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Lobo A, Hansen JK, Hansen LN, Kjær ED. Differences among six woody perennials native to Northern Europe in their level of genetic differentiation and adaptive potential at fine local scale. Ecol Evol. 2018;8:2231--2239. <https://doi.org/10.1002/ece3.3824>
1. INTRODUCTION {#ece33824-sec-0001}
===============
Phenology is important for fitness of perennial species. Higher spring temperatures are expected to prolong the growing seasons due to earlier bud burst (Dragoni et al., [2011](#ece33824-bib-0010){ref-type="ref"}; Menzel & Fabian, [1999](#ece33824-bib-0034){ref-type="ref"}; Menzel et al., [2006](#ece33824-bib-0035){ref-type="ref"}; Richardson et al., [2010](#ece33824-bib-0045){ref-type="ref"}, [2013](#ece33824-bib-0046){ref-type="ref"}; Vitasse, Delzon, Dufrêne, et al., [2009](#ece33824-bib-0059){ref-type="ref"}), while lack of synchrony between phenology and occurrence of spring frost events increases risk of damage to early flushing plants (Duputié, Rutschmann, Ronce, & Chuine, [2015](#ece33824-bib-0012){ref-type="ref"}). Yet again, raised temperatures with no frost events and changes in daily temperature fluctuations could also result in later bud burst and influence time between bud development and growth cessation (Chmura et al., [2011](#ece33824-bib-0008){ref-type="ref"}; Rohde, Bastien, & Boerjan, [2011](#ece33824-bib-0048){ref-type="ref"}; Way, [2011](#ece33824-bib-0060){ref-type="ref"}). Several studies have documented substantial variation among populations in their phenology reflecting their geographic origin (Alberto, Derory, et al., [2013](#ece33824-bib-0004){ref-type="ref"}; Chuine & Beaubien, [2001](#ece33824-bib-0009){ref-type="ref"}; Salmela, Cavers, Cottrell, Iason, & Ennos, [2013](#ece33824-bib-0049){ref-type="ref"}). It is therefore fair to assume that genetic variation within species combined with divergent selection has played an important role for the ability of many tree species to thrive in a large distribution range across strong environmental gradients (Brousseau et al., [2016](#ece33824-bib-0006){ref-type="ref"}; Kawecki & Ebert, [2004](#ece33824-bib-0023){ref-type="ref"}; Pluess et al., [2016](#ece33824-bib-0042){ref-type="ref"}). However, species differ in their response and spring and autumn phenology may show different patterns. A study based on 59 tree species from similar climatic clines thus showed a relative clear pattern with respect to bud set in autumn but less clear pattern in spring bud burst (Alberto, Aitken, et al., [2013](#ece33824-bib-0003){ref-type="ref"}). The expectation of local adaptation often leads restoration and conservation programs to focus on gene pools at the local scale rather than the regional scale (Stanturf et al., [2015](#ece33824-bib-0056){ref-type="ref"}), but only few studies have actually assessed the variation at the local scale (within \~100 km).
Fast climate change calls for high phenotypic plasticity (Valladares et al., [2014](#ece33824-bib-0057){ref-type="ref"}) and genetic variation in relevant traits in plant populations (Franks, Weber, & Aitken, [2014](#ece33824-bib-0014){ref-type="ref"}) in order to ensure continued adaptation. Many tree species have revealed phenotypic plasticity and demonstrated their ability to thrive and reproduce when transferred to new areas (Kjær, Lobo, & Myking, [2014](#ece33824-bib-0025){ref-type="ref"}; Lascoux, Glémin, & Savolainen, [2016](#ece33824-bib-0033){ref-type="ref"}; Myking, Rusanen, Steffenrem, Kjær, & Jansson, [2016](#ece33824-bib-0036){ref-type="ref"}; Rehfeldt, Ying, Spittlehouse, & Hamilton, [1999](#ece33824-bib-0043){ref-type="ref"}). However, there are indications of declining plasticity in phenology in spring with continued global warming (Fu et al., [2015](#ece33824-bib-0015){ref-type="ref"}) and increased frequency of warm periods in the early spring may lead to frost backlashes (Jönsson & Bärring, [2011](#ece33824-bib-0022){ref-type="ref"}). The temperatures are likely to change in the future, but the seasonal variations in day length remain the same. Species will therefore respond differently as regards phenology to temperature changes depending on the species‐specific importance of day length versus heat sum in controlling flushing time (Alberto, Aitken, et al., [2013](#ece33824-bib-0003){ref-type="ref"}; Caffarra & Donnelly, [2011](#ece33824-bib-0007){ref-type="ref"}; Fu et al., [2015](#ece33824-bib-0015){ref-type="ref"}; Vitasse, Delzon, Bresson, Michalet, & Kremer, [2009](#ece33824-bib-0058){ref-type="ref"}; Vitasse, Delzon, Dufrêne, et al., [2009](#ece33824-bib-0059){ref-type="ref"}).
Variation among individual genotypes in their phenology triggers selection at the population level in favor of the new conditions, if the variation is in fitness traits and expressed with moderate or high heritability (Alberto, Aitken, et al., [2013](#ece33824-bib-0003){ref-type="ref"}; Falconer & Mackay, [1996](#ece33824-bib-0013){ref-type="ref"}). Genetic differentiation within and among population is therefore important for the species ability to adapt to climate change (Kubisch, Holt, Poethke, & Fronhofer, [2014](#ece33824-bib-0029){ref-type="ref"}). However, natural selection is not the only evolutionary force, because the combined actions of natural selection, gene flow and genetic drift drive the level of genetic variation within and among populations in fitness traits (Nadeau, Meirmans, Aitken, Ritland, & Isabel, [2016](#ece33824-bib-0037){ref-type="ref"}). The realized patterns therefore depend on effective population sizes, force of natural selection and limitations in pollen and seed dispersal (Aitken & Whitlock, [2013](#ece33824-bib-0002){ref-type="ref"}; Savolainen, Pyhäjärvi, & Knürr, [2007](#ece33824-bib-0052){ref-type="ref"}). Natural selection tends to increase the level of genetic variation among populations, but reduce the variation within populations leading to local adaptation. Fragmentation into small populations can create drift that also increase variation among population variation, but without supporting local adaptation. Small populations can rather reduce fitness due to inbreeding, at least in outcrossing species. Pollination and seed dispersal across the landscape among populations will counteract the fragmentation, but seed and pollen dispersal across environmental gradients will also counteract natural selection and thereby reduce the ability of populations to adapt to specific local conditions (Kubisch et al., [2014](#ece33824-bib-0029){ref-type="ref"}). The level of gene flow is expected to differ among woody species depending on occurrence, landscape features, mating system, pollen, and seed dispersal vectors. We therefore expect that the degree of local differentiation---and the degree to which this reflect local adaptation---will differ among woody plant species depending on their major life‐history traits.
1.1. Objectives of this study {#ece33824-sec-0002}
-----------------------------
On the above background, the objectives of this study were to use data on spring phenology to Assess the degree to which woody perennials reveal fine‐scale genetic differences among geographically and climatic close (differences in mean annual temperature around 1°C) populations, and test whether differences (if any) influence the fitness revealing signs of local adaptation.Estimate levels of genetic variation within and among populations in spring phenology and compare the levels of population differentiation (*Q* ~ST~) and additive genetic variation (*V* ~A~) among species.
Based on the results, we discuss the implications for dynamic conservation of genetic resources and the ability of woody plant species to adapt to future climate conditions.
2. MATERIALS AND METHODS {#ece33824-sec-0003}
========================
Six broadleaved species in Denmark were studied, of which three are insect pollinated---Common dogwood (*Cornus sanguinea* L.), European Crab Apple (*Malus sylvestris* (L.) Mill.) and Glaucous Dog Rose (*Rosa dumalis* (Bechst) Boulay), while three are wind pollinated---Downy birch (*Betula pubescens* Ehrh.), Common Hazel (*Corylus avellana* L.) and Sessile oak (*Quercus petraea* (Matt.) Liebl.). We used data from progeny trials with offspring from open pollinated trees based on seed collected from putative natural populations across Denmark (Figure [1](#ece33824-fig-0001){ref-type="fig"}). Description of the field trials, location of populations, and climate information at the population site and information on families within each population are provided in Appendix [S1](#ece33824-sup-0001){ref-type="supplementary-material"}.
![Locations of original populations for the different species in the study are spread across different climate zones (here represented by temperature minimum in May \[*T* ~min~\])](ECE3-8-2231-g001){#ece33824-fig-0001}
2.1. Assessment of phenotypic data and estimation of variance components {#ece33824-sec-0004}
------------------------------------------------------------------------
We used spring phenology assessed as bud burst scores using a scale from 0 to 8, where class 0 was closed winter buds and class 8 was fully opened leaves. The time of assessment was adapted for each species so that the trees in the trials were approximately in the middle of their flushing time (average values of bud burst score between 3 and 5). This was to ensure maximum variation among individuals in their bud burst ranking. Bud burst for *B. pubescens, C. sanguinea, M. sylvestris,* and *R. dumalis* were assessed in spring 2010, while assessments of *C. avellana* and *Q. petraea* were from spring 2013. Height was measured in the year of phenology assessment except for *R. dumalis*.
All analyzes were made per species, because they were tested in different common garden trials. The variation among trees in their phenological stage was separated into variance and covariance components reflecting genetic differences among and within populations as implemented in the software ASReml (Gilmour, Gogel, Cullis, & Thompson, [2009](#ece33824-bib-0017){ref-type="ref"}) from the model described below.
Population values were predicted by application of mixed model (1), and genetic variance component corresponding to each of the random effects in the model was estimated from the analysis; $$Y_{\mathit{ijkl}} = \mu + B_{i} + P_{j} + \lambda_{\mathit{ij}} + f_{k(j)} + \tau_{\mathit{ikj}} + \varepsilon_{\mathit{ijkl}}$$
where *Y* ~*ijkl*~ is the trait measured for tree *l*, μ is the overall mean of the trait, *B* ~*i*~ is the fixed block effect, *P* ~*j*~ is the fixed population effect, λ~*ij*~ is the fixed population by block interaction, *f* ~*k*(*j*)~ is the random effect of family within population, τ~*ijk*~ is the random effect of plot *k*, and ε~*ijkl*~ is the residual. *Q* ~ST~ values were estimated from model (1) as well having population effects as random.
The significance of the genetic variances of traits within sites and genotype by environment interaction across sites was tested using the log likelihood ratio (Gilmour et al., [2009](#ece33824-bib-0017){ref-type="ref"}). The significance of the populations was tested using the Satterthwaite approximation (Satterthwaite, [1946](#ece33824-bib-0051){ref-type="ref"}) in the procedure GLM in the statistical software program SAS (SAS Institute Inc. [2011](#ece33824-bib-0050){ref-type="ref"}).
2.2. Quantitative genetic analysis {#ece33824-sec-0005}
----------------------------------
We estimated the additive genetic variance (*V* ~A~) as 4σ~f~ ^2^ and narrow sense heritability within sites as; $\binom{\frown}{h}^{2} = 4\sigma_{f}^{2}/{(\sigma_{f}^{2} + \sigma_{\tau}^{2} + \sigma_{\varepsilon}^{2})}$, where σ~f~ ^2^ is the estimated family variance, σ~τ~ ^2^ is the estimated plot variance, and σ~ε~ ^2^ is the estimated within plot variance. The phenotypic variance was estimated as $\binom{\frown}{V}_{P} = \sigma_{f}^{2} + \sigma_{\tau}^{2} + \sigma_{\varepsilon}^{2}$. Families were considered to represent groups of half‐sibs. This assumption will overestimate the additive genetic variance, heritability, and expected response to selection, if the offspring within progeny groups on average are more related than half‐sibs (Falconer & Mackay, [1996](#ece33824-bib-0013){ref-type="ref"}). While the assumption of half‐sibs may provide a good fit for most trees species (Kjær, McKinney, Nielsen, Hansen, & Hansen, [2012](#ece33824-bib-0026){ref-type="ref"}; Larsen & Kjær, [2009](#ece33824-bib-0032){ref-type="ref"}), the situation may be more complicated for *R. dumalis,* where polyploidy may be present as the variance between families is biased due to a fraction of the dominance genetic variance, even if the families consist of pure half‐sibs (Roberts, Gladis, & Brumme, [2009](#ece33824-bib-0047){ref-type="ref"}).
The degree of differentiation among populations was estimated as the *Q* ~ST~ values (Spitze, [1993](#ece33824-bib-0055){ref-type="ref"}) using the formula;$$\binom{\frown}{Q}_{ST} = \binom{\frown}{V}_{POP}/{(\binom{\frown}{V}_{POP} + 2\binom{\frown}{V}_{A})}$$
where $\binom{\frown}{V}_{POP}$ is the estimated variance between populations, and $\binom{\frown}{V}_{A}$ is the estimated additive genetic variance.
Variance components, narrow sense heritability estimates, *Q* ~ST~ estimates, and their standard errors were estimated from model (1) using the software ASReml (Gilmour et al., [2009](#ece33824-bib-0017){ref-type="ref"}).
2.3. Support to the hypothesis of location adaptation {#ece33824-sec-0006}
-----------------------------------------------------
Climate data for the locations of origins were estimated with the ClimateEU v4.63 software package following the methodology described by (Hamann, Wang, Spittlehouse, & Murdock, [2013](#ece33824-bib-0019){ref-type="ref"}). Weighted regression analysis to test relationship between phenology of populations and climate at the population sites was carried out using procedure REG in SAS (SAS Institute Inc. [2011](#ece33824-bib-0050){ref-type="ref"}) having the predicted values of populations as dependent variable and climate variables as explanatory variables. One divided by the error variance for the predicted population values were used as weights. The climate variables tested for bud burst were monthly minimum temperatures and differences between monthly maximum and minimum temperatures in March, April, May and June, because these data are expected to provide good proxies for the risk of frost occurring after flushing. A backward selection approach (5% level) was used for the selection of climate variables that could best explain the variation in flushing time. The relationship between geographic distances and difference in average budburst was tested by a Mantel test as implemented in R version 3.2.2 (Dray & Dufour, [2007](#ece33824-bib-0011){ref-type="ref"}).
2.4. Fitness effects of spring phenology {#ece33824-sec-0007}
----------------------------------------
Genetic correlations between bud burst and height growth were estimated based on individual tree data to assess whether the height as a proxy for realized fitness in the present climate varied with the phenology. Genetic correlations could be estimated between survival and phenology based on plot means, because the plants were grown in family plots in all trials (see Appendix [S1](#ece33824-sup-0001){ref-type="supplementary-material"}). Genetic correlations were estimated according to Falconer and Mackay ([1996](#ece33824-bib-0013){ref-type="ref"}) as $\binom{\frown}{r}_{g} = \frac{\sigma_{\mathit{fxy}}}{\sqrt{\sigma_{\mathit{fx}}^{2}\sigma_{\mathit{fy}}^{2}}}$ , where σ~*fxy*~ is the family (within population) covariance between trait *x* and *y*, σ~*fx*~ ^2^ is the family (within population) variance for trait *x*, and σ~*fy*~ ^2^ is the family (within population) variance for trait *y*.
3. RESULTS {#ece33824-sec-0008}
==========
3.1. Presence of genetic variation within and among populations in bud burst {#ece33824-sec-0009}
----------------------------------------------------------------------------
Genetic variation both within and among populations was significant in all six species, but with large differences among the species. The additive genetic variance for bud burst (*V* ~A~) thus ranged from 0.07 in *C. sanguinea* to 0.34 in *Q. petraea* (Table [1](#ece33824-tbl-0001){ref-type="table-wrap"}), while the level of differentiation among populations (*Q* ~ST~) ranged from 0.04 in *M. sylvestris* to 0.27 in *C. sanguinea* (Table [2](#ece33824-tbl-0002){ref-type="table-wrap"}). The three wind‐pollinated species *B. pubescens, C. avellana* and *Q. petraea* showed the highest level of additive variance, while the insect‐pollinated species *C. sanguinea* and *R. dumalis* showed the highest level of population differentiation as measured by the *Q* ~ST~ (Figure [2](#ece33824-fig-0002){ref-type="fig"}).
######
Family variation and genetic parameters for bud burst in different species
Species Trait Mean *p*‐Value family *V* ~A~ *V* ~P~ *h* ^2^ *SE*
-------------------- ------------------------- ------ ------------------ --------- --------- --------- ------
*Betula pubescens* Bud burst 24 April 2010 3.27 .003 0.15 0.30 0.50 0.12
*Cornus sanguinea* Bud burst 6 May 2010 3.81 \<.0001 0.07 0.27 0.24 0.11
*Corylus avellana* Budburst 24 April 2013 3.09 \<.0001 0.13 0.27 0.47 0.09
*Malus sylvestris* Bud burst 28 April 2010 4.05 \<.0001 0.03 0.10 0.31 0.08
*Quercus petraea* Bud burst 24 May 2013 4.64 \<.0001 0.34 0.58 0.58 0.26
*Rosa dumalis* Bud burst 26 April 2010 3.58 \<.0001 0.09 0.21 0.41 0.14
*V* ~A~ = additive genetic variance, *V* ~P~ = *p* variance, *h* ^2^ = narrow sense heritability, *SE* = standard error for *h* ^2^.
John Wiley & Sons, Ltd
######
Level of population differentiation for bud burst in different species
Species Trait *Q* ~st~ *SE* *p*‐Value population Max. pop. diff.[a](#ece33824-note-0002){ref-type="fn"}
-------------------- ------------------------- ---------- -------- ---------------------- --------------------------------------------------------
*Betula pubescens* Bud burst 24 April 2010 0.13 0.08 \<.001 0.71
*Cornus sanguinea* Bud burst 6 May 2010 0.27 *0.14* \<.001 0.88
*Corylus avellana* Budburst 24 April 2013 *0.12* *0.05* \<.0001 0.86
*Malus sylvestris* Bud burst 28 April 2010 0.04 *0.03* .02 0.28
*Quercus petraea* Bud burst 24 May 2013 0.13 0.09 .0003 2.45
*Rosa dumalis* Bud burst 26 April 2010 0.18 0.09 \<.001 1.52
Maximum difference between populations in bud burst score.
QST = quantitative genetic differentiation among populations in bud burst, SE = standard error for QST
John Wiley & Sons, Ltd
 burst insect‐pollinated species,  wind‐pollinated species](ECE3-8-2231-g002){#ece33824-fig-0002}
3.2. Relationship of phenology with growth, fitness, distance between populations and climate at original population site {#ece33824-sec-0010}
-------------------------------------------------------------------------------------------------------------------------
The regression between bud burst of populations and population site minimum temperatures in March were significant (*p \< .05*), or close to significant (*p = .05*) for *B. pubescens* and *C. avellana* (Figure [3](#ece33824-fig-0003){ref-type="fig"}), and the regression between bud burst of populations and population site minimum temperatures in May was close to significant (*p = .05*) for *Q. petraea* (Figure [3](#ece33824-fig-0003){ref-type="fig"}). This corresponds to the general time for bud burst, where *B. pubescens* and *C. avellana* flush before *Q. petraea* in Denmark and indicate that spring frost event has played an important role in developing the pattern of variation among population in these species. Similar significant correlations were not observed in any of the three insect‐pollinated species. Mantel tests for correlation between geographic distance and difference in bud burst at population were nonsignificant in all the species (Figure [4](#ece33824-fig-0004){ref-type="fig"}). Survival was not significantly correlated to bud burst data in any of the species evaluated, and a significant additive genetic correlation between height and budburst was only observed in *B. pubescens (rg = *0.35; *SE 0.23*).
