Multiscale Modelling of Vascular Tumour Growth in 3D: The Roles of Domain Size and Boundary Conditions
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{"title"=>"Multiscale modelling of vascular tumour growth in 3D: The roles of domain size and boundary conditions", "type"=>"journal", "authors"=>[{"first_name"=>"Holger", "last_name"=>"Perfahl", "scopus_author_id"=>"9943873600"}, {"first_name"=>"Helen M.", "last_name"=>"Byrne", "scopus_author_id"=>"7005568690"}, {"first_name"=>"Tingan", "last_name"=>"Chen", "scopus_author_id"=>"7405543509"}, {"first_name"=>"Veronica", "last_name"=>"Estrella", "scopus_author_id"=>"55890935500"}, {"first_name"=>"Tomás", "last_name"=>"Alarcón", "scopus_author_id"=>"55213301100"}, {"first_name"=>"Alexei", "last_name"=>"Lapin", "scopus_author_id"=>"7102543897"}, {"first_name"=>"Robert A.", "last_name"=>"Gatenby", "scopus_author_id"=>"7004904379"}, {"first_name"=>"Robert J.", "last_name"=>"Gillies", "scopus_author_id"=>"7102089341"}, {"first_name"=>"Mark C.", "last_name"=>"Lloyd", "scopus_author_id"=>"14019667300"}, {"first_name"=>"Philip K.", "last_name"=>"Maini", "scopus_author_id"=>"7007021673"}, {"first_name"=>"Matthias", "last_name"=>"Reuss", "scopus_author_id"=>"57191862072"}, {"first_name"=>"Markus R.", "last_name"=>"Owen", "scopus_author_id"=>"7402497349"}], "year"=>2011, "source"=>"PLoS ONE", "identifiers"=>{"sgr"=>"79955030886", "doi"=>"10.1371/journal.pone.0014790", "pui"=>"361636038", "pmid"=>"21533234", "scopus"=>"2-s2.0-79955030886", "issn"=>"19326203", "isbn"=>"1932-6203 (Electronic)\\r1932-6203 (Linking)"}, "id"=>"6c5d9548-0b38-3aeb-aee1-ed09976fa3b7", "abstract"=>"We investigate a three-dimensional multiscale model of vascular tumour growth, which couples blood flow, angiogenesis, vascular remodelling, nutrient/growth factor transport, movement of, and interactions between, normal and tumour cells, and nutrient-dependent cell cycle dynamics within each cell. In particular, we determine how the domain size, aspect ratio and initial vascular network influence the tumour's growth dynamics and its long-time composition. We establish whether it is possible to extrapolate simulation results obtained for small domains to larger ones, by constructing a large simulation domain from a number of identical subdomains, each subsystem initially comprising two parallel parent vessels, with associated cells and diffusible substances. We find that the subsystem is not representative of the full domain and conclude that, for this initial vessel geometry, interactions between adjacent subsystems contribute to the overall growth dynamics. We then show that extrapolation of results from a small subdomain to a larger domain can only be made if the subdomain is sufficiently large and is initialised with a sufficiently complex vascular network. Motivated by these results, we perform simulations to investigate the tumour's response to therapy and show that the probability of tumour elimination in a larger domain can be extrapolated from simulation results on a smaller domain. Finally, we demonstrate how our model may be combined with experimental data, to predict the spatio-temporal evolution of a vascular tumour.", "link"=>"http://www.mendeley.com/research/multiscale-modelling-vascular-tumour-growth-3d-roles-domain-size-boundary-conditions", "reader_count"=>131, "reader_count_by_academic_status"=>{"Unspecified"=>7, "Professor > Associate Professor"=>11, "Researcher"=>35, "Student > Doctoral Student"=>8, "Student > Ph. D. Student"=>40, "Student > Postgraduate"=>3, "Student > Master"=>11, "Other"=>2, "Student > Bachelor"=>7, "Professor"=>7}, "reader_count_by_user_role"=>{"Unspecified"=>7, "Professor > Associate Professor"=>11, "Researcher"=>35, "Student > Doctoral Student"=>8, "Student > Ph. D. Student"=>40, "Student > Postgraduate"=>3, "Student > Master"=>11, "Other"=>2, "Student > Bachelor"=>7, "Professor"=>7}, "reader_count_by_subject_area"=>{"Unspecified"=>11, "Agricultural and Biological Sciences"=>27, "Chemical Engineering"=>1, "Chemistry"=>1, "Computer Science"=>12, "Engineering"=>22, "Biochemistry, Genetics and Molecular Biology"=>5, "Materials Science"=>1, "Mathematics"=>16, "Medicine and