A Geometrically-Constrained Mathematical Model of Mammary Gland Ductal Elongation Reveals Novel Cellular Dynamics within the Terminal End Bud
[['Mathematical models have been used to inform basic biological research for well over 100 years [1] .', 'Today , mathematical and computational models are particularly useful as a complementary approach to lab-based studies in developmental biology .', 'Such models have been developed to investigate morphogenetic phenomena [2 , 3] in a broad sense , and in particular , for developmental pattern formation ( e . g . [4 , 5] ) , vascular branching ( e . g . [6] ) , cell morphology ( e . g . [7] ) , angiogenesis ( e . g . [8 , 9] ) , as well as both normal ( e . g . [10–13] ) and cancerous ( e . g . [14–16] ) development and regeneration/renewal ( e . g . [17–19] ) in many tissue types , including breast [20 , 21] .', 'Branching morphogenesis specifically has been modeled ( see , e . g . [22] for a review ) in the lung [23] , kidney [24] , vasculature structures ( e . g . [6] ) , plants ( e . g . [25 , 26] ) , and to a certain extent , the mammary gland [11 , 27–30] ) .', 'In most of these studies , the modeling framework includes reaction-diffusion equations to account for soluble factors driving the growth ( e . g . VGEF , FGF ) or playing a regulatory role ( e . g . Wnt , TGFβ ) .', 'With respect to the normal mammary gland , several groups have modeled normal physiological processes .', 'Grant et al . developed a model of mammary ductal morphogenesis adapted from a cellular-automaton model of vascular morphogenesis [27] .', 'While this model was capable of approximating mammary branching gross morphology , it did not take into account the arrangement of the cells at small scales , or use any experimentally derived measurements .', 'Nelson et al . also investigated branching processes in the mammary gland with engineered epithelial tubes .', 'Although the process described does not directly mimic mammary development in vivo , they were able to mathematically model spacing of separate ductal units [31] .', 'At smaller scales of interest , Tang et al . created an agent based model of breast acinus formation in vitro , in which they were able to determine the proliferation and apoptotic dynamics required for proper lumen formation and DCIS development [11] , whereas Rejniak et al . have proposed a sophisticated single deformable cell based model to derive conditions for acinus structure and lumen stability [30] .', 'Although the works cited above have been successful in modeling particular aspects of normal mammary development ( i . e . ductal tree and acinus formation and some of the corresponding regulation ) , there are currently no published models of mammary ductal elongation , a critical process required for , and intimately coupled with , branching morphogenesis [32] .', 'In this work , we sought to fill this gap by generating a mathematical model of mammary ductal elongation that not only incorporates the structure responsible for the majority of pubertal development , the terminal end bud ( TEB ) [33] , but also accounts for all of the main biological processes occurring within the TEB that are required for its growth .', 'In the mammary gland , the rudimentary mammary duct formed during embryonic development remains relatively quiescent until puberty .', 'Ductal elongation during puberty is driven by , and dependent upon , TEBs [32] .', 'TEBs are bulbous , multi-layered structures that direct the growth of the duct throughout the fat pad during puberty .', 'Mammary gland organogenesis in the virgin animal is dependent largely on this process of ductal elongation .', 'During elongation TEBs are responsible for fat pad invasion , bifurcation of main ducts , lumen formation , basement membrane deposition , and the recruitment of periductal stroma and a blood supply .', 'These critical functions , and the geometrically-constrained manner in which they are accomplished , make the TEB an ideal structure on which to base a mathematical model for ductal elongation .', 'In addition , the ductal elongation rates in the murine mammary gland far exceed the allometric growth of other organs in the postnatal mouse , making this structure of unique interest .', 'Lastly , TEBs have been postulated as a major site susceptible to carcinogenesis , leading to tumors later in life [34 , 35] .', 'These characteristics make