantly linked to the rainy season , owning any net  and heads of household having secondary  or tertiary  education. The action score was highest 6–9 months  and nine to 16 months  following the campaign. Two factors decreased a household’s confidence to take action: the age of the head of household , and small families without children . Participation in the campaign, urban versus rural residence and gender of the head of the family had no statistically significant influence.The ability to take action to protect the family using ITN was the major measurement of attitudes towards net use and data were available for all ten surveys. When entered into a multivariable regression model, being confident to take action to protect the family with ITNs was positively linked to the number of messages recalled , responsible for 16 % of the variation in the response or 56 % of the overall coefficient of determination of the model (R-squared 0.29). On the other hand, intention to use net every night was only slightly influenced by BCC message recall without a dose–response relationship explaining merely 15 % of the overall variation explained by the model (R-squared 0.15). In contrast, both variables were linked to the confidence to take action although, again, the intention to use every night showed less of a dose–response , it was the rainy season and particularly the peak of the rains for use every night .Discussing net use within the family and intended use of nets every night were entered into a multivariable linear regression model , discussing net use in family  and intention to use every night ], the ‘net culture’ was better in the north—all other aspects being equal, i.e., adjusting for potential confounders such as wealth, seasonality, education, and level of exposure to messages. The models also identified some variables with a moderate negative effect, i.e., they reduced the likelihood that net use was discussed or an intention to use every night was expressed. These factors were non-literate heads of households for both variables [discussing net use in family  and intention to use every night ], young heads of households  and small families without young children  for discussion and urban residence  for use every night.Regarding regional differences, the north showed significantly better results for both variables meaning that for all three KAP measures for attitudes in this analysis [confidence to take action ; there was a clear dose–response relationship, showing knowledge increased with each additional message remembered, resulting in a proportion of 25 % of households with excellent knowledge if four of more messages were recalled. There were no major misconceptions to perceptions and beliefs.Statements on knowledge about ITN and perceptions and beliefs about their use were available for only 4309 (42 %) households from four states . In a multivariable linear regression model the number of messages recalled was the strongest predictor of knowledge  had access to an ITN, assuming that each net was used by two people. Of the de-facto population, 29.3 %  had used any net and 28.1 %  had used an ITN the previous night. This means that overall 62.9 %  of those with access used an ITN the previous night or 37.1 %  did not use an ITN. The latter is termed the ‘use gap’.Among the 49,033 individuals registered in the surveys 47,703 (97.5 %) stayed in the house the previous night and represent the p < 0.001) compared to less than one net for every three people . The model also confirms the strong impact of increasing confidence to take action (p < 0.001), expressing the intention to use nets every night , and discussing net use . Similarly, net use was 2.5 times more likely if the household had not reported any difficulties in hanging of nets . Adjusting for all other factors, net use was significantly higher during the mid-rains or peak of rainy season  while there was still an increase in the late rains, i.e. at the end of the rainy season , but this was only marginally significant. At the same time, net use was stronger in the north  and the change of use with seasons was significantly different in the north than in the south (p < 0.0001 for interaction term in model) suggesting a significant seasonal variation in the north and no or a weak one in the south.Multivariable logistic regression Table  among hop = 0.001) and older children five to 14 years . The highest use was seen in children under five  and pregnant women , followed by women of reproductive age but not currently pregnant . Heads of household, independent of age or gender, also had a significantly increased use rate , which was above that of older children, but lower than women in reproductive age. Finally, there was a significantly lower use in urban areas  but no independent effect of wealth quintiles, education or gender of the head of household.The relationship between age and gender and net use was explored by constructing different age-gender groups, and using men age 15–49 years of age as the reference point. The results show that this group had the lowest net use, followed by people aged 50 or more . However, there was no indication that these effects were additive as an outcome variable that included the combination of a high confidence with intention to use every night did not show a higher treatment effect that any of the two alone.Results are shown in Table This analysis of ten post-campaign surveys from Nigeria attempted to identify the impact of BCC message exposure on net hanging and use and describe the effects in as much detail as possible in order to inform the design and implementation of the second wave of LLIN mass campaigns in Nigeria. The main findings can be summarized as follows.Exposure to net-related messages in the previous 6 months was overall quite high with 58 % of household respondents stating that they had heard or seen any messages. Interestingly, results showed that different communication channels differed significantly not only in whom they reached, but also in what messages they prioritized.Exposure to the campaign channel was seen to increase over time, which could be an effect of people reading leaflets some time after the campaign or being reminded of the exposure by seeing posters in the months following the distribution. Recall of campaign BCC slightly favoured wealthier households and was significantly less in the north, which is most likely due to the fact that two of the five northern campaigns were integrated with polio-immunization where BCC focused more on the vaccination. Interestingly, campaign-related communication was strongly associated with the first year of campaigns, i.e., Kano and Anambra, suggesting that special effort was put into these from all stakeholders that may not have been present in the later campaigns. The campaign BCC  worked best to convey the “hang your net” message and was most intensive in the first year of campaigns. In the late 2000s it was hypothesized that lack of ITN use was attributed to difficulties in hanging ITNs and poor knowledge regarding their use . ResultaMessages from the campaign were reinforced by health workers during subsequent health facility visits, resulting in a still very strong association in health worker communication with the campaign and similarly strong increasing trend after the campaign. It clearly favoured households with better educated head of household—possibly because this group is more likely to seek treatment—and disadvantaged small households without any young children who either are also less likely to frequent the facility or were not seen as a target for net messages by the health workers. There was a marginally lower pick up of messages from health workers by older heads of household. Communication through mediators  also reinforced the net messages following the campaign and showed a strong increase with time since the campaign, but otherwise reached exactly those who were less favoured by the health system, namely older household heads and those with lower education. This channel was more associated with the northern campaigns and appeared to be more emphasized in the early campaigns (Kano and Anambra).p = 0.020) [Media communication mainly comprised radio messages, songs and community drama shows and was independent of the time after the campaign. It was, not surprisingly, the only channel that was associated with radio ownership. Media-related BCC favoured wealthier and better-educated families and marginally favoured female headed households (possibly due to radio listening during the day) and disadvantaged small families without young children. Mass media (mainly radio) was best in communicating the message “sleep under a net every night”. A previous study in Nigeria also found that respondents who listened to radio were approximately 1.6 times more likely to use an ITN  , and finAlthough social networks through family and friends are informal and not directly implemented as an external BCC activity, they not only had a significant role in providing households with net-related information, but also had a different reach compared to the other channels in that they were totally independent of net ownership, i.e., it happened in a similar way whether households owned campaign nets or not. It showed a peak in the three to 6 months after the campaign, declining thereafter and was linked to