Dentistry"=>10, "Design"=>1, "Pharmacology, Toxicology and Pharmaceutical Science"=>2, "Physics and Astronomy"=>22}, "reader_count_by_subdiscipline"=>{"Materials Science"=>{"Materials Science"=>1}, "Medicine and Dentistry"=>{"Medicine and Dentistry"=>10}, "Physics and Astronomy"=>{"Physics and Astronomy"=>22}, "Mathematics"=>{"Mathematics"=>16}, "Unspecified"=>{"Unspecified"=>11}, "Pharmacology, Toxicology and Pharmaceutical Science"=>{"Pharmacology, Toxicology and Pharmaceutical Science"=>2}, "Chemical Engineering"=>{"Chemical Engineering"=>1}, "Design"=>{"Design"=>1}, "Engineering"=>{"Engineering"=>22}, "Chemistry"=>{"Chemistry"=>1}, "Agricultural and Biological Sciences"=>{"Agricultural and Biological Sciences"=>27}, "Computer Science"=>{"Computer Science"=>12}, "Biochemistry, Genetics and Molecular Biology"=>{"Biochemistry, Genetics and Molecular Biology"=>5}}, "reader_count_by_country"=>{"United States"=>5, "Norway"=>1, "Ireland"=>1, "United Kingdom"=>3, "France"=>1, "Australia"=>1, "Switzerland"=>4, "Portugal"=>1, "Germany"=>2, "Spain"=>2}, "group_count"=>3}

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Figshare

  • {"files"=>["https://ndownloader.figshare.com/files/784424"], "description"=>"<p><b>A</b>) A basic parent vessel unit that includes eight vessels with equal pressure differences, but with four different mean pressures, arranged so that the nearest parent vessel in each case is at a different mean pressure, which should enable functional vessels to form within and between subdomains. Non-periodic means that we consider reflection boundary conditions in all directions. <b>B</b>) Periodic boundary conditions in the -direction. <b>C</b>) Periodic boundary conditions in the - and -directions.</p>", "links"=>[], "tags"=>["vascular"], "article_id"=>454782, "categories"=>["Mathematics", "Biological Sciences", "Cancer", "Biophysics"], "users"=>["Holger Perfahl", "Helen M. Byrne", "Tingan Chen", "Veronica Estrella", "Tomás Alarcón", "Alexei Lapin", "Robert A. Gatenby", "Robert J. Gillies", "Mark C. Lloyd", "Philip K. Maini", "Matthias Reuss", "Markus R. Owen"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0014790.g009", "stats"=>{"downloads"=>4, "page_views"=>7, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Multi_vessel_basic_vascular_unit_/454782", "title"=>"Multi-vessel basic vascular unit.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2011-04-13 01:19:42"}
  • {"files"=>["https://ndownloader.figshare.com/files/784058"], "description"=>"<p><b>-extent.</b> For the numerical experiments indicated in <a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0014790#pone-0014790-g005\" target=\"_blank\">Figure 5(A)</a>, on a domain, we show: <b>A</b>) the volume fraction of normal cells (red line, for simulations with normal cells only) or cancer cells (blue line, for simulations where a tumour is implanted) at their long-time value averaged over several simulations. Both cell densities increase as increases from 1, until a maximum is reached at . For larger values of the cell density decreases again. <b>B</b>) the associated vascular volume fractions.</p>", "links"=>[], "tags"=>["biophysics", "Computational biology", "mathematics", "computer science/applications"], "article_id"=>454412, "categories"=>["Mathematics", "Biological Sciences", "Cancer", "Biophysics"], "users"=>["Holger Perfahl", "Helen M. Byrne", "Tingan Chen", "Veronica Estrella", "Tomás Alarcón", "Alexei Lapin", "Robert A. Gatenby", "Robert J. Gillies", "Mark C. Lloyd", "Philip K. Maini", "Matthias Reuss", "Markus R. Owen"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0014790.g006", "stats"=>{"downloads"=>0, "page_views"=>1, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Enhancement_of_the_/454412", "title"=>"Enhancement of the", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2011-04-13 01:13:32"}