the TEB not only a good model for normal ductal morphogenesis , but also an ideal ground state for modeling breast tumorigenesis .', 'In order to begin to model this structure we base the equations on the following established TEB biology .', 'TEBs have two main cellular compartments .', 'The outer monolayer of the TEB is known as the cap cell layer , which differentiates into the myoepithelial cell layer of the mature duct [33 , 36 , 37] , and is a putative reservoir for mammary stem cells [33 , 38 , 39] .', 'Underlying the cap cell layer is a multi-cellular layer called the body cell layer .', 'The cells in the body cell layer give rise to the layer of luminal epithelial cells lining the interior of the duct [36 , 37] .', 'Apoptosis in the body cell layer has been postulated as the primary mechanism of lumen formation .', 'Finally , high levels of proliferation in these two compartments of the TEB produce the cells that form the mature duct .', 'We employed an investigative and integrative approach [19] using repeated iterations of biological measurements and corresponding parameter estimation to identify key processes required for ductal elongation .', 'Through mathematical output calculation , we show that evaluation of just five experimentally measurable parameters , ( cell type-specific proliferation , cell type-specific apoptosis , cell size and number , and duct tortuosity ) , are sufficient to predict ductal displacement and elongation rates accurately .', 'These data , and the associated modeling iterations required to interpret them , indicate that current common interpretations of developmental phenotypes can be inaccurate and require careful reconsideration .', 'This model serves as a baseline model upon which agent-based , cellular automaton , and other mathematical models can be built to explore both normal mammary gland and breast cancer development .'], ['In order to translate the current knowledge and concepts of mammary gland ductal development into a mathematical model for ductal elongation and displacement , we started the modeling process with a description of the time-invariant geometry of TEBs .', 'We defined the stereotypical morphology of a single TEB in a moving frame ( i . e . invading through the fat pad with the speed associated with the ductal elongation ) and divided the TEB into eight regions ( Fig 1A and S1 Fig ) .', 'Within the TEB structure , each region contains cells ( treated as a single population ) that are capable of proliferation and apoptosis .', 'We then used a population dynamics model to describe the fate ( proliferation or death ) of these cells .', 'In order to maintain a constant number of cells in each region , and a constant size of the regions , proliferation in a given region generates a flux of cells from that region into an adjacent one ( Eq 1 ) .', 'Ductal elongation is the net result of these regional fluxes: the number of newborn cells in the TEB generates cell fluxes ( based on local conservation laws ) , with deposition of these cells in the mature duct , thus increasing the length of the subtending duct as the TEB invades ( S1 Fig ) .', 'Thus , the net flux from the entire TEB into the mature duct dictates the rate of elongation .', 'We model each region i of the TEB as a stationary compartment that contains a time-invariant number of cells and write the corresponding balance of gain ( cell proliferation and influxes ) and loss ( cell death and outfluxes ) terms: Model 1: Base equation .', 'Ni : time-invariant cell number in Region i ri : average population growth ( proliferation ) rate in Region i di : average population death ( apoptotic ) rate in Region i φl→i , φi→l : cell influxes ( from adjacent regions l to i ) and outfluxes ( from i to l ) The derivation of the ductal elongation rate can be performed by considering the growth of either the outer myoepithelial ( basal ) layer of the duct ( Eq 1 for Regions i ∈ {1 , 2 , 3} ) or the inner ( luminal ) layer ( Eq 1 for Regions i ∈ {5 , 6 , 7} ) .', 'For the purposes of the initial model , we assumed that cap cells are entirely responsible for the formation of the myoepithelial layer of the mature duct , and that the body cells are entirely responsible for the formation of the luminal layer of the mature duct .', 'Such assumptions result in the only non-zero fluxes to be φ1→2 , φ2→3 and φ3→4 within the basal layer , and φ5→6 , φ6→7 and