urban and younger households with young children. Communication through this channel also appeared to have picked up some time after the start of the campaigns, which would suggest that it needed some momentum from the other channels to get started. In Bangladesh, interpersonal communication with community health workers, relatives, neighbours and friends was found to be an important source of acquiring malaria information , and resWhilst the campaign-related BCC activities were critical to reach high exposure levels, no single communication channel alone could reach more than one-fourth to one-third of the households, providing strong evidence that a multi-channel, multi-media approach is needed to reach these kinds of BCC exposure levels. The NMEP and its partners would be wise to continue to implement a multi-channel, multi-media approach to BCC for the support of a strong net culture that focuses on clear messages and is not limited to the distribution campaign, but extends through other channels in order to re-enforce the effects of BCC. In addition to the routine channels of campaign activities, health staff and media  other mechanisms, especially local leaders and social networks (family or friends) should be utilized or stimulated as these are able to reach groups not easily reached by the other channels.Some messages needed less intensive communication than others with “use net every night” and “hang your net” reaching good recall after exposure from only one or two information sources, while “value your net” appeared to have to be presented by at least three or four sources before similar recall levels were reached. There was also very strong evidence for a dose–response relationship between the number of information sources and the number of messages recalled, i.e., the more different channels and media people were exposed to, the more messages they could recall. Intensity of communication , has previously been suggested as an important factor which can affect ITN use . The detAnother factor that positively influenced BCC outcomes was seasonality; during the rainy season, particularly at the peak phase of rains, there was a consistently better outcome in the northern states, suggesting a stronger or faster net culture development at the same level of exposure and recall. The relationship between rains and net use in households with any nets varied between the north and the south; during the rains, net use was higher in the north compared to the south while in the dry season the picture was reversed. This could explain the differences that were found between the 2013 Demographic and Health Survey (DHS), with higher use among those with access in the south, and the 2010 MIS, which had described significantly better usage in the north. While the MIS was undertaken during the late rains, the DHS was done early in the year during the dry season. Adjusting for the seasonality of use, the data provide evidence that net use is overall better in the north, and in keeping with the finding of a consistently higher level of net culture in the north, hanging was significantly better there compared to the south and there was also a reduced likelihood of hanging in urban compared to rural areas.A critical question is: who is using the nets and are there preferential age or gender groups which have to be addressed by specific or additional BCC strategies or messages? This was assessed by age- and gender-specific use rates separately for households that have at least one net for every two people. Pregnant women and young children were most likely to use a net, indicating that messages on preferential use of young children, pregnant women and young mothers are being largely followed. Non-pregnant women of reproductive age were the second-most likely group to use an ITN. Older children and men 15–49 were least likely to have used a net the previous night. Low use of ITN among children aged five to 15 years has been attributed to available ITNs being used for younger siblings and/or pregnant women/mothers, in line with IEC/BCC messages targeted to at-risk communities . Low ITNIt has been suggested that in countries where there is only a small proportion of children not using ITNs, IEC/BCC efforts should shift from promoting net use among these target groups to focussing efforts on improving one of the other categories of non-use, such as increasing ownership or hanging nets , or promIt is seen that preferential net use by different groups is primarily due to the lack of nets and preference being given to children and women is very much the same for all age and sexes once there are enough nets for all household members. Among the study population, there was a three-fold increase in the likelihood of a person having used a net the previous night if there was at least one net for every two people compared to less than one net for every three people. ITN to occupant ratio has been seen as a the main drivers for ITN use, wherein children in ITN-owning households with better intra-household access to an ITN were significantly more likely to have slept under an ITN the night before . In ZambThere was the significantly reduced likelihood of net hanging, and even more so a household having all their nets hanging, if there was an excess of nets, i.e., more than one net for every two people. In fact, during the rains households with a positive action score had all their nets hanging in 83 % of cases if they had fewer than enough nets for all members and only 45 % if they had excess nets. This suggests that the household supply situation with nets needs to be taken into account when net use and the potential impact of BCC is assessed in order not to confuse none-use due to surplus nets with unwillingness to use.In the overall sample, 37 % of household members who had access to a net had not used it the previous night. Kilian et al. found that the use gap differed between northern and southern geopolitical zones in Nigeria, with a use gap of 11 and 36 %, respectively . In thisp = 0.036) the odds of using an ITN [p = 0.27) nets in a separate model that included only LLINs, but adjusted for the same factors.Analysis of knowledge and perceptions about ITNs and their use in four of the ten states also revealed that there is some misconception about the possible harm of the insecticide for young children and pregnant women. Whilst there have been improvements, 27.3 % of respondents  in a study in Zambia in 2000 believed that the chemical on the net could be dangerous to the foetus or pregnant woman, whilst in 2004 this proportion was 6.5 %. Misconceptions about malaria can impact on net use; in Nigeria, it was seen that pregnant women who did not have any misconceptions about malaria prevention had 1.56  covered a large part of the country (with the exception of the northeast), they were chosen purposively based on the need and implementation area of specific projects. This implies that the samples were representative for each state but not statistically representative for the country as a whole. This could reduce the external validity  of the findings, but only if states not included would be significantly different from other states within the same region. Fifth, while exposure to information through various communication channels was recorded in the surveys as perceived by the respondents, there was no information available at the time of analysis as to the intensity, quality or geographical spread of the BCC interventions. This mapping exercise is currently ongoing and should be considered in the final considerations of the findings. Finally, observational data sets, such as the post-campaign surveys, generally are not ideal to assess the impact of specific interventions as they do not have clearly defined control groups nor allow a before-after assessment in the same population. Therefore, whether a person or household actually was or was not exposed can be influenced by a number of other factors and, similarly the treatment effects can be confounded in observational data. This limitation was addressed by applying appropriate statistical methods that have been developed specifically for this purpose and can largely, but not completely, avoid over- or under-estimation of intervention effects. It must be kept in mind, however, that experimental designs that could more precisely answer the questions of BCC impact on net hanging and use and cover diverse regions and seasons would be prohibitively expensive and unrealistic.The findings allow the conclusion that the multi-channel BCC campaign, through the campaign as well as other media, was overall very effective in contributing to an increase in net culture and subsequent net hanging and use, particularly by vulnerable groups. While the data clearly point out that there are differences between eco-climatic or cultural zones, here north and south, independent of seasonal variations, wealth, education etc., they do not suggest that the relationship between increasing confidence to take action for malaria prevention and increasing net use is different in principle between these zones. However, whether more intensive BCC activities are needed in the south, or rather a qualitatively different approach to BCC, cannot be answered from the data available. That would need more qualitative studies of the socio-cultural factors underlying these differences.


