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Crawshaw JR, Flegg JA, Bernabeu MO, Osborne JM. Mathematical models of developmental vascular remodelling: A review. PLoS Comput Biol 2023; 19:e1011130. [PMID: 37535698 PMCID: PMC10399886 DOI: 10.1371/journal.pcbi.1011130] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 08/05/2023] Open
Abstract
Over the past 40 years, there has been a strong focus on the development of mathematical models of angiogenesis, while developmental remodelling has received little such attention from the mathematical community. Sprouting angiogenesis can be seen as a very crude way of laying out a primitive vessel network (the raw material), while remodelling (understood as pruning of redundant vessels, diameter control, and the establishment of vessel identity and hierarchy) is the key to turning that primitive network into a functional network. This multiscale problem is of prime importance in the development of a functional vasculature. In addition, defective remodelling (either during developmental remodelling or due to a reactivation of the remodelling programme caused by an injury) is associated with a significant number of diseases. In this review, we discuss existing mathematical models of developmental remodelling and explore the important contributions that these models have made to the field of vascular development. These mathematical models are effectively used to investigate and predict vascular development and are able to reproduce experimentally observable results. Moreover, these models provide a useful means of hypothesis generation and can explain the underlying mechanisms driving the observed structural and functional network development. However, developmental vascular remodelling is still a relatively new area in mathematical biology, and many biological questions remain unanswered. In this review, we present the existing modelling paradigms and define the key challenges for the field.
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Affiliation(s)
- Jessica R. Crawshaw
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford, United Kingdom
- School of Mathematics and Statistics, The University of Melbourne, Melbourne, Australia
| | - Jennifer A. Flegg
- School of Mathematics and Statistics, The University of Melbourne, Melbourne, Australia
| | - Miguel O. Bernabeu
- Centre for Medical Informatics, The Usher Institute, University of Edinburgh, Edinburgh, United Kingdom
- The Bayes Centre, The University of Edinburgh, Edinburgh, United Kingdom
| | - James M. Osborne
- School of Mathematics and Statistics, The University of Melbourne, Melbourne, Australia
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Alberding JP, Secomb TW. Simulation of Angiogenesis in Three Dimensions: Development of the Retinal Circulation. Bull Math Biol 2023; 85:27. [PMID: 36842140 DOI: 10.1007/s11538-023-01126-7] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/29/2022] [Accepted: 01/09/2023] [Indexed: 02/27/2023]
Abstract
A theoretical model is used to describe the three-dimensional development of the retinal circulation in the human eye, which occurs after the initial spread of vasculature across the inner surface of the retina. In the model, random sprouting angiogenesis is driven by a growth factor that is produced in tissue at a rate dependent on oxygen level and diffuses to existing vessels. Vessel sprouts connect to form pathways for blood flow and undergo remodeling and pruning. These processes are controlled by known or hypothesized vascular responses to hemodynamic and biochemical stimuli, including conducted responses along vessel walls. The model shows regression of arterio-venous connections on the surface of the retina, allowing perfusion of the underlying tissue. A striking feature of the retinal circulation is the formation of two vascular plexuses located at the inner and outer surfaces of the inner nuclear layer within the retina. The model is used to test hypotheses regarding the formation of these structures. A mechanism based on local production and diffusion of growth factor is shown to be ineffective. However, sprout guidance by localized structures on the boundaries of the inner nuclear layer can account for plexus formation. The resulting networks have vascular density, perfusion and oxygen transport characteristics consistent with observed properties. The model shows how stochastic generation of vascular sprouts combined with a set of biologically based response mechanisms can lead to the generation of a specialized three-dimensional vascular structure with a high degree of organization.
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Affiliation(s)
| | - Timothy W Secomb
- BIO5 Institute, University of Arizona, Tucson, AZ, 85724, USA.
- Department of Physiology, University of Arizona, Tucson, AZ, 85724, USA.
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3
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Stepien TL, Secomb TW. Spreading mechanics and differentiation of astrocytes during retinal development. J Theor Biol 2022; 549:111208. [DOI: 10.1016/j.jtbi.2022.111208] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/22/2021] [Revised: 05/27/2022] [Accepted: 06/21/2022] [Indexed: 11/30/2022]
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Campion S, Inselman A, Hayes B, Casiraghi C, Joseph D, Facchinetti F, Salomone F, Schmitt G, Hui J, Davis-Bruno K, Van Malderen K, Morford L, De Schaepdrijver L, Wiesner L, Kourula S, Seo S, Laffan S, Urmaliya V, Chen C. The benefits, limitations and opportunities of preclinical models for neonatal drug development. Dis Model Mech 2022; 15:dmm049065. [PMID: 35466995 PMCID: PMC9066504 DOI: 10.1242/dmm.049065] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/25/2022] Open
Abstract
Increased research to improve preclinical models to inform the development of therapeutics for neonatal diseases is an area of great need. This article reviews five common neonatal diseases - bronchopulmonary dysplasia, retinopathy of prematurity, necrotizing enterocolitis, perinatal hypoxic-ischemic encephalopathy and neonatal sepsis - and the available in vivo, in vitro and in silico preclinical models for studying these diseases. Better understanding of the strengths and weaknesses of specialized neonatal disease models will help to improve their utility, may add to the understanding of the mode of action and efficacy of a therapeutic, and/or may improve the understanding of the disease pathology to aid in identification of new therapeutic targets. Although the diseases covered in this article are diverse and require specific approaches, several high-level, overarching key lessons can be learned by evaluating the strengths, weaknesses and gaps in the available models. This Review is intended to help guide current and future researchers toward successful development of therapeutics in these areas of high unmet medical need.