φ7→8 within the luminal layer ( S1 Fig ) .', 'These fluxes correspond to the net production , by unit time , of newborn cells in each region , which result in the net cell fluxes Φbas ( Eq 2 ) corresponding to the elongation of Region 4 ( the basal layer ) and Φlum ( Eq 3 ) corresponding to the elongation of Region 8 ( the luminal layer ) .', 'Cell flux from the TEB to the mature duct .', 'Based on the assumption of homeostasis of the mature duct ( i . e . , apoptosis and proliferation processes balance in Regions 4 and 8 ) , the duct elongates only because cells are generated within the TEB , with time-invariant rates Φbas and Φlum for the basal and luminal layers , respectively .', 'These two values , by definition , must match since the two layers adhere to one another , and one layer does not outpace the other during the process of elongation , as observed experimentally .', 'The ductal elongation rate of each layer is finally found by equalizing the lateral surface area of the mature duct in a 2D cross-section to the surface monolayer covered by N adjacent cells and by taking the time derivative of these expressions ( Eq 4 ) .', 'L ( t ) : length of the mature duct N ( t ) : cell number along the mature duct', 'l', ': length of a single cell', 'Φ=dNdt', ': time-invariant rate of cells provided to the mature duct by the TEB', 'λ=dLdt', ': ductal elongation rate ( for a monolayer ) To determine the morphological characteristics that define the geometric framework of the model , manual measurements of each region and the individual cells within those regions were taken from histological sections of 5–6 week old paraffin embedded terminal end buds ( see Methods: Image Analysis and Measurement and Definition of Terminal End Bud Regions ) .', 'Regional measurements are represented in Fig 1A as a scaled model of the TEB .', 'Cellular dimensions are represented in Fig 1B and 1C as box plots .', 'In Regions 1–4 , the cells become thinner ( from 10 . 1μm to 3 . 0μm ) and the intra-nuclear distance increases ( from 7 . 6 to 21 . 5μm ) as they mature in Region 4 ( Fig 1B ) .', 'The cells in Regions 5–8 are relatively the same size ( Fig 1C ) .', 'Regions 1 and 5 are represented as hollow half-discs and make up the leading tip of the TEB .', 'The total length of these two regions is 85 . 8μm ( 75 . 7+10 . 1 ) , and the width of these two regions is 164 . 8μm ( 144 . 6+ ( 10 . 1x2 ) ) ( Fig 1A ) .', 'Regions 2 and 6 are represented as trapezoids , with the larger end joining Regions 1 and 5 and sharing the same total width .', 'The smaller end of Regions 2 and 6 join with Regions 3 and 7 at a total width of 91 . 3μm ( 70 . 9+ ( 10 . 2x2 ) ) ( Fig 1A ) .', 'Regions 3 and 7 maintain approximately this width for a distance of 216 . 5μm , which we defined as the length of duct refractory to ovarian hormone treatment ( S2 Fig ) .', 'The mature regions of the duct , Regions 4 and 8 , have a total width of 66 . 5μm ( 60 . 5+ ( 3 . 0x2 ) ) ( Fig 1A ) .', 'All measurements are summarized in Table A in S1 Text .', 'To determine the proliferation rate and cell cycle duration within each of the defined regions of the TEB , we conducted a dual pulse labeling experiment in which proliferating cells were labeled at time zero by incorporation of 5-ethynyl-2-deoxyuridine ( EdU ) .', 'Proliferating cells were subsequently labeled at two hour increments over 24 hours with a 5-bromo-2-deoxyuridine ( BrdU ) pulse , and glands were harvested two hours after BrdU pulse labeling ( see Methods Section: Determination of Proliferation Rates and Cell Cycle Dynamics ) .', 'Representative images from each time point are presented in Fig 2A .', 'In all regions , the number of EdU/BrdU double positive cells was highest at time 0 , and reached its lowest levels eight hours after EdU labeling , indicating an S -phase duration of about six hours ( +/- ~30 minutes ) .', 'In Regions 1 , 2 , 3 , and 5 , the number of double positive cells peaked 16 ( Regions 1 , 2and 5 ) , 18 ( Region 3 ) , or 22 ( Region 3 ) hours after EdU labeling , suggesting that cells in each region re-enter the cell cycle .', 'In Regions 6 and 7 , the number of double positive cells remained low , suggesting that these cells do not re-enter the cell cycle at high frequency ( Fig 2B ) .', 'Total proliferation levels did not change substantially over time in each region ( S3A Fig ) .', 'Additional time points out to 120 hours were also investigated ( S5 Fig ) .', 'To determine the duration of G2/M phase in the EdU labeled cells , we performed further EdU/BrdU pulse time points at 30 min , 1 hour and 1 . 