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Computational studies of particles and cells transport in microfluidic devices Arefi, Seyedmohammadamin 2021

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Computational Studies of Particles and Cells Transport inMicrofluidic DevicesbySeyedmohammadamin ArefiB.Sc., University of Tehran, 2012M.Sc., University of Tehran, 2015A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Chemical and Biological Engineering)The University of British Columbia(Vancouver)March 2021© Seyedmohammadamin Arefi, 2021The following individuals certify that they have read, and recommend to the Fac-ulty of Graduate and Postdoctoral Studies for acceptance, the thesis entitled:Computational Studies of Particles and Cells Transport in MicrofluidicDevicessubmitted by Seyedmohammadamin Arefi in partial fulfillment of the require-ments for the degree of Doctor of Philosophy in Chemical and Biological Engi-neering.Examining Committee:James J. Feng, Chemical and Biological Engineering and Mathematics, UBCSupervisorCheng Wei Tony Yang, Centre for Heart Lung InnovationCo-SupervisorEdmond Young , Mechanical Engineering, University of TorontoExternal ExaminerBoris Stoeber , Mechanical Engineering, UBCUniversity ExaminerGwynn Elfring , Mechanical Engineering, UBCUniversity ExaminerAdditional Supervisory Committee Members:Karen Cheung, Electrical and Computer Engineering, UBCSupervisory Committee MemberAnthony Wachs, Chemical and Biological Engineering, UBCSupervisory Committee MemberiiAbstractBio-mimicking microfluidic devices have been developed to provide a biologicallyrelevant framework for drug development and toxicological studies. This disserta-tion aims to describe three case studies in the area of particle and cell transport inbio-mimicking microfluidic devices using finite-element simulations.In the first study, we model the deposition of submicron particles on the ep-ithelial layer of a well-established lung-on-a-chip device. As main results of thestudy, our simulations predict enhanced submicron particle deposition during phys-ical exercising, subject to opposing effects of elevated air volume and breathingfrequency. Moreover, the deposition efficiency varies non-monotonically with par-ticle size, due to the distinct effects of gravitational settling and Brownian motion.In the second case study, we propose a biomechanical model for the passageof a tumor cell through the endothelial cells monolayer. Based on prior in vitroobservations, we assume that the tumor cell extends a protrusion between adja-cent endothelial cells that adheres to the extracellular matrix. Inside the protrusion,a contractile element composed of stress-fibers and focal-adhesions pulls the nu-cleus through the endothelial opening. We modeled the chemo-mechanics of thecontractile element as well as the elastic deformation and cytosolic flow duringtransmigration process. Using physiologically reasonable parameters, our modelshows that the contractile element can produce a force on the order of 70 nN, whichis sufficient to deform the endothelial cells for transmigration.The third project deals with the tenertaxis hypothesis. During immune reac-tion, leukocytes transmigrate either directly through the body of an individual en-dothelial cell (the transcellular route) or from the junction between adjacent en-dothelial cells (the paracellular route). What determines the usage of one routeiiiover the other is ambiguous. A recently proposed tenertaxis hypothesis claimsthat leukocytes choose the path with less mechanical resistance against leukocyteprotrusions. Our simulations results show that the required force to breach the en-dothelium through the transcellular route is greater than paracellular route (22 pNversus 15 pN). Moreover, experiments have demonstrated that manipulation of therelative strength of endothelial resistance can make the transcellular route prefer-able. Our simulations have confirmed this reversal, and thus tentatively confirmedthe hypothesis of tenertaxis.ivLay SummaryBio-mimicking microfluidic devices are widely used in biological studies to in-vestigate behaviours and functions of living cells and tissues. These devices arecomposed of micron size channels through which living cells can be cultured andvisualized. Moreover, the micro-environment of the cells can be precisely con-trolled. Lung-on-a-chip devices and on-chip microvascular networks are two ex-amples of microfludic devices that mimic pulmonary and vascular tissues. Ourresearch involves three case studies related to these microfludic devices.In the first case study, we investigate the deposition of nanoparticles inside alung-on-a-chip device. In the second study, we describe the underlying physics ofthe passage of tumour cells through the walls of capillaries in an on-chip microvas-cular networks. In the third case study, we investigate the passage of immunecells through the walls of capillaries during immune response to infections. Theoutcomes of our modeling can inform the design of bio-mimicking microfluidicdevices.vPrefaceThis dissertation entitled ”Computational studies of particles and cells transportin microfluidic devices” describes the research projects that I carried out duringmy PhD. The research projects that introduced in this dissertation were mainly de-signed and supervised by James J. Feng (Supervisor). In this preface, the contribu-tions and collaborations to the projects and published papers are briefly explained.A version of chapter 3, and sections 1.1, 2.1 and 2.2 have been published as:S. M. Amin Arefi, Cheng Wei Tony Yang, Don D. Sin, and James J. Feng. ”Sim-ulation of nanoparticle transport and adsorption in a microfluidic lung-on-a-chipdevice.” Biomicrofluidics 14, no. 4 (2020): 044117.” This research project wasdesigned by Cheng Wei Tony Yang, Don D. Sin, and James J. Feng. The author(S.M. Amin Arefi) conducted the model formulation, numerical simulations anddata collection. S. M. Amin Arefi, Cheng Wei Tony Yang, Don D. Sin, and JamesJ. Feng all contributed to the data analysis and the writing of the paper.A version of chapter 4 and sections 1.2, 2.1, 2.3, 2.4 and 2.5 have been pub-lished as: S.M. Amin Arefi, Daria Tsvirkun, Claude Verdier, and James J. Feng. ”Abiomechanical model for the transendothelial migration of cancer cells.” PhysicalBiology 17, no. 3 (2020): 036004. Claude Verdier, and James J. Feng conceivedthe project and S.M. Amin Arefi performed the numerical computations. S.M.Amin Arefi, Daria Tsvirkun, Claude Verdier, and James J. Feng. all contributed tothe writing of the paper.S.M. Amin Arefi and J.J. Feng both designed the research project in chapter 5.S.M. Amin Arefi conducted the numerical simulations and wrote this chapter forpurposes of this work.viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Transport and deposition of submicron particles in a LOAC device 31.2 Mesenchymal transmigration of cancer cells . . . . . . . . . . . . 51.3 Amoeboid transmigration of leukocytes and tenertaxis . . . . . . 72 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.1 Fluid flow simulation . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Modeling particle transport and deposition . . . . . . . . . . . . . 112.2.1 Particle tracking using Eulerian approach . . . . . . . . . 112.2.2 Particles tracking using Lagrangian approach . . . . . . 122.2.3 Adsorption models . . . . . . . . . . . . . . . . . . . . . 122.3 Solid hyperelasticity simulation . . . . . . . . . . . . . . . . . . 13vii2.4 Fluid-Solid coupling . . . . . . . . . . . . . . . . . . . . . . . . 142.5 Chemomechanical modeling of stress fibers and focal adhesions . 153 Case study I: Transport and deposition of submicron particles in aLOAC device . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.1 Problem setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.1.1 Geometric setup . . . . . . . . . . . . . . . . . . . . . . 193.1.2 Governing equations . . . . . . . . . . . . . . . . . . . . 203.1.3 Numerical setup . . . . . . . . . . . . . . . . . . . . . . 243.1.4 Parameters estimation . . . . . . . . . . . . . . . . . . . 253.2 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . 293.2.1 Eulerian model: particle deposition with constant unidirec-tional in-flow . . . . . . . . . . . . . . . . . . . . . . . . 303.2.2 Eulerian model: particle deposition in bidirectional pul-satile flow . . . . . . . . . . . . . . . . . . . . . . . . . . 353.2.3 Eulerian model: breathing patterns during exercising andsmoking . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.2.4 Lagrangian model: particle tracking . . . . . . . . . . . . 413.3 Summary and discussion . . . . . . . . . . . . . . . . . . . . . . 464 Case study II: Mesenchymal transmigration of cancer cells . . . . . 494.1 Problem setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.1.1 Model description . . . . . . . . . . . . . . . . . . . . . 504.1.2 Geometric setup . . . . . . . . . . . . . . . . . . . . . . 524.1.3 Governing equations . . . . . . . . . . . . . . . . . . . . 544.1.4 Boundary conditions . . . . . . . . . . . . . . . . . . . . 564.1.5 Parameters estimation . . . . . . . . . . . . . . . . . . . 574.1.6 Numerical techniques . . . . . . . . . . . . . . . . . . . 594.2 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . 614.2.1 Dry contact simulation . . . . . . . . . . . . . . . . . . . 614.2.2 Wet-contact simulation . . . . . . . . . . . . . . . . . . . 644.3 Parametric studies . . . . . . . . . . . . . . . . . . . . . . . . . . 694.4 Summary and discussion . . . . . . . . . . . . . . . . . . . . . . 71viii5 Case study III: Amoeboid transmigration of leukocytes and tenertaxis 755.1 Problem setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.1.1 Geometric setup . . . . . . . . . . . . . . . . . . . . . . 775.1.2 Parameters estimation . . . . . . . . . . . . . . . . . . . 795.1.3 Numerical setup . . . . . . . . . . . . . . . . . . . . . . 815.2 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . 835.2.1 Baseline results . . . . . . . . . . . . . . . . . . . . . . . 835.2.2 Effects of manipulating of the endothelium . . . . . . . . 875.3 Summary and discussion . . . . . . . . . . . . . . . . . . . . . . 906 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96ixList of TablesTable 3.1 Baseline values for the parameters used in our model. For thechannel length L, the experimental device of Huh et al. [1] hasL > 1 cm. In our model, L = 2 mm turns out to be sufficientfor most of the simulations, with the exception of the breathingpatterns of Sec. 3.2.3 (see discussions therein). . . . . . . . . . 26Table 4.1 Geometric parameters used in our model. Additional lengthsare marked and given in Fig. 4.3. . . . . . . . . . . . . . . . . 58Table 4.2 Physical parameters used in our model. . . . . . . . . . . . . . 58Table 4.3 Biochemical parameters used in our model. In evaluating ∆µ ,we have taken kB = 1.381×10−23 m2 kg/(s2 K) and T = 310 K. 59Table 5.1 Important baseline parameters used in our model. . . . . . . . 81xList of FiguresFigure 1.1 An invadopodium-based mechanism for extravasation proposedby Chen et al. [2]. Inside the TC, actomyosin assembles intostress fibers that propel a protrusion between the ECs. Adhe-sion molecules ,i.e. integrins, anchor the tips of the protrusiononto laminin of the ECM. The focal-adhesion proteins includetensin, talin and vinculin among others. Actomyosin contrac-tion in the stress fibers then pulls the TC past the constrictionbetween ECs. . . . . . . . . . . . . . . . . . . . . . . . . . . 6Figure 3.1 Schematic representation of the computational domain, mod-eled after the LOAC design of Huh et al. [1]. The upper chan-nel (in blue) is separated from the bottom channel (in pink) byan elastic membrane (in magenta). The top mimics the alveo-lus through which air flows and the bottom mimics the bloodvessel. The two side chambers are for the purpose of imposingvacuum to stretch the elastic membrane. The air flows alongthe −z direction, and θ denotes the angle between the air flowand gravity g. Most of the computations have θ = 90◦, butθ can be varied to allow different inclinations of the devicerelative to gravity. . . . . . . . . . . . . . . . . . . . . . . . . 20xiFigure 3.2 (a) A typical finite-element mesh used for the fluid-structureinteraction simulation that allows the membrane and all wallsto be elastically deformable. (b) A typical mesh used for com-puting the flow and Eulerian particle transport inside the airchannel, with all walls treated as rigid. . . . . . . . . . . . . . 24Figure 3.3 (a) shows the convergence of a velocity profile with increasingmesh refinement toward the analytical solution [3]. (b) showsthe line in the air channel along which the velocity profile istaken. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Figure 3.4 Verification of time step resolution in the Brownian force cal-culation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Figure 3.5 (a) A snapshot of the LOAC at the time of its maximum defor-mation when the membrane between the two chambers stretchesto 7% strain. The elastic modulus of the walls and the negativepressure in the side chamber are taken from the experiment [1].The color contour in the air chamber represents magnitude ofthe air velocity (left color bar, in m/s) whereas that inside thesolid indicates the level of the von Mises stress (right color bar,in Pa). (b) The cyclic stretching of the membrane produces amild oscillation of amplitude 2.7% in the surface number den-sity of deposited nanoparticles, with an average slightly abovethat for a rigid membrane. . . . . . . . . . . . . . . . . . . . 30Figure 3.6 Deposition of SPs under constant in-flow with the Langmuirmodel. (a) Time evolution of volume concentration distribu-tion c in the air channel shown on the mid-plane highlighted inthe left figure. The height of the rectangles has been stretchedby 4 times to facilitate visualization. After about 12 secondsthe channel is full of smoky air at the constant particle volumeconcentration c0 = 5.24×10−6. (b) Temporal evolution of thedistribution of the fraction of occupied sites θ on the substrate.(c) Time evolution of the averaged areal number density ρs ofdeposited particles on the substrate. (d) Time evolution of thedeposition efficiency ε . . . . . . . . . . . . . . . . . . . . . . 32xiiFigure 3.7 (a) Effect of the surface adsorption models on ρs(t), the aver-aged areal particle number density on the substrate. (b) Effectof diffusivity D on the number density ρs of deposited parti-cles. (c) Effect of slip velocity vs on the particle deposition onthe substrate. . . . . . . . . . . . . . . . . . . . . . . . . . . 34Figure 3.8 Deposition in bidirectional pulsatile flow. (a) Particle volumeconcentration c in the mid-plane (cf. Fig. 3.6a) at the end of thefirst 4 inhalations (left) and 4 exhalations (right). The height ofthe rectangles has been stretched by 4 times to facilitate visu-alization. (b) Averaged bulk concentration at the outlet, scaledby the inlet concentration c0, as a function of time during thefirst 4 cycles. (c) Variation of the average particle number den-sity ρs adsorbed onto the substrate over the first 4 cycles ofbreathing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Figure 3.9 (a) Effect of the airflow velocity on SP deposition, with thefrequency kept at the rest-state 15 cycles per minute or 4 s percycle. (b) Effect of the breathing frequency on SP deposition,with the airflow velocity kept at the rest-state V0 = 0.337 mm/s.The number density of deposited particles ρs is plotted againstthe number of breathing cycles. (c) Same as the above, withρs plotted against real time. (d) Effect of exercise on SP de-position, with the faster airflow (V0 = 1.91 mm/s) and higherbreathing frequency (20 cycles/min or 3 s per cycle). ρs isplotted against the number of breathing cycles. (e) Same asthe above, with ρs plotted against real time. . . . . . . . . . . 39Figure 3.10 Effect of breath-holding on particle deposition during smok-ing. For holding time th = 0, 1 s and 10 s, the number den-sity of deposited particles ρs is plotted against the number ofbreathing cycles in (a) and against real time in (b). . . . . . . 41Figure 3.11 Deposition efficiency ε for non-Brownian and Brownian parti-cles of different sizes. ε is calculated by dividing the numberof absorbed particles by the total number of particles released. 43xiiiFigure 3.12 Snapshots of the distribution of non-Brownian particles at t =2, 4 and 6 s, for particle diameter (a) d = 100 nm, and (b) d =500 nm. The color indicates the instantaneous particle speed,with blue particles (up = 0) being stationary and adsorbed ontothe substrate. . . . . . . . . . . . . . . . . . . . . . . . . . . 43Figure 3.13 Snapshots of the distribution of Brownian particles of diame-ter d = 100 nm at t = 2, 4 and 6 s. The color indicates the in-stantaneous particle speed, with blue particles (up = 0) beingstationary and adsorbed onto the substrate. Note the markedenhancement in deposition relative to the non-Brownian parti-cles of Fig. 3.12(a). . . . . . . . . . . . . . . . . . . . . . . . 44Figure 3.14 Effect of the orientation of gravity on the deposition efficiencyε for Brownian particles of different sizes. The angle θ isbetween the directions of air flow and gravity (see Fig. 3.1).Gravity is along the flow direction at θ = 0, perpendicular tothe flow at θ = 90◦ and against the flow at θ = 180◦. . . . . . 45Figure 4.1 Schematic showing the various components of the model. Apointed arrowhead means “activate” or “promote” whereas aflat arrowhead means “impede” or “inhibit”. . . . . . . . . . . 50Figure 4.2 (a) Schematic showing the geometric setup for the “dry-contact”simulation. Stress fibers AB and focal adhesions BC link thenucleus to the roof of the ECM cavity. The TC membrane andcytosol are ignored, and the critical condition of the break-through of the TC nucleus is determined from the direct butfrictionless contact between the nucleus and the endothelium.(b) Schematic illustrating the geometric setup for the “wet-contact” simulation, with the TC membrane and cytosol ex-plicitly accounted for. . . . . . . . . . . . . . . . . . . . . . . 52Figure 4.3 The geometric setup for the so-called “wet-contact” simula-tion, with the cytosol and the membrane of the tumor cell ex-plicitly represented. . . . . . . . . . . . . . . . . . . . . . . . 53xivFigure 4.4 Illustration of the boundary conditions. The green lines desig-nate fixed boundaries and the red lines show boundaries withdistributed forces. . . . . . . . . . . . . . . . . . . . . . . . . 