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Affiliation(s)
- Sarah Campion
- Pfizer Worldwide Research, Development, and Medical, Groton, CT 06340, USA
| | - Amy Inselman
- U.S. Food and Drug Administration, National Center for Toxicological Research, Division of Systems Biology, Jefferson, AR 72079, USA
| | - Belinda Hayes
- U.S. Food and Drug Administration, Center for Drug Evaluation and Research, Office of New Drugs, Silver Spring, MD 20993, USA
| | - Costanza Casiraghi
- Department of Experimental Pharmacology and Translational Science, Chiesi Farmaceutici S.p.A., 43122 Parma, Italy
| | - David Joseph
- U.S. Food and Drug Administration, Center for Drug Evaluation and Research, Office of New Drugs, Silver Spring, MD 20993, USA
| | - Fabrizio Facchinetti
- Department of Experimental Pharmacology and Translational Science, Chiesi Farmaceutici S.p.A., 43122 Parma, Italy
| | - Fabrizio Salomone
- Department of Experimental Pharmacology and Translational Science, Chiesi Farmaceutici S.p.A., 43122 Parma, Italy
| | - Georg Schmitt
- Pharma Research and Early Development, Roche Innovation Center Basel, Pharmaceutical Sciences, F. Hoffmann-La Roche, 4070 Basel, Switzerland
| | - Julia Hui
- Bristol Myers Squibb, Nonclinical Research and Development, Summit, NJ 07901, USA
| | - Karen Davis-Bruno
- U.S. Food and Drug Administration, Center for Drug Evaluation and Research, Office of New Drugs, Silver Spring, MD 20993, USA
| | - Karen Van Malderen
- Federal Agency for Medicines and Health Products (FAMHP), Department DG PRE authorization, 1210 Brussels, Belgium
| | - LaRonda Morford
- Eli Lilly, Global Regulatory Affairs, Indianapolis, IN 46285, USA
| | | | - Lutz Wiesner
- Federal Institute for Drugs and Medical Devices, Clinical Trials, 53175 Bonn, Germany
| | - Stephanie Kourula
- Janssen R&D, Drug Metabolism & Pharmacokinetics, 2340 Beerse, Belgium
| | - Suna Seo
- U.S. Food and Drug Administration, Center for Drug Evaluation and Research, Office of New Drugs, Silver Spring, MD 20993, USA
| | - Susan Laffan
- GlaxoSmithKline, Non-Clinical Safety, Collegeville, PA 19406, USA
| | | | - Connie Chen
- Health and Environmental Sciences Institute, Washington, DC 20005, USA
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5
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Zhang Y, Wang H, Oliveira RHM, Zhao C, Popel AS. Systems biology of angiogenesis signaling: Computational models and omics. WIREs Mech Dis 2021; 14:e1550. [PMID: 34970866 PMCID: PMC9243197 DOI: 10.1002/wsbm.1550] [Citation(s) in RCA: 16] [Impact Index Per Article: 5.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/29/2021] [Revised: 12/03/2021] [Accepted: 12/06/2021] [Indexed: 01/10/2023]
Abstract
Angiogenesis is a highly regulated multiscale process that involves a plethora of cells, their cellular signal transduction, activation, proliferation, differentiation, as well as their intercellular communication. The coordinated execution and integration of such complex signaling programs is critical for physiological angiogenesis to take place in normal growth, development, exercise, and wound healing, while its dysregulation is critically linked to many major human diseases such as cancer, cardiovascular diseases, and ocular disorders; it is also crucial in regenerative medicine. Although huge efforts have been devoted to drug development for these diseases by investigation of angiogenesis‐targeted therapies, only a few therapeutics and targets have proved effective in humans due to the innate multiscale complexity and nonlinearity in the process of angiogenic signaling. As a promising approach that can help better address this challenge, systems biology modeling allows the integration of knowledge across studies and scales and provides a powerful means to mechanistically elucidate and connect the individual molecular and cellular signaling components that function in concert to regulate angiogenesis. In this review, we summarize and discuss how systems biology modeling studies, at the pathway‐, cell‐, tissue‐, and whole body‐levels, have advanced our understanding of signaling in angiogenesis and thereby delivered new translational insights for human diseases. This article is categorized under:Cardiovascular Diseases > Computational Models Cancer > Computational Models
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Affiliation(s)
- Yu Zhang
- Department of Biomedical Engineering, Johns Hopkins University School of Medicine, Baltimore, Maryland, USA
| | - Hanwen Wang
- Department of Biomedical Engineering, Johns Hopkins University School of Medicine, Baltimore, Maryland, USA
| | - Rebeca Hannah M Oliveira
- Department of Biomedical Engineering, Johns Hopkins University School of Medicine, Baltimore, Maryland, USA
| | - Chen Zhao
- Department of Biomedical Engineering, Johns Hopkins University School of Medicine, Baltimore, Maryland, USA.,School of Pharmacy, Nanjing Medical University, Nanjing, Jiangsu, China
| | - Aleksander S Popel
- Department of Biomedical Engineering, Johns Hopkins University School of Medicine, Baltimore, Maryland, USA
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Soltani M. Capillary network formation and structure in a modified discrete mathematical model of angiogenesis. Biomed Phys Eng Express 2021; 8. [PMID: 34883475 DOI: 10.1088/2057-1976/ac4175] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/08/2021] [Accepted: 12/09/2021] [Indexed: 01/01/2023]
Abstract
Angiogenesis, as part of cancer development, involves hierarchical complicated events and processes. Multiple studies have revealed the significance of the formation and structure of tumor-induced capillary networks. In this study, a discrete mathematical model of angiogenesis is studied and modified to capture the realistic physics of capillary network formation. Modifications are performed on the mathematical foundations of an existing discrete model of angiogenesis. The main modifications are the imposition of the matrix density effect, implementation of realistic boundary and initial conditions, and improvement of the method of governing equations based on physical observation. Results show that endothelial cells accelerate angiogenesis and capillary formation as they migrate toward the tumor and clearly exhibit the physical concept of haptotactic movement. On the other hand, consideration of blood flow-induced stress leads to a dynamic adaptive vascular network of capillaries which intelligibly reflects the brush border effect . The present modified model of capillary network formation is based on the physical rationale that defines a clear mathematical and physical interpretation of angiogenesis, which is likely to be used in cancer development modeling and anti-angiogenic therapies.