  • {"files"=>["https://ndownloader.figshare.com/files/783920"], "description"=>"<p><b>A</b>) We consider two parallel vessels running in the -direction and centred with respect to the variable -extension, with equal mean pressures and opposite pressure drops. <b>B</b>) We analyse how the interference between neighbouring subsystems influences the behaviour in a larger domain: <b>Bi</b>) We consider a large () domain, where all 10 subdomains, each with two parent vessels, are allowed to interfere with each other and farfield communication is incorporated in this setting. <b>Bii</b>) Depicts a subsystem with two parent vessel pairs. Thus nearfield (but not farfield) interaction in the -direction is now included. <b>Biii</b>) The most restricted example, a single subdomain, thus preventing all communication in the -direction. In A) and B) we apply reflecting (zero-flux) boundary conditions. <b>C</b>) Different boundary conditions are considered: <b>Ci</b>) Periodic boundary conditions in the - and -direction. The grey subdomain is effectively surrounded by other networks. <b>Cii</b>) Periodic boundary conditions in the -direction only, which should produce similar behaviour to Bi).</p>", "links"=>[], "tags"=>["numerical"], "article_id"=>454281, "categories"=>["Mathematics", "Biological Sciences", "Cancer", "Biophysics"], "users"=>["Holger Perfahl", "Helen M. Byrne", "Tingan Chen", "Veronica Estrella", "Tomás Alarcón", "Alexei Lapin", "Robert A. Gatenby", "Robert J. Gillies", "Mark C. Lloyd", "Philip K. Maini", "Matthias Reuss", "Markus R. Owen"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0014790.g005", "stats"=>{"downloads"=>1, "page_views"=>6, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Schemes_of_numerical_studies_/454281", "title"=>"Schemes of numerical studies.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2011-04-13 01:11:21"}
  • {"files"=>["https://ndownloader.figshare.com/files/784543"], "description"=>"<p><b>A–C</b>) Simulations in a domain are compared to a domain for each combination of boundary conditions. We find that the domain size has only a weak effect, but the choice of boundary conditions makes a significant difference to the long term tumour cell and vessel volume fractions. The most efficient vascular network can be formed on the doubly-periodic domain, while we get the least efficient network for the non-periodic boundary conditions. <b>B</b>) and <b>D</b>) show the long-time values with 95% confidence intervals.</p>", "links"=>[], "tags"=>["vascular"], "article_id"=>454906, "categories"=>["Mathematics", "Biological Sciences", "Cancer", "Biophysics"], "users"=>["Holger Perfahl", "Helen M. Byrne", "Tingan Chen", "Veronica Estrella", "Tomás Alarcón", "Alexei Lapin", "Robert A. Gatenby", "Robert J. Gillies", "Mark C. Lloyd", "Philip K. Maini", "Matthias Reuss", "Markus R. Owen"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0014790.g010", "stats"=>{"downloads"=>1, "page_views"=>5, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Multi_vessel_basic_vascular_unit_simulations_/454906", "title"=>"Multi-vessel basic vascular unit simulations.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2011-04-13 01:21:46"}
  • {"files"=>["https://ndownloader.figshare.com/files/784307"], "description"=>"<p>To analyse the discrepancy (illustrated in <a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0014790#pone-0014790-g007\" target=\"_blank\">Figure 7</a>) between the results in the isolated small () subdomain and the large () simulation domain in more detail, different boundary conditions are implemented and their influence is studied. In particular, we compare a subset of the numerical experiments indicated in <a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0014790#pone-0014790-g005\" target=\"_blank\">Figure 5</a>: (Bi) non-periodic BC, ; (Biii) non-periodic BC, ; (Ci) double-periodic BC, ; (Cii) single-periodic BC, . For all three types of boundary condition, the simulations lead to similar final vessel densities for the small domains (), which still differ significantly from the large domain. Hence, for this initial subdomain vasculature and with the various possible choices of boundary conditions, the smallest domain cannot be representative of the larger domain.</p>", "links"=>[], "tags"=>["biophysics", "Computational biology", "mathematics", "computer science/applications"], "article_id"=>454663, "categories"=>["Mathematics", "Biological Sciences", "Cancer", "Biophysics"], "users"=>["Holger Perfahl", "Helen M. Byrne", "Tingan Chen", "Veronica Estrella", "Tomás Alarcón", "Alexei Lapin", "Robert A. Gatenby", "Robert J. Gillies", "Mark C. Lloyd", "Philip K. Maini", "Matthias Reuss", "Markus R. Owen"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0014790.g008", "stats"=>{"downloads"=>0, "page_views"=>1, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Simulations_with_different_boundary_conditions_/454663", "title"=>"Simulations with different boundary conditions.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2011-04-13 01:17:43"}