5 hours and performed immunofluorescent ( IF ) staining for phospho-histoneH3 ( pHH3 ) colocalization with EdU .', 'Levels of EdU+pHH3+ cells peaked in all regions two hours after EdU labeling .', 'By eight hours after EdU labeling , cells were no longer co-stained with pHH3 .', 'Together with the previous finding that the S phase is ~6 hours long , this indicates a G2/M phase duration of two hours +/- approximately 30 min ( S3B and S3C Fig ) .', 'To determine the level of apoptosis occurring within each region of the TEB , we performed IF staining for cleaved caspase III ( CC3 ) , a marker of mid-stage apoptosis ( See Methods Section: Immunofluorescent Staining Of Tissue ) .', 'Representative images from each time point are shown in Fig 3A .', 'The majority of apoptosis occurs in the inner regions of the TEB ( 4 . 0% ±0 . 5 in Region 5 , 5 . 3% ±0 . 4 in Region 6 , and 3 . 5% ±0 . 5 in Region 7 ) ( Fig 3B ) .', 'Although the location of apoptosis is consistent with published TUNEL assay data ( an indicator of late stage apoptosis ) [40] , the total levels are ~50% lower probably due to differences in apoptotic stage and the possibility of non-CC3-mediated cell death mechanisms being used .', 'Apoptosis levels in the outer basal layer were less than 1% ( 0 . 1% ±0 . 1 in Region 1 , 0 . 3% ±0 . 1 in Region 2 , and 0 . 5% ±0 . 3 in Region 3 ) .', 'Additionally , the time of CC3 positivity after EdU labeling was found to peak between 6 and 10 hours ( S4 Fig ) .', 'To calculate the rate at which the mammary duct elongates , measurements corresponding to the current scientific knowledge ( proliferation and apoptosis rates ( S1 Text , section A ) , cell cycle and apoptosis times , and regional cell numbers ( Table B in S1 Text ) were used to calculate net population growth rates in the basal regions 1 , 2 , and 3 and the luminal regions 5 , 6 , and 7 .', 'These growth rates were input into the base mathematical equation ( Model 1 ) ( Eqs 2–4 ) .', 'The predicted linear elongation rate for the basal layer was 1 . 24 ( ±0 . 09 ) mm/day , while the predicted elongation rate for the luminal layer was 0 . 78 ( ±0 . 07 ) mm/day .', 'This difference indicated that these initial parameters , which encompass much of the known biology of TEB , were not sufficient to model elongation because the results indicate that the two layers do not grow in a coordinated fashion , but rather that they elongate independently , which is not in accordance with observed biology .', 'This result indicated that our model is incomplete .', 'To address what process ( es ) were missing from our model , we sought to quantify a previously unappreciated experimental observation that cap cells migrate into the body cell layer [33] .', 'In 1983 , Williams and Daniel used time-lapse video to study TEBs and noted that cap cells “migrate from their peripheral location into the deeper regions of the end bud” [33] .', 'Indeed , we found many SMA and p63 positive cells within Regions 5 and 6 ( Fig 4A and 4B ) ; however , the fate of these cells was still unknown .', 'This cap cell migration introduces an additional flux of cells from the basal to the luminal regions into the model , and would therefore decrease the predicted elongation rate for the basal layer , while potentially increasing the predicted elongation rate for the luminal layer .', 'One hypothesis is that migrating cap cells contribute to the body cell layer and may represent mammary multipotent progenitors or stem cells [33 , 41] .', 'To address the fate of these cells , we performed staining for SMA and either BrdU or pHH3 .', 'Quantification of these images revealed that while SMA positive cells in Regions 1 and 2 have high levels of BrdU incorporation ( 37 . 2% ±4 . 5 and 31 . 9% ±3 . 9 respectively ) ( Fig 4C ) , SMA positive cells in Regions 5 and 6 had significantly lower levels ( 7 . 1% ± 1 . 8 and 4 . 0% ± 2 . 0 respectively , p<0 . 0001 ) ( Fig 4C ) , implying that cap cells present in the body cell layer are not as proliferative .', 'Additionally , staining for SMA and pHH3 indicate that cap cells in the body cell layer were not undergoing mitosis ( 0 . 0% in Region 5 and 6 versus 7 . 6% ± 1 . 0 in Region 1 and 8 . 0% ± 1 . 1 in Region 2 , p<0 . 