Myocardial fibrosis is an important risk factor for cardiac arrhythmias. Previous experimental and numerical studies have shown that the texture and spatial distribution of fibrosis may play an important role in arrhythmia onset. Here, we investigate how spatial heterogeneity of fibrosis affects arrhythmia onset using numerical methods. We generate various tissue textures that differ by the mean amount of fibrosis, the degree of heterogeneity and the characteristic size of heterogeneity. We study the onset of arrhythmias using a burst pacing protocol. We confirm that spatial heterogeneity of fibrosis increases the probability of arrhythmia induction. This effect is more pronounced with the increase of both the spatial size and the degree of heterogeneity. The induced arrhythmias have a regular structure with the period being mostly determined by the maximal local fibrosis level. We perform ablations of the induced fibrillatory patterns to classify their type. We show that in fibrotic tissue fibrillation is usually of the mother rotor type but becomes of the multiple wavelet type with increase in tissue size. Overall, we conclude that the most important factor determining the formation and dynamics of arrhythmia in heterogeneous fibrotic tissue is the value of maximal local fibrosis. Cardiac arrhythmias remain one of the largest causes of death in the industrialised world. In many cases, cardiac arrhythmias occur in a structurally abnormal heart where the properties of cardiac tissue have changed. The problem of substrate, that is, of identification of properties of cardiac tissue that predispose to cardiac arrhythmias, is of great interest. Structural changes of cardiac tissue, also called remodelling, occur as a result of many cardiac diseases. For example, in the ventricles of the heart, remodelling occurs after myocardial infarction and during heart failure12INaThe relation of fibrosis and arrhythmias has been investigated in several experimental studies. It has been shown that the inducibility of ventricular arrhythmias increases in a nearly linear fashion with the amount of fibrosis47Mathematical modelling has been widely used to study the effects of fibrosis on cardiac arrhythmias. In a series of papers111213In addition, a lot of research has focused on the effect of possible myocyte-fibroblast coupling on the conduction velocity and spiral wave dynamics172021Although some of these computational studies have investigated the effect of different types of fibrosis on wave propagation e.g., so far In this paper, we use a human ventricular cardiomyocyte model to study the effect of spatial heterogeneity of diffuse fibrosis on the onset of cardiac arrhythmias. For that, we developed a generic spatial model of fibrosis, which allowed us to change the degree of heterogeneity of fibrosis  and the characteristic size of the heterogeneity (size of tissue patches with a constant fibrosis level). Our study focuses on the question whether heterogeneity in the distribution of fibrosis is an important additional factor contributing to the onset of cardiac arrhythmia. In addition, we aim to answer how this depends on both the degree and the characteristic size of the heterogeneity. Finally we study how heterogeneous fibrosis affects spatiotemporal arrhythmia dynamics and characterise whether fibrillation is of the mother rotor or of the multiple wavelet type, and how this may depend on tissue characteristics.We used the ten Tusscher and Panfilov model23V is the transmembrane voltage for the cardiomyocyte, Cm is the membrane capacitance, and Iion is the sum of all ionic currents, which depends on V, on the gating variables, and on the concentrations of intracellular calcium. We used the set of parameters that corresponds to a slope of 1.1 of the restitution curve as described in23where Fibroblasts were assumed to be inexcitable and electrically disconnected from the myocytes, causing them to be passive obstacles for wave propagation.μm.For our spatial model of a 2D cardiac tissue with fibrosis, we used a rectangular grid with a size of 512 × 512 nodes, where each node can be occupied by either a cardiomyocyte or a fibroblast. Thus, we assume that the area occupied by a cardiomyocyte is the same as the area occupied by a fibroblast—a square with the size of 250 Following these assumptions the spatial model can be described as:i, k) is a position of a node occupied by a cardiomyocyte , ggap is the conductance of the gap junction channels that couple two neighbouring myocytes, and where .To generate a uniform fibrosis distribution with a mean level of fibrosis of σ and the spatial size of heterogeneity l. We divided the 2D tissue into squares of the size l × l. For each square, we assigned a certain local fibrosis level. We used 4 possible values to assign local fibrosis levels:For generating a heterogeneous distribution of fibrotic tissue, we used two extra parameters: the extent of heterogeneity σ is the difference between the maximal and the minimal possible local fibrosis levels. After assigning the local fibrosis level floc for a square, we set each node within that square to be a fibroblast with the probability floc or a myocyte with the probability (100% − floc).Thus, the meaning of σ implies the following restriction:As one can see, our choice of representing the degree of heterogeneity since the local fibrosis level can be neither smaller than 0% nor larger than 100%.The (pseudo-)ECG was computed using the following formula:where the transfer function z-axis above the centre of the 2D tissue.and the measuring point η. If a node occupied by a cardiomyocyte underwent an ablation procedure, the cardiomyocyte was replaced by an empty node, and the connectivity tensor η was recomputed in accordance with (3).The (pseudo-)ablations were modelled by changing the connectivity tensor The system of coupled ordinary differential 23The numerical solver was implemented with the C and C++ programming languages, using the CUDA toolkit for performing the majority of computations on graphical processing units. The auxiliary tools for manipulating the connectivity tensor, visualisation, and computing the ECG were implemented using the OCaml programming language. Computations were performed with single precision and run on an Intel Core i7-3930K machine with two GeForce GTX 780 Ti graphics cards.The random number generator was taken fromf = 25%. The bottom row shows the results for a heterogeneous distribution of fibrosis with the same mean level f, an extent of heterogeneity of σ = 25% and a heterogeneity size l = 16 mm. For the homogeneous distribution, the electrical activity vanished with the termination of pacing (shown in the top right part of We studied the induction of arrhythmias by high frequency pacing for homogeneous and heterogeneous distributions of diffuse fibrosis. We applied a burst pacing protocol of 10 stimuli with a period of 240 ms and monitored the induced activity. f), extent of heterogeneity and (σ) and