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Katsaounis D, Harbour N, Williams T, Chaplain MA, Sfakianakis N. A Genuinely Hybrid, Multiscale 3D Cancer Invasion and Metastasis Modelling Framework. Bull Math Biol 2024; 86:64. [PMID: 38664343 PMCID: PMC11045634 DOI: 10.1007/s11538-024-01286-0] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/15/2023] [Accepted: 03/22/2024] [Indexed: 04/28/2024]
Abstract
We introduce in this paper substantial enhancements to a previously proposed hybrid multiscale cancer invasion modelling framework to better reflect the biological reality and dynamics of cancer. These model updates contribute to a more accurate representation of cancer dynamics, they provide deeper insights and enhance our predictive capabilities. Key updates include the integration of porous medium-like diffusion for the evolution of Epithelial-like Cancer Cells and other essential cellular constituents of the system, more realistic modelling of Epithelial-Mesenchymal Transition and Mesenchymal-Epithelial Transition models with the inclusion of Transforming Growth Factor beta within the tumour microenvironment, and the introduction of Compound Poisson Process in the Stochastic Differential Equations that describe the migration behaviour of the Mesenchymal-like Cancer Cells. Another innovative feature of the model is its extension into a multi-organ metastatic framework. This framework connects various organs through a circulatory network, enabling the study of how cancer cells spread to secondary sites.
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Affiliation(s)
- Dimitrios Katsaounis
- School of Mathematics and Statistics, University St Andrews, North Haugh, St Andrews, UK.
| | - Nicholas Harbour
- School of Mathematical Sciences, University Nottingham, Nottingham, UK
| | - Thomas Williams
- School of Mathematics and Statistics, The University of Melbourne, Melbourne, Australia
| | - Mark Aj Chaplain
- School of Mathematics and Statistics, University St Andrews, North Haugh, St Andrews, UK
| | - Nikolaos Sfakianakis
- School of Mathematics and Statistics, University St Andrews, North Haugh, St Andrews, UK
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2
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Conte M, Loy N. Multi-Cue Kinetic Model with Non-Local Sensing for Cell Migration on a Fiber Network with Chemotaxis. Bull Math Biol 2022; 84:42. [PMID: 35150333 PMCID: PMC8840942 DOI: 10.1007/s11538-021-00978-1] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/23/2021] [Accepted: 11/23/2021] [Indexed: 11/29/2022]
Abstract
Cells perform directed motion in response to external stimuli that they detect by sensing the environment with their membrane protrusions. Precisely, several biochemical and biophysical cues give rise to tactic migration in the direction of their specific targets. Thus, this defines a multi-cue environment in which cells have to sort and combine different, and potentially competitive, stimuli. We propose a non-local kinetic model for cell migration in which cell polarization is influenced simultaneously by two external factors: contact guidance and chemotaxis. We propose two different sensing strategies, and we analyze the two resulting transport kinetic models by recovering the appropriate macroscopic limit in different regimes, in order to observe how the cell size, with respect to the variation of both external fields, influences the overall behavior. This analysis shows the importance of dealing with hyperbolic models, rather than drift-diffusion ones. Moreover, we numerically integrate the kinetic transport equations in a two-dimensional setting in order to investigate qualitatively various scenarios. Finally, we show how our setting is able to reproduce some experimental results concerning the influence of topographical and chemical cues in directing cell motility.
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Affiliation(s)
- Martina Conte
- Department of Mathematical Sciences, "G. L. Lagrange", Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129, Torino, Italy
| | - Nadia Loy
- Department of Mathematical Sciences, "G. L. Lagrange", Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129, Torino, Italy
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3
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Loy N, Preziosi L. Modelling physical limits of migration by a kinetic model with non-local sensing. J Math Biol 2020; 80:1759-1801. [DOI: 10.1007/s00285-020-01479-w] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/22/2019] [Revised: 12/24/2019] [Indexed: 01/30/2023]
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Giniūnaitė R, Baker RE, Kulesa PM, Maini PK. Modelling collective cell migration: neural crest as a model paradigm. J Math Biol 2020; 80:481-504. [PMID: 31587096 PMCID: PMC7012984 DOI: 10.1007/s00285-019-01436-2] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/01/2019] [Revised: 09/09/2019] [Indexed: 12/01/2022]
Abstract
A huge variety of mathematical models have been used to investigate collective cell migration. The aim of this brief review is twofold: to present a number of modelling approaches that incorporate the key factors affecting cell migration, including cell-cell and cell-tissue interactions, as well as domain growth, and to showcase their application to model the migration of neural crest cells. We discuss the complementary strengths of microscale and macroscale models, and identify why it can be important to understand how these modelling approaches are related. We consider neural crest cell migration as a model paradigm to illustrate how the application of different mathematical modelling techniques, combined with experimental results, can provide new biological insights. We conclude by highlighting a number of future challenges for the mathematical modelling of neural crest cell migration.