05 ) ( Fig 4C ) .', 'To determine if migrating cap cells undergo apoptosis , we stained for SMA and CC3 .', 'Quantification of these images found that indeed cap cells present in Regions 5 and 6 were undergoing cell death at a much higher rate than SMA positive cells in Regions 1 and 2 ( 46 . 9% ± 4 . 0 in Region 5 and 42 . 0% ± 4 . 1 in Region 6 versus 1 . 8% ± 0 . 4 in Region 1 and 2 . 3% ± 0 . 4 in Region 2 , p<0 . 0001 ) ( Fig 4D ) .', 'Overall , these data suggest that cap cells present in the body cell layer are no longer proliferative , and are actively undergoing apoptosis .', 'Thus , they do not appear to contribute substantially to the luminal layer of the mature duct .', 'These results are contrary to the current consensus that cap cells contribute substantially to the luminal lineage .', 'This result also indicated that the mathematical description of cap cell movement from the basal layer into the luminal layer , with subsequent elimination of those cells from the system , was an important missing component to our model .', 'We accounted for cap cells migrating into the body cell layer by considering a flux of cells φbas→lum ( = φ1→5 + φ2→6 + φ3→7 , see S1 Fig ) , which leads to the modified fluxes Φbas − φbas→lum and Φlum + φbas→lum to be considered for the evaluation of the basal and luminal elongation rates ( Model 2 ) , respectively ( S1 Text , section B ) .', 'When cap cell specific death is accounted for , our model revealed that the apoptotic index experimentally measured for the luminal layer is underestimated , resulting in inconsistencies in the model .', 'To address this issue , an apoptotic correction factor ( δ = 97% ) was introduced ( S1 Text , section C and Table C ) into the equation for the luminal layer ( Model 3 ) , yielding an apoptotic index of 8 . 5% .', 'Inclusion of the additional flux and apoptotic correction factor into our equations increased the predicted elongation rate for the luminal layer from 0 . 78 to 0 . 81 ( ±0 . 08 ) mm/day , and decreased the rate for the basal layer from 1 . 24 mm/day to 0 . 76 ( ±0 . 12 ) mm/day , bringing these two calculated elongation rates much closer together in accordance with the observation that the layers elongate in a coordinated fashion .', 'To validate the accuracy of our prediction we sought to measure ductal elongation in vivo .', 'To validate the model prediction that the gland elongates at a rate in the range of 0 . 76–0 . 81 mm per day , we measured the rate at which the TEBs are displaced during pubertal elongation in vivo .', 'To do this we measured outgrowths in both inguinal glands of FVB mice from the nipple to the ductal front .', 'The length of ductal outgrowth was then compared across each time point to provide the TEB displacement rate ( Fig 5 ) .', 'The rate of displacement from weeks 5 to 6 ( 0 . 57mm/day ) , 6 to 7 ( 0 . 25mm/day ) , and 7 to 8 ( 0 . 82mm/day ) averages 0 . 54 mm/day over the three week period , a rate significantly less than the predicted elongation rate of 0 . 76–0 . 81mm/day .', 'Our model prediction represents the linear elongation rate of a new duct formed as a straight tube ( or the speed at which the new duct forms in a straight line ) , which is different from the displacement rate that we measure here .', 'The displacement rate is not a measure of the total length of duct , but rather it is based on the shortest distance between the nipple and the ductal front and does not take into account the actual path ( i . e . turning ) of the TEB .', 'In order to compare our mathematical elongation rate prediction to the measured displacement rate , we must convert the linear output to its displacement equivalent .', 'Tortuosity refers to the characteristic of a winding or twisting path as opposed to a straight path .', 'The path of the TEB is tortuous due to both bifurcation events and natural turning .', 'Because bifurcation is a frequent event during elongation , we sought to determine the effect that bifurcation has on the straight line route of a TEB’s growth path .', 'We measured the angle at which a TEB is deflected off its path at the time of bifurcation by measuring the angle between the original path and the path after bifurcation ( Fig 6A ) ( see Methods: Image Analysis and Measurement ) .', 'We found that the average angle of deflection is 35 . 5° ( ±1 . 