spatial size of heterogeneity (l) we performed a large series of simulations for different combinations of parameter values . However, we observed that the outcome of our simulations, whether or not sustained electrical activity occurs, was highly stochastic in nature. The Results depended not only on the mean value of fibrosis and the pattern of heterogeneity, but also on the specific locations of randomly distributed fibrotic cells. Because of this stochasticity, we generated for each specific  parameter setting 50–100 different fibrosis patterns, by using different seeds for the random number generator before creating the pattern. We subsequently performed a statistical analysis of our simulation results, determining the boundary between parameter settings for which at least 25% of simulations results in persistent spiral wave activity and parameter settings for which fewer simulations result in persistent electrical activity.To study the effects of mean fibrosis level (σ—f parameter space when l = 12.5 mm. We can see that reentrant activity occurs for homogeneous fibrosis (σ = 0) if the mean fibrosis level f ≈ 28% and higher. With the increase of heterogeneity σ, the minimal value of the mean fibrosis level f required for reentry formation decreases linearly. In σ—l parameter space with fixed f = 25%. We see that reentrant activity is more difficult to trigger for smaller values of the size of heterogeneity l. The boundary separating the two regions has a hyperbolic shape.In the top panel of l on reentry generation, we calculated phase diagrams in the σ—f parameter space for two additional values of l. l, the boundary becomes less steep, whereas for a larger l, the slope becomes steeper (more negative). We can interpret these results as follows. If we consider the case of a very small value of l the difference between heterogeneous and homogeneous fibrosis disappears due to spatial averaging. Therefore, we expect that the probability of arrhythmia induction for this case is the same as for the case of homogeneous fibrosis and does not depend on the extent of heterogeneity σ. This corresponds to the horizontal green line in l = 8 mm (light green points) approaches that line. This indicates that heterogeneity at a small spatial scale has much less influence on the probability of spiral wave formation.To further investigate the effect of heterogeneity size l, then the patches of fibrosis can be considered independent, in the sense that each of them can be considered as a region with homogeneous fibrosis. The value of such local fibrosis ranges from (f − σ/2) to (f + σ/2). As the probability of reentry initiation increases with an increase in fibrosis level, the wavebreaks most often occur in the regions with the highest fibrosis density, which is (f + σ/2). For reentry initiation this percentage should be larger than 28%; therefore, we expect that for a very large value of l, the reentry in heterogeneous tissue will occur if f + σ/2 ≥ 28%. This line is shown with the dark blue colour in l = 64 mm (light blue points) are in agreement with this theoretical prediction.On the other hand, if we consider the case of a relatively large Overall, the phase diagrams demonstrate that a heterogeneous distribution increases the likelihood of arrhythmia onset relative to a homogeneous distribution. Furthermore, they show that the main mechanism of this dependency is the presence of localised tissue patches with larger level of fibrosis.The induced activation patterns can comprise either only a single spiral as in  or multiThe spatial patterns in 2718Because of the periodicity of the resulting patterns, it was possible to measure the period and investigate its dependency on the fibrosis distribution.f = 25% is given in l and saturates for l ≥ 20 mm. Similar to our earlier results, the excitation period for the small sized heterogeneity is the same as for homogeneous fibrosis with the same average level. To study what determines the saturated plateau level, we investigated how the period for a constant sized heterogeneity of l = 40 mm depends on the extent of heterogeneity σ and the average fibrosis level f. These results are given in f, so the rotation of the spiral waves is slowed down in the presence of fibrosis. This is consistent the earlier studies with14We investigated how the period of the activation patterns depends on the amount and distribution of fibrosis. The dependency of the period on the size of heterogeneity for a constant mean fibrosis level of σ/2  for the homogeneous case versus an average fibrosis level of f for the heterogeneous case. As (f + σ/2) corresponds to the maximal level of fibrosis in heterogeneous tissue, we can conclude that the period for heterogeneous fibrosis is mostly determined by the maximal local fibrosis level, or in other words, by the region where the spiral rotation is slowest.As the shapes of the graphs obtained for different heterogeneity values look similar to each other, we tried to characterise this similarity formally. We found that the graphs will reasonably well match one another if there are all shifted horizontally by σ/2 . This shAs we showed, despite irregular appearing of spatiotemporal wave patterns, the electrical activation patterns are highly periodic and regular. As the tissue is heterogeneous, these findings may indicate that fibrillation is driven by a single periodic source of excitation, and that the arrhythmia is of the mother rotor typeTo classify the type of the activation patterns, we first performed “local ablations”. For this we subdivided the 2D tissue into 64 equally sized squares of size 16 × 16 mm. During an individual local ablation experiment, we “removed” one of these squares, similar to performing a clinical point ablation procedure, replacing the active cardiomyocytes in this region of the tissue by passive “dead” cells.The results of applying this procedure for our representative examples are demonstrated in In the case of rotors, small, localised ablations are unlikely to remove them. Indeed to terminate spiral wave rotation it is necessary to connect the core of the rotor to the boundary of the mediumA few examples of the application of this line ablation procedure are given in 2, 150 cm2, 350 cm2, and 625 cm2. For a tissue size of 150 cm2 the probability of arrhythmia induction is approximately 50%, and from the blue-green ratio it follows that for the subset of cases that arrhythmia induction occurs 80% of them corresponds to mother rotor type fibrillation. For a smaller tissue of 40 cm2 we see that the probability of arrhythmia induction is significantly lower. In addition, it is no longer possible to induce multiple wavelet type fibrillation. If instead we consider larger sized tissues we observe the reverse phenomenon, with the probability of arrhythmia induction increasing and a larger fraction of induced fibrillation being of the multiple wavelet type. For example, for a tissue of 625 cm2, arrhythmia induction occurs in almost