56Figure 4.5 The effect of different mesh resolution on the contractile forcein the stress fibers. The solutions using the coarse, normal andfine resolutions are sufficiently close, and we have used thenormal resolution in all results reported later. . . . . . . . . . 61Figure 4.6 Snapshots showing the progression of transmigration in thedry-contact setup of the simulation. The color contours showthe local strain in the vertical direction. . . . . . . . . . . . . 62Figure 4.7 The nucleus velocity vn during the passage through endothelium. 63Figure 4.8 Dynamics of transmigration illustrated by the evolution of (a)the active myosin level in the SF η and the fraction of high-affinity integrins ξH/ξ0 in the FA; (b) the tension τ in the SF. . 64Figure 4.9 Snapshots illustrating the extravasation of a tumor-cell nucleusin a wet-contact simulation. The left color bar shows the levelof the vertical strain component in the solids while the rightcolor bar shows the velocity magnitude (m/s) in the cytosol. . 66Figure 4.10 Dynamics of transmigration illustrated by the temporal evolu-tion of the velocity of the centroid of the nucleus in time. Thedry-contact result is also shown for comparison. . . . . . . . . 67Figure 4.11 The temporal evolution of (a) the active myosin level η on thestress fibers and the fraction of high-affinity integrin ξH/ξ0 inthe focal adhesions; (b) the tension τ on the stress fibers. Theforce for the dry-contact case is also shown for comparison. Inlater times (t > 200 s), τ and ξH recover somewhat as in thedry-contact case, but this portion of the evolution is omittedfor a better view. . . . . . . . . . . . . . . . . . . . . . . . . 67Figure 4.12 Effect of the EC gap radius rg on extravasation of the TC nu-cleus in the dry-contact simulations. The outcome is depictedby the tension in the SF-FA assembly, and extravasation failsfor too small a gap (rg ≤ 0.9 µm). . . . . . . . . . . . . . . . 70xvFigure 4.13 Effect of the ECM modulus GECM on extravasation of the TCnucleus in the dry-contact simulations. (a) Temporal evolutionof the tension τ for 4 values of the ECM modulus. Extrava-sation fails if the ECM is too soft (GECM ≤ 100 Pa). (b) Asnapshot of the ECM deformation at t = 210 s for GECM = 100Pa shows great deformation of the soft ECM that accompaniesfailure of extravasation. The color contours indicate the verti-cal component of the stretching. . . . . . . . . . . . . . . . . 71Figure 4.14 Effect of time scale θ of the activation signal (see Eq. 4.1).Transmigration requires a minimum θ , which is between 10and 50 s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72Figure 5.1 Schematic of a leukocyte extending protrusions on the endothe-lium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77Figure 5.2 Geometric setup for the simulations of (a) the paracellular pro-trusion and (b) the transcellular protrusion. . . . . . . . . . . 78Figure 5.3 (a) Contour plot of the von-Mises stress (in Pa) when the rodreaches the basement membrane during the paracellular pro-trusion. To have a better view, half of the EC domain wasremoved in the figure. (b) Mid-plane of the domain. . . . . . . 83Figure 5.4 Progression of the rod in the paracellular route at the mid-plane(refer to Fig. 5.2). The color bar shows the level of von-Misesstress in Pa. . . . . . . . . . . . . . . . . . . . . . . . . . . 84Figure 5.5 Plot of the vertical contact force versus the rod displacement.The dashed arrow suggests how the rod will pass over an un-stable portion of the curve in a dynamic simulation with a pre-scribed force. . . . . . . . . . . . . . . . . . . . . . . . . . . 85Figure 5.6 Snapshots of the level of von-Mises stress (in Pa) during tran-scellular tunnel formation . . . . . . . . . . . . . . . . . . . 86Figure 5.7 Plot of resistance force versus rod displacement . . . . . . . . 87Figure 5.8 Plots of resistance force vs displacement during transcellularpenetration for the various elastic modulus of the EC body. . . 88xviFigure 5.9 Plots of resistance force vs rod displacement during paracellu-lar penetration for various elastic modulus of the ECs near thejunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89xviiAcknowledgmentsI would first like to thank my advisor, Prof. James J. Feng. He taught me how tobe an independent scholar and how to solve complex and challenging problems.His patience, guidance, sharpness, and approach to science have all contributed toshape who I am in my research career.I want to thank my two co-advisors, Profs. Claude Verdier and Dr. Tony Yang,who have worked closely with me through the years and given me invaluable ad-vice. Claude supported me a lot during my visit to University of Grenoble-Alpesand also helped me with experiments. And Tony always came to my aid by beinggenerous with his time and offering technical suggestions. I would also like tothank my supervisory committee, Prof. Karen Cheung and Prof. Anthony Wachs,for their input and guidance in turning this into a stronger dissertation. I wouldalso like to show my appreciation for the work of the members of the Feng-KeshetSnacks group whose lasting impact will continue to be felt in the years to come.Especially I would like to thank my lab mates, Mingfeng Qiu, Clinton H. Durney,Brian Merchant, Mohar Dey, and Lex Lee for creating an intellectually stimulatingenvironment where I could better develop my research ideas.To my friends: thank you so much, Mohamad Ali Bijarchi, Amirhossein Naseri,Hamed Helisaz, and Ali Adibnia for all your assistance and for giving me strongencouragement when, at times, I had to grapple with deep frustration and self-doubt. Thank you Alireza Zakeri (Hajagha Gol) for your companionship and goodvibes throughout, and I will always cherish our adventures in Vancouver and Eu-rope! My warm gratitude also goes to (in alphabetic order) Milad Bakhshizadeh,Mohammad Ali Ghazizadeh, Kasra Ghorbaninejad, Hosein Kamali, MohammadMahdavyfakhr, Ali Moallemi, Hamed Pouriyayevali, Ali Shademani, Javad Shari-xviiiatzadeh, Armin Taheri, Roozbeh Yousefnejad, and Hedayat Zarkoob for all theirhelp and emotional support. At times when projects did not work as they weresupposed to, having a conversation with the best friends (i.e., you) provided mewith the necessary motivation to keep on.Last but not least, I would like to thank my lovely parents, sisters, brother,and uncle, Hossein, for their moral support and encouragement for the pursuit ofeducation. Without the foundation they gave me, I would never have reached thisstage.xixChapter 1IntroductionMicrofluidic devices are widely used in clinical and biological studies as in vitromodels to investigate complex behaviours and functions of living cells [4]. In re-cent years, miniature models of human tissue or organs have been fabricated usingmicrofluidic chips to create physiologically-relevant interfaces between differenttissues and cells, and to mimic the mechanical and physiological properties of realorgans or tissues. These include models of the microvasculature, airway, lung,liver, gut, kidney, and the nervous system [1, 5–9]. The primary driving force be-hind such development is pharmaceutical and toxicological studies. Traditionalanimal testing is notoriously inaccurate in predicting drug efficacy and toxicity inhuman patients [10]. In engineered chip-based models, on the other hand, we canchoose which aspects of the real tissue or organ to represent relevantly and targetthe design to particular drug or pathological tests.Lung-on-a-chip (LOAC) devices and on-chip microvascular networks (µVNs)are two examples of microfludic devices that recapitulate pulmonary and vasculartissues. A representative LOAC device involves flow in the vascular and alveolarchannels, biological barriers of alveoli and communication between apposed en-dothelial and epithelial cell layers, and mechanical stretching of these layers thatmimics breathing and is used to study immune response to nanoparticles deposi-tion, bacteria and inflammatory cytokines in the alveolar tissue [1]. The µVNsdevices provide in vitro microvasculature with physiologically relevant endothelialbarrier function to investigate and visualize transmigration of cancer cells through1a layer of endothelial cells [11]. In both types of devices, the transport of micro-or nano-particles and cells is an important process that can be modeled and inves-tigated. This may represent, for example, how particulate pollutants in the air canenter the airways and be transported into the blood stream, and how immune andtumor cells can transmigrate through the endothelium of a blood vessel in an im-mune response or during cancer metastasis. In general two types of particles andcells transports can be considered in these bio-mimicking microfludic devices:• Transport of particles or cells by flow in the bulk• Passage of particles or cells through endothelial or epithelial barriers.The first typically involves advection and diffusion, as well as adhesion, adsorp-tion and desorption [12]. For the second, there are two main mechanisms for cellsto pass through endothelial or epithelial layers: (i) amoeboid locomotion and (ii)mesenchymal locomotion. During amoeboid migration, cells such as neutrophilsand lymphocytes squeeze their body to traverse pores while in mesenchymal mi-gration, cells such as cancer cells use traction force at the sites of focal adhesions(FAs) to relocate. The amoeboid mode of migration is characterized by strongfront-rear polarity, weak adhesions to the surrounding and fast locomotion of thecell ( ∼10 µm/min) whereas the mesenchymal mode is defined by slow locomo-tion ( >1 µm/min), weak polarization and strong integrin mediated adhesions ofthe cells to the extracellular matrix [13, 14]. In this research we focused on threecase studies related to different kinds of transports in bio-mimicking microfluidicdevices:• Submicron particle transport and deposition in the alveolar channel of aLOAC device• Mesenchymal transmigration of cancer cells through endothelial monolayerin a µVNs device.