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Affiliation(s)
- M Soltani
- Department of Mechanical Engineering, K. N. Toosi University of Technology, Tehran, Iran.,Department of Electrical and Computer Engineering, University of Waterloo, Waterloo, Ontario, Canada.,Centre for Biotechnology and Bioengineering (CBB), University of Waterloo, Waterloo, Ontario, Canada.,Advanced Bioengineering Initiative Center, Computational Medicine Center, K. N. Toosi University of Technology, Tehran, Tehran Province, Iran
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7
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Iizuka O, Kawamura S, Tero A, Uemura A, Miura T. Remodeling mechanisms determine size distributions in developing retinal vasculature. PLoS One 2020; 15:e0235373. [PMID: 33052908 PMCID: PMC7556457 DOI: 10.1371/journal.pone.0235373] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/13/2020] [Accepted: 09/16/2020] [Indexed: 11/18/2022] Open
Abstract
The development of retinal blood vessels has extensively been used as a model to study vascular pattern formation. To date, various quantitative measurements, such as size distribution have been performed, but the relationship between pattern formation mechanisms and these measurements remains unclear. In the present study, we first focus on the islands (small regions subdivided by the capillary network). We quantitatively measured the island size distribution in the retinal vascular network and found that it tended to exhibit an exponential distribution. We were able to recapitulate this distribution pattern in a theoretical model by implementing the stochastic disappearance of vessel segments around arteries could reproduce the observed exponential distribution of islands. Second, we observed that the diameter distribution of the retinal artery segment obeyed a power law. We theoretically showed that an equal bifurcation branch pattern and Murray’s law could reproduce this pattern. This study demonstrates the utility of examining size distribution for understanding the mechanisms of vascular pattern formation.
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Affiliation(s)
- Osamu Iizuka
- School of Medicine, Kyushu University, Fukuoka, Japan
| | | | - Atsushi Tero
- Institute of Mathematics for Industry, Kyushu University, Fukuoka, Japan
| | - Akiyoshi Uemura
- Department of Retinal Vascular Biology, Nagoya City University Graduate School of Medicine, Nagoya, Japan
| | - Takashi Miura
- Department of Anatomy and Cell Biology, Kyushu University Graduate School of Medical Sciences, Fukuoka, Japan
- * E-mail:
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8
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Szulc U, Dąbrowska E, Pieczyński J, Białkowski P, Narkiewicz K, Schmieder RE, Harazny J. How to measure retinal microperfusion in patients with arterial hypertension. Blood Press 2020; 30:4-19. [PMID: 32969283 DOI: 10.1080/08037051.2020.1823816] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/23/2022]
Abstract
PURPOSE Assessment and monitoring of changes in microcirculatory perfusion, perfusion dynamic, vessel structure and oxygenation is crucial in management of arterial hypertension. Constant search for non-invasive methods has led the clinical focus towards the vasculature of the retina, which offers a large opportunity to detect the early phase of the functional and structural changes in the arterial hypertension and can reflect changes in brain vasculature. We review all the available methods of retinal microcirculation measurements including angiography, oximetry, retinal vasculature assessment software, Optical Coherence Tomography Angiography, Adaptive Optics and Scanning Laser Doppler Flowmetry and their application in clinical research. MATERIALS AND METHODS To further analyse the applicability of described methods in hypertension research we performed a systematic search of the PubMed electronic database (April 2020). In our analysis, we included 111 articles in which at least one of described methods was used for assessment of microcirculation of the retina in hypertensive individuals. RESULTS Up to this point, the methods most commonly published in studies of retinal microcirculation in arterial hypertension were Scanning Laser Doppler Flowmetry followed shortly by Optical Coherence Tomography Angiography and retinal vasculature assessment software. CONCLUSIONS While none of described methods enables the simultaneous measurement of all microcirculatory parameters, certain techniques are widely used in arterial hypertension research, while others gain popularity in screening.