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Affiliation(s)
- Rasa Giniūnaitė
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Woodstock Road, Oxford, OX2 6GG, UK.
| | - Ruth E Baker
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Woodstock Road, Oxford, OX2 6GG, UK
| | - Paul M Kulesa
- Stowers Institute for Medical Research, 1000 E 50th Street, Kansas City, MO, 64110, USA
| | - Philip K Maini
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Woodstock Road, Oxford, OX2 6GG, UK
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5
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Colombi A, Scianna M, Painter KJ, Preziosi L. Modelling chase-and-run migration in heterogeneous populations. J Math Biol 2019; 80:423-456. [PMID: 31468116 PMCID: PMC7012813 DOI: 10.1007/s00285-019-01421-9] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/07/2018] [Revised: 08/12/2019] [Indexed: 12/12/2022]
Abstract
Cell migration is crucial for many physiological and pathological processes. During embryogenesis, neural crest cells undergo coordinated epithelial to mesenchymal transformations and migrate towards various forming organs. Here we develop a computational model to understand how mutual interactions between migrating neural crest cells (NCs) and the surrounding population of placode cells (PCs) generate coordinated migration. According to experimental findings, we implement a minimal set of hypotheses, based on a coupling between chemotactic movement of NCs in response to a placode-secreted chemoattractant (Sdf1) and repulsion induced from contact inhibition of locomotion (CIL), triggered by heterotypic NC–PC contacts. This basic set of assumptions is able to semi-quantitatively recapitulate experimental observations of the characteristic multispecies phenomenon of “chase-and-run”, where the colony of NCs chases an evasive PC aggregate. The model further reproduces a number of in vitro manipulations, including full or partial disruption of NC chemotactic migration and selected mechanisms coordinating the CIL phenomenon. Finally, we provide various predictions based on altering other key components of the model mechanisms.
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Affiliation(s)
- A Colombi
- Department of Mathematical Sciences "G. L. Lagrange" - Excellence Department 2018-2022, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129, Turin, Italy
| | - M Scianna
- Department of Mathematical Sciences "G. L. Lagrange" - Excellence Department 2018-2022, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129, Turin, Italy
| | - K J Painter
- Department of Mathematics and Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Edinburgh, Scotland, EH14 4AS, UK.
| | - L Preziosi
- Department of Mathematical Sciences "G. L. Lagrange" - Excellence Department 2018-2022, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129, Turin, Italy
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6
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Kinetic models with non-local sensing determining cell polarization and speed according to independent cues. J Math Biol 2019; 80:373-421. [PMID: 31375892 DOI: 10.1007/s00285-019-01411-x] [Citation(s) in RCA: 16] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/13/2018] [Revised: 07/26/2019] [Indexed: 12/25/2022]
Abstract
Cells move by run and tumble, a kind of dynamics in which the cell alternates runs over straight lines and re-orientations. This erratic motion may be influenced by external factors, like chemicals, nutrients, the extra-cellular matrix, in the sense that the cell measures the external field and elaborates the signal eventually adapting its dynamics. We propose a kinetic transport equation implementing a velocity-jump process in which the transition probability takes into account a double bias, which acts, respectively, on the choice of the direction of motion and of the speed. The double bias depends on two different non-local sensing cues coming from the external environment. We analyze how the size of the cell and the way of sensing the environment with respect to the variation of the external fields affect the cell population dynamics by recovering an appropriate macroscopic limit and directly integrating the kinetic transport equation. A comparison between the solutions of the transport equation and of the proper macroscopic limit is also performed.
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7
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Bernardi S, Colombi A, Scianna M. A particle model analysing the behavioural rules underlying the collective flight of a bee swarm towards the new nest. JOURNAL OF BIOLOGICAL DYNAMICS 2018; 12:632-662. [PMID: 30051763 DOI: 10.1080/17513758.2018.1501105] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/28/2017] [Accepted: 07/09/2018] [Indexed: 06/08/2023]
Abstract
The swarming of a bee colony is guided by a small group of scout individuals, which are informed of the target destination (the new nest). However, little is known on the underlying mechanisms, i.e. on how the information is passed within the population. In this respect, we here present a discrete mathematical model to investigate these aspects. In particular, each bee, represented by a material point, is assigned its status within the colony and set to move according to individual strategies and social interactions. More specifically, we propose alternative assumptions on the flight synchronization mechanism of uninformed individuals and on the characteristic dynamics of the scout insects. Numerical realizations then point out the combinations of behavioural hypotheses resulting in collective productive movement. An analysis of the role of the scout bee percentage and of the phenomenology of the swarm in domains with structural elements is finally performed.