• Amoeboid transmigaration of leukocytes through endothelial monolayer invascular channel of a LOAC device.21.1 Transport and deposition of submicron particles in aLOAC deviceThe rapid development of technology is extending the use of submicron particles(SPs) especially nanoparticles (1 up to 500 nm in diameter [15]) to ever-expandingnew fields of applications (e.g., biomedicine, agriculture, food, renewable energy,electronics, and cosmetics) and is accelerating their release to the environment,posing serious risks to human health [16–18]. The human body can interact withnanoparticles via skin contact, inhalation, or ingestion [16, 17]. Although somedefense mechanisms exist, the alveolar tissue is the most vulnerable contact sitebecause it is not as well protected against environmental damages as the skin andgastrointestinal tract [16, 18]. In particular, the COVID-19 pandemic is caused byparticles of the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2)being inhaled into the lower respiratory tract to initiate active infection [19]. Othersources of air-borne SPs include cigarette and marijuana smoke, industrial and oc-cupational exposure, and indoor and outdoor air pollution [16, 20–23]. Exposure toenvironmental SPs can lead to the development and progression of infectious dis-eases such as COVID-19, as well as other respiratory diseases such as asthma andchronic obstructive pulmonary disease [19, 24–27]. Hence, the study of air-borneSPs and their effects on human health continues to be an active area of research.Much of our understanding of particle transport and deposition in the airwaysand lungs stem from pharmacological studies [28]. Apart from in vivo, ex vivoand in vitro experimentation [29–31], there have also been numerous computersimulations of the transport and deposition process, using simplified [32, 33] orCT-scanned geometries of the tracheobronchial system [12, 34, 35]. The focus ofsuch studies is the transport and deposition of SPs in a certain portion of the air-way or over the large-scale geometry of the whole lung. These studies have offeredinsights into how the flow of air affects the deposition in macroscopic geometricfeatures of the lungs (e.g., recirculation zones and bifurcations) [32]. In contrast,there has been little study of SP deposition further downstream on alveolar ep-ithelia, which happens at a much smaller length scale and in a more varied andcomplex geometry that cannot be easily captured by CT scan [36]. A mechanisticunderstanding of SP deposition and particulate/microbe-alveolar interaction and its3downstream inflammation reaction in lung diseases [37] calls for a more detailedlocal model.For this purpose, the microfluidic lung-on-a-chip (LOAC) device provides anideal platform [1, 38]. The LOAC is among the first “organs-on-chips”, engineeredmicro-physiological systems that aim to recapitulate certain aspects of the real or-gan for disease modeling or drug development applications [6, 39–42]. A par-ticularly successful LOAC design [1, 9] features an alveolar chamber separatedfrom a microvascular channel by a perforated elastic membrane. Different fromthe traditional monolayer cultures and co-culture systems with static air and fluidincubation, the LOAC design allows controlled air and liquid flow through the twochannels, and the membrane can be cyclically stretched to a strain comparableto physiological levels during normal breathing. This LOAC model reproducesclosely in vivo conditions of the alveoli and allows well-controlled tests for SPtransport and deposition. More recent LOAC designs have striven to capture otheraspects of the human airways [43–46].Despite these advances, wide-spread adoption of the LOAC device (and otherorgan chips) has not occurred [38]. For one, there has been a lack of communi-cation between the device researchers and the end users to define the barriers toimplementation. There has also been little quantitative data on the transport anddeposition of SPs in LOAC. Most importantly, these devices have been designedmostly by trial and error, with little theoretical guidance and high failure rates[1, 9]. As a result, low intra-laboratory and inter-laboratory reproducibility hasbeen a challenge for clinical translation of these devices.Using LOAC as a surrogate for the human alveolus, we have developed com-puter simulations to examine the transport and deposition of SPs (a) under differentexercise and breath-holding patterns, and (b) for a range of particle sizes with vary-ing relative importance of Brownian and gravitational forces. Our objective is tounderstand the transport of SPs in an LOAC device and show that in silico exper-iments can inform and potentially accelerate the design and application of suchdevices for analyzing particulate/microbe-alveolar interaction. We have based oursimulations on the well-known design of Huh et al. [1], which features a relativelysimple but physiologically relevant geometry.41.2 Mesenchymal transmigration of cancer cellsRecent development in µVNs devices has allowed in vitro reproduction and highresolution visualization of the transmigration. Chen et al. [5, 47] developed assaysof a highly realistic vascular network on a perfused microfluidic chip that shed lighton the mechanisms and events during mesenchymal transmigration of cancer cells.Transmigration of cancer cells through endothelial layer is a key step in metastasisof cancer. Metastasis is a cascade of processes in which tumor cells (TCs) firstexfoliate from the primary tumor, and invade the surrounding extracellular matrix(ECM) until they reach a nearby blood vessel. Through a process known as in-travasation, a TC breaches the blood vessel wall to enter the blood circulation, thusbecoming a circulating tumor cell (CTC) [48]. After being carried to a target tissueor organ, the CTC extravasates from the blood vessel. During extravasation, a TCforms protrusive structures that extend through the endothelium and basal mem-brane of the blood vessel and lead to mesenchymal transmigration of the TC. It isfollowed by invasion of the tissue outside, and establishment of a new colony inthe target site. Although both intravasation and extravasation entail breaching theendothelial barrier of the blood vessel, the two differ significantly in the signalingpathways and physical process [49–51]. We will focus only on extravasation in thisresearch.TCs may extravasate through two distinct mechanisms: paracellular transmi-gration between endothelial cells (ECs) and transcellular transmigration throughan EC [52]. Most cancer cells follow the paracellular route in vitro, althoughit is unclear what factors determine the preferred route of transmigration in vivo[50]. Henceforth we will focus on the paracellular route. Through decades offocused research, much has been learned about the biochemical signaling that con-trols extravasation. For example, the adhesion to ECs is known to be controlledby cell-surface proteins such as CD44, MUC1, E-selectin, ICAM-1, VCAM-1 andintegrins [52, 53]. The adhesion triggers remodeling of the cytoskeleton of theTC and formation of protrusions, through the CDC42/Rac1 pathway, activating itfor deformation and transmigration [54]. Finally, the TC also signals to the ECs,causing them to soften and retract from each other and disrupting the EC junctions.This involves, among other pathways, the phosphorylation and disassembly of the5Active integrinsEndothelial cells Extracellular matrixTumor cellStress fibersFocal-adhesion proteinsFigure 1.1: An invadopodium-based mechanism for extravasation proposedby Chen et al. [2]. Inside the TC, actomyosin assembles into stressfibers that propel a protrusion between the ECs. Adhesion molecules,i.e. integrins, anchor the tips of the protrusion onto laminin of the ECM.The focal-adhesion proteins include tensin, talin and vinculin amongothers. Actomyosin contraction in the stress fibers then pulls the TCpast the constriction between ECs.VE-cadherin–β -catenin complex [55, 56], with the result of opening the EC junc-tion to facilitate TC transmigration. A more detailed description of the signalingpathways can be found in a review by Reymond et al. [50].In contrast to our knowledge of the biochemical signaling, much less is knownabout the mechanical forces involved in extravasation. By the experimental obser-vation through the µVNs, TCs are seen to extend protrusions through the junctionsbetween ECs, which then adhere to the ECM outside. Antagonizing a key integrinβ1 in the TCs causes them to remain round and fail to protrude in-between theECs, and extravasation is abolished [2]. Based on these observations, Chen et al.