  • {"files"=>["https://ndownloader.figshare.com/files/783342"], "description"=>"<p>This figure shows the connections between the different modelling layers. In the subcellular layer the cell cycle protein concentrations and the p53 and VEGF concentrations are modelled via systems of coupled ordinary differential equations. The local external oxygen concentration influences the duration of the cell cycles. Cells consume oxygen, and produce VEGF in the case of hypoxia. Extracellular VEGF also influences the emergence of endothelial sprouts and their biased random walk towards hypoxic regions. If endothelial sprouts connect to other sprouts or the existing vascular network, new vessels form. Vessel diameter is influenced by the local oxygen concentration and flow-related parameters, such as pressure and wall shear stress. The vascular network delivers oxygen throughout the tissue.</p>", "links"=>[], "tags"=>["overview"], "article_id"=>453703, "categories"=>["Mathematics", "Biological Sciences", "Cancer", "Biophysics"], "users"=>["Holger Perfahl", "Helen M. Byrne", "Tingan Chen", "Veronica Estrella", "Tomás Alarcón", "Alexei Lapin", "Robert A. Gatenby", "Robert J. Gillies", "Mark C. Lloyd", "Philip K. Maini", "Matthias Reuss", "Markus R. Owen"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0014790.g001", "stats"=>{"downloads"=>1, "page_views"=>5, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Multiscale_model_overview_interaction_diagram_/453703", "title"=>"Multiscale model overview (interaction diagram).", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2011-04-13 01:01:43"}
  • {"files"=>["https://ndownloader.figshare.com/files/784913"], "description"=>"<p>We reconstructed the vascular network by applying the following strategy. 3D multiphoton fluorescence microscopy images (A) taken from mouse models <i>in vivo</i> formed the basis of our geometrical reconstruction. These images were transferred to OpenInventor and Matlab for image analysis. Based on the data we reconstructed the vascular graph model that describes the connectivity of the vascular network. B) We assigned inflow (red points) and outflow nodes (blue points) at various pressures in order to obtain a persistent and stable network. The vascular graph is characterised by the spatial coordinates of the nodes and the connections between them.</p>", "links"=>[], "tags"=>["biophysics", "Computational biology", "mathematics", "computer science/applications"], "article_id"=>455266, "categories"=>["Mathematics", "Biological Sciences", "Cancer", "Biophysics"], "users"=>["Holger Perfahl", "Helen M. Byrne", "Tingan Chen", "Veronica Estrella", "Tomás Alarcón", "Alexei Lapin", "Robert A. Gatenby", "Robert J. Gillies", "Mark C. Lloyd", "Philip K. Maini", "Matthias Reuss", "Markus R. Owen"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0014790.g013", "stats"=>{"downloads"=>1, "page_views"=>7, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Image_reconstruction_/455266", "title"=>"Image reconstruction.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2011-04-13 01:27:46"}
  • {"files"=>["https://ndownloader.figshare.com/files/783448"], "description"=>"<p>The flowchart shows the temporal sequence of the computational steps in our simulation.</p>", "links"=>[], "tags"=>["overview"], "article_id"=>453807, "categories"=>["Mathematics", "Biological Sciences", "Cancer", "Biophysics"], "users"=>["Holger Perfahl", "Helen M. Byrne", "Tingan Chen", "Veronica Estrella", "Tomás Alarcón", "Alexei Lapin", "Robert A. Gatenby", "Robert J. Gillies", "Mark C. Lloyd", "Philip K. Maini", "Matthias Reuss", "Markus R. Owen"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0014790.g002", "stats"=>{"downloads"=>2, "page_views"=>8, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Multiscale_model_overview_flowchart_/453807", "title"=>"Multiscale model overview (flowchart).", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2011-04-13 01:03:27"}