{#ece33824-fig-0003}
{#ece33824-fig-0004}
4. DISCUSSION {#ece33824-sec-0011}
=============
Our results show that fine‐scale local genetic differentiation in fitness traits such as phenology indeed can be present and to a larger extent than previously anticipated in the Danish gene conservation strategy (Graudal, Kjær, & Canger, [1995](#ece33824-bib-0018){ref-type="ref"}). However, the analysis across the six different species did not provide unique patterns of local adaptation, as the absence of significant regression between bud burst time and spring temperature in the three insect‐pollinated species suggests that the variation among population is not always simply reflecting local adaptation. It is rather more likely a result of natural selection and neutral processes simultaneously acting as predicted by theory (Nadeau et al., [2016](#ece33824-bib-0037){ref-type="ref"}).The fact that populations in all studied six species were significantly differentiated in their spring phenology suggests that local populations of woody perennials more often than not are genetically differentiated, and this can be the case even if they only are separated by 10 to 35 km (Figure [4](#ece33824-fig-0004){ref-type="fig"}) and growing in areas with low variation in altitude and where the spring temperature varies only between 1°C and 2.5°C (Figure [3](#ece33824-fig-0003){ref-type="fig"}). This finding should be compared to the prediction of Northern Europe becoming 2--4°C warmer by the year 2100 (Nikulin, Kjellström, Hansson, Strandberg, & Ullerstig, [2011](#ece33824-bib-0039){ref-type="ref"}). However, the observed high level of additive genetic variation in the studied species indicates that the native populations of woody species do possess substantial evolutionary potential. Divergent natural selection is possible within few generations, if the timing of phenology creates large differences in fitness. This is supported by the fact that the observed patterns in our study for *B. pubescens*,*C. avellana,* and *Q. petraea* (Figure [3](#ece33824-fig-0003){ref-type="fig"}) are at least partly due to past natural selection in favor of local adaption.
The presence of local adaptation and evolutionary potential in woody species are often attributed to their life‐history traits including pollination mechanisms and associated genetic processes, and it is therefore interesting that the patterns were different for the wind‐ and insect‐pollinated species in our study. The result that the three wind‐pollinated, continuous distributed species (*B. pubescens, C. avellana, Q. petraea)* were characterized by having high levels of genetic variation within populations (right circle of Figure [2](#ece33824-fig-0002){ref-type="fig"}) supports the general assumption that wind‐pollinated, widely distributed species have allocated genetic variation within populations to a larger extent than insect‐pollinated species with small and scattered populations (Petit & Hampe, [2006](#ece33824-bib-0041){ref-type="ref"}; Smith & Donoghue, [2008](#ece33824-bib-0053){ref-type="ref"}). The results correspond to the expectation that long‐distance gene flow through wind pollination will maintain connectivity among populations (lowering *Q* ~ST~) and counteract loss of genetic variation (maintaining high *h* ^2^) within populations (Sork, [2016](#ece33824-bib-0054){ref-type="ref"}). High levels of gene flow among populations across landscapes have been reported in the wind‐pollinated *Quercus* (Gerber et al., [2014](#ece33824-bib-0016){ref-type="ref"}). The insect‐pollinated *C. sanguinea* was in the opposite corner of *Q. petraea* in Figure [2](#ece33824-fig-0002){ref-type="fig"}, reflecting a much higher proportion of genetic variation located among populations. *C. sanguinea* in general occurs in scattered and small populations in Denmark (Ødum, [1968](#ece33824-bib-0040){ref-type="ref"}). Seed dispersal by birds (Krüsi & Debussche, [1988](#ece33824-bib-0028){ref-type="ref"}) may generate long‐distance gene flow across the landscape, but the observed level and distribution of genetic diversity in the present study point toward genetic drift rather than natural selection as important driver behind the observed population differentiation. The true difference between *Q. petraea* and *C. sanguinea* in level of genetic variation may be even higher, because the assumption of half‐sib relationship within seed from single tree collection is likely to create a bias toward overestimation of *V* ~A~ in the rose as discussed in M&M above. *M. sylvestris* is again a different case in Figure [2](#ece33824-fig-0002){ref-type="fig"}, with low level of genetic differentiation but also relatively low additive genetic variance. Pollinating insects have been reported to visit primarily trees in immediate vicinity, but long‐distance pollination events are also likely to occur (Larsen & Kjær, [2009](#ece33824-bib-0032){ref-type="ref"}; Reim et al., [2015](#ece33824-bib-0044){ref-type="ref"}) and seeds moved by birds and deer that feed on the wild apples also create long‐distance gene flow (Larsen & Kjær, [2009](#ece33824-bib-0032){ref-type="ref"}). Studies based on neutral SSR markers revealed low level of population differences within Denmark (*F* ~ST~ = 0.03) (Larsen, Asmussen, Coart, Olrik, & Kjær, [2006](#ece33824-bib-0031){ref-type="ref"}), which is very close to the estimated *Q* ~ST~ = 0.04 for flushing time in the present. Therefore, the observed variation among populations in *M. sylvestris* may have been generated by random processes without significant effects of divergent selection for flushing time.
The studied populations have in general maintained a high level of genetic variation and thereby possess a high ability to adapt to new conditions if exposed to natural selection in favor of---or against---early flushing. The question is how fast species can adapt under the anticipated rapid increase in temperature over the next 100 years, and how important such changes are for the fitness of the species. In *B. pubescens*, the maximum difference in minimum temperature in March of the populations tested in this study was 2.4°C (Figure [3](#ece33824-fig-0003){ref-type="fig"}). Even with a conservative estimate of a generation time of 50 years (Hynynen et al., [2010](#ece33824-bib-0021){ref-type="ref"}), the results suggest that *B. pubescens* possesses the resilience at the population level to change phenology corresponding to the predicted increase in temperature. A similar conclusion can probably be drawn for *Q. petraea* also (Figure [3](#ece33824-fig-0003){ref-type="fig"}). For many of the other studied species, the generation time is possibly less than 20 years (e.g., *R. dumalis*,*M. sylvestris*,*C. avellana,* and *C. sanguinea*), and for these species, it therefore also seems reasonable to assume that populations can evolve in spring phenology with a speed that can match the predicted increase in temperature. The response will only be realized if early bud burst has substantial influence on fitness (survival and reproduction) or because of directional selection implemented in domestication programmes. The estimates in the present study refer to the latter situation, because the heritability was estimated in managed progeny test and the realized heritability in natural population may be substantially lower due to higher environmental and developmental heterogeneity. In Denmark, all major woody plant species are included in domestication programs based on breeding seed orchards in order to support effective and rapid selection for continued fitness (Kjær et al., [2009](#ece33824-bib-0024){ref-type="ref"}).
Our finding on genetic differences among geographically close populations (cf. Table [2](#ece33824-tbl-0002){ref-type="table-wrap"}) supports that selectively divergent phenotypes can be maintained even if there are options for gene flow among these populations as suggested by Aitken and Bemmels ([2016](#ece33824-bib-0001){ref-type="ref"}). Kremer et al. ([2012](#ece33824-bib-0027){ref-type="ref"}) argue that the positive effect of gene flow on adaptation of trees to climate change is larger than the negative effect due to the introduction of genetic variation in adaptive traits. Species are different with respect to their genetic differentiation within and among their populations as revealed in our study. Assisted migration for genetic enrichment can serve as a supplement to genetic management based on native populations; especially, the species associated with small population sizes and limited gene flow. Populations at the low latitudinal limit of the species range have in general maintained high biological diversity over time as shown for, for example, *Abies alba* (Bergmann, Gregorius, & Larsen, [1990](#ece33824-bib-0005){ref-type="ref"}; Larsen, [1986](#ece33824-bib-0030){ref-type="ref"}) and southern populations should therefore be considered as a source of gene pool for assisted migration of genotypes under the future climate predictions (Hampe & Petit, [2005](#ece33824-bib-0020){ref-type="ref"}). The populations in the present study are close to the northern distribution range of the species (except for *B. pubescens*) and genetic diversity may therefore be lower compared to populations closer to the refugial areas. But we still observed substantial level of genetic variation in the fitness‐related trait. For species occurring in extremely small fragmented populations assisted migration at a more localized landscape level may still be desirable to counteract inbreeding within these populations due to random drift. Adaptive potential of a species is not only determined by the presence of genetic variation within and among populations as studied here, but also by the presence of phenotypic plasticity in adaptive traits (Nicotra et al., [2010](#ece33824-bib-0038){ref-type="ref"}). Final conclusions on the adaptive potential of species to climate change will therefore also depend on the role of plasticity in local adaptation in the studied species.
DATA ACCESSIBILITY {#ece33824-sec-0013}
==================
Data for this study is available at University of Copenhagen---Electronic Research Data Archive (UCPH‐ERDA).
CONFLICT OF INTEREST {#ece33824-sec-0014}
====================
None declared.
AUTHOR CONTRIBUTION {#ece33824-sec-0015}
===================
Albin Lobo was responsible for the design of study, analysis and interpretation of data, and writing of the manuscript. Jon Kehlet Hansen was responsible for supervision of the work, design of study, and writing and approving of manuscript. Lars Nørgaard Hansen was responsible for field data collection, and revising and approving the final version of the manuscript before submission. Erik Dahl Kjær was responsible for overall supervision of the study, conception and design of work, data interpretation, and writing and final approval of the manuscript.
Supporting information
======================
######
######
Click here for additional data file.
We would like to thank the Danish Nature Agency for providing access to all trials and supporting the huge effort of data collection. The authors would also like to thank the Villum Foundation for financial support for the data collection and the analysis as part of the Trees for future forests project (VKR‐023063). We also thank Dr. Lene Rostgaard Nielsen for the critical reading of this manuscript.
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Introduction {#s1}
============
Understanding cell migration mechanisms is a critical issue in many biophysical phenomena, including angiogenesis, tumor growth, metastasis, and wound healing [@pcbi.1002926-Condeelis1]--[@pcbi.1002926-Li1]. Cell migration is a complex multifaceted process, triggered by chemotaxis and haptotatic responses from the extracellular environment [@pcbi.1002926-Lamalice1]. Initially, a thin lamellipodium protrudes due to actin polymerization at the leading edge, followed by actin depolymerization at the lamellipodium base [@pcbi.1002926-Insall1]--[@pcbi.1002926-Watanabe1]. Focal adhesions (FAs) are assembled between the lamellipodium base and the extracellular matrix (ECM). FAs are composed of FA molecules (such as FAK, paxillin, vinculin, Zyxin, VASP, and talin), and transmembrane proteins, especially integrins α~v~β~3~ and α~v~β~5~ that link the ECM to the cytoskeleton via FA molecules [@pcbi.1002926-Critchley1], [@pcbi.1002926-Wozniak1]. Afterwards, contractile bundles of actin filaments, called stress fibers (SFs), extend from nascent FAs and some of which connect to the nucleus [@pcbi.1002926-Salmon1]. The corresponding motor activity exerts force on the FA\'s fore and aft [@pcbi.1002926-Kaverina1], enabling the generation of a traction force and the release of FAs in the rear of the cell, creating the cell body\'s forward movement.
The following individual processes of these steps of cell migration have been studied extensively in the literature: actin polymerization and depolymerization [@pcbi.1002926-Ponti1]--[@pcbi.1002926-Watanabe1], focal adhesion dynamics [@pcbi.1002926-Sarvestani1], [@pcbi.1002926-Gallant1], and motor activity of contractile myosin [@pcbi.1002926-Kaunas1], [@pcbi.1002926-Stachowiak1]. Furthermore, both experiments and computational models from those prior works mostly involve 2-dimensional migration on a flat substrate. However, it still remains a challenge to elucidate how these mechanisms work together to mimic 2-D cell migratory behaviors, which have been observed in existing experimental works. The current work is motivated by two experimental works; one on Chinese Hamster Ovary (CHO) cell migration on 2-D ([Figure S1](#pcbi.1002926.s001){ref-type="supplementary-material"}-A) fibronectin coated substrate [@pcbi.1002926-Palecek1], and the other on cells spreading on 2-D ([Figure S1](#pcbi.1002926.s001){ref-type="supplementary-material"}-B) fibronectin coated micropatterns on chips [@pcbi.1002926-Tseng1]. Cell migration experiments have indicated that three separate variables, such as substratum ligand density, cell integrin expression level and integrin--ligand binding affinity, significantly affect changes in cell migration speed. For example, when cells migrate on various fibronectin coating concentrations, the cell migration speed takes a maximum at a particular ligand density (∼1140 molecules/µm^2^) with a biphasic curve [@pcbi.1002926-Palecek1]. On the other hand, cell spreading experiments have revealed that interactions between a cell\'s cytoskeleton and micropatterned geometries impinge on cell morphology and mechanics [@pcbi.1002926-Tseng1]. For example, when cell spreading occurs on a crossbow pattern, the cell exhibits locally high traction forces at three corners of the pattern, which may be due to concentrated ventral SFs.
Explaining complex interactions with 3-D ECM structure ([Figure S1](#pcbi.1002926.s001){ref-type="supplementary-material"}-C&D) entails a proper model mechanism of cell spreading because the cell morphology in 3-D ECM is strikingly different from that on 2-D ECM surfaces as the cell is elongated with the highest directionality and highest velocity of migration in 3-D ECM, but the cell forms peripheral lamellae with an increased random migration on 2-D plastic or fibronectin-coated substrates [@pcbi.1002926-Pankov1]. To this end, we have built a computational 3-D cell migration model on 2-D curved ECM surfaces and discovered that the cell migration speed differs depending on the diameter of a sprout, and explained the mechanism [@pcbi.1002926-Kim1]. It is interesting to note that there is an optimal sprout diameter that creates the highest speed of cell migration. In a similar way as on 2-D curved surfaces, we first aim to look at 3-D cell migration model on 2-D planar surfaces with various fibronectin coating concentrations to understand relationship between the migratory speed and ligand surface density. After verifying our 3-D model with 2-D cell migratory mechanism, we then aim to look at 3-D cell spreading model on various 2-D fibronectin-coated patterns. This entails a) deformation mechanics of both cell membrane and nucleus, b) 3-D interactions between transmembrane integrins and ECM ligands, leading to focal adhesion formation, c) SF formation and traction generation, and d) lamellipodium protrusion at the leading edge of the cell. Integration of these key mechanisms is pivotal for elucidating the aforementioned migratory and spreading behaviors.
Several prior works have incorporated multiple force-generating systems in their cell migratory models [@pcbi.1002926-Kapustina1]--[@pcbi.1002926-Walcott1]. These works, however, have considered only frictional forces with the substrate rather than focal adhesion (FA) dynamics [@pcbi.1002926-Wong1], [@pcbi.1002926-Cirit1], which generate a mechanical traction force due to a gradient in degraded ligand matrix density during the formation and rupture of ligand-receptor bonds [@pcbi.1002926-Sarvestani1], [@pcbi.1002926-Wong1], interplay between Rac-mediated membrane protrusion and adhesions at the leading edge [@pcbi.1002926-Cirit1]. To explain these mechanisms, a model having ligand-receptor bonds distributed across the cell membrane is necessary. Thereby, we have applied FA dynamics to our cell migratory model. Furthermore, our 3-D computational cell spreading model differs from other existing 2-D models [@pcbi.1002926-Novak1]--[@pcbi.1002926-Grosberg1] in that we incorporate aforementioned FA dynamics, cell membrane and nuclear remodeling, actin motor activity, and lamellipodia protrusion. Additionally, our model can predict 3-D spatiotemporal behavior of cell spreading on 2-D micropatterns as well as spatiotemporal distribution of two kinds of actin stress fibers (SFs), one is a SF connected to the nucleus and the other is a ventral SF, in 3-D intracellular domain.
To our knowledge, neither a cell migration or a spreading model integrating focal adhesion dynamics, cell membrane and nuclear remodeling, actin motor activity, and lamellipodia protrusion has been published that reflects 3-D spatiotemporal dynamics of both cell spreading and migration, all interfaced with a 2-D planar surface and fibronectin coated patterns. In the following, numerical simulations demonstrate the diverse migration and spreading behaviors in relation to the various ligand densities of migrating 2-D surfaces and micropatterns, respectively.
Results {#s2}
=======
First, we aim to verify our model against 2-D cell migration on fibronectin coated substrates under five different fibronectin coating concentrations [@pcbi.1002926-Tseng1]. After this verification, we further aim to verify our model against 2-D cell spreading on micro-patterned structures. We simulate binding kinetics between integrin receptors and extracellular matrix protein ligands (eg. collagen, fibronectin and laminin), model the formation of SFs, and predict how the forces acting on the cell deform the nucleus and the cytoskeleton, resulting in diverse patterns of the cell profile and migratory motion. Simulations of cell migration and spreading were performed respectively for five different ligand surface densities on the planar surface and three different fibronectin coated micropatterns. Fibronectin was considered for both those two sets of simulations. Fibronectin ligand surface densities are summarized in [Table 1](#pcbi-1002926-t001){ref-type="table"}.
10.1371/journal.pcbi.1002926.t001
###### Ligand surface density (Fibronectin).
{#pcbi-1002926-t001-1}
Cell migration Cell spreading
-------------------------------------------- ---------------- ---------------- ----- ------ ------ -----
Ligand surface density \[molecules/µm^2^\] 19.4 192 568 1140 1522 475
The molecular mass of Fibronectin is 480 kDa, the corresponding ligand surface density was converted using the relationship between plating concentration and ligand surface density of Fibronectin [@pcbi.1002926-Rajagopalan1].
At the initial state of each simulation, both cell and nuclear membranes were assumed to be round. Since the migration model is stochastic, simulations were repeated multiple times from the same initial conditions. [Table 2](#pcbi-1002926-t002){ref-type="table"} lists all the parameters used for the simulations with numerical values and their sources.
10.1371/journal.pcbi.1002926.t002
###### List of simulation parameters.
{#pcbi-1002926-t002-2}
Parameter Definition Value Sources
----------- ------------------------------------------------------------------------------------------------- ------------ -----------------------------------------------------
*A* Area \[µm^2^\]
Area of the *i*-th surface of the cell membrane \[µm^2^\]
Area of the *i*-th surface of the nucleus \[µm^2^\]
*A~L~* Equilateral triangular area of ligands surface element \[µm^2^\] 0.243 Current work
*A~SF~* Averaged SFs\' sectional area \[µm^2^\] 0.196 [@pcbi.1002926-Lu1]
*C~c~* Friction coefficients associated with the energy dissipation at the integrin node \[N s m^−1^\] 0.001 [@pcbi.1002926-Kapustina1], [@pcbi.1002926-Drury1]
*C~n~* Friction coefficients associated with the energy dissipation at the nuclear node \[N s m^−1^\] 0.001 [@pcbi.1002926-Kapustina1], [@pcbi.1002926-Drury1]
*F* Force \[N\]
*E* Elastic energy \[pJ\]
*E~SF~* Young\'s modulus value of SFs \[kPa\] 230 [@pcbi.1002926-Deguchi2]
*L* Length
Length of the *i*-th line on the surface of the cell membrane \[µm\]
Length of the *i*-th line on the surface of the nucleus \[µm\]
*L~b~* Stretched length of bonds between receptors and ligands
Length of the *i*-th single unit of SFs at the present time \[nm\]
Length of the *i*-th single unit of SFs at the previous time \[nm\]
*~N~* Number of nodes at each membrane 549 Current work
Number of contractile compartments in the *i*-th SFs
*P* Probability
*W* Total stored elastic energy
*c~L~* Ligand density on the lumen \[molecule µm^−2^\]
Distance between *i*-th integrin and *j*-th nuclear nodes
*h~c~* Critical height \[nm\] 300 Current work
*h~p~* Height from the surface to the *i*-th integrin node \[nm\]
*k~f~* Forward reaction rate \[molecule^−1^ s^−1^\] 1.0 Current work
Effective spring constant of area elements of the cell membrane \[N/m\] 1.0×10^−4^ [@pcbi.1002926-Tsubota1]
Effective spring constant of line elements of the cell membrane \[N/m\] 5.0×10^−5^ [@pcbi.1002926-Drury1], [@pcbi.1002926-Honarmandi1]
Effective spring constant of area elements of the nucleus \[N/m\] 1.0×10^−4^ [@pcbi.1002926-Tsubota1]
Effective spring constant of line elements of the nucleus \[N/m\] 5.0×10^−3^ [@pcbi.1002926-Tsubota2]
Effective spring constant of ligand-receptor bond \[pN/nm\] 1.0 [@pcbi.1002926-Dembo1]
*k~on~* Kinetic association rate \[s^−1^\]
*k~off~* Kinetic dissociation rate \[s^−1^\]
Kinetic dissociation rate at an unstressed state \[s^−1^\] Current work
Effective stiffness of the *i*-th single unit of SFs \[N/m\]
*n~b~* Number of bonds between receptors and ligands
Unit normal vector at the *i*-th integrin node
Unit normal vector at the local surface of the lumen
*t* Time \[s\]
***v*** Velocity vector \[nm/s\]
*v~m~* Sliding rate of non-muscle myosin II on the actin filaments \[nm/s\] [@pcbi.1002926-Ruppel1]--[@pcbi.1002926-Lodish1]
***x*** Location vector \[µm\]
*x~L,i~* Root of ligand-receptor bonds on the local surface of the lumen \[nm\]
***λ*** Equilibrium distance of an integrin \[nm\] 30 [@pcbi.1002926-Kanchanawong1]
**Sup**
*D* Drag or friction
*E* Elastic
*FA* Focal adhesion
*SF* Stress fiber
*c* cytoskeleton
*n* nucleus
*i* *i*-th node
*0* Previous time or initial state
*1* Present time
**Sub**
*b* bond
*r* rupture
Integrated cell migration model {#s2a}
-------------------------------
We model the geometric structure of a cell as a double mesh structure: the outer mesh representing the cell membrane and the inner mesh for the nucleus membrane. See [Figure 1-A](#pcbi-1002926-g001){ref-type="fig"}. Each mesh consists of *N* nodes connected elastically to adjacent nodes, forming a double elastic membrane. The inner and outer mesh nodes may be connected when SFs are formed between the nucleus and the cell membrane [@pcbi.1002926-Chancellor1], [@pcbi.1002926-Hale1]. Multiple transmembrane integrins are bundled together and placed at each node on the outer mesh. They can bind to ligands on the matrix substrate, forming a focal adhesion, to which a SF is connected ([Figure 2-A](#pcbi-1002926-g002){ref-type="fig"}). Furthermore, the model also includes ventral SFs which extend between two focal adhesions.