[2, 5] have hypothesized the following scenario for paracellular transmigration oftumor cells (Fig. 1.1). First, the TC extends a protrusion through the junction be-tween ECs. As the protrusion grows in size, it branches out at the front. As its tipsreach the basement membrane outside the endothelium, activated integrins helpform secure focal adhesion sites. As more of the TC body moves out through theEC junction, the relatively rigid nucleus is left behind inside the spherical-shapedmembrane. Finally, contraction of the actomyosin in the stress fibers (SFs) pulls thenucleus through the junction to complete the extravasation. More recently, Abidine6et al. [57] reported high-resolution images of the cancer cell during transmigration,and identified rapid reorganization of TC actin below the nucleus and close to theEC junction. This seems to support the above hypothesis, especially the idea ofstress fibers forming inside the TC protrusion.This picture for extravasation attributes the driving force to contraction ofthe stress fibers inside the “filopodium-like protrusions” extended from the TC[50, 58], termed invadopodia hereafter. But it also raises further questions. Aretypical stress fibers capable of generating the force necessary for overcoming theresistance to the passage? How do the size of the EC opening and the rigidityof the ECM affect the balance between the driving force and the resistance? Caoet al. [59] built a mechanical model on the transmigration of a hyperelastic TCnucleus through small constrictions. By specifying a constant force that pulls onthe nucleus, an elastic equilibrium is computed where elastic resistance due to thenuclear and endothelial deformations balances the pulling force. The main result isa critical pulling force for the passage, which for the geometric and mechanical pa-rameters adopted amounts to 38 nN. Because this is essentially a static calculation,it does not give a dynamic picture of the transmigration. In particular, althoughthe actomyosin apparatus is modeled as under biochemical control and responsiveto external deformation [60], the static nature of the computation implies that thisforce is essentially prescribed; the biochemical factors do not participate in the me-chanical equilibrium nor influence the final result. Besides, the cell membrane andcytosol of the TC are neglected, and the nucleus is assumed to compress the EC indirect solid-solid contact.1.3 Amoeboid transmigration of leukocytes andtenertaxisThe immune response requires recruitment of leukocytes to defend the body againstforeign microorganisms such as bacteria and viruses. Leukocytes recruitment in-volves several consecutive events: First they form weak adhesion and roll onthe endothelium surface. Then they bind firmly on the endothelial surface, passthrough the endothelium in a process called diapedesis, and breach the basementmembrane. Finally, they move toward the chemotactic stimulus in the interstitial7tissue [61]. Therefore, reaction to infections requires frequent crossing of leuko-cytes through layers of endothelial cells (ECs). Leukocytes can transmigrate eitherdirectly through the body of an individual EC (transcellular path) or through thejunction between ECs (paracellular path) [62]. It is still unclear what determinesthe path of transmigration.To address this question, Martinelli et al. [63] conducted several experimentsto examine the correlation between junctional integrity and the route of diapedesis.They did in-vitro experiments of diapedesis using different endothelial cells suchas rat brain endothelium and rat heart endothelium. The brain endothelium hasstronger junctional integrity than heart endothelium. They saw that transcellulardiapedesis is much higher than paracellular diapedesis in brain endothelium. Theyalso enhanced or disrupted endothelial junction using pharmacological agents andhormones. They noticed that disrupting junctional integrity led to a remarkableincrease (about two folds) in paracellular diapedesis accompanied by decrease intranscellular transmigration. They also used a shear flow on the endothelium asa mechanical modifying agent to promote the junctional strength and remodel-ing of cytoskeleton. This alteration causes a significant increase in transcellulardiapedesis. They concluded that strong junctional integrity is correlated with dom-inant transcellular route of transmigration, and the EC junction tightness and localstiffness are the major determinants of route of diapedesis. As a result, they hypoth-esized that leukocytes choose a path with the least mechanical resistance during thepassage [63].In this case study we aim to test the least-resistant route hypothesis. It wasshown that the elastic modulus of endothelial layer varies spatially. The elas-tic modulus is the highest on the top of the nucleus and it is the lowest nearthe edges[64]. Furthermore, the mechanical resistance of the endothelial junctionduring diapedessis depends on strength of intercellular junctional integrity and itstightness [65]. Thus, finding the least resistance route is basically a trade off be-tween mechanical resistance due to the deformation of the EC body and openingthe intercellular junction as a result of the extending of ILPs. Given these in vivomeasurements of the protrusion forces and EC resistances, can the leukocyte real-ize paracellular and transcelluar migrations at different locations? Will this choicebe consistent with the tenertaxis hypothesis? Will the hypothesis work in cases8whether the stiffness profile of the endothelium or junctional integrity has beenmodified by drugs or by mechanical perturbations. These are the questions that westrive to answer through modeling and computation.It was shown that leukocytes sense the local stiffness and also the strength ofjunctional integrity by extending and retracting invadosome-like protrusions (ILPs)in the ventral surface of the endothelial layer [62]. For lymphocytes and mono-cytes, live cell fluorescens imaging showed that they form between 10 to 100 pro-trusions on the surface Ecs before diapedesis. The diameter of ILPs can rangefrom 0.1 µm to 2 µm and their depth also varies between 0.1∼2 µm .The lifetimeof protrusions is between tens of seconds to several minutes and most of them re-tract after 10∼30 seconds. [62, 66] It was shown by Labernadie et al. [67] that thepolymerization of actin filaments in the core of ILPs pushes the membrane out andgenerates protrusive force on the surface of ECs. This protrusive force ranges from∼ 1 nN to 10 nN depending on the size of protrusions [68, 69].9Chapter 2MethodologyThis thesis will contain three computational projects that use a set of mathematicalformulations and computational tools. This chapter will provide an overview ofthese. Each problem will also have its own particular setup and techniques, andthese will be deferred to the following chapters that are devoted to each problem inturn.For the purpose of computational study of particle transport in the LOAC de-vice we require the simulation of fluid flow in the channel and particle tracking anddeposition. Eukaryotic cells are composed of intracellular fluid and solid compo-nents such as the cell membrane and the nucleus. Thus, to achieve the numericalstudy of the cell-cell interactions during transmigration, we mainly deal with cel-lular chemomechanics modeling as well as simulation of fluid-solid interactions(FSI). To test the tenertaxis hypothesis, we will mostly need computation of elasticdeformations. All computations are done by finite elements using the COMSOLframework. Therefore, in this chapter we represent the required materials to modelfluid flow and particle tracking and deposition, solid deformation and fluid-solidcoupling, chemomechincs of SFs and FAs and finally the numerical implementa-tion.102.1 Fluid flow simulationUnder physiologic conditions inside a cell and also typical designs and perfusionconditions of LOACs [1, 33], the fluid flow is laminar with a Reynolds numberup to several hundreds. Therefore, we treat the air in the LOAC channel and thecytosol inside migrating cells, as viscous Newtonian fluids, and model their flowsby the Navier-Stokes equations:∇ · v = 0, (2.1)ρ(∂v∂ t+ v ·∇v)=−∇p+µ∇2v, (2.2)where p is pressure, v is the velocity vector, ρ is the density of the fluid and µ isa constant fluid dynamic viscosity. For the airflow in the LOAC, we have com-puted steady-state as well as pulsatile flows that reflect different breathing patterns(e.g., breathing at normal frequency, fast breathing typical of physical exercise,and breath-holding patterns that mimic cigarette and marijuana smoking). The fre-quency and pulsatile flow rate are tuned to physiological data and experimentaldesigns in the literature.2.2 Modeling particle transport and deposition2.2.1 Particle tracking using Eulerian approachFor finer SPs of negligible inertia (diameter d < 100 nm) [70], we adopt an Eulerianapproach that does not account for individual particles but represents the particulatephase by a concentration field c(r, t) that obeys a convection-diffusion equation tobe solved together with the Navier-Stokes equation for the airflow:∂c∂ t+ v ·∇c = D∇2c, (2.3)where D is the diffusivity of the particles. In our simulations the volume fractionof solid particles is so low as not to affect the air flow (see Table. 