  • {"files"=>["https://ndownloader.figshare.com/files/785227"], "description"=>"<p>We increased the endothelial tip cell chemotaxis coefficient by a factor of 10 (so that ), and generated a “normal” vascular network from the parent vessel configuration in <a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0014790#pone-0014790-g009\" target=\"_blank\">Figure 9(C)</a> (i.e. with doubly-periodic boundary conditions) by filling the domain with normal cells and allowing angiogenesis and vessel remodelling to proceed (Video S4 and Video S5). We then implanted a small tumour into the new “normal” network. The tumour stimulates further sprouting, so that the tumour vasculature is more dense than the corresponding normal network. <b>A</b>) The generated vascular network and pressure distribution for normal tissue. <b>B</b>) A short time after implantation the first endothelial sprouts appear and migrate into the tumour. The appearance of new vessels within the healthy tissue around a tumour is an effect that can often be observed in gliomas. <b>C</b>)<b>–E</b>) A time-series of the growing tumour in the normal vasculature, the normal cells that surround the tumour are faded out (Video S6). On the right hand side the changes to the normal vasculature due to the implanted tumour are shown. The original normal vasculature is depicted in grey whereas new vessels that emerge due to the angiogenic response are coloured red (Video S7). In D) one can see an effect of the doubly periodic boundary conditions where tumour cells that departed through the upper boundary enter again via the lower boundary.</p>", "links"=>[], "tags"=>["angiogenesis", "enhanced"], "article_id"=>455583, "categories"=>["Mathematics", "Biological Sciences", "Cancer", "Biophysics"], "users"=>["Holger Perfahl", "Helen M. Byrne", "Tingan Chen", "Veronica Estrella", "Tomás Alarcón", "Alexei Lapin", "Robert A. Gatenby", "Robert J. Gillies", "Mark C. Lloyd", "Philip K. Maini", "Matthias Reuss", "Markus R. Owen"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0014790.g015", "stats"=>{"downloads"=>1, "page_views"=>4, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Tumour_angiogenesis_with_enhanced_chemotaxis_/455583", "title"=>"Tumour angiogenesis with enhanced chemotaxis.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2011-04-13 01:33:03"}
  • {"files"=>["https://ndownloader.figshare.com/files/392933", "https://ndownloader.figshare.com/files/392970", "https://ndownloader.figshare.com/files/393005", "https://ndownloader.figshare.com/files/393026", "https://ndownloader.figshare.com/files/393063", "https://ndownloader.figshare.com/files/393116", "https://ndownloader.figshare.com/files/393145", "https://ndownloader.figshare.com/files/393172"], "description"=>"<div><p>We investigate a three-dimensional multiscale model of vascular tumour growth, which couples blood flow, angiogenesis, vascular remodelling, nutrient/growth factor transport, movement of, and interactions between, normal and tumour cells, and nutrient-dependent cell cycle dynamics within each cell. In particular, we determine how the domain size, aspect ratio and initial vascular network influence the tumour's growth dynamics and its long-time composition. We establish whether it is possible to extrapolate simulation results obtained for small domains to larger ones, by constructing a large simulation domain from a number of identical subdomains, each subsystem initially comprising two parallel parent vessels, with associated cells and diffusible substances. We find that the subsystem is not representative of the full domain and conclude that, for this initial vessel geometry, interactions between adjacent subsystems contribute to the overall growth dynamics. We then show that extrapolation of results from a small subdomain to a larger domain can only be made if the subdomain is sufficiently large and is initialised with a sufficiently complex vascular network. Motivated by these results, we perform simulations to investigate the tumour's response to therapy and show that the probability of tumour elimination in a larger domain can be extrapolated from simulation results on a smaller domain. Finally, we demonstrate how our model may be combined with experimental data, to predict the spatio-temporal evolution of a vascular tumour.