{#pcbi-1002926-g001}
{#pcbi-1002926-g002}
[Figure 1-B](#pcbi-1002926-g001){ref-type="fig"} shows the free body diagram of the *i*-th node of the cytoskeleton, called the *i*-th integrin node, where a bundle of integrins is formed. Double membranes in the integrated cell migration model move in Lagrangian approach. Acting on this node are force vectors due to frictional dissipative force , focal adhesion force , elastic energy force , SF force , and lamellipodium force . The equation of motion for each integrin node is given bywhere is the velocity vector of the *i*-th integrin node. Similarly, the equation of motion for each node of the nucleus is given bywhere , and are frictional dissipative force, elastic energy force and SF force at the *i*-th nuclear node, respectively, and is the velocity of the *i*-th nuclear node. The velocities and are expressed aswhere and represent coordinates of the *i*-th integrin node and the *i*-th nuclear node, respectively.
Most of the frictional dissipative term arises from the rupture of stretched ligand-receptor bonds; when they rupture, the stored strain energy is released and dissipated. Similarly, also arises from the energy stored in SFs that, when F-actin is depolymerized, the stored strain energy is released and dissipated. These dissipative forces can be written aswhere and are friction coefficients associated with the energy dissipation at the integrin node and the nuclear node, respectively. In the literature these coefficients are estimated as 0.001 Ns/m [@pcbi.1002926-Kapustina1], [@pcbi.1002926-Drury1], [@pcbi.1002926-Bausch1]. comes from the binding and rupture of ligand-receptor bonds and cannot easily be measured [@pcbi.1002926-Filippov1].
It should be noted that the sum of forces is zero because the motion is quasi-static in time ([Text S1](#pcbi.1002926.s006){ref-type="supplementary-material"}, [Figure S2](#pcbi.1002926.s002){ref-type="supplementary-material"}), thus [Equations (1)](#pcbi.1002926.e023){ref-type="disp-formula"}--[(4)](#pcbi.1002926.e037){ref-type="disp-formula"} can be simplified to the following two force balance equations:
Focal adhesion dynamics {#s2b}
-----------------------
Formation of a focal adhesion is described by a stochastic process due to binding kinetics between receptors and ligands on the surface of ECM. Monte Carlo simulation methods have been established for various ligand-receptor binding kinetics in the literature [@pcbi.1002926-Hammer1]--[@pcbi.1002926-Pawar1]. We apply a similar technique to cell migration and spreading on planar surfaces. First we represent the 2-D planar surface and a micropatterned geometry using a mesh of triangles, over which ligands are distributed ([Figure S3](#pcbi.1002926.s003){ref-type="supplementary-material"}). Each focal adhesion consists of a bundle of ligand-receptor bonds ([Figure 2-B](#pcbi-1002926-g002){ref-type="fig"}), each of which ruptures and binds stochastically.
Let be the probability with which a single receptor binds to a ligand on the substrate during a time interval . where is the forward reaction rate (1 molecule^−1^ s^−1^), represents the density of bound ligands, the original density of the ligands (molecules area^−1^), and the area associated with the integrin node under consideration. Note that represents the number of unbound ligands available for bonding in the vicinity of the integrin node. In simulations, a triangular mesh of approximate side lengths of 0.75 µm were used for area . (See [Figure S3](#pcbi.1002926.s003){ref-type="supplementary-material"}).
Similarly, existing ligand-receptor bonds may rupture with probability during a time interval ,where is the kinetic dissociation rate at a distance from the force equilibrium location. Here, is the equilibrium distance of an integrin when it is unstressed (20--30 nm) [@pcbi.1002926-Kanchanawong1], represents the stretched distance from the equilibrium (See [Figure 2-B](#pcbi-1002926-g002){ref-type="fig"}). We utilized the Bell\'s model to run stochastic simulation of bond rupturing and bonding, Bell\'s equation for the kinetic dissociation rate is defined by [@pcbi.1002926-Bell1] where is the kinetic dissociation rate (1 s^−1^) under unstressed conditions with an equilibrium distance , is a force applied to the bond, is the transition distance (0.02 nm), is the Boltzmann constant, and *T* is absolute temperature [@pcbi.1002926-Bell1].
The number of ligand-receptor bonds, i.e. the size of each focal adhesion, can be simulated with these binding and rupture probabilities. Let be the number of ligand-receptor bonds at the *i-th* integrin node, and be the number of ligands on the *j-th* local surface near the *i-th* integrin node. The initial value of is calculated by multiplying and . The number of bonds and available ligands vary stochastically. By drawing a random number, , between 0 and 1:
If *P~ran~* ~1~\<, then one bonding occurs, update and .
Similarly, the rupture of ligand-receptor bonds can be simulated by drawing a random number, :
If *P~ran~* ~2~\<, then one rupture occurs, update and .
Above bonding-rupture tests continue in subsequent time until the bond breaks completely ().
Once is known, the focal adhesion force of the *i*-th integrin node is computed aswhere is an effective spring constant for a single ligand-receptor bond (∼1 pN/nm) [@pcbi.1002926-Dembo1], and is a unit normal vector representing the *i*-th integrin node\'s direction on the cell membrane (See [Figure 2-B](#pcbi-1002926-g002){ref-type="fig"}). This focal adhesion force acts between the *i*-th integrin node and the point on the ECM surface where the extension of the unit normal vector intersects with the ECM surface. From [Figure 2-B](#pcbi-1002926-g002){ref-type="fig"} this intersection position, that is, the root location of receptor and ligand bonds (), is given bywhere is the bond length, is the unit normal vector of the ECM surface, and is the gap between the *i*-th integrin node and the ECM surface, as shown in [Figure 2-B](#pcbi-1002926-g002){ref-type="fig"}. These expressions are valid only when and the gap is less than a critical height () of 300 nm (\<10 ): . The latter condition is to restrict the formation of receptor-ligand bonds within the upper limit .
Comparison to 2-D cell migration experiments {#s2c}
--------------------------------------------
The first set of cell migration simulations was aimed at comparing the integrated model against the experimental data published previously. Palecek et al. [@pcbi.1002926-Palecek1] performed CHO cell migration experiments in 2-D planar plates under various fibronectin coating concentrations. They found that the observed cell migration speed significantly depends on substratum ligand level, cell integrin expression level and integrin--ligand binding affinity. Interestingly, CHO cell migration speed exhibits a biphasic dependence on extracellular-matrix ligand concentration regardless of integrin expression level (the α~5~β~1~ receptor on fibronectin) [@pcbi.1002926-Palecek1]. The simulation results, too, showed similar behaviours of the biphasic dependence on fibronectin coating concentrations.
[Figure 3-A](#pcbi-1002926-g003){ref-type="fig"} show samples of trajectories and morphologies of simulated cell migrations along the planar surface of five different fibronectin surface densities of 19.4, 192, 568, 1140 and 1522 molecules µm^−2^ for three hours (see Videos S1, S2, S3, S4 and S5). The ligand densities used for the simulations matched those of the available experiment data; ligand surface densities of fibronectin were converted from fibronectin plate concentrations (µg ml^−2^) using the relationship between plating concentration and ligand surface density of fibronectin [@pcbi.1002926-Rajagopalan1]. First the total path length of each trajectory was obtained and was divided by the travelling time, 3 hours, to obtain the time-averaged cell migration speed. In the experiments, the speed of CHO cell migration was monitored in every 15 minutes, and was time averaged over the entire migration period (12 h) for each of fibronectin concentrations. [Figure 3-B](#pcbi-1002926-g003){ref-type="fig"} compares the average migration speed between the experiment and simulations. Here an error bar indicates a SE (standard error) of means.
![Cell migration along the planar surface of fibronectin.\
A) Simulated trajectories of cell migrations on fibronectin coated substrates under five different ligand surface densities of 19.4, 192, 568, 1140 and 1522 molecules/µm^2^. The black lines indicate trajectories of nuclei for the first three hours, B) comparison of average cell migration speeds: the simulation model vs. experiment data by Palecek *et al.* [@pcbi.1002926-Palecek1]. Average speed and standard error of mean (N = 5) are shown for the five different ligand surface densities, and C). linear regression (R^2^ = 0.767) of simulated migration speed vs. experimental migration speed.](pcbi.1002926.g003){#pcbi-1002926-g003}
The experimental data show that the cell migration speed is the lowest when migrating in the lowest ligand density, increases with increasing the ligand density, reaches a maximum value at the ligand density of 1140 molecules µm^−2^, and then decreases as the ligand density becomes too dense ([Figure 3-B](#pcbi-1002926-g003){ref-type="fig"}) [@pcbi.1002926-Palecek1], [@pcbi.1002926-Rajagopalan1]. The simulated cell migration speed, too, shows a trend similar to the experiments: slow for a very low ligand density, the fastest at the particular ligand density of 1140 molecules µm^−2^, then slower again for the highest simulated ligand density. Both experiments and simulations attain the fastest speed at the particular ligand density of 1140 molecules µm^−2^. Overall both the simulation and experiment show an excellent agreement over the ligand density range of 10∼1500 \[molecules µm^−2^\]. Statistical analysis of linear regression was performed by comparing the experiment and the simulation in terms of the mean values of time-averaged cell migration speed for the same ligand density. As shown in [Figure 3-C](#pcbi-1002926-g003){ref-type="fig"}, good correlations were found between the two with *R* ^2^ = 0.767. Therefore, the model validates and, in turn, is validated by showing that cell migration speeds are strongly dependent on ligand density.
Comparison to 2-D cell spreading experiments {#s2d}
--------------------------------------------
The second set of cell spreading simulation was intended to compare the integrated model against the recent experimental data published by Tseng *et al.* [@pcbi.1002926-Tseng1]. They developed a method to micropattern ECM proteins on poly-acrylamide gels in order to impinge on cell morphology and mechanics simultaneously, and have reported that measured traction forces differ considerably depending on the shape of micropatterns. In particular, in the case of the crossbow shaped micropatterns, concentrated cell traction forces are repeatedly located in the bottom part of the vertical bar. The simulation of the integrated model also showed similar spreading cell morphologies on micropatterned models and traction force distributions on the cell surface ([Figure 4-A, B and C](#pcbi-1002926-g004){ref-type="fig"}).
{#pcbi-1002926-g004}
[Figure 4](#pcbi-1002926-g004){ref-type="fig"} shows spreading cell morphologies with traction force contours and oriented SFs on three micropatterned geometries (a disk, a "pacman" shape, and a crossbow shape), after 60 minutes of spreading time for all shapes (see Videos S6, S7 and S8). Initially, all cell models start spreading from a spherical shape. The dimensions of micropatterns used for the simulations matches those of experiments for quantitative comparisons regarding contour plots of traction forces (or ) and spatial distributions of SFs inside of the cell; we obtained traction stress per a cell (unit: Pa) by dividing summations of tangential component of at *i*-th integrin node by a total area of ventral cell surface where focal adhesions are formed ([Figure 4-D and E](#pcbi-1002926-g004){ref-type="fig"}). Outside of the micropatterns, it was assumed that the ligand density was zero such that focal adhesion and lamellipodia protrusive forces only existed within the micropatterns.
Both experiments and simulations reveal similar trends in terms of concentrated traction forces on local areas of the ventral cell surface ([Figure 4-A, B and C](#pcbi-1002926-g004){ref-type="fig"}) as well as the order of higher traction stress per a cell among the three micropatterns ([Figure 4-D and E](#pcbi-1002926-g004){ref-type="fig"}). For the disk shaped micropattern, a few concentrated traction stress areas were observed at the ridge of the disk ([Figure 4-A](#pcbi-1002926-g004){ref-type="fig"}, two yellow circles). However, locations of concentrated traction forces on the disk shaped micropattern stochastically varied with time (see Video S6). This time-varying inconsistent distribution of stress on the pattern may be due to the smooth ridge of the shape, which gives a short length of receptor-ligand bonds such that the traction energy dissipates quickly. In the case of the "pacman" shaped micropattern, two sites of concentrated traction stress ([Figure 4-B](#pcbi-1002926-g004){ref-type="fig"}, two yellow circles) with SFs connected to the nucleus ([Figure 4-B](#pcbi-1002926-g004){ref-type="fig"}, black arrows a, b) and an oriented ventral SF was observed in between the sharp edges of the "pacman" mouth, as seen in experimental observations ([Figure 4-B](#pcbi-1002926-g004){ref-type="fig"}, black arrow **c**) although additional concentrated traction forces were located in the smooth ridge of the shape like the disk shaped micropattern. Interestingly, this behaviour was visualized to be persistent over time (see Video S7). In the case of the crossbow shaped micropattern, ventral SFs were aligned along the top roof and the bottom bar, as seen in experimental observations ([Figure 4-C](#pcbi-1002926-g004){ref-type="fig"}, black arrows e, f, g, h), and three sites of concentrated traction stress were observed at right and left end tips of the top roof and a bottom part of the vertical bar ([Figure 4-C](#pcbi-1002926-g004){ref-type="fig"}, three yellow circles). In addition, the strongest traction stress resulted from the contractile activity of SFs ([Figure 4-C](#pcbi-1002926-g004){ref-type="fig"}, black arrow d) at the bottom part of the vertical bar. As the activity of actin SFs are stronger, the length of receptor-ligand bonds is stretched more at the leading edge, which results in stronger traction stress. The animation of cell spreading simulation on the crossbow shaped micropattern, too, shows concentrated traction force at theses three sites (see Video S8).
Since a cell tends to migrate toward the stiffer gel region from the more compliant one [@pcbi.1002926-Schwarz1], the cell may sense locally increased tension at the sharp edge of the micropatterns as the fibronectin bundles are anchored to the plate [@pcbi.1002926-Khatau1]. Thereby, larger areas of FAs are formed at the corners of the micropatterns while smaller areas of FAs are observed at the round boundary. From the agreement between simulation and experimental results on these micropatterned shapes, the model validates and, in turn, is validated by showing persistent high stress concentrations at sharp geometrically patterned edges.
Discussion {#s3}
==========
Coupling of focal adhesion dynamics and motor activity {#s3a}
------------------------------------------------------
It has been reported that nascent adhesions (smaller than ∼0.25 µm) initiate the adhesion of protrusions of the leading edge of the cell, followed by the disassembly of a subpopulation of nascent adhesions within a minute and growth of the remainder into focal complexes (∼0.5 µm in size) and then focal adhesions (1--5 µm in size) within 5 minutes [@pcbi.1002926-Choi1]. Afterwards, focal adhesions either disassemble or mature within the ventral surface of the cell membrane within 10--20 minutes [@pcbi.1002926-Gupton1], [@pcbi.1002926-Gardel1]. Furthermore, it is known that the maturation and turnover of focal adhesions involves protein recruitment and elongation, followed by protein disengagement and shrinkage [@pcbi.1002926-Gardel1]. In the current integrative cell migration model, the disengagement of actin stress fibers from integrins bound to the ECM is assumed to occur when a force-transmitting structural linkage ruptures ( = 0) (see [Figure 2-B](#pcbi-1002926-g002){ref-type="fig"}). With the onset of motor activity after actin polymerization, the generated force is transmitted to the focal adhesions, and receptor-ligand bonds at the focal adhesions are subsequently stretched, resulting in an increases in both traction force and rupture probability for a receptor-ligand bond according to Bell\'s law [@pcbi.1002926-Bell1]. As shown in [Figure 5-A](#pcbi-1002926-g005){ref-type="fig"}, the situation differs at the leading and trailing edges, in large part due to the location of the nucleus closer to the rear of the cell. Note that the angle between the inclined stress fiber and the horizontal plane of the substrate at the trailing edge is higher than that at the leading edge of the cell. If we assume that the stress fibers all exert comparable levels of force then the normal force component will be larger at the trailing edge and therefore have a higher probability of rupture, thereby allowing forward motion of the cell. To test this hypothesis, 266 stress fibers connected to the nucleus at the leading edge and 245 stress fibers connected to the nucleus at the trailing edge were monitored and statistically analysed during three hours of simulated cell migration on the plate with fibronectin density of 200 molecules/µm^2^ ([Figure 5-A](#pcbi-1002926-g005){ref-type="fig"}, Video S9). Consistent with this hypothesis, we found the lifetime of stress fibers at the trailing edge to be less than that at the leading edge of the cell; 32.00±2.78 s at the leading edge and 24.92±2.17 s at the trailing edge ([Figure 5-B](#pcbi-1002926-g005){ref-type="fig"}). Therefore, we propose that increased magnitude of normal force on the adhesion site at the trailing edge plays a key role in accelerating the rupture of receptor-ligand bonds, leading to an increase in cell migration speed.
{#pcbi-1002926-g005}
Lifetime of actin stress fiber {#s3b}
------------------------------
Our modelled stress fiber lifetime physically represents a contractile SF period which is related with the turnover time of the three main dynamic components consisting of SF-actin, alpha-actinin, and myosin. However, it should be noted that there is total lifetime of stress fiber which includes multiple periods of the lifetime of its constituent until it fully disappears. Recently, Hotulainen and Lappalainen [@pcbi.1002926-Hatulainen1] have observed highly dynamic associations and dissociations of these components in the SF by FRAP analysis. They found recovery times for actin, alpha-actinin, and myosin light chain (MLC) in bleached regions of the SF were 323 s, 123 s, and 223 s, respectively (see fig 7A in [@pcbi.1002926-Hatulainen1]). Interestingly, all components of the SF (see fig. 7A in [@pcbi.1002926-Hatulainen1], white boxes) disappeared at the time of +4 s (depolymerization occurs) after SF\'s contractile motion got started at the time of −20 s. Thus, it seems to us that this time period of 24 s may be related with contractile period of the SF among full periods of the SF (actin polymerization, SF contractile motion, and actin depolymerization). Additionally, time periods for actin polymerization and actin depolymerization in our model were set to be 180 s and 1--5 s, respectively, and time period for SF contractile motion in the model was determined to be ∼30 s. Summation over the full period yields ∼215 s, which is within a similar range of the recovery times for the three main components of a SF.