all cases, and in 75% of these cases it will be of the multiple wavelet type.These methods for determining the type of fibrillation can easily be implemented as an automatic procedure, allowing us to perform a large number of such ablation procedures and subsequent classifications. We used this approach to determine how fibrillation type depends on value of heterogeneity, heterogeneity size, and the tissue size. Note, that while the larger tissue sizes studied here appear to lie outside the physiologically relevant domain , we did not include tissue anisotropy in our current simulations. Adding anisotropy, that is, slower propagation in the transversal direction, is equivalent to an decrease in effective tissue size2, corresponds to an anisotropic tissue of size 10 cm × 5 cm = 50 cm2. On a similar note the 350 cm2 re-scale to 116 cm2 and the 625 cm2 to 207 cm2. We estimate the surface area of the left human ventricle to be around 160 cm2 (see Appendix A in Suppementary Information).Considering a typical propagation velocity anisotropy of 1:3, an isotropic tissue of size 10 cm × 15 cm = 150 cmFibrosis is an important risk factor for arrhythmogenesis since it affects propagation of the excitation wave. It was shown that the fibrosis texture, that is, the layout of fibroblasts, is a more important factor for the onset of arrhythmias than an average level. In this work, we investigated the contribution of the heterogeneity in the distribution of fibrosis in cardiac tissue on arrhythmia formation. We have shown that the heterogeneity of fibrosis promotes the onset of arrhythmias. This effect depends on both the spatial size and the degree of the heterogeneity: a larger size and larger degree of heterogeneity make formation of arrhythmias more probable.σ—l parametric space , most of arrhythmias were of the mother rotor type (2 (≈208 cm2 in the anisotropic case) we still observe mother rotor type fibrillation in 30% of the cases. Interestingly, the multiple wavelet type of fibrillation never occurred for small tissues of 40 cm2 (≈13 cm2 in the anisotropic case). Therefore, in most realistic cases, one would expect fibrillation to be of the mother rotor type. This may explain why ablations of rotors may remove atrial fibrillationThe strong periodicity of the activation patterns may indicate the presence of a mother rotor28tor type . Even foIn the current study, arrhythmias were induced by a burst pacing protocol. Another commonly used method of triggering arrhythmia is the S1S2 protocol. Our choice for the burst pacing was mainly because this way it is possible to see if arrhythmia can be induced for a particular fibrosis layout. In contrast, the S1S2 protocol always leads to arrhythmia initiation; fibrosis may only contribute to generation of additional excitation sources which may be due to a mechanism that is different from the mechanism of formation of the initial source. Nevertheless, the S1S2 protocol is required for studies targeting the transition from tachycardia to fibrillation which will be addressed in the subsequent research.Although the probability of the onset of arrhythmias is certainly affected by the stimulation frequency, we expect that this dependency does not significantly change the main results of our paper. Indeed, the results regarding the period of the arrhythmia and its nature were studied after pacing was stopped and should, therefore, not be affected by the initiation procedure. Clearly, as it is known that higher pacing frequencies independently lead to an increase in the likelihood of arrhythmia induction, an increase in the frequency of burst pacing is expected to shift the boundaries on the phase diagrams in 4041per se and ionic heterogeneity are additional factors enhancing the drift of spiral waves4345As it is the case for any modelling study, our model represents a simplified description of fibrotic cardiac tissue and as such it has several limitations. A first major model simplifications is to consider solely the effects of structural remodelling. Under many disease conditions the structural remodelling is accompanied by electrical remodelling. Often, the degree of fibrosis is in correlation with the degree of electrical remodelling as well as the extent of ionic heterogeneity. Both electrical remodelling in vitro, while others assume that this coupling occurs in cardiac tissue and is facilitated by tissue remodelling. The effect of this coupling requires additional study, outside the scope of our current research. However, we can suggest that in accordance with experiments byA second important limitation in the current model is the assumption that myocytes and fibroblasts are not electrically coupled, while some experimental studies indicate that this under some conditions this coupling be the caseIt should be noted, however, that while the above model simplifications can be interpreted as too strong, they, in fact, are crucial to allow us to study the effect of fibrosis as obstacles to wave propagation in isolation, without other tissue remodelling factors or other characteristics of fibrosis itself obscuring the results.A third limitation was that we considered only one type of fibrosis: diffuse fibrosis. Other types of fibrosis are: compact, interstitial, and patchyHeterogeneity in distribution of diffuse fibrosis was modelled by introduction of rectangular patches with different levels of local fibrosis. The square shape is not physiologically based, however, we assume that a particular shape of the fibrotic region has a little impact on arrhythmogenesis.A fourth limitation was the absence of anisotropy in our model. There are evidences that fibrosis results in increase of anisotropy ratio for a tissue48Finally, taking into account the 3D nature of ventricular cardiac tissue would allow for additional 3D mechanisms of spiral breakup that were previously shown to occur in relation to reduced excitability and fibrosisμm, a fibroblasts in vivo form membrane extensions, which increase its size and surface area2122In addition, we assumed that the fibroblasts and the myocytes are similar in size. In real tissue, the shape and the size of a fibroblast is a debate point. Although, an isolated fibroblast is a rounded cell with a diameter of 7–9 Overall, we expect that improving upon our model simplifications will mostly affect our outcomes in a quantitative manner, shifting the ratios between mother rotor versus multiple wavelet dynamics, or shifting the phase diagram boundaries separating reentry from non-reentry. We are confident that our qualitative results that heterogeneity of fibrosis is an additional proarrhythmic factor and that this proarrhythmicity stems mostly from the maximum local fibrosis densities occurring within the heterogeneous tissue, are also valid under alternative model settings.How to cite this article: Kazbanov, I. V. et al. Effects of Heterogeneous Diffuse Fibrosis on Arrhythmia Dynamics and Mechanism. Sci. Rep.6, 20835; doi: 10.1038/srep20835 (2016).