</p></div>", "links"=>[], "tags"=>["multiscale", "vascular", "tumour", "roles", "conditions"], "article_id"=>137597, "categories"=>["Mathematics", "Biological Sciences", "Cancer", "Biophysics"], "users"=>["Holger Perfahl", "Helen M. Byrne", "Tingan Chen", "Veronica Estrella", "Tomás Alarcón", "Alexei Lapin", "Robert A. Gatenby", "Robert J. Gillies", "Mark C. Lloyd", "Philip K. Maini", "Matthias Reuss", "Markus R. Owen"], "doi"=>["https://dx.doi.org/10.1371/journal.pone.0014790.s001", "https://dx.doi.org/10.1371/journal.pone.0014790.s002", "https://dx.doi.org/10.1371/journal.pone.0014790.s003", "https://dx.doi.org/10.1371/journal.pone.0014790.s004", "https://dx.doi.org/10.1371/journal.pone.0014790.s005", "https://dx.doi.org/10.1371/journal.pone.0014790.s006", "https://dx.doi.org/10.1371/journal.pone.0014790.s007", "https://dx.doi.org/10.1371/journal.pone.0014790.s008"], "stats"=>{"downloads"=>27, "page_views"=>14, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/Multiscale_Modelling_of_Vascular_Tumour_Growth_in_3D_The_Roles_of_Domain_Size_and_Boundary_Conditions/137597", "title"=>"Multiscale Modelling of Vascular Tumour Growth in 3D: The Roles of Domain Size and Boundary Conditions", "pos_in_sequence"=>0, "defined_type"=>4, "published_date"=>"2011-04-13 02:06:37"}
  • {"files"=>["https://ndownloader.figshare.com/files/783629"], "description"=>"<p>The tumour cells and vasculature are depicted in the left column (see Video S1), the vasculature and normal cells in the middle column (Video S2) and the vessel network in the right column. The figure shows a realisation of a domain with a cube of tumour cells implanted in healthy tissue with two straight initial vessels. Most tumour cells become quiescent and then die. Thereafter the following steps occur: <b>A</b>) Vessels emerge near the initial tumour and form a well-vascularised tumour; far from the two initial vessels most normal cells have died, leaving two cylindrical shaped cell populations around the vessels. <b>B</b>) The tumour grows along the direction of maximum oxygen supply and displaces the normal cells. <b>C</b>) An important step in the tumour development – the first bridge between the upper and lower network is built. <b>D</b>) Once a connection has been made between the upper and lower vessels, the tumour is able to colonise the lower part of the simulation domain.</p>", "links"=>[], "tags"=>["biophysics", "Computational biology", "mathematics", "computer science/applications"], "article_id"=>453989, "categories"=>["Mathematics", "Biological Sciences", "Cancer", "Biophysics"], "users"=>["Holger Perfahl", "Helen M. Byrne", "Tingan Chen", "Veronica Estrella", "Tomás Alarcón", "Alexei Lapin", "Robert A. Gatenby", "Robert J. Gillies", "Mark C. Lloyd", "Philip K. Maini", "Matthias Reuss", "Markus R. Owen"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0014790.g003", "stats"=>{"downloads"=>0, "page_views"=>1, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Tumour_growth_in_healthy_tissue_/453989", "title"=>"Tumour growth in healthy tissue.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2011-04-13 01:06:29"}
  • {"files"=>["https://ndownloader.figshare.com/files/783797"], "description"=>"<p><b>A,B</b>) the time courses of tumour and vessel volume fractions for 25 realisations of our multiscale model of vascular tumour growth performed in a domain. One can clearly see the highly stochastic nature of the process. Sometimes there is a long lag-phase before the tumour starts to grow exponentially (see red line in A) and B)). <b>C,D</b>) depict how the mean behaviour is located in the domain that is spanned by all simulations.</p>", "links"=>[], "tags"=>["realisations", "angiogenesis"], "article_id"=>454156, "categories"=>["Mathematics", "Biological Sciences", "Cancer", "Biophysics"], "users"=>["Holger Perfahl", "Helen M. Byrne", "Tingan Chen", "Veronica Estrella", "Tomás Alarcón", "Alexei Lapin", "Robert A. Gatenby", "Robert J. Gillies", "Mark C. Lloyd", "Philip K. Maini", "Matthias Reuss", "Markus R. Owen"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0014790.g004", "stats"=>{"downloads"=>0, "page_views"=>3, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Multiple_realisations_of_angiogenesis_simulations_/454156", "title"=>"Multiple realisations of angiogenesis simulations.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2011-04-13 01:09:16"}