It should be noted that most nonmotile cell types contain thick, non-dynamic stress fibers, whereas most motile cell types contain very few and thin stress fibers [@pcbi.1002926-Hatulainen1] or few and large stress fibers on the soft substrata [@pcbi.1002926-Pelham1]. In case of nonmotile cells, most SFs are known to form at the ventral surface of the cell, and its movements are very slow. However, in case of motile cells, it is possible to assemble ventral SFs by the interaction with preassembled dorsal SFs and transverse arcs within the period of 27 min (see fig.5 in [@pcbi.1002926-Hatulainen1]). During the course of the assembly of ventral SF in motile cells, three major processes (actin polymerization, SF contractile motion, and actin depolymerization) are periodically repeated due to the turnover of actin in either dorsal SF or transverse arcs and SFs\' alignments were dynamically varied due to actin motor activity. Thus, it should be emphasized that there exist three main highly dynamic processes of the SF. In addition, it has been known that rapid SF depolymerization occur because of cell shortening [@pcbi.1002926-Sato1] or SF detachment via localized application of trypsin at focal adhesions [@pcbi.1002926-Sato2], [@pcbi.1002926-Deguchi1].
Note that for the sake of video visualization of the processes of actin polymerization and bundling, the frame-to-frame time scale is 360 s while the simulation time step used is 0.001--0.01 s. Because the frame rate is greater than the SFs dynamic period (∼215 s), the simulated SF dynamics may appear discontinuous, when they are, in fact, not.
What is important to maximum cell migration speed? {#s3c}
--------------------------------------------------
Although there are differences in cell migration speeds between the model and experiment, we are interested in similar trends across a range of the ligand density, and linear regression between the cell migration speed of both the model and experiment with identical ligand density confirms good agreement between the model and experimental data. Additionally, we also simulated cell migration models in which SFs are disconnected from the nuclear membrane on the substrates under five ligand surface densities ([Figure S4](#pcbi.1002926.s004){ref-type="supplementary-material"}), which resulted in lower cell migration speed than cell migration model with SFs connected to the nuclear membrane ([Figure 3-B](#pcbi-1002926-g003){ref-type="fig"}). Thus, our simulated results reveal that these SFs connected to the nucleus play an important role in cell migration. In the literature [@pcbi.1002926-Chancellor1], the authors also demonstrated that nesprin-1 depleted endothelial cells showed decreased migration speed with no SFs connected to the nuclear membrane. Furthermore, Khatau, *et al.* [@pcbi.1002926-Khatau1] highlighted the interplay between cell shape, nuclear shape, and cell adhesion mediated by the perinuclear actin cap. We also found that the cell migration speed is limited by ligand density and integrin density ([Figure S5](#pcbi.1002926.s005){ref-type="supplementary-material"}). They work together to promote adhesion of the cell, and in turn, cell speed. This example shows how either value alone is enough to act as a bottle neck and limit the migration speed. If the ligand density is high (950 molecules/µm^2^), but the integrin density is insufficient (≤137 molecules/µm^2^), the cell speed will be limited. Similarly, if the integrin density is high (205 molecules/µm^2^) but the ligand density is insufficient (200 molecules/µm^2^), then the migration speed is again limited ([Figure S5](#pcbi.1002926.s005){ref-type="supplementary-material"}). We believe that the integration of focal adhesion dynamics (receptor-ligand bonds) and actin motor activity is important to observe and predict maximum cell migration speeds. In addition, as cell\'s contacting area on the substrate becomes larger, the numbers of focal adhesion sites such that ventral SFs anchored at FAs is increased. That is to say, two resultant forces from focal adhesions and actin SFs are increased and they are important to capture the maximum cell migration speed dependent on substrate geometry as well as ligand surface density.
[Figure 6-A](#pcbi-1002926-g006){ref-type="fig"} shows samples of trajectories and morphologies of simulated cell migrations along the planar surface of fibronectin surface density of 1140 molecules/µm^2^ for three hours under nine different cases of polymerization times with 60, 180, and 300 s (rows) and depolymerization times with 1, 10, and 30 s (columns). First, simulated data were compared with different depolymerization times for the three values (rows) of polymerization times of 60, 180, and 300 s. Cell migration speed at each value (row) of polymerization time increases as the depolymerization time becomes larger ([Figure 6-B](#pcbi-1002926-g006){ref-type="fig"}). In the case of the polymerization time of 60 s, especially, the morphologies of cells were observed to be round. This phenomenon results from faster actin motor activity with the inclusion of a shorter polymerization process. Thereby, the occurrence of more frequent actin motor activity prevents the cell from stretching more than the other cases of polymerization times of 180 and 300 s. On the other hand, as the polymerization time becomes larger, the cell tends to stretch more and its morphology is changed to wider crescent-shape from the rounded shape. Next, simulated data were compared with different polymerization times for three values (columns) of depolymerization times of 1, 10, and 30 s ([Figure 6-B](#pcbi-1002926-g006){ref-type="fig"}). As for cases of depolymerization times of 1 and 10 s, cell migration speed increases as polymerization time decreases. In our model, a shorter polymerization process represents faster FA component (integrin and vinculin) renewal within FAs due to increased level of myosin II activation per FA. Contraction could pull these components out of FAs. It has been reported that faster turnover rates of vinculin and integrin due to further increase in actomyosin contractility are correlated with faster cell migration speed at the intermediated ligand surface density [@pcbi.1002926-Gupton1]. However, in case of depolymerization time of 30 s, cell migration speed takes a maximum at an intermediated value of polymerization time of 180 s, which suggest that a balance between adhesion strength and myosin II activity is required for optimal cell migration [@pcbi.1002926-Gupton1].
{#pcbi-1002926-g006}
Model {#s4}
=====
Membrane stiffness and elastic forces {#s4a}
-------------------------------------
The elastic forces, and , are obtained by using the virtual work theory in structural mechanics. To this end, the elastic energy stored in the cell and nucleus membranes are obtained. Two types of elastic energy are considered. One is the elastic energy associated with distance changes between surface nodes [@pcbi.1002926-Tsubota1], [@pcbi.1002926-Tsubota2]: where is the length of the *i*-th line of the cell membrane mesh, and is that of the nucleus. Both are updated at every time-step. and are their relaxed (zero force) lengths. and are effective stiffness constants of the line elements of the cell membrane (5.0×10^−5^ N/m) [@pcbi.1002926-Drury1], [@pcbi.1002926-Honarmandi1] and nucleus (5.0×10^−3^ N/m) [@pcbi.1002926-Zeng1], respectively. Similarly, the elastic energy associated with area changes is given by where is the *i*-th mesh area of the cell membrane and is that of the nucleus. and are their relaxed values. Parameters and are effective stiffness constants of area elements of the cell membrane (1.0×10^−4^ N/m^2^) and nucleus (1.0×10^−4^ N/m^2^), respectively [@pcbi.1002926-Tsubota2].
Elastic forces and can be obtained by differentiating the total energy, where and indicate total stored energies of the cell membrane and nucleus, respectively, and , , and are obtained analytically.
Actin motor activity {#s4b}
--------------------
An actin SF is a bundle of actin microfilaments assembled by actin-myosin II interactions. It is known that at least one end of each SF is connected to focal adhesion molecules, such as vinculin, talin, paxillin, zyxin, and FAK [@pcbi.1002926-Kanchanawong1], and the other end of a SF can be connected to the nuclear membrane [@pcbi.1002926-Chancellor1], transmitting a force to the nucleus. In the model, the *i*-th integrin node is connected to the *j*-th nuclear node by a SF. Its connection to the *j*-th nuclear node is determined by the nearest distance from the *i*-th integrin node to the nucleus. In addition, the *i*-th integrin node is connected to the *k*-th integrin node by a ventral SF. To consider the alignment of the ventral SF which is preferentially parallel to the stronger elastic resistance direction [@pcbi.1002926-Schwarz1], [@pcbi.1002926-Borau1], its connection to the *j*-th integrin node is established by the lower principal direction of Lagrange strain tensor [@pcbi.1002926-Bower1] at the cortical surface bound to the *i*-th integrin node.
The stiffness of a SF is variable. According to the literature, the stiffness increases with a contractile agonist (histamine) and decreases with a relaxing agonist (isoproterenol) [@pcbi.1002926-Wang1]. These characteristics must be reflected in the formulation of the SF stiffness:where is Young\'s modulus of SFs (230 kPa) directly measured from isolated smooth muscle cells [@pcbi.1002926-Deguchi2], is the average cross-sectional area of SFs (250 nm in radius [@pcbi.1002926-Lu1]), and is the length of a single compartment of the *i*-th SF. As shown in [Figure 2-B](#pcbi-1002926-g002){ref-type="fig"}, a SF consists of contractile compartments, each of which consists of two half 'I bands' (F-actin filaments) and an 'A band' (myosin II) in F-actin filaments [@pcbi.1002926-Ruppel1], [@pcbi.1002926-Golomb1]. represents the unstressed length of the *i*-th contractile compartment, which slides at a rate at both ends. Therefore, where [equation (17b)](#pcbi.1002926.e137){ref-type="disp-formula"} is the discretized form of [equation (17a)](#pcbi.1002926.e136){ref-type="disp-formula"}, and indicates the length of a single unit of the *i*-th SF at the previous time (*t*−) [@pcbi.1002926-Lodish1]. Similarly, the elastic energy stored in the *i*-th SF is given bywhere is the number of contractile compartments in the *i*-th SF, represents the distance between *i*-th integrin and *j*-th nuclear nodes for a SF connected to the nucleus or between *i*-th integrin and *j*-th integrin nodes for a ventral SF. It should be noted that physically means the length of SFs under tension and represents the length of a single unstressed bundle of SFs (See [Figure 2-B](#pcbi-1002926-g002){ref-type="fig"}). Using the virtual work theory, forces due to actin SFs\' motor activity at the *i*-th integrin and *j*-th nuclear nodes or at *i*-th integrin and *j*-th integrin nodes (ventral SFs) are given by These forces are generated when focal adhesions have been formed and F-actin filaments are fully polymerized. It has been known that SF assembly occurs over several minutes [@pcbi.1002926-Kaunas2]--[@pcbi.1002926-Ridley1], but SF disassembles rapidly within seconds [@pcbi.1002926-Costa1]--[@pcbi.1002926-Kumar1]. In addition, it takes several minutes to form FAs from focal complexes (FCs). These observations suggest that myosin motor activities in SFs are switched off during the remodelling of the actin cytoskeleton (polymerization) and SF turnover. In our simulations, time for full formation of F-actin is set to be 180 s, and time for the complete disassembly of F-actin is set to 1 s, based on the above reference information.
Actin motor activity is assumed not to start until the other end of a SF is connected to the nucleus. Time for polymerization of F-actin appears to be the waiting time before actin motor activity takes place, during which time an adhesion complex (AC) becomes a fully developed FA. The myosin II\'s sliding rate is known to fluctuate (i.e. is non-uniform) unlike myosin I which slides with a uniform rate. Furthermore, the sliding rate of myosin II is adjusted by sensing the transmitted focal adhesion force from the ECM [@pcbi.1002926-Walcott1]. To incorporate these characteristics into the model, force-velocity relation of muscle myosin II, first proposed by A.V. Hill [@pcbi.1002926-Hill1], is adopted as the following equation:where is the sliding rate of myosin in the absence of load (10 nm/s) [@pcbi.1002926-Lodish1], is the isometric force of myosin, or stall force, and is a parameter for the force-velocity relationship for myosin. Initially, the length of sarcomere unit is 800 nm ( = 800 nm at *t* = 0 s), which contracts until 60% of the initial length has contracted. As the contraction takes place at both sides of each sarcomere unit, the minimum time required for 60% contraction is calculated as 16 s with . Furthermore, an additional condition for terminating actin motor activity is also considered when integrin nodes are broken from FA formations. Afterwards, the depolymerization of actin SFs occurs in 1 s. During this period, formations of nascent ACs are inhibited. In summary, actin motor activity consists of three evolving periods, polymerization (180 s), motor activity (\>16 s) and depolymerization (1 s) [@pcbi.1002926-Kaunas2]--[@pcbi.1002926-Kumar1].
Lamellipodium force {#s4c}
-------------------
Lamellipodium force is a characteristic feature at the leading edge of migratory cells. It is believed to be the motor which push the cortical cytoskeleton forward during the process of cell migration. Normally, cells experience a small protrusive pressure that results from osmotic pressure or actin branches stimulated by activated arp2/3 [@pcbi.1002926-Jeon1]. Recently, time-averaged high protrusive force measured per pillar was 800 pN for NIH 3T3 fibroblasts and diseased cells [@pcbi.1002926-Mathur1]. Here, we assume lamellipodium protrusive force is due to constant actin polymerization rate [@pcbi.1002926-Kabaso1]. Thereby, we approximate the magnitude of the lamellipodium force at the *i*-th integrin node () is constant at 300 pN and exists at only leading edges of the cell. It should be noted that the magnitude of net force at the *i*-th integrin node is non-uniform because it is a vectorial sum of and the local membrane restoring forces from neighboring nodes.
Numerical methods of "integrated cell migration model" {#s4d}
------------------------------------------------------
Cell migration simulations were carried out using a fourth order Rosenbrock method [@pcbi.1002926-Press1] based on an adaptive time-stepping technique for integrating ordinary differential equations with the convergence criterion \<10^−4^. The ordinary differential equations were solved for the 6×*N* (*N* = 549 for both cell migration and spreading simulations) unknown variables associated with the mesh node position vectors for both cell membrane and nucleus membrane: (see [Figure 2-A](#pcbi-1002926-g002){ref-type="fig"}). For cell migration simulation the Rosenbrock method outperforms the standard Runge--Kutta method which requires a relatively large number of iterations [@pcbi.1002926-Press1]. Furthermore, the Rosenbrock method consumes less computing time by using adaptive time-step control that ranges from 10^−3^ s to 10^−2^ s in the present work. Thus, it is suitable for simulating transient cell migratory behaviours over 10 hours.
The focal adhesion dynamics were computed based on the Monte-Carlo simulation. The model assumes a total of 164,700 integrin molecules on the cell membrane [@pcbi.1002926-Moore1] and 549 integrin nodes for both cell migration and spreading models with a cell radius of 8 µm. Therefore, the density of receptors over the cell membrane is 300 integrins/node for both models, among which some fraction of integrins bond to ligands; the number of ligand-receptor bonds varies stochastically in the range . Recall that is determined by drawing random numbers and and simulating binding and rupturing events stochastically using Bell\'s equation. Additionally, each integrin node represents a collection of integrins having the collective stiffness for receptor-ligand bonds (see [equation (11)](#pcbi.1002926.e082){ref-type="disp-formula"}).
The elastic force at the *i*-th node represents the resultant force acting on the *i*-th node that is calculated by vectorial addition of elastic forces from neighbouring nodes. To compute this, first the coordinates of each node are updated in each time cycle, and distances from each node to neighbouring nodes are computed along with the areas of the surrounding rectangles. The elastic forces are derived from these distances and areas for individual nodes.
The methods for building geometrical models for the simulation of cell migration have been well documented in the literatures [@pcbi.1002926-Kim2], [@pcbi.1002926-Kim3]. See also geometrical models of micropatterns, as shown in [Figure S3](#pcbi.1002926.s003){ref-type="supplementary-material"}. One practical issue in computing finite mesh geometric models is to check geometrical compatibility. As the coordinates of cell membrane and nuclear nodes are updated based on the equations of motion, geometrically incompatible situations occur occasionally in the configurations of the cell membrane mesh and that of the nucleus in relation to the curved ECM surface. For example, some cell membrane nodes intersect with the substrate, and the nucleus intersects with the cell membrane. These incompatible situations must be checked in every computational cycle, and necessary corrections must be made.
Supporting Information {#s5}
======================
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Schematics of A) 2-D cell migration in planar surface, B) 2-D cell migration and spreading on a micropatterned structure, C) 3-D cell migration in a rectangular channel and D) 3-D cell migration in 3-D ECM.
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Samples of A) cell migration speed and B) cell migration acceleration for three hours. Blue lines indicate time-averaged cell migration speed and acceleration of 4.24 nm/s and 3.18×10^−4^ nm/s^2^, respectively.
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Meshes of three micropattern models of A) disk, B) pacman and C) crossbow shapes; all meshes have triangular elements with approximate side lengths of 0.75 µm.
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Comparison of average cell migration speeds: cell migration model with SFs connected to the nuclear membrane vs. cell migration model with SFs disconnected to the nuclear membrane. Average speed and standard error of mean (N = 5) are shown for the five different ligand surface densities.
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Comparisons of average cell migration speeds: cell migration model with four different integrin densities of 34, 68, 137, and 205 molecules/µm^2^ on the cell surface on two different low and high ligand surface densities of 200 and 950 molecules/µm^2^. Average speed and standard error of mean (N = 5) are shown for the four different integrin surface densities and two ligand surface densities.
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Why the net force is zero in a dynamic moving system?
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Example of a simulated cell migration on the plate with the ligand density of 19.4 molecules/µm^2^. Cell and nuclear membranes are visualised with green and blue, respectively. Bold red lines in the cell indicate actin stress fibers, and black line indicates a trajectory of nuclear center. Six seconds of the video represents three hours.
(AVI)
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Example of a simulated cell migration on the plate with the ligand density of 192 molecules/µm^2^. Cell and nuclear membranes are visualised with green and blue, respectively. Bold red lines in the cell indicate actin stress fibers, and a black line indicates a trajectory of the nucleus center. Six seconds of the video represents three hours.
(AVI)
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Click here for additional data file.
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Example of a simulated cell migration on the plate with the ligand density of 568 molecules/µm^2^. Cell and nuclear membranes are visualised with green and blue, respectively. Bold red lines in the cell indicate actin stress fibers, and a black line indicates a trajectory of the nucleus center. Six seconds of the video represents three hours.
(AVI)
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Click here for additional data file.
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Example of a simulated cell migration on the plate with the ligand density of 1040 molecules/µm^2^. Cell and nuclear membranes are visualised with green and blue, respectively. Bold red lines in the cell indicate actin stress fibers, and a black line indicates a trajectory of nucleus center. Six seconds of the video represents three hours.
(AVI)
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Click here for additional data file.
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Example of a simulated cell migration on the plate with the ligand density of 1522 molecules/µm^2^. Cell and nuclear membranes are visualised with green and blue, respectively. Bold red lines in the cell indicate actin stress fibers, and a black line indicates a trajectory of nucleus center. Six seconds of the video represents three hours.
(AVI)
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Click here for additional data file.
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Example of a simulated cell spreading on the disk shaped micropattern with the ligand density of 475 molecules/µm^2^. Cell and nuclear membranes are visualised with green and blue, respectively. Bold red lines in the cell indicate actin stress fibers, and contours indicate traction forces on the ventral surface of cell membrane. Twelve seconds of the video represents sixty minutes.
(AVI)
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Click here for additional data file.
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Example of a simulated cell spreading on the pacman shaped micropattern with the ligand density of 475 molecules/µm^2^. Cell and nuclear membranes are visualised with green and blue, respectively. Bold red lines in the cell indicate actin stress fibers, and contours indicate traction forces on the ventral surface of cell membrane. Twelve seconds of the video represents sixty minutes.
(AVI)
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Click here for additional data file.
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Example of a simulated cell spreading on the crossbow shaped micropattern with the ligand density of 475 molecules/µm2. Cell and nuclear membranes are visualised with green and blue, respectively. Bold red in the cell lines indicate actin stress fibers, and contours indicate traction forces on the ventral surface of cell membrane. Twelve seconds of the video represents sixty minutes.
(AVI)
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Click here for additional data file.
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Example of a simulated cell migration on the plate with the ligand density of 200 molecules/µm^2^. Cell and nuclear membranes are visualised with green and blue, respectively. Bold red lines in the cell indicate actin stress fibers. Twenty seconds of the video represents three hours.
(WMV)
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Click here for additional data file.
MCK specifically acknowledges the assistance of Joo-Siong Sim for his help using high performance computing facilities.
[^1]: The authors have declared that no competing interests exist.
[^2]: Analyzed the data: MCK DMN RDK HHA. Wrote the paper: MCK DMN RDK HHA. Built a code for the model: MCK. Performed the simulations: MCK DMN. Conceived and designed the experiments: MCK DMN RDK HHA.