9° , n = 62 ) ( Fig 6B ) .', 'In addition , we evaluated the frequency of bifurcation during pubertal outgrowth .', 'This was determined by measuring the distance ( path length ) between bifurcation events ( Fig 6C ) .', 'The average distance between bifurcation forks was 1141μm ( ±99 . 1 , n = 23 ) ( Fig 6D ) .', 'Additionally , we compared path tracing measurements to corresponding displacement measurements between bifurcation events and found that the displacement measurements consistently underestimate the total distance by 6 . 1% ( ±0 . 9% , n = 23 ) ( Fig 6E ) .', 'We used the measurements of TEB path tortuosity to convert ( S1 Text , section D ) our previously predicted linear elongation rates ( 0 . 81 and 0 . 76 mm/day ) into the measured displacement rate using our ultimate model ( Model 4 ) ( Eqs 5–6 ) which accounts for the cap cells dropping into the body cell layer ( S1 Text , section D ) : Model 4 .', 'λbas=lbas2T ( ∑i=1 , 2 , 3 ( ri−di ) Ni−φbas→lum )', '( 5 )', 'λlum=llum2T ( ∑i=5 , 6 , 7 ( ri−di ( 1+δ ) ) Ni+φbas→lum )', '( 6 )', 'where , Tortuosity due to bifurcation ( a factor of 1 . 23 ) and tortuosity due to turning ( a factor of 1 . 06 ) are calculated using measurements determined in the previous experiment ( S1 Text , section D ) .', 'The predicted rate is converted to displacement rate by dividing our previous predictions of 0 . 81 and 0 . 76 mm/day by the total tortuosity ( T = 1 . 31 ) .', 'This conversion gives a displacement rate of 0 . 62mm/day and 0 . 58mm/day for the luminal and basal layers , respectively ( Table D in S1 Text ) .', 'The difference between the experimentally measured displacement rate of 0 . 54mm/day and our model’s prediction indicate that these parameters , including the addition of a novel cap cell flux with the luminal apoptotic correction factor , are sufficient to account for the kinetics of ductal elongation .', 'These results also indicate that displacement measurements can underestimate total duct length by 24% .', 'Our initial model , Model 1 , yielded single value predictions for the luminal and basal layers that were incompatible with known biology , and with our experimentally measured displacement rate of 0 . 54mm per day ( Model 1—Fig 7A ) .', 'With the addition of the cap cell flux ( Model 2—Fig 7A ) the model outputs become dependent on the value of this flux , which then allows us to perform a model fit analysis: for a fixed value of δ , φbas→lum ( and therefore , the corresponding cap cell fraction X , see S1 Text , section E ) can be determined by using the matching condition λbas = λlum .', 'Without accounting for cap cell specific death ( δ = 0 ) , the fit predicts an elongation rate that is incompatible with our experimentally measured rate ( Model 2—Fig 7A ) .', 'However , addition of the minimal apoptotic correction factor of δ = 0 . 97 ( S1 Text , section C ) yielded a cap cell flux value ( corresponding to X = X2 = 49% ) leading to an elongation rate still outside the error of our experimentally derived levels ( Model 3—Fig 7B ) .', 'When we account for tortuosity , the linear elongation rate is converted to a displacement rate ( Model 4—Fig 7C ) and we find agreement between the model fitting and our experimentally measured rate , however with a very large range of admissible values of the cap cell flux ( Fig 7C red error bar ) .', 'The ultimate model , when constrained by specific values estimated experimentally ( Model 4—Fig 7D ) for X = X* = 54% and for δ = 0 . 97 ( S1 Text , section E ) , yields values ( λlum = 0 . 62mm / day and λbas = 0 . 58mm / day , Fig 7D blue and black dots ) that fall within the error of the experimentally measured displacement rate .', 'When we compare these values to those found from model fitting with δ = 0 . 97 ( Fig 7D blue solid line and white dot ) we also find a rate within this error box .', 'Finally , when we apply an apoptotic correction factor ( δ = 1 . 