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Affiliation(s)
- Sara Bernardi
- a Department of Mathematical Sciences , Politecnico di , Torino , Italy
| | | | - Marco Scianna
- a Department of Mathematical Sciences , Politecnico di , Torino , Italy
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8
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Carrillo JA, Colombi A, Scianna M. Adhesion and volume constraints via nonlocal interactions determine cell organisation and migration profiles. J Theor Biol 2018; 445:75-91. [DOI: 10.1016/j.jtbi.2018.02.022] [Citation(s) in RCA: 16] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/18/2017] [Revised: 02/18/2018] [Accepted: 02/20/2018] [Indexed: 12/17/2022]
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9
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Bernardi S, Colombi A, Scianna M. A discrete particle model reproducing collective dynamics of a bee swarm. Comput Biol Med 2018; 93:158-174. [PMID: 29316459 DOI: 10.1016/j.compbiomed.2017.12.022] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/12/2017] [Revised: 12/19/2017] [Accepted: 12/21/2017] [Indexed: 01/10/2023]
Abstract
In this article, we present a microscopic discrete mathematical model describing collective dynamics of a bee swarm. More specifically, each bee is set to move according to individual strategies and social interactions, the former involving the desire to reach a target destination, the latter accounting for repulsive/attractive stimuli and for alignment processes. The insects tend in fact to remain sufficiently close to the rest of the population, while avoiding collisions, and they are able to track and synchronize their movement to the flight of a given set of neighbors within their visual field. The resulting collective behavior of the bee cloud therefore emerges from non-local short/long-range interactions. Differently from similar approaches present in the literature, we here test different alignment mechanisms (i.e., based either on an Euclidean or on a topological neighborhood metric), which have an impact also on the other social components characterizing insect behavior. A series of numerical realizations then shows the phenomenology of the swarm (in terms of pattern configuration, collective productive movement, and flight synchronization) in different regions of the space of free model parameters (i.e., strength of attractive/repulsive forces, extension of the interaction regions). In this respect, constraints in the possible variations of such coefficients are here given both by reasonable empirical observations and by analytical results on some stability characteristics of the defined pairwise interaction kernels, which have to assure a realistic crystalline configuration of the swarm. An analysis of the effect of unconscious random fluctuations of bee dynamics is also provided.
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Affiliation(s)
- Sara Bernardi
- Department of Mathematical Sciences, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy.
| | - Annachiara Colombi
- Department of Mathematical Sciences, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy.
| | - Marco Scianna
- Department of Mathematical Sciences, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy.
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10
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Colombi A, Scianna M. Modelling human perception processes in pedestrian dynamics: a hybrid approach. ROYAL SOCIETY OPEN SCIENCE 2017; 4:160561. [PMID: 28405352 PMCID: PMC5383809 DOI: 10.1098/rsos.160561] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 07/29/2016] [Accepted: 02/01/2017] [Indexed: 06/07/2023]
Abstract
In this paper, we present a hybrid mathematical model describing crowd dynamics. More specifically, our approach is based on the well-established Helbing-like discrete model, where each pedestrian is individually represented as a dimensionless point and set to move in order to reach a target destination, with deviations deriving from both physical and social forces. In particular, physical forces account for interpersonal collisions, whereas social components include the individual desire to remain sufficiently far from other walkers (the so-called territorial effect). In this respect, the repulsive behaviour of pedestrians is here set to be different from traditional Helbing-like methods, as it is assumed to be largely determined by how they perceive the presence and the position of neighbouring individuals, i.e. either objectively as pointwise/localized entities or subjectively as spatially distributed masses. The resulting modelling environment is then applied to specific scenarios, that first reproduce a real-world experiment, specifically designed to derive our model hypothesis. Sets of numerical realizations are also run to analyse in more details the pedestrian paths resulting from different types of perception of small groups of static individuals. Finally, analytical investigations formalize and validate from a mathematical point of view selected simulation outcomes.
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Affiliation(s)
| | - M. Scianna
- Author for correspondence: M. Scianna e-mail:
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11
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Simmons A, Burrage PM, Nicolau DV, Lakhani SR, Burrage K. Environmental factors in breast cancer invasion: a mathematical modelling review. Pathology 2017; 49:172-180. [PMID: 28081961 DOI: 10.1016/j.pathol.2016.11.004] [Citation(s) in RCA: 17] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/11/2016] [Revised: 11/07/2016] [Accepted: 11/13/2016] [Indexed: 12/17/2022]
Abstract
This review presents a brief overview of breast cancer, focussing on its heterogeneity and the role of mathematical modelling and simulation in teasing apart the underlying biophysical processes. Following a brief overview of the main known pathophysiological features of ductal carcinoma, attention is paid to differential equation-based models (both deterministic and stochastic), agent-based modelling, multi-scale modelling, lattice-based models and image-driven modelling. A number of vignettes are presented where these modelling approaches have elucidated novel aspects of breast cancer dynamics, and we conclude by offering some perspectives on the role mathematical modelling can play in understanding breast cancer development, invasion and treatment therapies.