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\section{Introduction}
Reducing optical losses is of paramount importance for further developing photovoltaic (PV) devices. This holds especially true for the market-dominating single-junction c-Si solar cells, for which optical losses constitute one of the main limitations to reach their efficiency limit~\cite{shockley_detailed_nodate,richter_n-type_2017}. While a common approach involves employing various uniform anti-reflective (AR) coatings combined with a chemical texturing of the c-Si wafer, resulting in the formation of micron-sized pyramidal features~\cite{seidel_anisotropic_1990}, in some cases, alternative approaches to suppress the optical losses are needed. For example, in industrial solar cells, the micron-sized textures are realized on relatively thick c-Si wafers, with a current standard of 160~$\mu$m. However, a transition to the wafer thickness below the standard value and switching to the foil-like thinner c-Si can allow for lower material consumption. This can reduce the manufacturing cost and facilitate the acceleration of the expansion of PV manufacturing~\cite{liu_revisiting_2020} to keep up with estimates for global installed PV capacity~\cite{haegel_terawatt-scale_2019}. However, for such thin devices, the use of micron-sized textures becomes very challenging due to c-Si wafer handling issues, and novel approaches have to be identified to reduce the optical losses.
In response to that need, various nanophotonic concepts~\cite{garnett_photonics_2021} applicable to c-Si-based solar cell stacks were proposed to enhance light harvesting through improved light in-coupling on the front surface. Examples of possible solutions include plasmonic structures~\cite{green_harnessing_2012,Catchpole:08,doi:10.1021/nl8022548}, periodically arranged silicon~\cite{hou_efficient_2020,he_enhanced_2016} and dielectric~\cite{spinelli_effect_2015, saive2020, spinelli_broadband_2012} nanoscatterers, and biomimetic structures~\cite{hou_biomimetic_2017}. Moreover, double-sided AR and light trapping (LT) nanostructure gratings introduced at the front and rear sides of the solar cells were also suggested. For example, such a concept was investigated for thin-film c-Si~\cite{wang_absorption_2012} and thin-film hydrogenated nanocrystalline silicon (nc-Si:H)~\cite{isabella_advanced_2018} solar cells.
Nevertheless, while a plethora of nanophotonic structures was proposed and investigated in recent years, it is of paramount importance to analyze their performance in a full solar module architecture under realistic irradiation conditions. For example, such analysis was performed for different solar module architectures where sufficient optical properties were achieved using strategies involving textured interfaces and/or flat ARCs~\cite{singh_comparing_2021,C7EE01232B,tucher_energy_2019, lehr_energy_2020,gota2020energy,jovst2018textured}. Since the solar cell is not always illuminated with light at normal incidence in the realistic scenario, one has to go beyond the analysis of the ability of the structure to enhance the short-circuit current density under the standard test conditions and ensure that the proposed AR and/or LT nanostructure designs are robust concerning irradiation impinging on the solar module at increasing angles of incidence. Additionally, the absorption of photons by the solar cell absorber depends on the sun's position. It is influenced by the cloud coverage effect on irradiation received by the module and the module orientation. For solar cells with double-sided photonic nanostructures, it is also appealing to consider and assess the possible power output of the solar modules with nanostructured solar cells in the case of a bifacial module architecture. This module configuration allows for harvesting the photons that can be absorbed when the sunlight either hits the module from its back or is reflected from the ground and can be absorbed in the solar cell.
Here, we study the energy yield of relatively thin wafer-based c-Si solar modules, for which the solar cell stack is coated with double-sided nanostructure gratings. Suitably designed square arrays of dielectric high-index TiO$_2$ nanodisks are used as AR and LT photonic nanostructures. Their geometrical parameters are subject to optimization. Results are compared to modules containing more traditional planar thin-film anti-reflective layers. We consider mono- and bifacial modules and introduce the nanodisk array on the front surface of the c-Si-based heterojunction (HJT) solar cell for the former module configuration and both the front and rear side for the latter. The nanodisk arrays are initially optimized at normal incident light by full-wave optical simulations concerning the short-circuit current associated with the reflected portion of the light (details in Sec. \textbf{Calculation of reflectance, transmittance, and absorptance}~\ref{sec:calc}). The optimal design of these nanodisk arrays depends on whether they are employed on the front side, where they serve the purpose of suppressing reflectance, or on the rear side, where they facilitate the light trapping. Therefore, when optimizing the nanodisk arrays, different designs are found depending on the mechanisms through which they contribute to enhancing the absorptance in the c-Si wafer. To estimate the annual energy yield (EY) of the solar module architectures with nanodisk arrays, the optical response from the optimized nanodisk arrays when placed on the front and rear solar cell contact layers is simulated using full-wave optical tools depending on the angle of incidence. This is the primary information fueled afterward into the EY modeling framework. The annual EY is assessed for monofacial and bifacial module architectures with and without TiO$_2$ nanodisk arrays at locations with different climate conditions. The influence of albedo radiation is also considered, which is especially relevant to consider in bifacial module configuration~\cite{russell_influence_2017}. Our key contribution is to show that for the wafer-based c-Si cells with thicknesses for which standard chemical texturing becomes impractical, and alternative AR and LT structures are of interest, the nanodisk arrays that we suggest outperform to a considerable extent the traditional design that relies only on the planar anti-reflective coatings.
In passing, we note that the design of the nanodisk arrays for front and rear solar cell contacts proposed here is exemplary, and it is neither restricted to a particular solar cell stack nor the materials used.
\section{Module architectures and numerical methods}\label{methods}
\subsection{Investigated module architectures and material properties}
The four different architectures studied in this contribution are schematically depicted in Fig.~\ref{fig:schematic}. The annual EY calculations were performed for two reference monofacial and bifacial module configurations and two configurations with AR and LT nanodisk arrays introduced on top of the front and rear ITO contact layers, respectively. The c-Si absorber thickness of all architectures was varied between 5 and 80~$\mu$m, which is a typical thickness range between thin PV and conventional wafer-based c-Si PV.
\begin{figure}[h]
\centering
\includegraphics[width=\textwidth]{Figure1.pdf}
\caption{Schematic representation of the four solar module architectures discussed in this paper: (a) Monofacial standard reference module with optimized transparent conductive ITO layer on the front, serving as both anti-reflective coating and front contact, and silver back contact. Front ITO layer is preceded with window glass and encapsulation (EVA) layers covered with an anti-reflective coating. For all considered solar module configurations, the window layers are identical. (b) Monofacial module with optimized anti-reflective nanodisk array on top of the front ITO contact, and silver back contact. (c) Bifacial standard reference module with front and rear ITO contacts with symmetric window layers on both sides of the solar module. (d) Bifacial module with optimized anti-reflective and light trapping nanodisk arrays on top of the front and rear ITO contacts with symmetric window layers on both sides of the solar module, respectively.}
\label{fig:schematic}
\end{figure}
In the case of flat reference c-Si HJT solar cell stacks, the front hydrogenated amorphous silicon (a-Si:H) (passivation intrinsic and n$^{+}$ doped, the thickness can be found in~\cite{slivina_insights_2019}) and conducting ITO (75~nm) layers were considered. For the rear side of the HJT solar cell stack, the a-Si:H (passivation intrinsic and p$^{+}$ doped) layer was slightly thicker than the a-Si:H layer on the front, while the ITO layer was thinner than its counterpart on the front side.
The configurations containing optimized AR nanodisk array had a reduced front ITO thickness of 10~nm with TiO$_2$ nanodisks arranged in a square lattice of 320~nm pitch, with individual nanodisk having a radius of 125~nm and a height of 90~nm. The bifacial module configuration with added optimized LT TiO$_2$ nanodisk square array (individual nanodisk with a radius of 215~nm and height of 395~nm, 565~nm pitch) had the same rear ITO thickness as the reference architectures. These values for the geometrical parameters are the results of an optimization of the AR and LT nanodisk arrays discussed in Sec. \textbf{Calculation of reflectance, transmittance, and absorptance}~\ref{sec:calc}.
In the case of the monofacial modules, HJT c-Si solar cell had 300~nm thick silver back contact while the front side contacting metallic grid and its effect on the optical performance of the module is neglected. The window module layers comprise encapsulating EVA (400~$\mu$m) and glass (4~mm) with thin-film anti-reflective MgF$_2$ coating (130~nm). In the case of the bifacial architecture, the rear side of the solar cell typically has the same contacting scheme as the front side (no conformal metallic layer). The modules in the bifacial configuration were considered to have the same window layers as the ones introduced on the front side. In such a configuration, the module can absorb the light that is incident on its front and from its rear side. Additionally, one can harvest albedo radiation since the transmitted portion of the light is reflected from the ground, and thus, can be reabsorbed in the silicon, significantly boosting the annual EY. When a bifacial solar module is tilted, albedo radiation can also impact the annual EY due to the light reflected from the ground and incident of the front of the module. We note that when a monofacial solar module is tilted, albedo radiation that is incident on the front of the module can also be considered.
Refractive indices of c-Si, TiO$_2$, ITO, and Ag used in the calculations were taken from literature~\cite{schinke_uncertainty_2015,Rutile,holman_infrared_2013,jiang_realistic_2016}. Refractive index data for the front and rear composite (passivation intrinsic and doped) a-Si:H layers were obtained using ellipsometry, and corresponding $n$ and $k$ values provided by Meyer Burger Research AG are plotted in Fig.~\ref{fig:indices}. For the window layers, a non-absorbing optically thick glass layer was considered to have a nondispersive refractive index of $n=1.5$, and EVA and MgF$_2$ refractive index data was taken from~\cite{mcintosh_optical_2009} and~\cite{siqueiros_determination_1988}, respectively.
\begin{figure}[h]
\centering
\includegraphics[width=7cm]{Figure2.pdf}
\caption{Refractive index $n$ and extinction coefficient $k$ of a-Si:H composite layers. The measured data was fitted to a Tauc-Lorentz model. }
\label{fig:indices}
\end{figure}
\subsection{Simulation framework}
The numerical simulations of the EVA-cell interface for both AR and LT TiO$_2$ nanodisk arrays were performed using the finite element method (FEM) with commercial software \textit{JCMsuite}~\cite{pomplun2007adaptive}. The annual EY was calculated using a comprehensive modeling framework enabling the quick simulation of various and sophisticated PV architectures under realistic irradiation conditions discussed in detail in~\cite{schmager_methodology_2019,GithubEY}.
\subsection{Electrical parameters}
The electrical parameters corresponding to a typical c-Si HJT solar cell used in the annual EY calculation are summarized in Table~\ref{tab:electric}. The shadowing by electrical connections for all considered architectures is disregarded.
\begin{table}[h]
\centering
\caption {Electrical parameters of the solar cell} \label{tab:electric}
\begin{tabular}{c c}
\hline
Shunt resistance, $R_{\text{sh}}$ [$\Omega\cdot\text{cm}^2$] & 5000 \\
Series resistance, $R_{\text{s}}$ [$\Omega\cdot\text{cm}^2$] & 0.7 \\
Reverse-blocking current, $J_{\text{0}}$ [$\text{A}/\text{cm}^2$] & 2$\cdot10^{-13}$ \\
Ideality factor, $n$ & 1.1 \\
Temperature coefficient of $J_{\text{SC}}$, $t_{J_{\text{SC}}}$ [$\%/\text{K}$] & 0.05 \\
Temperature coefficient of $V_{\text{OC}}$, $t_{V_{\text{OC}}}$ [$\%/\text{K}$] & -0.25 \\
\hline
\end{tabular}
\end{table}
A device characterized by these properties would have a $J_{\text{SC}}$ around 38.3~mA/cm$^2$. This yields $V_{\text{OC}}=0.734$~V at temperature $\text{T}=25^\circ$C from the following equation:
\begin{equation}\label{Voc_RT}
V_{\text{OC}}=nV_\text{th}\text{ln}\left(\frac{J_{\text{SC}}}{J_{\text{0}}}+1\right),
\end{equation}
where the thermal voltage $V_\text{th}=kT/q=0.0257$~V, and the values of ideality factor and reverse-blocking current can be found in Table~\ref{tab:electric}.
\subsection{Calculation of reflectance, transmittance, and absorptance}\label{sec:calc}
At the EVA-cell interfaces with introduced AR and LT nanodisk arrays, reflectance and transmittance into all scattering directions at each wavelength and incidence polar and azimuth angles is calculated as the ratio of the scattered reflected or transmitted power to the power of the incident field. At the considered interfaces, both c-Si and EVA are assumed to be semi-infinite. For a given azimuth angle $\phi_\text{in}$, reflectance and transimittance values form matrices of the size ($N_{\theta_\text{in}}$, $N_{\theta_\text{r,t}}$, $N_{\lambda}$), where the entries correspond to all polar angles of incidence, scattering angles, and wavelengths, respectively. The polar angle $\theta_\text{in}$ is varied from $\ang{0}$ to $\ang{89}$ with $\ang{5}$ step, and the results are then interpolated at intervals of $\ang{1}$. In case of azimuth angle $\phi_\text{in}$, the symmetry of the nanodisk coating is exploited, and only calculations for angles between $\ang{0}$ and $\ang{45}$ with $\ang{15}$ step are performed. The calculated matrices for different $\phi_\text{in}$ values are subsequently averaged. Total reflectance and transmittance for a certain wavelength and incident polar and azimuth angles are calculated according to:
\begin{equation}\label{eq:Refl}
R=\frac{\sum\limits_{\bm{k}_\text{r}}|\Tilde{\bm{E}}(k_{\text{r,x}},k_{\text{r,y}})|^2\cdot \text{cos}(\theta_\text{r})}{|\bm{E}_0|^2\text{cos}(\theta_\text{in})},
\end{equation}
\begin{equation}\label{eq:Trans}
T=\frac{\sum\limits_{\bm{k}_\text{t}}n_{\text{out}}|\Tilde{\bm{E}}(k_{\text{t,x}},k_{\text{t,y}})|^2\cdot \text{cos}(\theta_\text{t})}{n_{\text{in}}|\bm{E}_0|^2\text{cos}(\theta_\text{in})},
\end{equation}
where $\bm{k_\text{r,t}}$ are the wave vectors of reflected and transmitted fields, $\theta_\text{r,t}=\Re{(\pm k_\text{z}/k_\text{r,t})}$ are the scattering angles, $\bm{E}_0$ is the amplitude of an incident plane wave with a mixed TE-TM polarization, and $n_\text{in}$ and $n_\text{out}$ are the refractive indices of the media where the incident and scattered waves propagate, respectively. We use the angular spectrum representation of the fields, and $\Tilde{\bm{E}}(k_{\text{\{r,t\},x}},k_{\text{\{r,t\},y}})$ in equations~\ref{eq:Refl} and~\ref{eq:Trans} is calculated by means of the Fourier transform of the electric fields in real space obtained from full-wave simulations. Absorptance in each of the thin film layers and nanodisk array was calculated by integrating the divergence of the Poynting vector across the absorber volume, thus yielding absorbed power which is normalized to the power of the incident plane wave. Similarly to reflectance and transmittance, for a given azimuth angle, absorptance values form a matrix of the size ($N_m$, $N_{\theta_\text{in}}$, $N_{\lambda}$), where index $m$ runs over all absorbing layers in the front or rear solar cell stack. The angular dependent simulations were performed for optimized AR and LT nanodisk arrays. The optical performance of the nanodisk arrays was optimized with respect to the short-circuit current associated with reflectance at a normal light incidence, which is calculated using the following equation:
\begin{equation}\label{eq:JscR}
J_{\text{SC,R}}=\int_{\lambda_1}^{\lambda_2}e \frac{SI_\text{AM1.5}(\lambda)R(\lambda)}{E_\text{ph}}d\lambda,
\end{equation}
where $e$ is the electron charge, $E_\text{ph}=hc/\lambda$ is the energy of a photon, and $SI_\text{AM1.5}(\lambda)$ is the spectral irradiance. For this calculation, air mass 1.5 global (AM1.5G) tilted irradiance raw data was taken from~\cite{gueymard1995smarts2}, and the total reflectance $R(\lambda)$ was interpolated accordingly. The short-circuit current due to reflectance $J_{\text{SC,R}}$ was minimized for front nanodisk array and maximized for the rear LT nanodisk array.
Within the EY modeling framework, where the optical response of the entire architecture is computed, the light propagation in multi-layer thin-film stacks is treated coherently, for which the transfer matrix method is employed. When AR and LT nanodisk arrays are considered instead of those thin-film layers, the corresponding output matrices for reflectance, transmittance, and absorptance are integrated into the modeling framework. For thicker layers, such as the c-Si substrate of the cell and window layers of the module, the assumption of coherence breaks down. The Beer-Lambert law can describe the absorption of the light in those layers:
\begin{equation}\label{eq:Beer}
I(z,\lambda)=I_\text{0}\cdot\text{e}^{-\alpha(\lambda)z},
\end{equation}
where $I_\text{0}$ is the initial intensity, $\alpha$ is the absorption coefficient of the considered medium, and $z$ is the distance traveled in it.
\section{Results and discussion}
\subsection{Energy yield of solar modules}
We analyzed the annual EY of the four solar module architectures introduced in Fig.~\ref{fig:schematic} for three cities in the United States of America located in different climate zones~\cite{peel2007updated}. Two of the chosen cities, Anchorage, AK and Honolulu, HI, have highly contrasting irradiation conditions. The former is a cold and cloudy region (Boreal climate) and the latter a hot and sunny one (Tropical climate). The additionally chosen Kansas City, MO, has a temperate climate that receives an annual solar irradiance between Anchorage and Honolulu. By covering different climate zones, we aimed to highlight the robustness of the nanodisk arrays performance and their ability to improve the annual EY for all types of irradiation conditions, albeit with small differences that most likely originate from the spectral features of the nanodisk arrays. The solar modules were considered to face south, and the tilt angles $\theta_\text{m}$ were optimized for each location. This resulted in $\theta_\text{m}$ values to be $\ang{38}$ for Anchorage, $\ang{30}$ for Kansas City, and $\ang{17}$ for Honolulu, respectively.
Figure~\ref{fig:changeEY} demonstrates the relative improvement of the annual EY when the nanodisk arrays are used for light management instead of the optimized planar layers. The increase of the annual EY is shown as a function of the c-Si absorber thickness. We considered the case when the AR array is introduced on the front surface of the c-Si HJT cell in the case of the monofacial module architecture, and both AR and LT nanodisk arrays are applied to the front and rear surfaces of the cell in the case of the bifacial module architecture. For this calculation, no albedo was considered. The relative increase of the annual EY reached up to 11.0\,\%$_\text{rel}$ and 43.0\,\%$_\text{rel}$ at the minimal wafer thickness of 5~$\mu$m for monofacial and bifacial architectures with nanodisks, respectively.
\begin{figure}[h]
\centering
\includegraphics[width=13cm]{Figure3.pdf}
\caption{Relative increase of the annual EY for three locations in case of (a) the monofacial solar module (comparing (b) to (a) from Fig.~\ref{fig:schematic}) and (b) the bifacial solar module (comparing (d) to (c) from Fig.~\ref{fig:schematic}) with varying thickness of the c-Si absorber.}
\label{fig:changeEY}
\end{figure}
As expected, for the monofacial case, the module with an AR nanodisk array (Fig.~\ref{fig:schematic}(b)) outperforms the standard flat architecture with an optimized ITO coating (Fig.~\ref{fig:schematic}(a)), with this effect becoming even more apparent when reducing the c-Si absorber thickness. However, the front side AR array with a relatively small size of the individual disks does not have strong LT properties. Its contribution to light harvesting stems mainly from its broadband AR performance. When instead of a full metallic contact, the window encapsulation and glass layers are introduced, to take advantage of the solar irradiation which can hit the module on its back, one has to take care of an appropriate light trapping structure, which would also act as a decent AR coating. For this purpose, an LT nanodisk array was designed (Fig.~\ref{fig:schematic}(d)). With the individual disk parameters being significantly larger when compared to the nanodisk array used as the AR structure at the front interface, the light which is not absorbed in the silicon and reaches the rear interface of the cell is effectively scattered in multiple directions, thus improving the LT properties of the cell. From Fig.~\ref{fig:changeEY} (b) it can be seen that it translates into an even more significant increase of the annual EY than for monofacial module architecture when comparing to a reference bifacial module architecture (Fig.~\ref{fig:schematic}(c)).