55 ) equivalent to the 11% apoptotic index reported by Humphreys et al [40] ( Fig 7D blue dotted line ) , we find a flux value by model fitting that exactly matches for both TEB layers ( white dot ) , and lies within the same error box .', 'These results illustrate that our final model predictions , based on experimentally estimated values of X and δ , provides values of the elongation rates that 1 ) lie within the error range of our measurement and 2 ) are consistent with values forced by fitting but ignoring the biological cause , thus strengthening our final model and our modeling approach’s validity .'], ['The TEB is the driving force for mammary duct invasion and elongation into the fat pad .', 'Here , we exploited the geometric constraints on TEB size to develop a mathematical model for ductal elongation and displacement .', 'This iterative modeling approach identified a set of experimentally measurable parameters required for accurate modeling .', 'Our base model ( Model 1 ) incorporating only region-specific cell number , proliferation rate ( BrdU ) and apoptotic index ( CC3 ) , predicted two different elongation rates for the basal and luminal layers .', 'This prediction was inconsistent with the biological observation that the two cellular compartments grow in a coordinated fashion , and required identification and evaluation of additional processes and parameters .', 'Parameters were added sequentially to the model until the predicted elongation rates of the two cellular compartments could be matched ( Models 2 and 3 ) , thereby demonstrating that our investigative approach of ductal elongation helped identify the major biological processes involved , similar to the approach by Hoehme et al . [19] in the context of liver regeneration .', 'Once the predicted elongation rate was established , conversion to an observed displacement rate ( Model 4 ) required measurement of average branch angle and duct tortuosity .', 'The results from this process have a significant impact on how developmental phenotypes in elongating structures are interpreted generally , as well as for our understanding of cellular dynamics within the TEB .', 'Cap cells have long been referred to as putative reservoir for mammary stem cells , and have been credited with contributing to both the basal and luminal lineages [33 , 38 , 42 , 43] .', 'In 1983 , Williams and Daniel described time lapse video of cap cells migrating into the body cell layer and out of view .', 'They hypothesized that cap cells that migrate into the body cell layer differentiate into luminal cells , and that they are examples of multipotent progenitor cells [33] .', 'An alternative hypothesis is that migrating cap cells rejoin the basal layer as the duct elongates .', 'Numerous microscopy analyses have shown the presence of cap cells in the body cell layer [44–46]; however , none of these studies have investigated the fate of cap cells once actually there .', 'More recent lineage tracing experiments have led to conflicting conclusions about the contribution of cap cells to the luminal lineage .', 'The most recent studies by Rios et al maintained the historical view that cap cells in the TEB give rise to luminal cells [36 , 37] .', 'However their lineage tracing data did not address whether this contribution is due to migration of cap cells into the body cell layer , or due to asymmetric division directly from the cap cell layer into the body cell layer .', 'The failure of Model 1 along with the conflicting positions on cap cell biology led us to investigate Williams and Daniel’s observation further .', 'We demonstrate that the migrating cap cells are no longer proliferative by both decreased BrdU labelling and low pHH3 , and in fact , are actively undergoing programmed cell death at a high rate .', 'Using the measurements of proliferation and the number of cap cells present in the body cell layer , we were able to calculate the percentage of newborn cap cells that migrate into the body cell layer to die .', 'If we assume that cap cells are dropping in from only Regions 1 and 2 we calculate that a striking 54% of the daughter cells actually migrate into the body cell layer to undergo apoptosis ( 39% if region 3 is included 3 ) ( S1 Text , section E ) , and thus are not used productively for duct formation .', 'Together with recent lineage tracing data [37] , these data strongly suggests that few , if any , cap cells contribute to the body cell layer , or rejoin the basal layer .', 'The high levels of CC3+ cap cells present in the body cell layer forced us to reexamine the dynamics within the body cell layer and adjust the mathematical model .', 'Although several groups have previously investigated cell death in the body cell layer and accepted it as a major mechanism of lumen formation [40 , 44 , 47 , 48] , none have confirmed the lineage of these dying cells .', 'Our studies have shown the majority of CC3+ cells to be of cap cell origin , leaving only a small percentage of true body cells undergoing caspase-3-mediated cell death .', 'When this phenomenon was introduced into the mathematical model in Model 3 , a correction factor of 97% was needed for the luminal layer in order to keep the mathematical model consistent ( S1 Text , section C ) .', 'This correction factor represents an underestimation of total cell death by CC3 positivity .', 'These model manipulations provide mathematical evidence that true body cell clearance levels are higher than our biological measurements have indicated .', 'Given our results , it is possible that CC3-mediated apoptosis is a mechanism for removal of cap cells specifically , rather than a mechanism of luminal cell clearance .', 'Previous work indicated that cap cells , when detached from the cap cell layer , undergo apoptosis in response to anoikis .', 'In Netrin-1 KO mice , unattached cap cells die at a rate 17-fold higher than attached cap cells .', 'This coincides with a 2-fold increase in the number of cap cells present in the body cell layer [49] .', 'Additionally , targeted disruption of BIM , a pro-apoptotic factor , leads to a 5 . 6-fold increase in p63 positive cells in the TEB [44] .', 'It remains unclear whether non-CC3-mediated cell death mechanisms in the body cell layer may play a role in lumen formation .', 'In our ultimate model , Model 4 , the net increase in cell number , cell migration , and the geometric constraints within the TEB creates cellular fluxes into the mature duct , which results in forward movement of the TEB as it elongates through the fat pad .', 'However , the ultimate displacement within the fat pad is modulated by changes in direction due to branching and turning ( tortuosity ) .', 'The behavior of our final model has illustrated a significant problem with regard to the use of traditional measurements and terminology for charact
['Introduction', 'Results', 'Discussion', 'Methods']
['Mathematics is often used to model biological systems .', 'In mammary gland development , mathematical modeling has been limited to acinar and branching morphogenesis and breast cancer , without reference to normal duct formation .', 'We present a model of ductal elongation that exploits the geometrically-constrained shape of the terminal end bud ( TEB ) , the growing tip of the duct , and incorporates morphometrics , region-specific proliferation and apoptosis rates .', 'Iterative model refinement and behavior analysis , compared with biological data , indicated that the traditional metric of nipple to the ductal front distance , or percent fat pad filled to evaluate ductal elongation rate can be misleading , as it disregards branching events that can reduce its magnitude .', 'Further , model driven investigations of the fates of specific TEB cell types confirmed migration of cap cells into the body cell layer , but showed their subsequent preferential elimination by apoptosis , thus minimizing their contribution to the luminal lineage and the mature duct .']
['Our paper describes a mathematical model of mammary ductal elongation during pubertal development .', 'We make several conclusions that will be of interest to scientists studying mammary gland biology , epithelial tube formation , and branching morphogenesis .', 'First , our model indicates that a common measurement of developmental outgrowth ( ‘percent fat pad filled’ ) underestimates the total growth and leads to mischaracterization of mutant phenotypes .', 'Second , we show that cap cells , a population enriched with putative mammary stem cells , do not contribute to the luminal lineage as previously hypothesized .', 'Further , we find that a high percentage of proliferation in these cells is not used productively to actually form the mammary duct .', 'We believe our model has future application to other branching organs and also for the modeling of disease states in the breast .']
['cell death', 'cell motility', 'medicine and health sciences', 'reproductive system', 'cell cycle and cell division', 'cell processes', 'mathematical models', 'developmental biology', 'research and analysis methods', 'specimen preparation and treatment', 'staining', 'lipids', 'fats', 'mathematical and statistical techniques', 'exocrine glands', 'breast tissue', 'biochemistry', 'cell staining', 'cell biology', 'anatomy', 'apoptosis', 'cell migration', 'biology and life sciences', 'mammary glands']