  • {"files"=>["https://ndownloader.figshare.com/files/784182"], "description"=>"<p>For the numerical experiments indicated in <a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0014790#pone-0014790-g005\" target=\"_blank\">Figure 5(Bi)–(Biii)</a>, we show: <b>A</b>) The tumour cell volume fraction over time. After the first death of some tumour cells the tumour recovers and the increasing amount of oxygen supported by the vascular system promotes further growth. After the first phase of rapid growth, the tumour cells can grow fastest in the smallest domain. We obtain a different final tumour cell fraction (it is lowest on the smallest domain). <b>C</b>) The vessel volume fraction. Note that the largest domain has the most efficient oxygen supply to the tissue and thus the highest tumour cell to vessel ratio. We have also plotted the 95% confidence bands for the mean dynamics in time (A,C) by applying a bootstrapping method <a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0014790#pone.0014790-Sachs1\" target=\"_blank\">[31]</a>. From these confidence intervals we conclude that the differences between the mean values in the long-time behaviour are statistically significant. <b>B</b>) <b>and D</b>) show the confidence intervals of the long-time values of the tumour and vessel cell volume fractions.</p>", "links"=>[], "tags"=>["kinetics", "coupled"], "article_id"=>454547, "categories"=>["Mathematics", "Biological Sciences", "Cancer", "Biophysics"], "users"=>["Holger Perfahl", "Helen M. Byrne", "Tingan Chen", "Veronica Estrella", "Tomás Alarcón", "Alexei Lapin", "Robert A. Gatenby", "Robert J. Gillies", "Mark C. Lloyd", "Philip K. Maini", "Matthias Reuss", "Markus R. Owen"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0014790.g007", "stats"=>{"downloads"=>1, "page_views"=>4, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Growth_kinetics_in_isolated_and_coupled_subsystems_/454547", "title"=>"Growth kinetics in isolated and coupled subsystems.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2011-04-13 01:15:47"}
  • {"files"=>["https://ndownloader.figshare.com/files/784807"], "description"=>"<p>In order to study the influence of domain size when tumour cells are very likely to die (i.e. therapy, implantation), we implanted tumour cells in a hostile environment in which most of the tumour cells die within the first hours post implantation. Increasingly hostile environments (which mimic anti-tumour therapies) are initiated by reducing the oxygen permeability coefficient from to , and finally . For each value of we ran 120 realisations, and the points indicate the estimated mean probability of tumour elimination, , for five domain sizes (). The bars indicate the 95% confidence intervals obtained by boostrapping. If the probability of elimination in each subdomain is independent of the others, then we should have . In each case we plot the extrapolated elimination probability based on . The extrapolated values are in good agreement with the results from direct simulations — with the predicted mean lying within the 95% confidence intervals in most cases.</p>", "links"=>[], "tags"=>["elimination", "probability", "dependency"], "article_id"=>455169, "categories"=>["Mathematics", "Biological Sciences", "Cancer", "Biophysics"], "users"=>["Holger Perfahl", "Helen M. Byrne", "Tingan Chen", "Veronica Estrella", "Tomás Alarcón", "Alexei Lapin", "Robert A. Gatenby", "Robert J. Gillies", "Mark C. Lloyd", "Philip K. Maini", "Matthias Reuss", "Markus R. Owen"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0014790.g012", "stats"=>{"downloads"=>0, "page_views"=>1, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Tumour_elimination_probability_dependency_on_domain_size_/455169", "title"=>"Tumour elimination probability dependency on domain-size.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2011-04-13 01:26:09"}