If additionally, one considers albedo radiation, the advantage of the bifacial module architecture with AR and LT nanodisk arrays becomes even more apparent. The influence of albedo is shown in Fig.~\ref{fig:barEY}. Here, the annual EY of the four module architectures with a selected median c-Si absorber thickness of 40~$\mu$m is shown for the sandstone and grass ground surface compared to EY of the modules without albedo.
\begin{figure}[h]
\centering
\includegraphics[width=\textwidth]{Figure4.pdf}
\caption{Energy yield of the four module architectures for different locations when albedo irradiation is taken into account. For all solar module architectures, the c-Si absorber thickness was 40~$\mu$m.}
\label{fig:barEY}
\end{figure}
Since the interfaces of the standard reference modules are flat, without albedo, the monofacial standard reference module outperforms the bifacial reference. While for the former, the silver layer reflects the light reaching it back into the cell, for the latter, a lot of light is lost due to transmittance when no LT structure is introduced. This difference is bigger with a smaller module tilt angle since less irradiation can hit the module from the back. However, as soon as albedo radiation is considered, the bifacial standard reference outperforms its monofacial counterpart. While for monofacial architecture albedo radiation does not make a significant difference, one can see a robust improvement in the annual EY for the bifacial case. The increase of the annual EY for the monofacial architecture varies depending on the module tilt and is stronger for locations with a greater $\theta_\text{m}$ (Anchorage). The relative improvement reaches up to around 0.8\,\%$_\text{rel}$ with sandstone and 3.0\,\%$_\text{rel}$ with grass as a ground surface for both the standard reference monofacial module and monofacial module with AR nanodisk array with insignificant difference between them. In the case of the bifacial module architecture, a stronger increase of the annual EY is expected for the sunnier locations (Honolulu). It reaches up to 8.2\,\%$_\text{rel}$ with sandstone, and 28.9\,\%$_\text{rel}$ with grass ground surface in case of a standard reference bifacial module. For a bifacial module with AR and LT structures, the relative increase is 8.3 \,\%$_\text{rel}$ and 30.1\,\%$_\text{rel}$ with sandstone and grass ground surface, respectively.
\subsection{Optical performance of solar modules}\label{sec:opt}
To better understand the mechanism behind the annual EY improvement when the nanodisk arrays are introduced on top of the front and back contacts of a solar cell, one can look at the optical properties of the cell interfaces with the optimized nanodisk arrays. Figure~\ref{fig:Reflectance} shows reflectance at normal incidence of the EVA-cell interfaces for the front and rear HJT solar cell contacts with AR and LT nanodisk arrays. For both nanodisk arrays and illumination directions, reflectances of the corresponding reference flat interfaces are plotted for comparison. The graphs (a) and (b) in Fig.~\ref{fig:Reflectance} correspond to the optical response of the front EVA-cell interface. In this case, the objective was to minimize short-circuit current density corresponding to reflectance $J_{\text{SC,R}}$ introduced in Eq.~\ref{eq:JscR}, which resulted in an AR nanodisk array with reflectance shown in Fig.~\ref{fig:Reflectance}(a).
\begin{figure}[h]
\centering
\includegraphics[width=12cm]{Figure5.pdf}
\caption{Reflectance at normal incidence for the front and rear EVA-cell interfaces in case of both AR and LT nanodisk arrays. Inset sketches depict the corresponding interface and illumination direction. Interfaces are the same for the pairs of graphs (a)-(b) and (c)-(d), respectively.}
\label{fig:Reflectance}
\end{figure}
This design was based on the previous work on helicity preserving TiO$_2$ nanodisk array for the front interface of a c-Si HJT solar cell, where efficient and broadband backscattering suppression was achieved due to the ability of the system to suppress cross-talk between opposite handednesses of the electromagnetic field upon light-matter interaction~\cite{slivina_insights_2019}. The requirement for a system to be helicity preserving is to possess a high enough degree of rotational symmetry ($n\ge3$) along the illumination direction. For normal light incidence, for which the optimization of AR nanodisk array was performed, the illumination direction is along the symmetry axis of an individual nanodisk, which essentially means that $n\rightarrow\infty$ in this case. The resulting reflectance of the AR nanodisk array is lower than the one of the reference optimized flat AR coating and exceeds it slightly only in the wavelength region around $\lambda=600$~nm, for which standard AR coating of the solar cell is typically optimized. However, as shown in Fig.~\ref{fig:Reflectance}(b), the LT properties of this nanodisk array are not as good as its AR properties. When the light is impinging from the c-Si absorber, only the long-wavelength response is relevant since the short-wavelength photons are absorbed before reaching this interface. This nanodisk coating transmits the light reflected from the rear of the stack particularly strongly at longer wavelengths and is not possessing better LT properties than the standard flat reference. This optical response confirms that the front nanodisk array contributes to the light harvesting of the considered solar module architectures mainly by its AR properties.
The graphs (c) and (d) in Fig.~\ref{fig:Reflectance} demonstrate the optical response of the optimized LT nanodisk array. Here, the structure parameters strongly differ from those of the helicity preserving AR nanodisk array. The larger and more sparsely spaced LT nanodisks allow for improved harvesting of the long-wavelength photons reaching the rear of the solar cell. The impinging light is effectively scattered into multiple directions since many diffraction orders are allowed. Taking into account the absorption depth of c-Si, for the minimal considered absorber thickness of 5~$\mu$m, the photons that can reach the rear contact of the solar cell have wavelengths $\lambda\geq690$~nm. The optimized reflectance of the LT nanodisk array for the spectral region of interest is shown in Fig.~\ref{fig:Reflectance}(c). Its LT performance exceeds that of the flat reference solar cell stack at all wavelengths. Nevertheless, as can be seen from Fig.~\ref{fig:Reflectance}(d), though this nanodisk array outperforms its flat reference counterpart in terms of AR properties in the longer wavelength range, overall, its main contribution to the improved absorption in c-Si, and, consequently, the annual EY of the solar module, is due to its superior LT properties. This way, in the bifacial module case, when nanodisk arrays with decoupled AR and LT properties are present on both sides of the solar cell stack, one can achieve a solid overall solar module performance boost.
Another critical aspect of the solar module's optical performance is parasitic absorption. Here, all discussed results are for a selected median c-Si absorber thickness of 40~$\mu$m and at normal light incidence. Figures~\ref{fig:AMono} and~\ref{fig:ABi} show absorptance in all layers of the monofacial and bifacial module stacks, respectively, except for the glass layer, which was assumed to be non-absorbing. Additionally, the absorptance of the rear a-Si:H layer in monofacial module configuration and of rear and front a-Si:H layers in the bifacial module configuration for forward and backward illumination direction, respectively, was negligible, and, thus, it is not shown. Moreover, the LT TiO$_2$ nanodisk array, which is considered for the bifacial module configuration, does not introduce any parasitic absorption since the light absorbed in these nanodisks has short wavelengths and does not reach the rear cell interface while it is absorbed in c-Si. The short-circuit current densities indicated on the top of all graphs in Fig.~\ref{fig:AMono} and Fig.~\ref{fig:ABi} were calculated assuming AM1.5G spectrum using the modified Eq.~\ref{eq:JscR} with absorptance of c-Si instead of reflectance.
\begin{figure}[h]
\centering
\includegraphics[width=12cm]{Figure6.pdf}
\caption{Absorptance in the different layers of (a) the monofacial standrad reference module and (b) the monofacial module with AR nanodisk array on top of the front ITO contact at normal light incidence. For both solar module architectures, the c-Si absorber thickness was 40~$\mu$m.}
\label{fig:AMono}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[width=12cm]{Figure7.pdf}
\caption{Absorptance in the different layers of the bifacial solar modules at normal light incidence: (a) in case of forward illumination direction for the bifacial standard reference module, (b) in case of forward illumination direction for the bifacial module with AR and LT nanodisk coatings, (c) in case of backward illumination direction for the bifacial standard reference module, (d) in case of backward illumination direction for the bifacial module with AR and LT nanodisk coatings. For both solar module architectures, the c-Si absorber thickness was 40~$\mu$m.}
\label{fig:ABi}
\end{figure}
In the case of the monofacial standard reference module (Fig.~\ref{fig:AMono}(a)), the front transparent conductive ITO layer serves as an AR coating but also introduces some parasitic absorption. However, when the AR nanodisk array is introduced (Fig.~\ref{fig:AMono}(b)), ITO thickness for the optimized front solar cell contact is reduced considerably (from 75 to 10~nm), and parasitic absorption in this layer is significantly reduced. The nanodisks themselves have a tiny contribution to the parasitic absorption, considering that EVA absorbs all the light impinging on the solar module at the wavelengths between 300 and 360~nm and TiO$_2$ absorbs only at wavelengths shorter than $\lambda=380$~nm. On the other hand, while the AR nanodisk array increases the absorptance in c-Si thanks to a better in-coupling of the incident light, it also increases parasitic absorption in the front a-Si:H layer. However, it should be noted that in~\cite{holman_current_2012} it was shown that the carriers which are absorbed in an intrinsic a-Si:H layer can still contribute to the short-circuit current, and thus the parasitic absorption loss in this layer represents an upper bound and can have a less of an impact in reality. Thus, introducing the TiO$_2$ AR nanodisk array leads to a broadband enhancement of absorptance in the silicon absorber layer.
In the case of the bifacial standard reference module and forward illumination direction (Fig.~\ref{fig:ABi}(a)), the parasitic absorption for the wavelengths below $\lambda=700$~nm is similar to the one of the monofacial module reference. However, the parasitic absorption is slightly lower in the long-wavelength range since more light is transmitted through the glass and encapsulation window layers. When AR and LT nanodisk arrays are introduced (Fig.~\ref{fig:ABi}(b)), the front nanodisks improve absorptance in silicon the same way as in the case of the monofacial module. The rear nanodisk array additionally enhances aborptance at longer wavelengths. However, due to the strong scattering enabled by LT nanodisks, parasitic absorption is also increased for longer wavelengths. When the light is impinging on the rear side of the bifacial solar module (Fig.~\ref{fig:ABi}(d)), the absorptance in c-Si is also improved in the case when AR and LT arrays are introduced in comparison to the reference flat module (Fig.~\ref{fig:ABi}(c)), even though LT nanodisks are not optimal in terms of their AR properties and introduce dips in silicon absorptance due to the sharp spectral features which can be seen in Fig.~\ref{fig:Reflectance}(d).
\section{Conclusions}
We have numerically studied the annual energy yield (EY) under realistic irradiation conditions for monofacial and bifacial crystalline silicon (c-Si) heterojunction (HJT) solar module architectures with AR and light trapping (LT) titanium dioxide (TiO$_2$) nanodisk square arrays introduced on top of the front and rear ITO layers and compared their power outputs with the ones of the corresponding standard reference flat solar modules. We have shown that while reducing the silicon absorber thickness, the relative increase of the annual EY is reaching up to 11.0\,\%$_\text{rel}$ and 43.0\,\%$_\text{rel}$ for monofacial and bifacial modules with nanodisk coatings, respectively. This improvement is comparable for the locations with different climate conditions. Moreover, in the case of bifacial module architecture, taking into account the albedo radiation produces an additional boost of the module performance.
The designed dielectric nanodisk arrays for the front and rear contacts of c-Si HJT solar cell have both a significant impact on the light absorption in the c-Si wafer. At the same time, their AR and LT properties are decoupled. The front AR nanodisk array has a relatively small individual disk size and lattice constant, and its broadband backscattering suppression is related to the helicity preservation condition. In contrast, the rear LT nanodisk array has larger features and array pitch and allows for efficient scattering into multiple scattering directions. Furthermore, these AR and LT nanodisk square array designs are not restricted to a specific material or a particular photovoltaic solar cell stack and, thus, can be investigated for different solar module configurations.
\begin{acknowledgement}
The authors acknowledge support by the state of Baden-W{\"u}rttemberg through bwHPC and the German Research Foundation (DFG) through grant no INST 40/575-1 FUGG (JUSTUS 2 cluster). The authors are also grateful to JCMwave for their free provision of the FEM Maxwell solver JCMsuite.
\end{acknowledgement}
|
\section{Results}
Evaluation is performed on 2 test datasets, each containing 5 keyframes. Similarly to the training data, keyframes 1-4 all have interpolation sequences from keyframe $N$ to keyframe $N+1$ and keyframe 5 is a single image. The metric we use is the mean absolute error in mm of the depth measurement at each pixel. During the interpolation sequences, we mask the pixels which do not have associated ground truth so they are not considered in the error measurement and additionally we discard frames for which less than 10\% of the frames have ground truth measurements.
\subsection{Test Dataset 1}
The overall numerical errors on test dataset 1 from each method are shown in Fig. \ref{tab:dataset_1_results} and the plots of error per frame for keyframes with motion (1-4) are shown in Fig. \ref{fig:test_dataset_1_keyframe_1_2} and Fig. \ref{fig:test_dataset_1_keyframe_3_4}.
\begin{table*}
\centering
\small
\begin{tabular}{
p{0.17\textwidth} |
>{\centering}p{0.1\textwidth} |
>{\centering}p{0.1\textwidth} |
>{\centering}p{0.1\textwidth} |
>{\centering}p{0.1\textwidth} |
>{\centering}p{0.1\textwidth} |
p{0.05\textwidth}
}
& Keyframe 1 & Keyframe 2 & Keyframe 3 & Keyframe 4 & Keyframe 5 & Average \\
\hline
Congcong Wang & 6.30 & \textbf{2.15} & 3.41 & 3.86 & 4.80 & 4.10 \\
J.C. Rosenthal & 8.25 & 3.36 & 2.21 & \textbf{2.03} & \textbf{1.33} & \textbf{3.44} \\
KeXue Fu & 30.49 & 18.32 & 19.73 & 19.30 & 16.86 & 20.94 \\
Trevor Zeffiro & 7.91 & 2.97 & \textbf{1.71} & 2.52 & 2.91 & 3.60 \\
Wenyao Xia & \textbf{5.70} & 7.18 & 6.98 & 8.66 & 5.13 & 6.73 \\
Zhu Zhanshi & 14.64 & 7.77 & 7.03 & 7.36 & 11.22 & 9.60 \\
Huoling Luo & 29.68 & 16.36 & 13.71 & 22.42 & 15.43 & 19.52 \\
Xiran Zhang & 12.53 & 6.13 & 3.60 & 3.34 & 5.07 & 6.13 \\
Xiaohong Li & 34.42 & 20.66 & 17.84 & 27.92 & 13.00 & 22.77 \\
Lalith Sharan & 30.63 & 46.51 & 45.79 & 38.99 & 53.23 & 43.03 \\
\hline
Dimitris Psychogyios 1 & 7.73 & 2.07 & 1.94 & 2.63 & 0.62 & 3.00 \\
Dimitris Psychogyios 2 & 7.41 & 2.03 & 1.92 & 2.75 & 0.65 & 2.95 \\
Sebastian Schmid & 7.61 & 2.41 & 1.84 & 2.48 & 0.99 & 3.07 \\
\end{tabular}
\caption{\label{tab:dataset_1_results} The mean absolute depth error in mm for test dataset 1. The best method submitted during the challenge period is shown in bold.}
\end{table*}
\begin{figure*}
\captionsetup[subfigure]{labelformat=empty}
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{figures/results/dataset_1_keyframe_1.png}
\caption{Keyframe 1}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{figures/results/dataset_1_keyframe_2.png}
\caption{Keyframe 2}
\end{subfigure}
\caption{\label{fig:test_dataset_1_keyframe_1_2}Mean absolute error plots for each frame of test dataset 1, keyframes 1 and 2. }
\end{figure*}
\begin{figure*}
\captionsetup[subfigure]{labelformat=empty}
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{figures/results/dataset_1_keyframe_3.png}
\caption{Keyframe 3}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{figures/results/dataset_1_keyframe_4.png}
\caption{Keyframe 4}
\end{subfigure}
\caption{\label{fig:test_dataset_1_keyframe_3_4}Mean per-pixel error plots for each frame of test dataset 1, keyframes 3 and 4.}
\end{figure*}
\subsection{Test Dataset 2}
The overall numerical errors on test dataset 2 from each method are shown in Fig. \ref{tab:dataset_2_results} and the plots of error per frame for keyframes with motion (1-4) are shown in Fig. \ref{fig:test_dataset_2_keyframe_1_2} and Fig. \ref{fig:test_dataset_2_keyframe_3_4}.
\begin{table*}
\centering
\small
\begin{tabular}{
p{0.17\textwidth} |
>{\centering}p{0.1\textwidth} |
>{\centering}p{0.1\textwidth} |
>{\centering}p{0.1\textwidth} |
>{\centering}p{0.1\textwidth} |
>{\centering}p{0.1\textwidth} |
p{0.05\textwidth}
}
& Keyframe 1 & Keyframe 2 & Keyframe 3 & Keyframe 4 & Keyframe 5 & Average \\
\hline
Congcong Wang & 6.57 & 2.56 & 6.72 & 4.34 & 1.19 & 4.28 \\
J.C. Rosenthal & 8.26 & 2.29 & 7.04 & \textbf{2.22} & \textbf{0.42} & 4.05 \\
KeXue Fu & 23.71 & 15.46 & 26.43 & 9.60 & 10.92 & 17.22 \\
Trevor Zeffiro & 5.39 & \textbf{1.67} & \textbf{4.34} & 3.18 & 2.79 & \textbf{3.47} \\
Wenyao Xia & 13.80 & 6.85 & 13.10 & 5.70 & 7.73 & 9.44 \\
Zhu Zhanshi & 14.41 & 12.55 & 16.30 & 27.87 & 34.86 & 21.20 \\
Huoling Luo & 20.83 & 11.27 & 35.74 & 8.26 & 14.97 & 18.21 \\
Xiran Zhang & \textbf{3.20} & 3.30 & 6.75 & 4.79 & 3.91 & 4.39 \\
Xiaohong Li & 24.58 & 16.80 & 29.92 & 11.37 & 19.93 & 20.52 \\
Lalith Sharan & 35.46 & 50.09 & 25.24 & 62.37 & 70.45 & 48.72 \\
\hline
Dimitris Psychogyios 1 & 4.85 & 0.65 & 1.62 & 0.77 & 0.41 & 1.67 \\
Dimitris Psychogyios 2 & 4.78 & 1.19 & 3.34 & 1.82 & 0.36 & 2.30 \\
Sebastian Schmid & 4.33 & 1.10 & 3.65 & 1.69 & 0.48 & 2.25 \\
\end{tabular}
\caption{\label{tab:dataset_2_results} The mean absolute depth error in mm for test dataset 2. The best method submitted during the challenge period is shown in bold.}
\end{table*}
\begin{figure*}
\captionsetup[subfigure]{labelformat=empty}
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{figures/results/dataset_2_keyframe_1.png}
\caption{Keyframe 1}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{figures/results/dataset_2_keyframe_1.png}
\caption{Keyframe 2}
\end{subfigure}
\caption{\label{fig:test_dataset_2_keyframe_1_2}Mean per-pixel error plots for each frame of test dataset 2, keyframes 1 and 2. }
\end{figure*}
\begin{figure*}
\captionsetup[subfigure]{labelformat=empty}
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{figures/results/dataset_2_keyframe_3.png}
\caption{Keyframe 3}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{figures/results/dataset_2_keyframe_4.png}
\caption{Keyframe 4}
\end{subfigure}
\caption{\label{fig:test_dataset_2_keyframe_3_4}Mean per-pixel error plots for each frame of test dataset 2, keyframes 3 and 4. }
\end{figure*}
\subsection{Overall}
The overall winner of the challenge was the method with the lowest mean error across the two test datasets, which was Trevor Zeffiro of Rediminds Inc. Second place was awarded to Jean-Claude Rosenthal.