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Affiliation(s)
- Alex Simmons
- School of Mathematical Sciences, and ARC Centre of Excellence for Mathematical and Statistical Frontiers, Queensland University of Technology, Gardens Point, Brisbane, Qld, Australia
| | - Pamela M Burrage
- School of Mathematical Sciences, and ARC Centre of Excellence for Mathematical and Statistical Frontiers, Queensland University of Technology, Gardens Point, Brisbane, Qld, Australia
| | - Dan V Nicolau
- School of Mathematical Sciences, and ARC Centre of Excellence for Mathematical and Statistical Frontiers, Queensland University of Technology, Gardens Point, Brisbane, Qld, Australia; Mathematical Institute, University of Oxford, Oxford, United Kingdom; Molecular Sense Ltd, Oxford, United Kingdom
| | - Sunil R Lakhani
- The University of Queensland, Centre for Clinical Research and School of Medicine and Pathology Queensland, The Royal Brisbane and Women's Hospital, Brisbane, Qld, Australia
| | - Kevin Burrage
- School of Mathematical Sciences, and ARC Centre of Excellence for Mathematical and Statistical Frontiers, Queensland University of Technology, Gardens Point, Brisbane, Qld, Australia; Department of Computer Science, University of Oxford, United Kingdom.
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12
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Colombi A, Scianna M, Preziosi L. Coherent modelling switch between pointwise and distributed representations of cell aggregates. J Math Biol 2016; 74:783-808. [PMID: 27423897 DOI: 10.1007/s00285-016-1042-0] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/20/2016] [Revised: 06/19/2016] [Indexed: 02/03/2023]
Abstract
Biological systems are typically formed by different cell phenotypes, characterized by specific biophysical properties and behaviors. Moreover, cells are able to undergo differentiation or phenotypic transitions upon internal or external stimuli. In order to take these phenomena into account, we here propose a modelling framework in which cells can be described either as pointwise/concentrated particles or as distributed masses, according to their biological determinants. A set of suitable rules then defines a coherent procedure to switch between the two mathematical representations. The theoretical environment describing cell transition is then enriched by including cell migratory dynamics and duplication/apoptotic processes, as well as the kinetics of selected diffusing chemicals influencing the system evolution. Finally, biologically relevant numerical realizations are presented: in particular, they deal with the growth of a tumor spheroid and with the initial differentiation stages of the formation of the zebrafish posterior lateral line. Both phenomena mainly rely on cell phenotypic transition and differentiated behaviour, thereby constituting biological systems particularly suitable to assess the advantages of the proposed model.
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Affiliation(s)
- A Colombi
- Department of Mathematical Sciences "G. L. Lagrange", Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129, Torino, Italy
| | - M Scianna
- Department of Mathematical Sciences "G. L. Lagrange", Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129, Torino, Italy.
| | - L Preziosi
- Department of Mathematical Sciences "G. L. Lagrange", Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129, Torino, Italy
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Yu JS, Bagheri N. Multi-class and multi-scale models of complex biological phenomena. Curr Opin Biotechnol 2016; 39:167-173. [PMID: 27115496 DOI: 10.1016/j.copbio.2016.04.002] [Citation(s) in RCA: 20] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/09/2015] [Revised: 03/28/2016] [Accepted: 04/01/2016] [Indexed: 02/06/2023]
Abstract
Computational modeling has significantly impacted our ability to analyze vast (and exponentially increasing) quantities of experimental data for a variety of applications, such as drug discovery and disease forecasting. Single-scale, single-class models persist as the most common group of models, but biological complexity often demands more sophisticated approaches. This review surveys modeling approaches that are multi-class (incorporating multiple model types) and/or multi-scale (accounting for multiple spatial or temporal scales) and describes how these models, and combinations thereof, should be used within the context of the problem statement. We end by highlighting agent-based models as an intuitive, modular, and flexible framework within which multi-scale and multi-class models can be implemented.
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Affiliation(s)
- Jessica S Yu
- Chemical & Biological Engineering, Northwestern University, Evanston, IL, United States
| | - Neda Bagheri
- Chemical & Biological Engineering, Northwestern University, Evanston, IL, United States.
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