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Affiliation(s)
- Urszula Szulc
- Department of Human Physiology and Pathophysiology, University Warmia and Mazury, Olsztyn, Poland
| | - Edyta Dąbrowska
- Department of Hypertension and Diabetology, Faculty of Medicine, Medical University of Gdańsk, Gdańsk, Poland.,First Department of Cardiology, Faculty of Medicine, Medical University of Gdańsk, Gdańsk, Poland
| | - Janusz Pieczyński
- Department of Ophthalmology, University Warmia and Mazury, Olsztyn, Poland
| | - Paweł Białkowski
- Department of Ophthalmology, Provincial Specialist Hospital, Olsztyn, Poland
| | - Krzysztof Narkiewicz
- Department of Hypertension and Diabetology, Faculty of Medicine, Medical University of Gdańsk, Gdańsk, Poland
| | - Roland E Schmieder
- Clinical Research Center, Department of Nephrology and Hypertensiology, University Erlangen-Nuremberg, Erlangen, Germany
| | - Joanna Harazny
- Department of Human Physiology and Pathophysiology, University Warmia and Mazury, Olsztyn, Poland.,Clinical Research Center, Department of Nephrology and Hypertensiology, University Erlangen-Nuremberg, Erlangen, Germany
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9
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Fry BC, Harris A, Siesky B, Arciero J. Blood flow regulation and oxygen transport in a heterogeneous model of the mouse retina. Math Biosci 2020; 329:108476. [PMID: 32920096 DOI: 10.1016/j.mbs.2020.108476] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/12/2020] [Revised: 07/24/2020] [Accepted: 09/05/2020] [Indexed: 11/25/2022]
Abstract
Elevated intraocular pressure is the primary risk factor for glaucoma, yet vascular health and ocular hemodynamics have also been established as important risk factors for the disease. The precise physiological mechanisms and processes by which flow impairment and reduced tissue oxygenation relate to retinal ganglion cell death are not fully known. Mathematical modeling has emerged as a useful tool to help decipher the role of hemodynamic alterations in glaucoma. Several previous models of the retinal microvasculature and tissue have investigated the individual impact of spatial heterogeneity, flow regulation, and oxygen transport on the system. This study combines all three of these components into a heterogeneous mathematical model of retinal arterioles that includes oxygen transport and acute flow regulation in response to changes in pressure, shear stress, and oxygen demand. The metabolic signal (Si) is implemented as a wall-derived signal that reflects the oxygen deficit along the network, and three cases of conduction are considered: no conduction, a constant signal, and a flow-weighted signal. The model shows that the heterogeneity of the downstream signal serves to regulate flow better than a constant conducted response. In fact, the increases in average tissue PO2 due to a flow-weighted signal are often more significant than if the entire level of signal is increased. Such theoretical work supports the importance of the non-uniform structure of the retinal vasculature when assessing the capability and/or dysfunction of blood flow regulation in the retinal microcirculation.
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Affiliation(s)
- Brendan C Fry
- Department of Mathematics and Statistics, Metropolitan State University of Denver, P.O. Box 173362, Campus Box 38, Denver, CO 80217, USA.
| | - Alon Harris
- Department of Ophthalmology, Icahn School of Medicine at Mount Sinai Hospital, One Gustave L. Levy Place, Box 1183, New York, NY 10029, USA.
| | - Brent Siesky
- Department of Ophthalmology, Icahn School of Medicine at Mount Sinai Hospital, One Gustave L. Levy Place, Box 1183, New York, NY 10029, USA.
| | - Julia Arciero
- Department of Mathematical Sciences, Indiana University-Purdue University Indianapolis, 402 N. Blackford St, LD 270, Indianapolis, IN 46202, USA.
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Painter KJ. Mathematical models for chemotaxis and their applications in self-organisation phenomena. J Theor Biol 2019; 481:162-182. [DOI: 10.1016/j.jtbi.2018.06.019] [Citation(s) in RCA: 51] [Impact Index Per Article: 10.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/17/2018] [Revised: 06/20/2018] [Accepted: 06/22/2018] [Indexed: 01/31/2023]
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11
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Krause AL, Beliaev D, Van Gorder RA, Waters SL. Lattice and continuum modelling of a bioactive porous tissue scaffold. MATHEMATICAL MEDICINE AND BIOLOGY-A JOURNAL OF THE IMA 2019; 36:325-360. [PMID: 30107530 DOI: 10.1093/imammb/dqy012] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/28/2017] [Revised: 01/18/2018] [Accepted: 07/16/2018] [Indexed: 12/29/2022]
Abstract
A contemporary procedure to grow artificial tissue is to seed cells onto a porous biomaterial scaffold and culture it within a perfusion bioreactor to facilitate the transport of nutrients to growing cells. Typical models of cell growth for tissue engineering applications make use of spatially homogeneous or spatially continuous equations to model cell growth, flow of culture medium, nutrient transport and their interactions. The network structure of the physical porous scaffold is often incorporated through parameters in these models, either phenomenologically or through techniques like mathematical homogenization. We derive a model on a square grid lattice to demonstrate the importance of explicitly modelling the network structure of the porous scaffold and compare results from this model with those from a modified continuum model from the literature. We capture two-way coupling between cell growth and fluid flow by allowing cells to block pores, and by allowing the shear stress of the fluid to affect cell growth and death. We explore a range of parameters for both models and demonstrate quantitative and qualitative differences between predictions from each of these approaches, including spatial pattern formation and local oscillations in cell density present only in the lattice model. These differences suggest that for some parameter regimes, corresponding to specific cell types and scaffold geometries, the lattice model gives qualitatively different model predictions than typical continuum models. Our results inform model selection for bioactive porous tissue scaffolds, aiding in the development of successful tissue engineering experiments and eventually clinically successful technologies.
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Affiliation(s)
- Andrew L Krause
- Mathematical Institute, Andrew Wiles Building, University of Oxford, Radcliffe Observatory Quarter, Woodstock Rd, UK
| | - Dmitry Beliaev
- Mathematical Institute, Andrew Wiles Building, University of Oxford, Radcliffe Observatory Quarter, Woodstock Rd, UK
| | - Robert A Van Gorder
- Mathematical Institute, Andrew Wiles Building, University of Oxford, Radcliffe Observatory Quarter, Woodstock Rd, UK
| | - Sarah L Waters
- Mathematical Institute, Andrew Wiles Building, University of Oxford, Radcliffe Observatory Quarter, Woodstock Rd, UK
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12
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Best BT. Single-cell branching morphogenesis in the Drosophila trachea. Dev Biol 2018; 451:5-15. [PMID: 30529233 DOI: 10.1016/j.ydbio.2018.12.001] [Citation(s) in RCA: 13] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/19/2018] [Revised: 11/23/2018] [Accepted: 12/01/2018] [Indexed: 12/20/2022]
Abstract
The terminal cells of the tracheal epithelium in Drosophila melanogaster are one of the few known cell types that undergo subcellular morphogenesis to achieve a stable, branched shape. During the animal's larval stages, the cells repeatedly sprout new cytoplasmic processes. These grow very long, wrapping around target tissues to which the terminal cells adhere, and are hollowed by a gas-filled subcellular tube for oxygen delivery. Our understanding of this ramification process remains rudimentary. This review aims to provide a comprehensive summary of studies on terminal cells to date, and attempts to extrapolate how terminal branches might be formed based on the known genetic and molecular components. Next to this cell-intrinsic branching mechanism, we examine the extrinsic regulation of terminal branching by the target tissue and the animal's environment. Finally, we assess the degree of similarity between the patterns established by the branching programs of terminal cells and other branched cells and tissues from a mathematical and conceptual point of view.