After AVEC, the HR standard deviation and variability  were significantly greater than after correction by the Polar tool . Furthermore, the HR data strings with deleted values seemed to be closer to the original data than were those with inserted means. We concluded that our method of error correction is more suitable for dog HR and HR variability than is the customized Polar error correction, especially because AVEC decreases the likelihood of Type I errors, preserves the natural variability in HR, and does not lead to a time shift in the data.Dog heart rate (HR) is characterized by a respiratory sinus arrhythmia, and therefore makes an automatic algorithm for error correction of HR measurements hard to apply. Here, we present a new method of error correction for HR data collected with the Polar system, including (1) visual inspection of the data, (2) a standardized way to decide with the aid of an algorithm whether or not a value is an outlier , and (3) the subsequent removal of this error from the data set. We applied our new error correction method to the HR data of 24 dogs and compared the uncorrected and corrected data, as well as the algorithm-supported visual error correction (AVEC) with the Polar error correction. The results showed that fewer values were identified as errors after AVEC than after the Polar error correction ( Heart rate (HR) and heart rate variability (HRV) are frequently used to measure stress responses in humans and nonhuman animals. The sympathetic activity is reflected by HR, with a high HR indicating high arousal. In contrast, HRV gives information about autonomic flexibility and reflects the capacity for regulated emotional responding  and of the German Society of Psychology (Ethische Richtlinien der DGPs und des BDP). Ethical review was done by the animal-welfare committee of the Faculty of Life Sciences, University of Vienna .Participation in our study was voluntary; dog owners were informed that they could stop the test situation at any time and were also asked to sign an information and consent form. Data collection was conducted according to the standards of the SD: 3.64 ± 1.28 years; mean weight ± SD: 34.25 ± 15.05 kg), measured during the second meeting, were used to investigate the effects of error correction procedures.Twenty-four owner–dog dyads participated in two separate meetings. HR and HRV data from intact male pet dogs  on a beat-to-beat basis, with an accuracy of ±1 ms. The Polar system includes a chest belt with electrodes to measure HR. The information is then transmitted via a wireless connection to a data-logger watch, where the data are saved and transferred to a computer for permanent storage and analysis.The owner applied the HR monitor belt to the dog’s chest and additionally fixed it with a standard dog harness. The Polar belt was worn for about 1 h. To improve signal transduction through the fur, an ultrasound gel (Henry Schein) was used to wet the skin. HR and HRV were first measured during 5 min of owner–dog play and 5 min of rest, which served as an adaptation period to the Polar belt. Then the dog experienced two staged threat situations in a counterbalanced order, one with and one without the owner present. After the second threat, the dog was pacified by the previously threatening person by talking to it in a friendly manner and offering cheese. Every test situation was followed by a recovery period of 15 min, during which the dog could move around freely. Error correction trials for HR and HRV were carried out on the data from the threat situations and the following 5 min of the recovery period.x/y plot , and k given by  Erf  nor between the orders of testing  in the 24 dogs analyzed. In Threat Situation 1, the HR data of one dog included more than 5 % errors; the HR data of three dogs had more than 3 % errors. For Threat 2, the HR data of one dog had more than 3 % errors after the AVEC method. Since one of the dogs from Threat 1 had to be excluded from further analysis, only the results from the second threat will be used for further comparisons.When using our AVEC method for the two test situations, the percentages of errors differed neither between the threats with and without the owner present  for HR data in dogs was mainly preferable over the standard Polar error correction tool because it decreases the likelihood of committing Type I errors, in two ways: first, AVEC results in significantly lower error rates, and thereby allows more data strings to be kept in the analysis, and second, it distorts the data variability significantly less, particularly with regard to HRV. In our present data set, the HR data of only two dogs would have been below the 5 % error rate with the Polar correction method, whereas with AVEC all 24 data sets remained in the analysis. Our results agree with conclusions from the literature indicating that HRV parameters especially are sensitive to only a few errors . Also, the researcher has to make the final decision regarding whether or not a value is an outlier. However, we consider this a strength rather than a weakness. Also, behavioral data, if they are available, can be taken in account to support a decision. For example, the researcher may wish to check whether extreme HR episodes are linked to specific behaviors or environmental events.Although our method was specifically developed for dog HR, it may also be appropriate for other species. For example, we have successfully applied it to humans and wolves, but the difference from the customized Polar error correction may be less than in the case of dogs.We conclude that systems such as Polar are an easy-to-use alternative to conventional ECG systems, which are expensive and mainly designed for lab settings (von Borell et al.,


This paper presents studies on the characteristics of gas molecular mean free path in nanopores by molecular dynamics simulation. Our study results indicate that the mean free path of all molecules in nanopores depend on both the radius of the nanopore and the gas-solid interaction strength. Besides mean free path of all molecules in the nanopore, this paper highlights the gas molecular mean free path at different positions of the nanopore and the anisotropy of the gas molecular mean free path at nanopores. The molecular mean free path varies with the molecule’s distance from the center of the nanopore. The least value of the mean free path occurs at the wall surface of the nanopore. The present paper found that the gas molecular mean free path is anisotropic when gas is confined in nanopores. The radial gas molecular mean free path is much smaller than the mean free path including all molecular collisions occuring in three directions. Our study results also indicate that when gas is confined in nanopores the gas molecule number density does not affect the gas molecular mean free path in the same way as it does for the gas in unbounded space. These study results may bring new insights into understanding the gas flow’s characteristic at nanoscale. For nanoscale gas flows, the flow flux is larger than the value based on classical fluid mechanics theory. Previous investigations ,2 found λ / H, where λ is the molecular mean free path and H is the representative length of the domain. The Knudsen number is widely used to divide the gas flow into four regimes: Kn < 0.001 is the continuum regime, 0.001 < Kn < 0.1 is the slip regime in which the Navier-Stokes equations with the slip boundary condition are appropriate, 0.1 < Kn < 10 is the transition regime and 10 < Kn is the free molecular regime. According to the definition of Knudsen number, the gas’s molecular free path is very important in judging the flow regime. However, the gas molecular mean free path at macroscale and not its effective mean free path is usually used to calculate the Kn number. Thus, it is important to study the effective molecular free path of gas at the nanoscale.When the system size becomes comparable to the mean free path of the molecules, the continuum assumption should break down and the classical Navier-Stokes (N-S) equations with no-slip boundary conditions cannot be directly applied . The KnuThe mean free path concept is also central to most models of the transport phenomenon in gases, such as the momentum, energy and mass exchanges which are essential to the gas flow and heat transfer character at nanoscale space. The most important difference between gas flow at nanopore and macroscale tubes is that the boundary of the nanopore limits the molecular motions in the radial direction. Another important characteristic of gas flow at nanoscale is the uneven gas molecule number density distribution in the whole flow space ,6. Simplr is described as:Stops  theoretiThe study shows that the mean free path of gas is reduced. Dongari and his co-workers  studied Experimental