3.1).112.2.2 Particles tracking using Lagrangian approachFor larger particles, it becomes inaccurate to assume that they will follow the air-flow perfectly, and an explicit particle tracking scheme will be adopted followingearlier studies [33, 36, 70, 71]. For each particle, a Langevin equation is writtenout that includes gravity, a drag force approximated by a drag coefficient and aBrownian force:mdudt=pi8ρd2|v−u|(v−u) ·CD(Re)+m(1− ρρd)g+ fB, (2.4)where m = pi6ρdd3 is the mass of particle of density ρd and diameter d, u is itsvelocity, and CD is a drag coefficient [72] that depends on the particle Reynoldsnumber Re = ρ|v−u|d/µ . The Brownian force is written as [71, 73]fB = G√6piµd kBT∆t, (2.5)with G being a vector whose components are independent Gaussian random vari-ables of zero mean and unit variance, kB is the Boltzmann constant, T the absolutetemperature, and ∆t the time step of the time integration.2.2.3 Adsorption modelsIn the Eulerian modeling for the particulate phase, we have tested three kineticmodels for the surface adsorption and desorption. The simplest is a rapid uptakemodel, which assumes that once a particle contacts the substrate, it is instanta-neously adsorbed, with no desorption or surface saturation. This amounts to posingc= 0 as a boundary condition for the bulk air in contact with the substrate. To allowfinite adsorption and desorption rates, we adopt the Langmuir and Frumkin mod-els. The well-known Langmuir model relates adsorption to the bulk concentrationas well as surface coverage [74]:dcsdt= kadsc(Γs− cs)− kdescs, (2.6)where kads and kdes are the adsorption and desorption rate constants, respectively.c is the bulk concentration of particles at the substrate (y = 0), cs is the areal con-12centration of deposited particles, and Γs is the maximum areal concentration onthe surface based on maximum available sites for deposition. The ratio θ = cs/Γsis identified as the fraction of occupied sites. The Frumkin model generalizes theLangmuir model to account for attractive or repulsive interaction among adsorbedparticles in a monolayer on the substrate [74]:dcsdt= kadsce(−α csΓs ) (Γs− cs)− kdescs, (2.7)where α is the interaction parameter, and a positive α represents repulsive inter-action among the absorbed particles. Setting α to zero will recover the Langmuirmodel. The outcomes of different adsorption models will be compared and con-trasted. For large particles using Lagrangian tracking, we follow previous studiesin assuming rapid uptake on the substrate such that a particle is instantly adsorbedupon collision, with no desorption or saturation [33, 36, 70].2.3 Solid hyperelasticity simulationVarious elastic components in our simulations such as solid components of theLOAC device that made of Polydimethylsiloxane (PDMS) have a nonlinear stress-strain behaviour. Moreover, strain-stiffening behaviour is the hallmark of biologi-cal materials such as the TC cell membrane, the TC nucleus, the EC, and the ECM[59, 75]. Therefore, the elastic components are modeled as a hyperelastic materialobeying the Neo-Hookean constitutive equation:σ = GJ−35(FFT − I3tr(FFT))−K(J−1)I (2.8)where Fi j = ∂xi/∂X j is the deformation gradient tensor, with X and x being the un-deformed and current positions of a material point, and J = detF . The coefficientsG and K are the shear and bulk modulus, respectively, connected via the Poissonratio ν : K = 2G31+ν1−2ν . Finally, in all the hyperelastic bodies, the governing equationfor solving for mechanical deformation is given by:∇ ·σ = 0. (2.9)132.4 Fluid-Solid couplingThe elastic deformation of the LOAC during breathing-like stretching in the chan-nel and the transmigration of tumor cells both involve fluid-solid coupling. Thesolid is described in a Lagrangian frame while the governing equations of fluidflow usually are solved in a fixed domain defined in an Eulerian frame of refer-ence. Using Eulerian approach is straightforward for undeformed fluid domain;however, in FSI problems due to the deformation of solid, the fluid domain altersits shape and the interfaces of fluid and solid are moving boundaries. One of themost common techniques to handle moving boundaries in FSI problems is the ar-bitrary Lagrangian-Eulerian (ALE) method. The ALE formulation considers anarbitrary moving coordinate in addition to the Lagrangian and Eulerian frame ofreferences. In FSI problems, fluid grids at the fluid-solid interface relocate basedon Lagrangian movement of material points [76, 77]. Moreover, large deformationof solid may cause distortion and inverted mesh elements in fluid domain. Remesh-ing is a common solution to overcome this issue [76]. Coupling of the fluid-solidis reached by setting the fluid-solid interface boundary conditions [78]:1. Kinematic condition: velocity of the fluid and solid are matched at the inter-face.2. Dynamic condition: shear and normal stresses are matched at the interface.3. Geometric condition: fluid domain and solid domain are always in directcontact and there is no gap between these domains at the interfaces.The solution of FSI problems generally can be achieved by a monolithic or a par-titioned approach. In the monolithic approach, coupled system of fluid and solidequations are solved together and interface conditions are handled implicitly. Inthe partitioned approach, fluid and solid equations are solved separately and theinterface conditions are satisfied explicitly using iteration loop between solvers.Although monolithic methods need large RAM memory capacity, they have nostability problems[79]. We adopted monolithic approach in our simulations andthe fully coupled system of equations were solved.142.5 Chemomechanical modeling of stress fibers and focaladhesionsStress fibers are the contractile elements in the cells that anchored to focal adhe-sions and generate the contractile force for relocation of the nucleus during trans-migration [2].The SF contraction promotes the stabilization and growth of the FA [80]. To-gether, the assembly exerts a force on the top of the nucleus and pulls the nucleusand the whole cell through the narrow constriction against the elastic resistance dueto the deformation of the nucleus and the endothelium. To represent the coupleddynamics of the SF and FA, we adapt the models of Deshpande et al. for the SFcontractility and FA growth [81, 82]. In the original model, the FA is being shearedwith force and strain tangential to the substrate. In our setup, FA is being stretchedby the SF, with force and strain perpendicular to the substrate. The adapted modelcan predict stiffness-dependent actomyosin force generation and FA formation dueto the normal force on integrins.The state of the stress fibers is represented by the fraction of activated myosinmotors η on the SF, which ranges from 0 to 1. The kinetics of η depends on acti-vation by an external signal (e.g., an influx of Ca2+ ions [83]) and force-regulateddeactivation:dηdt= (1−η)kae− tθ − kd(η− ττmax), (2.10)where ka and kd are the kinetic rates for myosin activation and deactivation, e−t/θrepresents an exponentially decaying calcium signal [81, 83], and the deactivationis suppressed by the tension τ in a well-documented positive feedback [84, 85],τmax being the maximum tension when the myosin is fully activated (η = 1). Sincethe SF is under tension as a result of actomyosin contraction, the terms “tension”and “contractile force” are synonyms in the current context. The kinetic equationabove is complemented by a constitutive equation for the SF, relating the tensionτ by a Hill-like function to the contraction or extension rate of the stress fibers v f ,15similar to force generation in muscle cells [81, 83]:τ =0, v fv0 <−ηkv,ητmax(1+ kvηv fv0), − ηkv ≤v fv0≤ 0,ητmax,v fv0> 0,(2.11)where v f < 0 signifies contraction, v0 is the zero-load speed of a myosin motor[86] and kv is the fractional contraction rate at which the stress fibers can no longersustain a tension [83]. The FA dynamics are governed by the conversion from low-affinity integrins to high-affinity integrins that can bond to the ECM. The integrin-ligand tension Fb shifts the thermodynamic equilibrium between the two in favorof the high-affinity integrins and FA growth [82, 87, 88]:kBT lnξHξL= µL−µH −Φ+Fb db, (2.12)where kB is the Boltzmann constant, T is temperature, and ξH and ξL are the high-and low-affinity integrins per unit area, with the total number of integrins ξH +ξL =ξ0 being conserved. µL and µH are the reference chemical potentials for the low-and high-affinity integrins. Φ, the stretch energy stored in each integrin bond, isa piecewise quadratic function of the bond extension db, and Fb = ∂Φ/∂db is thetension in each bond, which adds up to yield the total tension τ in the SF:τ = ξHFbAb, (2.13)where Ab is the area of the FA. The stretch energy Φ is expressed as a piecewisequadratic potential in the integrin bond length db, and the force Fb on each high-16affinity integrin is piecewise linear:Φ=12λd2b , db ≤ dm,−λd2m +2λdmdb− 12λd2b , dm < db ≤ 2dm,λd2m, db > 2dm,(2.14)Fb =λdb, db ≤ dm,2λdm−λdb, dm < db ≤ 2dm,0, db > 2dm.