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Affiliation(s)
- Benedikt T Best
- Director's Research Unit, European Molecular Biology Laboratory (EMBL), Meyerhofstr. 1, 69117 Heidelberg, Germany; Collaboration for Joint PhD degree from EMBL and Heidelberg University, Faculty of Biosciences, Germany
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13
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Deconstructing the principles of ductal network formation in the pancreas. PLoS Biol 2018; 16:e2002842. [PMID: 30048442 PMCID: PMC6080801 DOI: 10.1371/journal.pbio.2002842] [Citation(s) in RCA: 19] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/25/2017] [Revised: 08/07/2018] [Accepted: 07/16/2018] [Indexed: 12/18/2022] Open
Abstract
The mammalian pancreas is a branched organ that does not exhibit stereotypic branching patterns, similarly to most other glands. Inside branches, it contains a network of ducts that undergo a transition from unconnected microlumen to a mesh of interconnected ducts and finally to a treelike structure. This ductal remodeling is poorly understood, both on a microscopic and macroscopic level. In this article, we quantify the network properties at different developmental stages. We find that the pancreatic network exhibits stereotypic traits at each stage and that the network properties change with time toward the most economical and optimized delivery of exocrine products into the duodenum. Using in silico modeling, we show how steps of pancreatic network development can be deconstructed into two simple rules likely to be conserved for many other glands. The early stage of the network is explained by noisy, redundant duct connection as new microlumens form. The later transition is attributed to pruning of the network based on the flux of fluid running through the pancreatic network into the duodenum. In the pancreas of mammals, digestive enzymes are transported from their production site in acini (clusters of cells that secrete the enzymes) to the intestine via a network of ducts. During organ development in fetuses, the ducts initially form by the coordinated polarization of cells to form small holes, which will connect and fuse, to constitute a meshwork. This hyperconnected network further develops into a treelike structure by the time of birth. In this article, we use methods originally developed to analyze road, rail, web, or river networks to quantify the network properties at different developmental stages. We find that the pancreatic network properties are similar between individuals at specific time points but eventually change to achieve the most economical and optimized structure to deliver pancreatic juice into the duodenum. Using in silico modeling, we show how the stages of pancreatic network development follow two simple rules, which are likely to be conserved for the development of other glands. The early stage of the network is explained by noisy, redundant duct connection as new small ductal holes form. Later on, the secretion of fluid that runs through the pancreatic network into the duodenum leads to the widening of ducts with the greatest flow, while nonnecessary ducts are eliminated, akin to how river beds are formed.
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Angiogenic Factors produced by Hypoxic Cells are a leading driver of Anastomoses in Sprouting Angiogenesis-a computational study. Sci Rep 2018; 8:8726. [PMID: 29880828 PMCID: PMC5992150 DOI: 10.1038/s41598-018-27034-8] [Citation(s) in RCA: 27] [Impact Index Per Article: 4.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/16/2017] [Accepted: 05/29/2018] [Indexed: 12/21/2022] Open
Abstract
Angiogenesis - the growth of new blood vessels from a pre-existing vasculature - is key in both physiological processes and on several pathological scenarios such as cancer progression or diabetic retinopathy. For the new vascular networks to be functional, it is required that the growing sprouts merge either with an existing functional mature vessel or with another growing sprout. This process is called anastomosis. We present a systematic 2D and 3D computational study of vessel growth in a tissue to address the capability of angiogenic factor gradients to drive anastomosis formation. We consider that these growth factors are produced only by tissue cells in hypoxia, i.e. until nearby vessels merge and become capable of carrying blood and irrigating their vicinity. We demonstrate that this increased production of angiogenic factors by hypoxic cells is able to promote vessel anastomoses events in both 2D and 3D. The simulations also verify that the morphology of these networks has an increased resilience toward variations in the endothelial cell's proliferation and chemotactic response. The distribution of tissue cells and the concentration of the growth factors they produce are the major factors in determining the final morphology of the network.
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15
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Suzuki T, Minerva D, Nishiyama K, Koshikawa N, Chaplain MAJ. Study on the tumor-induced angiogenesis using mathematical models. Cancer Sci 2017; 109:15-23. [PMID: 28892582 PMCID: PMC5765301 DOI: 10.1111/cas.13395] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/01/2017] [Revised: 08/29/2017] [Accepted: 09/06/2017] [Indexed: 12/01/2022] Open
Abstract
We studied angiogenesis using mathematical models describing the dynamics of tip cells. We reviewed the basic ideas of angiogenesis models and its numerical simulation technique to produce realistic computer graphics images of sprouting angiogenesis. We examined the classical model of Anderson-Chaplain using fundamental concepts of mass transport and chemical reaction with ECM degradation included. We then constructed two types of numerical schemes, model-faithful and model-driven ones, where new techniques of numerical simulation are introduced, such as transient probability, particle velocity, and Boolean variables.