  • {"files"=>["https://ndownloader.figshare.com/files/784666"], "description"=>"<p>To study the influence of domain size on tumour elimination (e.g. after therapy), we implanted tumour cells in a hostile environment (in which is reduced from 3800 to 3116, hence reducing nutrient delivery to the tissue). Columns 1 and 2 show the number of tumour cells only and the numbers of all cell types respectively, for five independent simulations in an isolated -domain. Columns 3 and 4 show equivalent results from one simulation in one large domain. <b>Row A</b>) At most of the initial tumour cells have died and only a few survived in both cases. At this early stage both isolated and coupled simulations lead to similar results, as the coupled subdomains in the case can be viewed as stochastically independent. <b>Row B</b>) At more tumour cells have died and only one isolated subdomain is occupied by tumour cells. The cells are restricted in their motion in the -direction in the case of isolated subdomains, whereas the tumour can spread easily in the unrestricted case. <b>Row C</b>) At competition between tumour cells and normal cells means that tumour growth slows down – this effect is stronger in the isolated subdomain. <b>Row D</b>) At the tumour is eliminated in 4/5 isolated subdomains, but in the larger domain the subdomains are not independent of one another and the tumour has spread into all -layers. In such a case one overpredicts the efficacy of therapies if one simulates in 2D or in a small three-dimensional subdomain.</p>", "links"=>[], "tags"=>["implications", "regrowth"], "article_id"=>455026, "categories"=>["Mathematics", "Biological Sciences", "Cancer", "Biophysics"], "users"=>["Holger Perfahl", "Helen M. Byrne", "Tingan Chen", "Veronica Estrella", "Tomás Alarcón", "Alexei Lapin", "Robert A. Gatenby", "Robert J. Gillies", "Mark C. Lloyd", "Philip K. Maini", "Matthias Reuss", "Markus R. Owen"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0014790.g011", "stats"=>{"downloads"=>0, "page_views"=>7, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Stochasticity_of_results_and_implications_on_growth_regrowth_and_therapy_/455026", "title"=>"Stochasticity of results and implications on growth, regrowth and therapy.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2011-04-13 01:23:46"}
  • {"files"=>["https://ndownloader.figshare.com/files/785040"], "description"=>"<p><b>A</b>)<b>–D</b>) show the temporal evolution of a tumour in a real vascular network embedded in normal tissue (see Video S3). As initial condition we have taken a vascular network from multiphoton fluorescence microscopy and embedded it in a cellular automaton domain. In the first column the tumour expands radially, and degrades the healthy tissue (second column). The predicted adaptations of the vascular system are shown in the third column where the experimentally derived network is shown in light red, while the new vessels are coloured in red.</p>", "links"=>[], "tags"=>["tumour", "experimentally", "derived", "vascular"], "article_id"=>455391, "categories"=>["Mathematics", "Biological Sciences", "Cancer", "Biophysics"], "users"=>["Holger Perfahl", "Helen M. Byrne", "Tingan Chen", "Veronica Estrella", "Tomás Alarcón", "Alexei Lapin", "Robert A. Gatenby", "Robert J. Gillies", "Mark C. Lloyd", "Philip K. Maini", "Matthias Reuss", "Markus R. Owen"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0014790.g014", "stats"=>{"downloads"=>0, "page_views"=>1, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Proof_of_concept_tumour_growth_in_an_experimentally_derived_vascular_network_/455391", "title"=>"Proof-of-concept: tumour growth in an experimentally derived vascular network.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2011-04-13 01:29:51"}

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Relative Metric

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