measurements are optimal for these studies; it is however, very difficult to carry out direct measurements on the gas molecular mean free path when it is confined in nanopores. Thus, theoretical analysis based on the kinetic theory is usually carried out to obtain the mean free path. Some assumptions should be made to calculate the gas free path in nanopores by kinetic theory because the interactions between gas molecules and the solid wall must be carefully considered. Besides theoretical calculation, simulation methods such as MD simulation allow us to assess some fundamental molecular properties such as the inter-molecular collisions and the distances between two collisions. MD is one very useful method to get the molecular free path at nanoscale space. This paper presents our numerical experiments on simple gas confined at nanopores, which are the most common and most easily processed. We highlight the profile of gas mean free path and radial mean free path and their relationships with the radius of the nanopores and the gas-solid interaction strength.c equals 0.2, 0.4 and 0.6, were applied to study its effects on the gas molecular mean free path.In this paper, we present our study results on nanopores of three different radii, which are 16.06, 21.4 and 32.0 nm. For each nanopore, three different gas-solid interaction strengths, that is Bk is the Boltzmann constant, d is the diameter of gas molecule and P is the gas pressure. Substituting the corresponding value in our studies we get that the mean free path of argon at the same temperature and density as our simulation cases but at macroscale space is 35.85 nm.According to the gas kinetic theory, the mean free path can be expressed as Equation (2) for gas at macroscale.ψ(r) is the possibility a molecule has a free path between r and r + dr, we get the mean free path of all molecules in the nanopore, which is represented by allλ in the following section.Through calculating the integration described as Equation (3), in which allλ to the kinetic theoretical value of the gas mean free path which is represented by λ0. From allλ is less than its counterpart value at macroscale space. 0) and the actual Knudsen number which is based on the effective mean free path. From allλ. According to allλ. A smaller nanopore has a larger ratio of surface area to its volume and the wall surface will take more constraints on the gas molecule’s movements so the mean free path naturally becomes smaller. When the gas-solid interaction becomes stronger more gas molecules will adhere to the wall of the nanopore; those molecules will experience shorter travel distance between two successive collisions and hence decrease the mean free path of all molecules in the nanopore.Another conclusion drawn from δ apart from the wall surface, their width is 0.125δ and others have the width of 4δ. The purpose of taking different widths is to obtain detailed information about the zone only 2δ apart from the wall surface. To get the gas molecular mean free path in each annulus the calculation of Equation (3) is done over each annulus. In this section we present the profile of mean free path including gas molecule’s collisions in all directions, which is expressed as all−directionλ.In this section, we present our study results on the profile of the gas molecular mean free path along the radius of the nanopore. To get the profile, we divide the nanopore into many concentric annuluses and the width of each annulus depends on its distance from the wall surface of nanopore. For annuluses which are only 2all−directionλ to the unconfined mean free path λ0 whose value is 35.85 nm. all−directionλ with its distance from the center of the nanopore. First, the largest value of the mean free path occurs at the center of the nanopore and it is about the radius of the nanopore. This phenomenon is physically reasonable because the molecules at the center of nanopore have the largest space to move; Second, the mean free path decreases along the nanopore’s radius direction but with different decreasing rates in different regions. For molecules in the region from the center of the nanopore to 2 nm apart from the wall surface, all−directionλ only declines about 5 nm. However, for molecules which are near the wall surface, their mean free path sharply decreases; Thirdly, like the mean free path of all gas molecules in the nanopore, the gas-solid interaction strength also affects the profile of all−directionλ. When the gas-solid interaction strength becomes stronger all−directionλ in the region that suffers no direct effects from the wall surface, becomes larger. This so-called region which suffers no direct effects from the wall surface in this paper means the zone in which gas molecules only experience gas-gas collisions and no collisions between molecules and the wall surface occurs. Stronger gas-solid interactions would cause more gas-wall collisions and hence decrease gas-gas collisions, which make all−directionλ in the region which suffers no direct effects from the wall surface become larger. This phenomenon can be further illustrated by the molecular collision times data recorded in our simulations. c is 0.2 the molecules in the region that suffers no direct effects from the wall surface experience the most gas-gas collisions, so we set it as the comparison standard. For the nanopores with the same radius the value in c is 0.2. Our simulation results are shown in all−directionλ within the zone of 0.7 nm apart from the wall surface. all−directionλ for different nanopores is nearly the same, which shows the gas-solid interaction strength plays a key role in determining all−directionλ at the wall surface.To further study the variation of mean free path near the wall surface of the nanopore, According to its definition, the value of the Knudsen number should have the same variation tendency with the mean free path. Substituting the mean free path to the definition of Kn, the line of Kn = 0.1 is shown in rλ in this paper is defined as the mean distance in the radial direction between gas molecule’s two successive collisions, which includes both gas molecule-gas molecule and gas molecule-wall surface collisions.In macroscale tubes, the gas molecular mean free path is isotropic due to a lack of surface effects on the molecule’s motion. But for a nanopore, the gas molecule’s motion is limited in the radial direction, so is the gas molecular mean free path still isotropic? In this section we will give our simulation results on the radial gas molecular mean free path in nanopores. The radial gas molecular mean free path expressed as rλ to the unconfined mean free path λ0 whose value is 35.85 nm. In the region far from the wall surface rλ, like all−directionλ, also increases with the distance from the wall surface, but it changes linearly with its distance from the center of the nanopore. This change tendency along the radius is different from that of all−directionλ. The linear fit of relationship between rλ / λ0 and r / R in the region far away from the wall surface is also presented in r / λλ0 = A + B × (r / R). The values of A and B only vary with the nanopore’s radius. For the same nanopore, even the gas-solid interaction strength is different from the relationship between rλ / λ0 and r / R can be described by the same function.rλ and all−directionλ. This phenomenon clearly illustrates that the gas molecules’ motions are anisotropic when they are confined in nanopores. The most important point is that the mean free path will affect the momentum and energy exchange, so it could be deduced that the momentum and energy exchange characteristics are also anisotropic in nanopores. The ratio of rλ to all−directionλ increases when the nanopore becomes larger but is not been affected by the gas-wall interaction strength. So it seems that the anisotropy of molecular transportation is only affected by the size of the nanopore.Comparing rλ is much smaller than that of all−directionλ in the same zone. Comparing rλ and all−directionλ show nearly the same change tendency with the distance from the wall surface. This phenomenon illustrates that in the zone directly affected by the wall surface the wall surface’s effects dominate both rλ and all−directionλ.It is well known that the gas number density shows an uneven distribution near the wall surface when it is confined in nanopores. The above results show that the gas molecular mean free path also varies with its distance from the center of the nanopore, so it is necessary to study the relationship between the gas number density and the gas molecular mean free path. In this paper we take the nanopore whose radius is 16.06 nm as example to study the relationship between the gas number density and the molecular mean free path. To carefully study both the variations of gas number density