\section{Introduction}
Reconstruction of the surgical scene is an important problem in computer assisted surgery (CAS). It is a fundamental part of SLAM, which is a requirement for augmented reality (AR) \cite{haouchine_image_2013}, as well as a key building block of perception in automation and safety systems \cite{shademan_supervised_2016} and has even found use in diagnostic tools \cite{mahmood_polyp_2019}.
There are several methods of estimating depth in endoscopic images. The most commonly used is depth from stereo, whereby pixels are matched between the two views of a calibrated stereo camera and triangulated to provide depth measurements. The matching can be achieved through classical methods such as semi-global block matching or more recently with deep learning \cite{ye_selfsupervised_2017}. These methods are popular because they require no modification of the scene and the data required for their use can be captured with a simple recording system attaching to the endoscope. When a high quality calibration is obtained they can provide accurate results however requiring clinical teams to calibrate scopes is not a practical solution at scale and calibrations themselves can be invalidated by changing parameters such as focal lengths during the procedure \cite{pratt_practical_2014}. Calibration can be avoided by using only a single monocular view, whereby structural cues are inferred from the contents of a single image. Solutions in this space are almost entirely based on machine learning but promising results have been demonstrated in sinus endoscopy \cite{liu_self_2018}. Using structured light projectors within the endoscope has been demonstrated \cite{lin_endoscopic_2017} however obtaining reconstruction at the camera frame rate is challenging and requires specialized processing hardware.
\begin{figure}
\captionsetup[subfigure]{labelformat=empty}
\begin{subfigure}[c]{0.24\textwidth}
\includegraphics[width=\textwidth]{figures/image2.png}
\end{subfigure}
\hfill
\begin{subfigure}[c]{0.24\textwidth}
\includegraphics[width=\textwidth]{figures/image3.png}
\end{subfigure}
\caption{\label{fig:example}An example image captured by the endoscope and the corresponding depth map.}
\end{figure}
The development and evaluation of methods to accurately estimate depth in surgical images is heavily dependent on the availability of high quality datasets. The mainstream computer vision community has seen great success in algorithm development following the release of several Middlebury Stereo datasets between 2001 and 2014 \cite{scharstein_high_2003} as well as more specialized datasets from autonomous driving \cite{sun_scalability_2019, geiger_vision_2013}. Few datasets of this type exist in the surgical domain and are typically much smaller or focus on phantom or non-realistic environments \cite{stoyanov_real_2010} largely due to challenges with obtaining ground truth data.
The stereo correspondence and reconstruction of endoscopic data (SCARED) sub-challenge was organized to help address this gap by creating a high quality dataset using porcine cadavers and a structured light projection system.
\section{Supplemental Material on Dataset inaccuracies}
Inaccuracies were present in the provided datasets. Many of these inaccuracies were expected, such as calibration errors for the endoscope and synchronization issues between the video and kinematics in the warping of depth data from the initial frame to later frames. Subsequent efforts to estimate and correct for these efforts were made by late submissions to the challenge and their results are explained here for completeness.
\subsection{Sebastian Schmid and Tom Kurmann}
The original dataset relies on the forward kinematics for forward projection of the point cloud. This limits the accuracy of the point clouds to the accuracy of the forward kinematics. These may suffer from positioning errors, noise and video synchronization issues making them a potential source of error in the training data. In order to minimize this error, we propose to compute the forward propagation of the point clouds using the imaging data. To do so, we perform a 4 step pipeline which detects keypoints in subsequent frames, matches them and then computes the camera's pose. Using the camera pose, the point cloud can then be projected into the image to obtain the disparity maps. More precisely, our pipeline is defined as follows:
\begin{description}
\item[$\bullet$] Extract SIFT features from the keyframe and a subsequent frame.
\item[$\bullet$] Match the features between subsequent and keyframe. The aim is to find as many feature pairs as possible.
\item[$\bullet$] Use the matched features to estimate the pose of the endoscope, substituting the given forward kinematics. This can be formulated as a perspective-n-point problem, where the projections of a set of $n$ 3D points onto an image plane are known and the pose of the camera has to be calculated. The pose is expressed as rotation $\mathbf{R}$ and translation $\mathbf{t}$.
\item[$\bullet$] Apply $\mathbf{R}$ and $\mathbf{t}$ to the point cloud of the keyframe to get a point cloud for the subsequent frame.
\end{description}
\subsection{Dimitris Psychogyios}
\subsubsection{Calibration Errors}
The ground truth depthmap is given as a 3-channel image. Each nonzero ground truth pixel $(u,v)$ stores the 3D coordinates of a point in space which projects to $(u,v)$ pixel in the left image. Using the provided camera calibration parameters to project a 3D point, stored in ground truth’s $(u,v)$, ends up in coordinates $(u+du, v+dv)$. This effect could potentially be caused by either discretization in the depth map or calibration error. Calibration files in Datasets 4 and 5 seem to have inaccuracies. Using the provided calibration parameters to rectify frames included in those two datasets, caused large errors in the rectification as shown in Fig. \ref{fig:supplemental_calib_error}. Matches in the rectified image pairs do not end up on the same scan-lines. Estimating the Essential matrix, using Shi and Tomashi features\cite{shi1994good} and LK tracking \cite{lucas1981iterative} between the two views and using it refine the extrinsics, resulted in rectifications that had the same problem. Rectification based only on the Fundamental matrix is adequate. This leads to the conclusion that the error is mainly in the intrinsics.
\begin{figure*}
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{figures/supplemental/calib1.png}
\caption{}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{figures/supplemental/calib2.png}
\caption{}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{figures/supplemental/calib3.png}
\caption{}
\end{subfigure}
\caption{\label{fig:supplemental_calib_error}(a) Stereo anaglyph of dataset 4, keyframe 1 showing rectified frames, using the provided calibration parameters, with sparse feature matching overlay. Corresponding features do not lie on the same horizontal scanlines. (b) The same image pair where the extrinsics are estimated from feature matches. (c) Uncalibrated rectification based only on visual matches.}
\end{figure*}
\subsubsection{Video/Kinematics Offsets}
Frames in `rgb.mp4' files and the interpolated ground truth depth images, based on robot kinematics, are not time synchronized. Overlaying the provided ground truth sequences over the corresponding videos, shows that the video is lagging.
\subsubsection{Ground Truth-RGB misalignment}
There are misalignments between the ground truth and the RGB data in datasets 8 and 9 (Figure \ref{fig:offset_error}).
\begin{figure*}
\captionsetup[subfigure]{labelformat=empty}
\begin{subfigure}[b]{0.48\textwidth}
\includegraphics[width=\textwidth]{figures/supplemental/offset1.png}
\caption{Scanline Offset}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.48\textwidth}
\includegraphics[width=\textwidth]{figures/supplemental/offset2.png}
\caption{Disparity Offset}
\end{subfigure}
\caption{\label{fig:offset_error} An example from dataset 9, keyframe 3. Ground truth depth map is shifted with respect to the RGB image. This is also visible between the disparity which is based on the ground truth data and the output of one of the networks.}
\end{figure*}
\section{Participating Methods}
The 10 methods described in this section were submitted during the challenge time window and were the only methods considered when deciding the challenge winner.
\subsection{Fraunhofer Institute for Telecommunications, Heinrich Hertz Institute}
The submission from The Fraunhofer Institute for Telecommunications, Heinrich Hertz Institute in Germany was from Jean-Claude Rosenthal, Zhenglei Hu, Niklas Gard and Peter Eisert. This stereo pipeline consists of six steps whereas steps 2-4 are developed at Fraunhofer HHI. (1) Histogram equalization: They apply equalization using OpenCV implementation of CLAHE \cite{zuiderveld_contrast_1994} as a pre-processing step. (2) Stereo geometry analysis: They detect and match sparse feature point correspondences based on binary feature point descriptors using the hamming distance. From matched feature points they compute stereo alignment errors for 5 parameters as indicated. \cite{zilly_semantic_2011}. (3) Rectification: They correct stereo alignment errors using the matched feature points by computing a linearized F-matrix and apply a homography to rectify stereo images. They warp the right image w.r.t. to the left view \cite{zilly_joint_2010}. Remaining alignment errors are $\pm$0.25 pixels. (4) Disparity estimation: They perform a quasi-dense disparity estimation using a spatio-temporal method called patch recursive matching (patchRM) \cite{Waizenegger2016Realtime3B}. Disparity estimation fully works in parallel on a graphic card with CUDA support. The procedure makes use of a statistical approach on sub-pixel level to estimate new correspondences. These correspondences are distributed into the local neighborhood where new correspondences are determined within the next iteration. This independent propagation of new estimated correspondences guarantees that the whole disparity map is constantly updated locally while propagating the results into the spatial-temporal consistent global map. (5) 3D reconstruction and depth filtering: They generate an almost complete raw depth z-map from step 4 which uses a depth range histogram to remove outliers and mismatches during this process. Therefore, they make use of pre-known ground truth data from training sets to derive a heuristic with reasonable endoscopic depth ranges using (a) the smallest “near range” 14.676 mm (D3k5) and (b) the largest “far range” is 245.554mm (D6k5) as working distances. (6) Depth Completion: They complete the final depth map as some depth values have been marked invalid in step 4 and 5. Therefore, they use a deep neural network. It is a 2-step process which is independent of the RGBD/depth sensor. First, object occlusion boundaries and surface normals are estimated using a CNN network in the RGB image. Second, in a global surface optimization step depth values are used as a soft constraint to solve for missing depths near estimated object boundaries using the surface normals \cite{zhang2018deep}. Remarks: The pipeline has a short initialization phase of 15-25 frames for step 2 and 3. Afterwards, the estimated homography matrix to rectify the stereo images is considered static for the rest of the sequence. Therefore, these steps are no longer executed except the warping of the right image w.r.t. to the left view.
\begin{table}
\centering
\small
\begin{tabular}{
p{0.17\textwidth} |
>{\centering}p{0.1\textwidth} |
>{\centering}p{0.1\textwidth} |
>{\centering}p{0.1\textwidth} |
>{\centering}p{0.1\textwidth} |
>{\centering}p{0.1\textwidth} |
p{0.05\textwidth}
}
& Keyframe 1 & Keyframe 2 & Keyframe 3 & Keyframe 4 & Keyframe 5 & Average \\
\hline
Congcong Wang & 6.30 & \textbf{2.15} & 3.41 & 3.86 & 4.80 & 4.10 \\
J.C. Rosenthal & 8.25 & 3.36 & 2.21 & \textbf{2.03} & \textbf{1.33} & \textbf{3.44} \\
KeXue Fu & 30.49 & 18.32 & 19.73 & 19.30 & 16.86 & 20.94 \\
Trevor Zeffiro & 7.91 & 2.97 & \textbf{1.71} & 2.52 & 2.91 & 3.60 \\
Wenyao Xia & \textbf{5.70} & 7.18 & 6.98 & 8.66 & 5.13 & 6.73 \\
Zhu Zhanshi & 14.64 & 7.77 & 7.03 & 7.36 & 11.22 & 9.60 \\
Huoling Luo & 29.68 & 16.36 & 13.71 & 22.42 & 15.43 & 19.52 \\
Xiran Zhang & 12.53 & 6.13 & 3.60 & 3.34 & 5.07 & 6.13 \\
Xiaohong Li & 34.42 & 20.66 & 17.84 & 27.92 & 13.00 & 22.77 \\
Lalith Sharan & 30.63 & 46.51 & 45.79 & 38.99 & 53.23 & 43.03 \\
\hline
Dimitris Psychogyios 1 & 7.73 & 2.07 & 1.94 & 2.63 & 0.62 & 3.00 \\
Dimitris Psychogyios 2 & 7.41 & 2.03 & 1.92 & 2.75 & 0.65 & 2.95 \\
Sebastian Schmid & 7.61 & 2.41 & 1.84 & 2.48 & 0.99 & 3.07 \\
\end{tabular}
\caption{\label{tab:dataset_1_results} The mean absolute depth error in mm for test dataset 1. The best method submitted during the challenge period is shown in bold.}
\end{table}
\subsection{University of Western Ontario}
The submission from the Robarts Institute at the University of Western Ontario, Canada was from Wenyao Xia. This method used multi-scale cost volume filtering with mean-squared-error loss wtth post-processing with guided median filter. A regression model was also trained to correct rectification errors.
\subsection{Norwegian University of Science and Technology}
The submission from the Norwegian University of Science and Technology was from Congcong Wang. This method used a variational disparity estimation method on the coarsest level of a multiscale pyramid \cite{wang2018liver}. The disparity map was upsampled using a modified bilateral filter.
\subsection{Rediminds Inc.}
The submission from the Rediminds Inc. USA was from Trevor Zeffiro. This method trained a Pyramid Stereo Matching network (PSMNet) using an Gaussian based interpolation strategy to remove holes in the training data.
\subsection{Unknown}
The unaffiliated submission is from KeXue Fu. This method uses unsupervised training of a DNN to predict the disparity and then reproject using the stereo calibration parameters.
\subsection{Harbin Institute of Technology}
The submission from the Harbin Institute of Technology, China was from Zhu Zhanshi. This method used a 6 level UNet with multi-scale loss on appearance matching, disparity smoothness and left-right consistency.
\subsection{Shenzhen Institute for Advanced Technology (1)}
The first submission from the Shenzhen Institute for Advanced Technology, China was from Huoling Luo and Fucang Jia. This method used a modified VGG encoder with skip connections to a dual branch decoder to directly predict a disparity map. 4 neighborhood smoothing is used as post-processing.
\subsection{Johns Hopkins University}
The submission from the Johns Hopkins University, USA was from Xiran Zhang. This method used the Pyramid Stereo Matching network (PSMNet), discarding inaccurate pixels in the training data using SGBM as a filter and ICP to generate a new point cloud which is reprojected to the create high quality training images. L1 loss is used to regularize the results.
\subsection{Shenzhen Institute for Advanced Technology (2)}
The second submission from the Shenzhen Institute for Advanced Technology, China was from Xiaohong Li and Fucang Jia. This method used a CycleGAN to synthesize a depth map in a 2 stage process directly from a rectified stereo pair.
\subsection{Heidelberg University}
The submission from the Heidelberg University, Germany was from Lalith Sharan. This method used a UNet trained to minimize a mean absolute error loss with a smoothness regularization.
\section{Post-Challenge Methods}
The 3 methods described in this section were submitted in the months after the challenge finished so were not considered in the final challenge ranking.
\subsection{Wellcome/EPSRC Centre for Interventional and Surgical Sciences (WEISS) UCL (1)}
\label{ssect:deep_pruner}
The first submission from UCL is from Dimitris Psychogyios. This method used Deep Pruner \cite{duggal2019deeppruner}. The model was pretrained on the Sceneflow dataset \cite{mayer2016large}. Outlier points were manually removed from ground truth and stereo frames were rectified using OpenCV. Disparity images were generated in the left rectified frame of reference. During training, a data augmentation scheme consisted of random crops of size 256x512 and color normalization was used. They ignored the interpolation sequences as well as datasets 4 and 5 due to high calibration errors. This left 25 samples, of which they allocated 18 for training and 7 for evaluation. Finally, inferred disparities were reconstructed in 3D using the provided calibration parameters. Training was performed using a batch size of 1 for 290 epochs. The Adam optimizer \cite{kingma2014adam} with a learning rate of $10^{- 4}$ , $\beta_{1} = 0.9$ and $\beta_{2} = 0.999$.
\subsection{Wellcome/EPSRC Centre for Interventional and Surgical Sciences (WEISS) UCL (2)}
The second submission from UCL is also from Dimitris Psychogyios. This method makes use of Hierarchical deep Stereo Matching network (HSM)\cite{yang2019hierarchical}. It followed the same process as illustrated in Subsection IV-A with the exception of a batchsize of 2 and 350 epochs of training.
\subsection{University of Bern}
The third late submission was from Sebastian Schmid and Tom Kurmann at University of Bern. Similar to previous methods such as pwc-net \cite{pwc_net_2018} and gwc-net \cite{guo2019group} they base their method on a 3D cost volume optimization for stereo matching. They extend the previous method by using two parallel pipelines to predict both left and right disparities during training. This allows them to incorporate photometric losses which enforce a correct reconstruction of the right image with the left image and disparities and vice-versa. They optimize the computationally expensive 3D cost volume and subsequent 3D convolution network by quantizing the cost volume. Rather than using a disparity stride of 1 when generating the volume, they use a dynamic stride which is based on the distribution of the ground truth disparities. They use k-means clustering of the ground truth disparities to extract n-centroids, which are used as the disparities in the cost volume. Without a loss of accuracy, they can reduce the number of disparities by 40\%, resulting in a decrease in inference time of 40\% (79.1ms). Furthermore, this reduces the memory footprint, allowing for larger batch sizes or higher resolution during training. Their loss function is the weighted sum of the L1-loss of the prediction and ground truth, the photometric loss and a disparity smoothness term.
\section{Data}
The SCARED dataset consists of 7 training datasets and 2 test datasets captured using a da Vinci Xi surgical robot. Each dataset corresponds to a single porcine subject and contains between 4 and 5 keyframes. A keyframe is a single unique view of the scene where the a structured light pattern was projected into the field of view of the camera and a dense stereo reconstruction was computed \cite{scharstein_high_2003}. As the endoscope was moved using the da Vinci Xi to observe a new keypoint, the camera's pose relative to the fixed robot base was recorded using the robot's internal measurements of the position of each joint in its arms. As the scene remained static during these sequences, it was possible to obtain approximate depth maps for intermediate frames by transforming the vertices from the original keypoint and reprojecting them into the new view.
To create the depth maps at each keypoint, the white light image of the stereo camera was first captured. Then the illuminator was turned off, and a projector was used to project a series of patterns onto the scene. The scene remained totally still during this period. The illuminator used was a AAXA P300 Neo Pico projector which provides a small form factor of $15\times3\times9$ cm and a $1280\times720$ pattern. The patterns projected were 10 bit Gray code patterns which provided a unique sequence for each pixel in the image using the method provided in \cite{scharstein_high_2003} (see Fig. \ref{fig:gray_code_pattern}). Due to challenges with positioning the projector as well, multiple projector positions had to be used to cover the entire scene. With a unique encoding for each pixel, stereo matching is straightforward and triangulation can be used to recover the scene depth.
\begin{figure*}
\captionsetup[subfigure]{labelformat=empty}
\begin{subfigure}[c]{0.3\textwidth}
\includegraphics[width=\textwidth]{figures/projector.png}
\caption{(a)}
\end{subfigure}
\hfill
\begin{subfigure}[c]{0.3\textwidth}
\includegraphics[width=\textwidth]{figures/pattern_view.png}
\caption{(b)}
\end{subfigure}
\hfill
\begin{subfigure}[c]{0.3\textwidth}
\includegraphics[width=\textwidth]{figures/pattern.png}
\caption{(c)}
\end{subfigure}
\caption{\label{fig:gray_code_pattern}(a) The projector used in this data collection. (b) The view of the endoscope while the pattern is being projected. (c) A single example of a Gray code pattern used to create the dataset.}
\end{figure*}
|
1.. Introduction
================
Breathing is one of the essential functions for the survival of most living beings. Many processes to measure the respiration rate have been proposed: using a stretch sensor or impedance meter to detect chest expansion \[[@b1-sensors-14-15371]--[@b3-sensors-14-15371]\], a pulse oximeter and extracting the respiration rate from the raw data \[[@b4-sensors-14-15371]\], an accelerometer to detect chest expansion and contraction \[[@b5-sensors-14-15371],[@b6-sensors-14-15371]\], measuring airflow pressure by oral or nasal cannula, and many others. In case of sleep apnea diagnosis for example, polysomnography (PSG), the commonly used test, employs nasal cannula and chest belts. The main problem common to all these techniques is the presence of a device directly in contact with the subject. As an example, for children this may induce rejection behavior leading them to remove the device. This is also true in the case of sleeping subjects, where the presence of the device can significantly disrupt the falling asleep and sleep. There have been several attempts to devise contactless methods. Electromagnetic waves can sense chest movement by Doppler effect \[[@b7-sensors-14-15371]--[@b10-sensors-14-15371]\] or by analyzing the signal backscattered by breathing movements \[[@b11-sensors-14-15371]\] and ultrasound waves telemeters permits detection of small body displacements during respiration \[[@b12-sensors-14-15371],[@b13-sensors-14-15371]\]. These solutions give good results but remain indirect because they do not analyze the true air flow and therefore cannot easily detect obstructive sleep apnea (OSA). Contactless direct air flow analysis is far less common in the literature. A microphone to detect exhalation sounds is proposed in \[[@b14-sensors-14-15371]\], a high precision single point infrared sensor is use in \[[@b15-sensors-14-15371]\] and a US patent \[[@b16-sensors-14-15371]\] was granted for a method based on phase differences between two ultrasonic waves traveling in opposite directions. Nevertheless, the first method is perturbed by ambient noises and so the microphone needs to be placed close to the face of the subject to obtain good results. Concerning the second method, the sensor has to be fixed in an accurate position while the third method is complicated to implement because the two waves must travel exactly the same distance in absence of air flow.