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Affiliation(s)
- Takashi Suzuki
- Center for Mathematical Modeling and Data Science, Osaka University, Osaka, Japan
| | - Dhisa Minerva
- Center for Mathematical Modeling and Data Science, Osaka University, Osaka, Japan
| | - Koichi Nishiyama
- International Research Center for Medical Sciences, Kumamoto University, Kumamoto, Japan
| | - Naohiko Koshikawa
- Division of Cancer Cell Research, Kanagawa Cancer Center Research Institute, Yokohama, Japan
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Boujelben A, Watson M, McDougall S, Yen YF, Gerstner ER, Catana C, Deisboeck T, Batchelor TT, Boas D, Rosen B, Kalpathy-Cramer J, Chaplain MAJ. Multimodality imaging and mathematical modelling of drug delivery to glioblastomas. Interface Focus 2016; 6:20160039. [PMID: 27708763 DOI: 10.1098/rsfs.2016.0039] [Citation(s) in RCA: 30] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/17/2022] Open
Abstract
Patients diagnosed with glioblastoma, an aggressive brain tumour, have a poor prognosis, with a median overall survival of less than 15 months. Vasculature within these tumours is typically abnormal, with increased tortuosity, dilation and disorganization, and they typically exhibit a disrupted blood-brain barrier (BBB). Although it has been hypothesized that the 'normalization' of the vasculature resulting from anti-angiogenic therapies could improve drug delivery through improved blood flow, there is also evidence that suggests that the restoration of BBB integrity might limit the delivery of therapeutic agents and hence their effectiveness. In this paper, we apply mathematical models of blood flow, vascular permeability and diffusion within the tumour microenvironment to investigate the effect of these competing factors on drug delivery. Preliminary results from the modelling indicate that all three physiological parameters investigated-flow rate, vessel permeability and tissue diffusion coefficient-interact nonlinearly to produce the observed average drug concentration in the microenvironment.
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Affiliation(s)
- Ahmed Boujelben
- School of Petroleum Engineering , Heriot-Watt University , Edinburgh EH14 4AS , UK
| | - Michael Watson
- School of Petroleum Engineering , Heriot-Watt University , Edinburgh EH14 4AS , UK
| | - Steven McDougall
- School of Petroleum Engineering , Heriot-Watt University , Edinburgh EH14 4AS , UK
| | - Yi-Fen Yen
- Martinos Center for Biomedical Imaging, Massachusetts General Hospital , Harvard Medical School , Charlestown, MA 02129 , USA
| | - Elizabeth R Gerstner
- Department of Neurology, Massachusetts General Hospital , Harvard Medical School , Boston, MA , USA
| | - Ciprian Catana
- Martinos Center for Biomedical Imaging, Massachusetts General Hospital , Harvard Medical School , Charlestown, MA 02129 , USA
| | - Thomas Deisboeck
- Martinos Center for Biomedical Imaging, Massachusetts General Hospital , Harvard Medical School , Charlestown, MA 02129 , USA
| | - Tracy T Batchelor
- Department of Neurology, Massachusetts General Hospital , Harvard Medical School , Boston, MA , USA
| | - David Boas
- Martinos Center for Biomedical Imaging, Massachusetts General Hospital , Harvard Medical School , Charlestown, MA 02129 , USA
| | - Bruce Rosen
- Martinos Center for Biomedical Imaging, Massachusetts General Hospital , Harvard Medical School , Charlestown, MA 02129 , USA
| | - Jayashree Kalpathy-Cramer
- Martinos Center for Biomedical Imaging, Massachusetts General Hospital , Harvard Medical School , Charlestown, MA 02129 , USA
| | - Mark A J Chaplain
- School of Mathematics and Statistics , University of St Andrews , St Andrews KY16 9SS , UK
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Fu X, Gens JS, Glazier JA, Burns SA, Gast TJ. Progression of Diabetic Capillary Occlusion: A Model. PLoS Comput Biol 2016; 12:e1004932. [PMID: 27300722 PMCID: PMC4907516 DOI: 10.1371/journal.pcbi.1004932] [Citation(s) in RCA: 21] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/13/2015] [Accepted: 04/20/2016] [Indexed: 12/14/2022] Open
Abstract
An explanatory computational model is developed of the contiguous areas of retinal capillary loss which play a large role in diabetic maculapathy and diabetic retinal neovascularization. Strictly random leukocyte mediated capillary occlusion cannot explain the occurrence of large contiguous areas of retinal ischemia. Therefore occlusion of an individual capillary must increase the probability of occlusion of surrounding capillaries. A retinal perifoveal vascular sector as well as a peripheral retinal capillary network and a deleted hexagonal capillary network are modelled using Compucell3D. The perifoveal modelling produces a pattern of spreading capillary loss with associated macular edema. In the peripheral network, spreading ischemia results from the progressive loss of the ladder capillaries which connect peripheral arterioles and venules. System blood flow was elevated in the macular model before a later reduction in flow in cases with progression of capillary occlusions. Simulations differing only in initial vascular network structures but with identical dynamics for oxygen, growth factors and vascular occlusions, replicate key clinical observations of ischemia and macular edema in the posterior pole and ischemia in the retinal periphery. The simulation results also seem consistent with quantitative data on macular blood flow and qualitative data on venous oxygenation. One computational model applied to distinct capillary networks in different retinal regions yielded results comparable to clinical observations in those regions.