and molecular mean free path in the zone near the wall surface we give the details in the zone 0.7 nm apart from the nanopore’s wall surface in As far as the radial gas molecular mean free path is concerned, c is 0.2 as an example to compare the difference between simulation results when the body force is added on or absent from every gas molecule. rλ and all−directionλ. Similar results are also obtained for other nanopore radii that we studied. This result is physically sound because the gas molecular mean free path is only dependent on the gas molecules’ relative motions and has nothing to do with molecular mass motion, which is generated by the body force.In our simulation runs one constant force was added on every molecule in the system in order to generate macroscopic gas flow. In this section we will study the effects by the body force on our simulation results. We take the nanopore whose radius is 16.06 nm and the value of In our present paper, molecular dynamics simulations are used to study the gas molecular free path in nanopores. The physical model for MD simulations is illustrated in r is the distance between two molecules, the energy parameter for argon is ε = 1.67 × 10−21 J and δ is the molecular length scale with value of 3.405 × 10−10 m [The MD simulation is based on the integration of Newton’s second law by appropriate integral methods such as the Verlet method. In MD method, the force between two molecules is described by the potential function, for argon Equation (4) gives the function. 10−10 m .∆P is the pressure difference between two ends of the nanopore, r is the radius of the nanopore and N is the total number of gas molecules in the nanopore.The purpose of our studies on the gas molecular mean free path is to further understand the characteristics of gas flow at nanoscale, so the flow should exist. In this paper, the flow is generated by adding one constant body force on every molecule in the simulation domain. The value of body force is based on the following equation:ε = 2.82 × 10−21 J and δ = 3.405 × 10−10 m where c is one variable parameter which represents the strength of interactions between gas molecules and wall atoms, when the value of c is larger the gas molecule-wall atom interaction strength is stronger. So for gas molecules without interaction with the wall, the force on every gas molecule is described as Equation (7) and for gas molecules interacting with the wall, is described as Equation (8).The potential function between the gas and solid atoms also uses equation like Equation (4) that is:In our present studies, the initial positions of gas molecules are arranged according to FCC structure with the density of argon gas at the mean pressure of two ends of the nanopore at 300 K. The velocities of gas molecules were randomly initialized according to the Maxwell velocity distribution at 300 K and the total kinetic temperature is equal to the kinetic energy in the calculation domain:N is the total number of gas molecules in the nanopore, which ranges from 2124 to 8592 when the diameter of the nanopore varies from 32.12 to 64 nm in this paper. gasT is always equal to 300 K and Bk is the Boltzmann constant, m is the mass of gas molecule and vi,x, vi,y, vi,z are gas molecule velocities in the x, y and z directions. Values for the initial force acted on each gas molecule were determined in terms of their initial positions ir(0).In Equation (9) x direction. In the radial direction of nanopore, no special boundary conditions are applied to gas molecules, because the movements of gas molecules are restricted by the wall of the nanopore.Due to the limitation of computational ability, it is impossible to simulate all of the molecules in the real flow system, so periodic boundary condition is applied to the flow direction, that is the c∆t is time difference between two successive collisions and currentv is the current speed of the molecule.According to the definition of mean free path, each molecule’s travel distance between two successive collisions is required to be recorded. But when the periodic boundary condition is applied, it is not appropriate to directly measure the travel distance of molecule as the difference between its current position and the position of its last collision. Instead, we calculate the molecule’s travel distance between two successive collisions asψ(r), which is the possibility a molecule has a free path between r and r + dr. Once ψ(r) is obtained, the effective gas molecular mean free path equals the value described as Equation (3).The molecule’s travel distance between two successive collisions obtained through the above mentioned way is recorded to calculate the mean free path distribution colr. In the present paper, the value of colr is set as δ. It should be noted that the molecule’s collisions with wall surface are also considered, we also judge the gas molecule-wall collision happens when the distance between the molecule and wall surface is less than colr. The validation of the colr is tested through the calculation of the mean free path of unconfined argon gas presented in In calculating the gas molecular mean free path, how to determine the collision is taking place is very important. The L-J potential is used in our studies, we judge the collision to take place if two molecules are closer together than a distance −15 s.In our MD study, an equilibrium molecular dynamics simulation for gas with initially assigned number density at 300 K was conducted for the first 1,000,000 run steps and then the body force was added to every gas molecules in the nanopore. The temperature of gas needs to be controlled at 300 K, so the velocities of each gas molecules are controlled to maintain the desired temperature 300 K. In the present study, the Verlet method was used to solve the motion equation of all molecules in the simulation system and the atom list method was used to reduce simulation run time. The simulation run time step was set as 1.0 × 10To confirm the validity of our methodology for calculating the gas molecular mean free path using Equation (3), the unconfined argon gas molecular mean free path is calculated first. To simulate the unconfined gas, the cubic spatial geometry with periodic boundary conditions applied in three directions is adopted. To ensure the correctness of the simulation the side of the cubic is larger than the mean free path and two different cubic lengths are used to test the validity of our simulation results. These two different cubic lengths are 42.4 and 53.0 nm. The molecular density in the cubic is equal to the argon’s density at 0.2 MPa and 300 K.The MD simulations in the present paper are based on our own developed solver. To firstly test the validity of our MD solver, the total energy including potential energy and kinetic energy per molecule and temperature profile in the case of unconfined argon gas at 0.2 MPa is presented in ψ(r) and the choice of the value colr are valid.For the unconfined argon, our MD data relaxes to a mean free path value of 36.0 nm, which differs from the kinetic theoretical value only +0.55%. Our MD simulation also counts the number of multiple collisions and two body collisions, the results show that for unconfined argon gas at 0.2 MPa and 300 K the multiple collisions account for only 0.07% of total collisions, so the argon gas at 0.2 MPa can be regarded as ideal gas and the effects on the mean free path by the multiple collisions can be ignored. This illustrates that our method of calculating the mean free path through the integration of rλ and all−directionλ vary with the distance from the nanopore’s center. The lowest value of mean free path occurs at the wall surface of the nanopore. For the same nanopore, the relationship between rλ / λ0 and r / R can be described by the same linear function. Near the wall surface the gas-solid interaction plays a key role in determining both rλ and all−directionλ, the radius of the nanopore has only minor effects on them. For gas confined in nanopores, the gas number density does not affect the gas molecular mean free path in the same way as it does in the unbounded space.The gas molecular free path in naonpores with different radius and gas-wall interaction strengths was studied using molecular dynamics simulations and several conclusions have been obtained. The mean free path of all molecules in the nanopore is smaller than its counterpart value at macroscale space and is related with the nanopore’s radius and the gas-wall interaction strength. Smaller nanopore size and stronger gas-solid interactions would cause a shorter mean free path of all molecules. The radial gas molecular mean free path is smaller than the molecular mean free path including all direction’s collisions, which means the gas molecular motions in the nanopore are anisotropic. Both