This study proposes a new ultrasonic contactless device \[[@b17-sensors-14-15371]\] to measure the presence or the absence of respiration and in the first case its relative intensity and rate. This sensor can be used alone in low cost systems or in conjunction with other remote sensors to increase reliability in sleep apnea diagnostic applications.
The physical principle of this measure is to "illuminate" the subject\'s head with an acoustic wave emitted by a transducer, then recovering and analyzing the reflected wave. Under these conditions, any movement, both in the subject himself and in the exhaled airflow, induce a frequency shift in the signal which is the Doppler effect. Every part of the head which participate to the reflection has approximately the same relative speed component in the direction of the receiver. Moreover, this speed is mostly low except in the case of very rapid movements. Hence, the surface of the skin returns a wave that remains nearly coherent and gives a single (or narrow) low Doppler frequency shift. In contrast the exhaled air flow returns a wave with a wide Doppler shift due to the presence of turbulence. The relative speeds are also higher than in the case of the movement which leads to a higher and broader Doppler shift. By frequency filtering and averaging, it therefore becomes possible to obtain and discriminate air flow signal and head and shoulder movement signal.
2.. Experimental Section
========================
The apparatus developed in the context of this work, allows monitoring the breathing of a subject lying on his back. These conditions correspond to the situation encountered in breath monitoring during sleep, like sleep apnea diagnosis. A 40 kHz ultrasound transmitter illuminates an area widely including the subject\'s head. One receiver, tuned to the same frequency, recovers the signal reflected from the scene. After reflection on the local environment, the incident wave is subject to various transformations related to physical and chemical characteristics of the media encountered, but also, to the movements of the neighboring objects. The result is a combination of a level attenuation, a frequency shift, (*i.e.*, Doppler effect), and a more complex effect resulting in a modification of spectral energy repartition *versus* time when the subject breathes out. The phenomenon, that we call "spectrum widening", characterizes precisely the breath events and represents the core of this work.
[Figure 1](#f1-sensors-14-15371){ref-type="fig"} shows the block diagram of the device. We used inexpensive 40 kHz ultrasonic transducers with a 6 dB beamwidth of 55°. The emitted level was set to about 100 dB/0.0002 μbar. The transmitter was placed above patient head at a distance of 50 cm. We found that transmitter to head distance was not critical due to the ultrasound level and large beamwidth used. Receiver sensitivity is −65 dB *versus* 1 V/μbar. The receiver placement is a compromise between the signal strength and the need to allow the subject to move without hitting the sensor. The best signal is obtained when the receiver is close to the subject and directed to the nostrils. We used a distance of 30 cm for all data presented in this work. This distance could be increased with better receiver sensitivity and higher transmitted level but there is a physical limit where the exhaled air flow effect on the environment becomes insignificant.
2.1.. Acquisition and Shaping of the Received Signal
----------------------------------------------------
The signal received by the ultrasonic receiver is centered around the source frequency, 40 kHz. To digitize that signal by respecting the Nyquist--Shannon sampling theorem, a high sampling frequency and a fast A/D converter would be necessary. As the received signal is narrow band, spanning about 2 kHz below and above 40 kHz, frequency translation to a much lower intermediate frequency is possible. We have chosen to translate it around 4 kHz using a frequency mixer. The received signal is mixed with a 44 kHz sinusoidal signal to generate 4 kHz and 84 kHz beat products. The high frequency product is removed with a low pass anti-aliasing filter before digitization. The block diagram is shown in [Figure 2](#f2-sensors-14-15371){ref-type="fig"}. This solution presents the technical advantage to allow a low frequency digitization as, for example, with an inexpensive computer sound card.
To visualize the spectral widening, we have used a time/frequency representation based on a short time Fourier transform (STFT) as: $$S{({t^{\prime},\nu})} = {\int{f\left( t \right)W\left( {t - t^{\prime}} \right)e^{2\pi i\nu t}\text{d}t}}$$with *S*(*t′*,*v*) the STFT of the signal to be analyzed *f*(*t*) computed for each window centered at *t* = *t′*, *v* the frequency parameter, W(*t* − *t′*) the windowing function centered at *t* = *t′* and *i* stands for the imaginary unit satisfying *i*^2^ = −1.
For computation, a 44.1 kHz sampled signal is used with a Kaiser-Bessel windowing function \[[@b18-sensors-14-15371]\] of size 65,536. This respects the Heisenberg inequality and allows a good compromise between time and frequency resolutions.
2.2.. Physical Principles
-------------------------
The observed phenomenon ([Figure 3](#f3-sensors-14-15371){ref-type="fig"}) is the result of several combined physical effects whose principal is the Doppler effect. We can nevertheless separate two main parts: the overall effect generated by the mean speed of expired airflow (mean speed *v* ≈ 1 m·s^−1^ and Doppler frequency shift Δ*f* ≈ 100 Hz) and local effects caused by turbulence which produce greater frequency extensions (several hundreds of Hz above and/or below the transmit frequency).
The breathing exhalation flow can be assumed turbulent near its source (*i.e.*, nose or mouth). In such a flow, unsteady vortices appear on many scales and interact with each other. The friction between the exhaled flow and the static surrounding environment increases and interacts with the ultrasound wave.
2.3.. Sound-Vorticity Interaction
---------------------------------
The phenomenon of scattering of sound by vorticity is known since the 1950\'s. Obukhov \[[@b19-sensors-14-15371]\] showed that an incident ultrasonic plane wave passing through an area with flow vorticity is scattered by it.
Inside a flow, we can show that any velocity field $\overset{\rightarrow}{u}$ can be decomposed into modes as: $$\overset{\rightarrow}{u} = {\overset{\rightarrow}{u}}_{p} + {\overset{\rightarrow}{u}}_{v} + {\overset{\rightarrow}{u}}_{s}$$with: $$\begin{matrix}
{\overset{\rightarrow}{\nabla} \cdot {\overset{\rightarrow}{u}}_{p} = 0\ \text{and}\ \overset{\rightarrow}{\nabla} \land {\overset{\rightarrow}{u}}_{p} = \overset{\rightarrow}{0},} \\
{\overset{\rightarrow}{\nabla} \cdot {\overset{\rightarrow}{u}}_{s} \neq 0\ \text{and}\ \overset{\rightarrow}{\nabla} \land {\overset{\rightarrow}{u}}_{s} = \overset{\rightarrow}{0},} \\
{\overset{\rightarrow}{\nabla} \cdot {\overset{\rightarrow}{u}}_{v} = 0\ \text{and}\ \overset{\rightarrow}{\nabla} \land {\overset{\rightarrow}{u}}_{v} = \overset{\rightarrow}{\Omega}\left( {\overset{\rightarrow}{r},t} \right) \neq \overset{\rightarrow}{0},} \\
\end{matrix}$$where *u⃗~p~* is a potential flow, *u⃗~s~* identify a sound wave and Ω⃗(*r⃗*,*t*) the vorticity.
Chu \[[@b20-sensors-14-15371]\], in 1958, showed the existence of a coupling between these different modes. More recently, Lund and Rojas \[[@b21-sensors-14-15371]\] have derived a convenient expression that connects the acoustic scattered pressure, *p~scat~* directly to the Fourier transform of the vorticity field: $$p_{\text{scat}}\left( {\overset{\rightarrow}{r},\nu} \right) = p_{\text{inc}}\frac{- \cos\left( \theta_{s} \right)\sin\left( \theta_{s} \right)}{1 - \cos\left( \theta_{s} \right)}\frac{2i\pi^{3}\nu}{c^{2}}\frac{e^{i{\overset{\rightarrow}{k}}_{d} \cdot \overset{\rightarrow}{r}}}{r}\Omega\left( {\overset{\rightarrow}{q},\nu - \nu_{0}} \right)$$with: *p~inc~* the pressure amplitude of the plane incident wave of frequency *v*~0~ and wave vector ${\overset{\rightarrow}{k}}_{i} = \frac{2\pi\nu_{0}}{c}\hat{i}$,θ*~s~* the diffusion angle ([Figure 4](#f4-sensors-14-15371){ref-type="fig"}),*v* the scattered wave frequency of wave vector ${\overset{\rightarrow}{k}}_{d} = \frac{2\pi\nu}{c}\hat{d}$,$\overset{\rightarrow}{q} = {\overset{\rightarrow}{k}}_{d} - {\overset{\rightarrow}{k}}_{i}$ the scattered wave vector,Ω the vorticity component perpendicularly to the plane.
This shows a linear relationship between the acoustic scattered pressure *p~scat~*, the incident pressure *p~inc~* and the wave frequency *v*, thus the system would benefit from increasing the transmit level and the wave frequency. However, practical transducers of high frequency tend to have a reduced beamwidth which could limit the ability to detect breathing for all head positions.
The angular term of the Lund formula that modulates the acoustic scattering intensity will be denoted by: $$L\left( \theta_{s} \right) = \frac{- \cos\left( \theta_{s} \right)\sin\left( \theta_{s} \right)}{1 - \cos\left( \theta_{s} \right)}$$
[Figure 5](#f5-sensors-14-15371){ref-type="fig"}, shows the evolution of *L*(θ*~s~*) and *L*^2^(θ*~s~*). This latter term modulates the acoustic scattering intensity. Note that the apparent divergence of this expression for small scattering angles is an artifact due to the conditions of validity of the Lund formula. Low angles would allow better scattered signal, but are not practical in our application. The optimum angle, experimentally determined, is in accordance with the second local maximum.
This shows that it is theoretically possible to measure the vorticity of a flow by diffusion of ultrasounds that corresponds to what is done by interpreting the spectrum of the acoustic received signal.
More physically, when the wave excites a vortex, this latter is advected by the oscillating velocity field of small amplitude *u⃗~p~*, the sound wave, and begins to oscillate. As an unsteady vortex is a sound source, the oscillating vortex will emit sound. Thus, the excitation by the incident wave will be scattered by the vorticity, in the same way that light is scattered by matter.
2.4.. Influence of Experimental Conditions
------------------------------------------
- Oral or Nasal Breathing
Typically, the phenomenon is similar whatever the type of breathing (nasal or oral). We can just observe a slightly lower level of the signal in case of nasal breathing ([Figure 6](#f6-sensors-14-15371){ref-type="fig"}). Position of the Subject Head, Relative to the Sensor
The [Figure 7](#f7-sensors-14-15371){ref-type="fig"} shows the change in the signal intensity according of the inclination of the subject\'s head. The signal level is maximum in front of the sensor (3) and gradually decreases with the angle of incidence. This level variation is not a problem for the application of apnea detection as the signal to noise ratio is acceptable for all positions. We can observe a high level Doppler offset of the incident wave frequency when the subject changes his position.
2.5.. Influence of Subject or Surrounding Movements
---------------------------------------------------
During rapid subject movements ([Figure 8](#f8-sensors-14-15371){ref-type="fig"}, mark 1), like sleeping position changes, the breathing signal is embedded in the Doppler shift. In this situation, it is difficult to verify the presence, or absence, of breathing activity.
2.6.. Obtaining a Breathing Signal
----------------------------------
In order to characterize respiration, we need a scalar signal from which breathing intensity and respiratory rate can be calculated.
The received signal is first bandpass filtered by an infinite impulse filter (IIR). The bandpass is chosen to avoid the noisy "movement region" near the transmit frequency. One can use either the low or high frequency "breathing zone" or both. Breathing signal intensity on these two zones depends on sensor orientation relative to the subject. In our experimental configuration, signal strength is greater on the low frequency side ([Figure 3](#f3-sensors-14-15371){ref-type="fig"}). For a 4 kHz carrier received after down-mixing, we use a 3500--3900 Hz bandpass elliptic IIR filter with 60 dB stop band attenuation and 1 dB bandpass ripple ([Figure 9](#f9-sensors-14-15371){ref-type="fig"}). To further improve rejection, we apply this filter twice. The 3900 Hz highpass frequency is chosen to exclude the frequency components close to the carrier frequency because the signal level in this zone varies with head position and head movements. The choice of the lower frequency is a compromise. On very deep exhalations, spectral widening can extent down to 3000 Hz but average exhalations do not produce significant signal down to 3500 Hz. Lowering the lowpass frequency further than 3500 Hz would not add more information and would introduce unwanted noise on average exhalations. This would decrease the average signal to noise ratio.
Adult respiratory rate can vary from 0.2 to 0.5 Hz, children and newborn can show rate up to 1 Hz. Thus we choose a 10 Hz sample rate for our breathing signal. The root mean square (RMS) value of the bandpass filtered signal is computed every 100 ms (1/10 Hz). This RMS signal is low-pass filtered with a Butterworth filter to give the breathing signal. The optimal low pass frequency was determined by the residual analysis method \[[@b22-sensors-14-15371]\]. The residual *Res*, defined mathematically as the root mean square value of the difference between filtered and unfiltered data, was plot over a set of 100 cutoff frequencies *f~c~* evenly spaced from 0.1 Hz up to the Nyquist frequency (5 Hz for 100 ms sample time in our case): $$Res\left( f_{c} \right) = \sqrt{\frac{1}{N_{s}}{\sum\limits_{i = 0}^{N}\left( {x_{i} - {x^{\prime}}_{i}} \right)}^{2}}$$where: *N~s~* is the number of samples,*x~i~* is the *i*th unfiltered RMS value,$x_{i}^{'}$ is the *i*th filtered RMS value.
If we make the assumption that data is contaminated by a white noise, the residual would vary linearly with the cutoff frequency. This is the case for our breathing data above 2.5 Hz and up to 4.5 Hz. If we extrapolate this linear region up to the point it intercepts the y-axis, we can estimate the RMS value of the noise (dash red line in [Figure 10](#f10-sensors-14-15371){ref-type="fig"}). The optimal frequency is chosen at the point where noise contribution to the residual equals signal contribution. This is a compromise between the amount of noise allowed to pass through the filter and the signal distortion. This analysis, repeated other a range of breathing data obtained from adults at rest leads to optimal cutoffs between 1.2 to 1.5 Hz. The lowpass frequency was set to 1.5 Hz for all data presented in this work ([Figure 11](#f11-sensors-14-15371){ref-type="fig"}). A typical breathing signal is shown in [Figure 12](#f12-sensors-14-15371){ref-type="fig"}.
2.7.. Obtaining a Signal Representing the Subject Motion
--------------------------------------------------------
The breathing signal is disturbed in presence of large and rapid movements because the received spectrum is also widened during such events. Little head or shoulder motions as they occurred during sleep have no effect on the breathing signal but when the patient changes sleeping position or moves his arms, the breathing signal should not be taken into account. Thus we need a movement signal to detect situations where the breathing signal is too perturbed to give meaningful information.
We derived a movement signal by tracking energy variation (using standard deviation) in a very narrow window below and above the center frequency. This was done by computing the Fourier transform of the received signal other a 371 ms time slice (16,384 samples at 44,100 Hz) and keeping only bins of interest. [Figure 13](#f13-sensors-14-15371){ref-type="fig"} shows the two movement zones of approximately 12 Hz apart the center frequency from where the signal movement is computed. A 25 Hz zone centered on the center frequency was excluded because the signal value in this area can vary widely due to patient position. [Figure 14](#f14-sensors-14-15371){ref-type="fig"} shows a movement signal and a breathing signal. The movement signal increases when the patient changes position. When the movement signal increases the breathing signal is meaningless.
3.. Results and Discussion
==========================
Several validations were performed with our sensor to determine its performance. A first one, qualitative, consisted of positioning a breathing subject under the sensor and observing in real time the spectrogram (time/frequency representation). Under these conditions, we can verify the perfect correspondence between the spectrogram and the breathing activity during normal breathing, simulated apnea or movements.
A second validation, more quantitative, was to equip the subject with a polygraph currently used in sleep laboratories to diagnose sleep apnea syndrome (Cid102L, Cidelec^®^, Sainte Gemmes sur Loire, France). It consists in a pressure sensor connected to a nasal cannula and to record both signals over a period of about 1 h which corresponds to about 1000 breathing cycles ([Figure 15](#f15-sensors-14-15371){ref-type="fig"}). During this period, the subject breathes through the mouth, the nose and simulates apnea and performs movements compatible with sleep. This experience showed the synchronization of the pressure signal provided by the nasal cannula with our signal spectrogram. However, the presence of the cannula in the nostrils derives a significant part of the exhaled air flow and therefore reduces the sensitivity of our device.
4.. Conclusion
==============
The sensor presented in this work represents a good solution for contactless breathing monitoring. Its applications span from real-time monitoring to sleep apnea diagnosis. Its contactless feature makes it suitable for breathing monitoring of infants or premature, and more generally monitoring of subjects at risk of having respiratory distress.
In future works, we plan to use several receivers evenly spaced around the subject head to allow us to remove the angular dependence of the breathing signal. Using a Fleish pneumotachograph we should be able to calibrate our system in order to obtain absolute measurements. In a second experiment, we will try a different signal processing methodology based on pattern matching classification of the different breathing states. To do this, and complete this study, a large measurement campaign on real patients is planned in collaboration with the CHITS (Toulon Hospital).
We thank the members of the Centre Hospitalier Intercommunal Toulon La Seyne, Sainte Musse Hospital for their contribution to this project:
Daniel D\'amore, pulmonologist, Marie-Françoise Mateo-Champion, neurophysiologist, the staff of the sleep laboratory and Jean-Philippe Suppini, head of the Clinical Research Unit.
We would like to thank Cidelec^®^ Company, 20 rue des Métiers 49130 Sainte Gemmes sur Loire, France, <http://www.cidelec.net/fr/en.html> who lent us the polygraph to establish a comparison with our device.
All the authors contributed equally to this work.
The authors declare no conflict of interest.
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"Background\n==========\n\nTotally Implantable Access Ports (TIAP)\n--------------------------------(...TRUNCATED) |
"Citation {#SECID0EDAAC}\n========\n\nZhang M, Li T-H, Wang C-Q, Zeng N-K, Deng W-Q (2019)Phylogenet(...TRUNCATED) |
"Introduction\n============\n\nUnderstanding a story text requires the reader to form a mental repre(...TRUNCATED) |
"\\section*{Appendix. #1}\n\\renewcommand{\\thesection.\\arabic{equation}}}{A\\arabic{equation}}\n (...TRUNCATED) |
"\\section{Introduction}\n\\label{sec:introduction}\n\nCurrently, there is great interest in develop(...TRUNCATED) |
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📖 unlearn_dataset
The unlearn_dataset serves as a benchmark for evaluating unlearning methodologies in pre-trained large language models across diverse domains, including arXiv, GitHub.
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To load the dataset:
from datasets import load_dataset
dataset = load_dataset("llmunlearn/unlearn_dataset", name="arxiv", split="forget")
- Available configuration names and corresponding splits:
arxiv:forget, approximate, retaingithub:forget, approximate, retaingeneral:evaluation, retain
🛠️ Codebase
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⭐ Citing our Work
If you find our codebase or dataset useful, please consider citing our paper:
@article{yao2024machine,
title={Machine Unlearning of Pre-trained Large Language Models},
author={Yao, Jin and Chien, Eli and Du, Minxin and Niu, Xinyao and Wang, Tianhao and Cheng, Zezhou and Yue, Xiang},
journal={arXiv preprint arXiv:2402.15159},
year={2024}
}
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