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Affiliation(s)
- Xiao Fu
- The Biocomplexity Institute, Indiana University, Bloomington, Indiana, United States of America
- Department of Physics, Indiana University, Bloomington, Indiana, United States of America
| | - John Scott Gens
- The Biocomplexity Institute, Indiana University, Bloomington, Indiana, United States of America
- Department of Physics, Indiana University, Bloomington, Indiana, United States of America
| | - James A. Glazier
- The Biocomplexity Institute, Indiana University, Bloomington, Indiana, United States of America
- Department of Physics, Indiana University, Bloomington, Indiana, United States of America
- Department of Intelligent Systems Engineering, Indiana University, Bloomington, Indiana, United States of America
| | - Stephen A. Burns
- School of Optometry, Indiana University, Bloomington, Indiana, United States of America
| | - Thomas J. Gast
- School of Optometry, Indiana University, Bloomington, Indiana, United States of America
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18
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Mathematical and computational models of the retina in health, development and disease. Prog Retin Eye Res 2016; 53:48-69. [PMID: 27063291 DOI: 10.1016/j.preteyeres.2016.04.001] [Citation(s) in RCA: 22] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/19/2015] [Revised: 04/05/2016] [Accepted: 04/05/2016] [Indexed: 12/12/2022]
Abstract
The retina confers upon us the gift of vision, enabling us to perceive the world in a manner unparalleled by any other tissue. Experimental and clinical studies have provided great insight into the physiology and biochemistry of the retina; however, there are questions which cannot be answered using these methods alone. Mathematical and computational techniques can provide complementary insight into this inherently complex and nonlinear system. They allow us to characterise and predict the behaviour of the retina, as well as to test hypotheses which are experimentally intractable. In this review, we survey some of the key theoretical models of the retina in the healthy, developmental and diseased states. The main insights derived from each of these modelling studies are highlighted, as are model predictions which have yet to be tested, and data which need to be gathered to inform future modelling work. Possible directions for future research are also discussed. Whilst the present modelling studies have achieved great success in unravelling the workings of the retina, they have yet to achieve their full potential. For this to happen, greater involvement with the modelling community is required, and stronger collaborations forged between experimentalists, clinicians and theoreticians. It is hoped that, in addition to bringing the fruits of current modelling studies to the attention of the ophthalmological community, this review will encourage many such future collaborations.
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19
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Santos-Oliveira P, Correia A, Rodrigues T, Ribeiro-Rodrigues TM, Matafome P, Rodríguez-Manzaneque JC, Seiça R, Girão H, Travasso RDM. The Force at the Tip--Modelling Tension and Proliferation in Sprouting Angiogenesis. PLoS Comput Biol 2015; 11:e1004436. [PMID: 26248210 PMCID: PMC4527825 DOI: 10.1371/journal.pcbi.1004436] [Citation(s) in RCA: 42] [Impact Index Per Article: 4.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/01/2015] [Accepted: 07/08/2015] [Indexed: 12/24/2022] Open
Abstract
Sprouting angiogenesis, where new blood vessels grow from pre-existing ones, is a complex process where biochemical and mechanical signals regulate endothelial cell proliferation and movement. Therefore, a mathematical description of sprouting angiogenesis has to take into consideration biological signals as well as relevant physical processes, in particular the mechanical interplay between adjacent endothelial cells and the extracellular microenvironment. In this work, we introduce the first phase-field continuous model of sprouting angiogenesis capable of predicting sprout morphology as a function of the elastic properties of the tissues and the traction forces exerted by the cells. The model is very compact, only consisting of three coupled partial differential equations, and has the clear advantage of a reduced number of parameters. This model allows us to describe sprout growth as a function of the cell-cell adhesion forces and the traction force exerted by the sprout tip cell. In the absence of proliferation, we observe that the sprout either achieves a maximum length or, when the traction and adhesion are very large, it breaks. Endothelial cell proliferation alters significantly sprout morphology, and we explore how different types of endothelial cell proliferation regulation are able to determine the shape of the growing sprout. The largest region in parameter space with well formed long and straight sprouts is obtained always when the proliferation is triggered by endothelial cell strain and its rate grows with angiogenic factor concentration. We conclude that in this scenario the tip cell has the role of creating a tension in the cells that follow its lead. On those first stalk cells, this tension produces strain and/or empty spaces, inevitably triggering cell proliferation. The new cells occupy the space behind the tip, the tension decreases, and the process restarts.
Our results highlight the ability of mathematical models to suggest relevant hypotheses with respect to the role of forces in sprouting, hence underlining the necessary collaboration between modelling and molecular biology techniques to improve the current state-of-the-art. Sprouting angiogenesis—a process by which new blood vessels grow from existing ones—is an ubiquitous phenomenon in health and disease of higher organisms, playing a crucial role in organogenesis, wound healing, inflammation, as well as on the onset and progression of over 50 different diseases such as cancer, rheumatoid arthritis and diabetes. Mathematical models have the ability to suggest relevant hypotheses with respect to the mechanisms of cell movement and rearrangement within growing vessel sprouts. The inclusion of both biochemical and mechanical processes in a mathematical model of sprouting angiogenesis permits to describe sprout extension as a function of the forces exerted by the cells in the tissue. It also allows to question the regulation of biochemical processes by mechanical forces and vice-versa. In this work we present a compact model of sprouting angiogenesis that includes the mechanical characteristics of the vessel and the tissue. We use this model to suggest the mechanism for the regulation of proliferation within sprout formation. We conclude that the tip cell has the role of creating a tension in the cells that follow its lead. On those first cells of the stalk, this tension produces strain and/or empty spaces, inevitably triggering cell proliferation. The new cells occupy the space behind the tip, the tension decreases, and the process restarts. The modelling strategy used, deemed phase-field, permits to describe the evolution of the shape of different domains in complex systems. It is focused on the movement of the interfaces between the domains, and not on an exhaustive description of the transport properties within each domain. For this reason, it requires a reduced number of parameters, and has been used extensively in modelling other biological phenomena such as tumor growth. The coupling of mechanical and biochemical processes in a compact mathematical model of angiogenesis will enable the study of lumen formation and aneurisms in the near future. Also, this framework will allow the study of the action of flow in vessel remodelling, since local forces can readily be coupled with cell movement to obtain the final vessel morphology.