(2.15)where λ is the stiffness of the bond and dm is the maximum bond length.17Chapter 3Case study I: Transport anddeposition of submicron particlesin a LOAC deviceIn this chapter1 we define our problem setup for the simulation of transport anddeposition of submicron particles (SPs) in a LOAC device and then represent ourresults and discussion. Compared with the existing literature of large-scale simula-tions of the human lung and airways, our work introduces the following novelties:• Adsorption-desorption kinetics. Previous work predicted the depositionefficiency, the ratio of particulates captured to the total amount released [33],solely based on the incidence of particle impingement, tacitly assuming in-stantaneous adsorption [70]. We introduce adsorption-desorption kinetics bythe Langmuir and Frumkin models to allow saturation on the solid surfaceand particle-particle interactions.• Physiologic breathing patterns. Earlier work distinguished the breath inresting and active states via the Reynolds numbers [33], with limited atten-tion to the pulsatile nature of tidal breathing [32, 36, 89]. We will study1The results of this chapter were published in: ”Arefi, S.M. Amin , Cheng Wei Tony Yang, DonD. Sin, and James J. Feng. ”Simulation of nanoparticle transport and adsorption in a microfluidiclung-on-a-chip device.” Biomicrofluidics 14, no. 4 (2020): 044117.”18pulsatile flows corresponding to normal tidal breathing in a state of restand intensified breathing—in terms of air volume per cycle and frequency—corresponding to physical exercising. Furthermore, we will examine the ef-fect of “puff profiles” of tobacco and marijuana smoking [90–94].• Effect of particle sizes and gravity. We represent finer particles by a con-tinuum Eulerian model, where particle size is reflected by the diffusivity.For larger particles, we track them using a discrete Lagrangian model sub-ject to gravitational settling and Brownian motion. Thus we will be able todelineate the deposition of SPs across different sizes.3.1 Problem setupModeling of air-borne particles depends on the particle size and concentration. Inmost cases, the concentration of particulates is low such that the flow of air isminimally affected by the particles [95, 96]. This assumption results in the so-called one-way coupling in which the air flow affects the solid particles, but notvice versa. In the context of using LOAC for studying SPs (e.g., virus particles,ambient pollutants and aerosols for drug delivery), the particle size can range fromnanometers to a micron. For example, the SARS-CoV-2 virus is 60–140 nm indiameter with 9–12 nm spikes [19]. Cigarette smoke contains particulates with sizeranging in the hundreds of nanometers, as well as some particles in the ultrafineregion (diameter d < 100 nm) [97, 98]. To cover a wide range of particle sizes,therefore, we will treat the particulate phase using two separate methods: Eulerianapproach for ultrafine particles and Lagrangian approach for coarser SPs. TheEulerian model is used to investigate the effects of exercise and breath-holdingpatterns. The Lagrangian model is used to investigate the effects of Brownianmotion, gravitational forces and spatial orientation of the device.3.1.1 Geometric setupThe computational domain is schematically illustrated in Fig. 3.1. This study fo-cuses on the air-particulate two-phase flow through the upper channel; blood flowin the lower channel is ignored. The particles are non-hygroscopic, and can deposit19Figure 3.1: Schematic representation of the computational domain, modeledafter the LOAC design of Huh et al. [1]. The upper channel (in blue)is separated from the bottom channel (in pink) by an elastic membrane(in magenta). The top mimics the alveolus through which air flows andthe bottom mimics the blood vessel. The two side chambers are for thepurpose of imposing vacuum to stretch the elastic membrane. The airflows along the −z direction, and θ denotes the angle between the airflow and gravity g. Most of the computations have θ = 90◦, but θ canbe varied to allow different inclinations of the device relative to gravity.and adsorb onto the membrane that forms the lower wall of the upper channel, ac-cording to a suitable adsorption kinetics. The lower wall represents the alveolarepithelium, and will be called the “substrate” hereafter. The other walls do notcarry epithelial cells and thus do not adsorb the particles.3.1.2 Governing equationsIn this section we briefly mention to the governing equations that required for thesimulations in this chapter. Although these equations were described in chapter 2,they are represented in here to make this chapter more self-contained.20Modeling fluid-solid interaction in the LOAC deviceUnder physiologic conditions and typical designs and perfusion conditions of LOACs[1, 33], the Reynolds number is less than one so the airflow is laminar and it is gov-erned by the standard Navier-Stokes equation:∇ · v = 0, (3.1)ρ(∂v∂ t+ v ·∇v)=−∇p+µ∇2v, (3.2)where v and p are the velocity and pressure of the air flow, and ρ and µ are theair density and viscosity, respectively. Due to the vacuum pressure in the sidechambers of the LOAC device, air and blood channels and the membrane deform.We treated the solid component of the LOAC which is made of PDMS as a Neo-Hookean hyperelstic material. The constitutive equation for this material isσ = GJ−35(FFT − I3tr(FFT))−K(J−1)I (3.3)where Fi j = ∂xi/∂X j is the deformation gradient tensor, with X and x being the un-deformed and current positions of a material point, and J = detF . The coefficientsG and K are the shear and bulk modulus, respectively, connected via the Poissonratio ν : K = 2G31+ν1−2ν . Finally, the governing equation of solid deformation is givenby:∇ ·σ = 0. (3.4)For PDMS the elastic modulus is about EPDMS = 1MPa and the Poisson’s ratio isνPDMS = 0.49 [99]. The air flow and the solid deformation are coupled throughfluid-solid interaction formulation stated in 2.4. In our LOAC model, the mem-brane stretching amounts only to a periodic change in the substrate area, up to7%. The surface deposition changes accordingly, with a maximum variation ofabout 5.4%. Since the effect is minor and the computational cost of fluid-structureinteraction is high, we have omitted membrane stretching and and the walls defor-mation for most of the simulations, and assumed that all the solid walls are rigid inthe LOAC.We have computed steady-state as well as pulsatile flows in the air channel21with rigid walls that reflect different breathing patterns (e.g., breathing at normalfrequency, fast breathing typical of physical exercise, and breath-holding patternsthat mimic cigarette and marijuana smoking). The frequency and pulsatile flowrate are tuned to physiological data and experimental designs in the literature.Modeling fine particles using Eulerian approachFor finer SPs of negligible inertia (diameter d < 100 nm) [70], we adopt an Eulerianapproach that does not account for individual particles but represents the particulatephase by a concentration field c(x,y,z, t) that obeys a convection-diffusion equationto be solved together with the Navier-Stokes equation for the airflow:∂c∂ t+ v ·∇c = D∇2c, (3.5)where D is the diffusivity of the particles. Typically the solid fraction is so lowas not to affect the air flow (see Table 3.1). It is worth mentioning that the Stokesnumber for 100 nm particles is much less than 0.1 and the particles have to followthe stream.Modeling coarser particles using Lagrangian approachFor larger particles, it becomes inaccurate to assume that they will follow the air-flow perfectly, and an explicit particle tracking scheme will be adopted followingearlier studies [33, 36, 70, 71]. For each particle, a Langevin equation is writtenout that includes gravity, buoyancy, a drag force approximated by a drag coefficientand a Brownian force:mdudt=pi8ρd2|v−u|(v−u) ·CD(Re)+m(1− ρρd)g+ fB, (3.6)where m = pi6ρdd3 is the mass of particle of density ρd and diameter d, u is itsvelocity, and CD = 24/Re+ 3.6Re−0.313 is a drag coefficient [72] that depends onthe particle Reynolds number Re = ρ|v−u|d/µ . The Brownian force is written as[71, 73]fB = G√6piµd kBT∆t, (3.7)22with G being a vector whose components are independent Gaussian random vari-ables of zero mean and unit variance, kB is the Boltzmann constant, T the absolutetemperature, and ∆t the time step of the time integration. We have confirmed thatthe Lagrangian scheme predicts two-dimensional random walk of particles thatare consistent with the continuum solution with a diffusivity given by the Stokes-Einstein relation: D = kBT/(3piµd). In the Lagrangian simulations, we also as-sume that all the solid walls are rigid in the LOAC.Adsorption modelsIn the Eulerian modeling for the particulate phase, we have tested three kineticmodels for the surface adsorption and desorption. The simplest is a rapid uptakemodel, which assumes that once a particle contacts the substrate, it is instanta-neously adsorbed, with no desorption or surface saturation. This amounts to posingc= 0 as a boundary condition for the bulk air in contact with the substrate. To allowfinite adsorption and desorption rates, we adopt the Langmuir and Frumkin mod-els. The well-known Langmuir model relates adsorption to the bulk concentrationas well as surface coverage [74]:dcsdt= kadsc(Γs− cs)− kdescs, (3.8)where kads and kdes are the adsorption and desorption rate