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Affiliation(s)
| | - António Correia
- Institute for Biomedical Imaging and Life Sciences (IBILI), Faculty of Medicine, University of Coimbra, Coimbra, Portugal
| | - Tiago Rodrigues
- Institute for Biomedical Imaging and Life Sciences (IBILI), Faculty of Medicine, University of Coimbra, Coimbra, Portugal
| | - Teresa M Ribeiro-Rodrigues
- Institute for Biomedical Imaging and Life Sciences (IBILI), Faculty of Medicine, University of Coimbra, Coimbra, Portugal
| | - Paulo Matafome
- Institute for Biomedical Imaging and Life Sciences (IBILI), Faculty of Medicine, University of Coimbra, Coimbra, Portugal
- Department of Complementary Sciences, Coimbra Health School (ESTeSC), Instituto Politécnico de Coimbra, Coimbra, Portugal
| | | | - Raquel Seiça
- Institute for Biomedical Imaging and Life Sciences (IBILI), Faculty of Medicine, University of Coimbra, Coimbra, Portugal
| | - Henrique Girão
- Institute for Biomedical Imaging and Life Sciences (IBILI), Faculty of Medicine, University of Coimbra, Coimbra, Portugal
| | - Rui D. M. Travasso
- CFisUC, Department of Physics, University of Coimbra, Coimbra, Portugal
- * E-mail:
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Treloar KK, Simpson MJ, McElwain DLS, Baker RE. Are in vitro estimates of cell diffusivity and cell proliferation rate sensitive to assay geometry? J Theor Biol 2014; 356:71-84. [PMID: 24787651 DOI: 10.1016/j.jtbi.2014.04.026] [Citation(s) in RCA: 52] [Impact Index Per Article: 5.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/28/2013] [Revised: 03/06/2014] [Accepted: 04/18/2014] [Indexed: 11/25/2022]
Abstract
Cells respond to various biochemical and physical cues during wound-healing and tumour progression. in vitro assays used to study these processes are typically conducted in one particular geometry and it is unclear how the assay geometry affects the capacity of cell populations to spread, or whether the relevant mechanisms, such as cell motility and cell proliferation, are somehow sensitive to the geometry of the assay. In this work we use a circular barrier assay to characterise the spreading of cell populations in two different geometries. Assay 1 describes a tumour-like geometry where a cell population spreads outwards into an open space. Assay 2 describes a wound-like geometry where a cell population spreads inwards to close a void. We use a combination of discrete and continuum mathematical models and automated image processing methods to obtain independent estimates of the effective cell diffusivity, D, and the effective cell proliferation rate, λ. Using our parameterised mathematical model we confirm that our estimates of D and λ accurately predict the time-evolution of the location of the leading edge and the cell density profiles for both assay 1 and assay 2. Our work suggests that the effective cell diffusivity is up to 50% lower for assay 2 compared to assay 1, whereas the effective cell proliferation rate is up to 30% lower for assay 2 compared to assay 1.
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Affiliation(s)
- Katrina K Treloar
- Mathematical Sciences, Queensland University of Technology (QUT), Brisbane, Australia; Tissue Repair and Regeneration Program, Institute of Health and Biomedical Innovation, QUT, Brisbane, Australia
| | - Matthew J Simpson
- Mathematical Sciences, Queensland University of Technology (QUT), Brisbane, Australia; Tissue Repair and Regeneration Program, Institute of Health and Biomedical Innovation, QUT, Brisbane, Australia.
| | - D L Sean McElwain
- Tissue Repair and Regeneration Program, Institute of Health and Biomedical Innovation, QUT, Brisbane, Australia
| | - Ruth E Baker
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, United Kingdom
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21
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Logsdon EA, Finley SD, Popel AS, Mac Gabhann F. A systems biology view of blood vessel growth and remodelling. J Cell Mol Med 2013; 18:1491-508. [PMID: 24237862 PMCID: PMC4190897 DOI: 10.1111/jcmm.12164] [Citation(s) in RCA: 98] [Impact Index Per Article: 8.9] [Reference Citation Analysis] [Abstract] [Key Words] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/30/2013] [Accepted: 09/16/2013] [Indexed: 12/29/2022] Open
Abstract
Blood travels throughout the body in an extensive network of vessels – arteries, veins and capillaries. This vascular network is not static, but instead dynamically remodels in response to stimuli from cells in the nearby tissue. In particular, the smallest vessels – arterioles, venules and capillaries – can be extended, expanded or pruned, in response to exercise, ischaemic events, pharmacological interventions, or other physiological and pathophysiological events. In this review, we describe the multi-step morphogenic process of angiogenesis – the sprouting of new blood vessels – and the stability of vascular networks in vivo. In particular, we review the known interactions between endothelial cells and the various blood cells and plasma components they convey. We describe progress that has been made in applying computational modelling, quantitative biology and high-throughput experimentation to the angiogenesis process.
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Affiliation(s)
- Elizabeth A Logsdon
- Institute for Computational Medicine and Department of Biomedical Engineering, Johns Hopkins University, Baltimore, MD, USA
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22
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Scianna M, Bell C, Preziosi L. A review of mathematical models for the formation of vascular networks. J Theor Biol 2013; 333:174-209. [DOI: 10.1016/j.jtbi.2013.04.037] [Citation(s) in RCA: 108] [Impact Index Per Article: 9.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/14/2012] [Revised: 02/15/2013] [Accepted: 04/30/2013] [Indexed: 02/08/2023]
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23
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Bentley K, Jones M, Cruys B. Predicting the future: Towards symbiotic computational and experimental angiogenesis research. Exp Cell Res 2013; 319:1240-6. [DOI: 10.1016/j.yexcr.2013.02.001] [Citation(s) in RCA: 23] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/24/2012] [Revised: 02/01/2013] [Accepted: 02/02/2013] [Indexed: 01/14/2023]
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