constants, respectively.c is the bulk concentration of particles at the substrate (y = 0), cs is the areal con-centration of deposited particles, and Γs is the maximum areal concentration onthe surface based on maximum available sites for deposition. The ratio θ = cs/Γsis identified as the fraction of occupied sites. The Frumkin model generalizes theLangmuir model to account for attractive or repulsive interaction among adsorbedparticles in a monolayer on the substrate [74]:dcsdt= kadsce(−α csΓs ) (Γs− cs)− kdescs, (3.9)where α is the interaction parameter, and a positive α represents repulsive inter-action among the absorbed particles. Setting α to zero will recover the Langmuirmodel. The outcomes of different adsorption models will be compared and con-23trasted. For large particles using Lagrangian tracking, we follow previous studiesin assuming rapid uptake on the substrate such that a particle is instantly adsorbedupon collision, with no desorption or saturation [33, 36, 70].3.1.3 Numerical setup(a)(b)Figure 3.2: (a) A typical finite-element mesh used for the fluid-structure in-teraction simulation that allows the membrane and all walls to be elas-tically deformable. (b) A typical mesh used for computing the flow andEulerian particle transport inside the air channel, with all walls treatedas rigid.We have used COMSOL Multiphysics® to compute the air flow and the SP trans-port. We deploy cuboid elements for solving the Navier-Stokes equations in 3D,with finer elements in regions of sharper gradient to ensure adequate resolution.Typical finite-element meshes are shown in Fig. 3.2. For simulations with rigidwalls, the CFD and particle-tracking modules of COMSOL are used, the latter in-corporating the particle inertia and gravitational forces directly.24Boundary and initial conditionsTo complete the statement of the mathematical problem, we pose the followingboundary and initial conditions. On the four side walls of the air-flow channel,we impose the no-slip boundary condition, with the possible exception of a slip onthe substrate due to a liquid lubricating film. The presence of a liquid film rich insurfactants atop the epithelium argues for a slip boundary condition for the airflow,and we will examine the effect of the slip velocity relative to the traditional no-slipboundary condition. We prescribe a uniform velocity v = V0 at the inlet and thestress-free condition at the outlet. Note that for simulating tidal breathing, the flowreversal requires a periodic change in the designation of the inlet and outlet. Moredetails will be given in the Results section.In the Eulerian treatment of the SPs, we typically impose a constant solid vol-ume fraction c = c0 at the inlet. An exception is in the exhalation phase of tidalbreathing, when we impose c = 0 at the inlet for reasons to be detailed in the Re-sults section. On the substrate (y = 0), we impose c = 0 for the rapid uptake model.With an adsorption model, the following balance in SP transport is posed:D∂c∂y∣∣∣∣y=0=dcsdt. (3.10)There is zero flux at the outlet and on the other walls. For Lagrangian particletracking, we release a large number of particles (typically 1000) at the inlet andtrace out their trajectories. As the particles are treated as points with no spatialdimension nor interactions among themselves, the trajectories do not interfere witheach other, and the results are equivalent to the compilation of many individualsimulations of single particles.3.1.4 Parameters estimationThe simulation requires a host of geometric and physical parameters, whose base-line values are tabulated in Table 3.1, along with justifications and sources. Forseveral of these, we have explored variations from the baseline over a range inparametric studies. We have evaluated these parameters from experimental mea-surements in the literature, or absent that, made estimations based on the most25relevant information available.Symbol Description Value SourcesH Air channel height 70 µm [1]W Air channel width 400 µm [1]L Air channel length 2 mm [1]d Particle diameter 100 nm [21, 97]D Diffusivity 6.8×10−10 m2 s−1 [100]c0 Volume concentration at inlet 5.24×10−6 EvaluatedΓs Maximum areal concentration 52.4 nm Evaluatedkads Adsorption rate 12600 s−1 Evaluatedkdes Desorption rate 40 s−1 EvaluatedV0 Uniform air velocity at inlet 0.337 mms−1 [100]ρ Air density (at 25°C) 1.18 kgm−3 [101]ρd Solid density 1180 kgm−3 [98]f Normal breathing frequency 15 min−1 [102]vs Slip velocity on membrane 1.37 µms−1 EvaluatedTable 3.1: Baseline values for the parameters used in our model. For thechannel length L, the experimental device of Huh et al. [1] has L > 1cm. In our model, L = 2 mm turns out to be sufficient for most of thesimulations, with the exception of the breathing patterns of Sec. 3.2.3(see discussions therein).Volume concentrationFor cigarette smoke, Rostami [12] reported a number concentration of 1010 particles/cm3.Taking an average particle of diameter d = 100 nm (volume of 5.24× 105 nm3),and noting the 1 cm3 = 1021 nm3, we can estimate the volume concentration ofnanoparticles in cigarette smoke asc0 = 1010×5.24×105/(1021) = 5.24×10−6. (3.11)26Maximum areal concentrationThe areal concentration of deposited particles cs is defined by the boundary condi-tion for the substrate:D∂c∂y∣∣∣∣y=0=dcsdt. (3.12)As c is the bulk volume concentration, cs is the amount of solid volume that is de-posited per unit surface area, with dimension of length. The maximum or saturatedareal concentration Γs is, therefore, the maximum of cs when the surface is entirelycovered by a monolayer of particles.Assuming that the particles are spheres of diameter d = 100 nm, and each occu-pies a square area of d2, we estimate the total number of particles in the monolayeras Ns = A/d2, A = 2000×400 µm2 being the entire area of the LOAC membrane.Therefore,Γs =pi6d3 · Ad2· 1A=pid6= 52.4nm. (3.13)Kinetic rate constantsWe have used experimental data of Heyder et al. [102, 103] for deposition in thehuman acinus to estimate the rate constants in Langmuir adsorption kinetics. Theamount of deposited particles depends on many factors, including air flow rate,bulk concentration of particles and particle size among others. In particular, thebulk concentration of 1010 particles/cm3 in cigarette smoke [100] is much higherthan that of 104 particles/cm3 in Heyder et al.’s experiment [103]. Thus, it wouldnot be reasonable to equate the amount of deposition in both cases. Instead, wehave assumed equal deposition fraction ε in both cases, such that the amount ofdeposition varies in proportion to the bulk concentration.Heyder et al. [103] reported a deposition fraction of ε = 0.21. To calculatethe number of deposited particles in the experiment, we assume that in a normalbreathing cycle, the aerosol of concentration 104 particles/cm3 fills the entire vol-ume of the acinus, from terminal bronchioles to alveolar sacs, generation 16–23,with a volume Va = 0.183 cm3 [104]. Given the deposition efficiency ε = 0.21, the27number of deposited particles in the experiment isNexp = 0.21×0.183×104 = 384. (3.14)Since the acinic area Aa = 23.5 cm2 [104] differs from the surface area in ourLOAC device, we need to convert the above into a surface density or a dimension-less fraction of occupied sites. Assuming that each particle occupies a square ofedge length that equals its diameter d = 100 nm, the maximum number of particlesthat can be accommodated on the acinic surface is Mexp = 23.5 cm2/104 nm2 =2.35× 1011. Thus, the experimental surface has a fraction of occupancy θexp =Nexp/Mexp = 1.63×10−9.As the bulk particulate concentration c0 in cigarette smoke is 106 times higherthan in the experiment of Heyder et al. [103], we expect the surface occupationfraction to be elevated by the same factor in our LOAC model:θmod = θexp×106 = 1.63×10−3. (3.15)Then we test ranges of the adsorption rate kads and desorption rate kdes with thegoal of producing this θmod value. First, we have found that kdes largely determinesthe saturation time for achieving a steady-state surface coverage in our simulation,and kdes = 40 s−1 produces saturation in less than 4 breathing cycles in accordancewith experimental observations [103]. Then we choose kads = 12600 s−1 such thatthe saturated surface coverageθsat =kadsc0kdes + kadsc0(3.16)equals the target value θmod = 1.63× 10−3, with c0 = 5.24× 10−6 being the bulkvolume density in the LOAC model.Slip velocityThe slip velocity on the substrate is based on the idea that the epithelial cells secretea surfactant-rich liquid layer. Consider a planar Poiseuille flow of air in a channelof height H = 70 µm, with a liquid layer of thickness 100 nm at the bottom [105].28Assuming the viscosity of water for the liquid [106] and a mean air velocity of V0 =0.337 mm/s, a stress balance on the interface gives a slip velocity vs = 1.37 µm/s.3.2 Results and discussionThe results of the simulations and their interpretations are reported in this section.To verify the accuracy of our numerical results, especially with respect to spatialand temporal resolution, we have conducted several validations of the flow field inthe LOAC geometry and the solution of advection-diffusion problems (Fig. 3.3).Also to ensure the accuracy of our discrete representation of the Brownian force(Eq. 3.7), we have refined the time step ∆t for Lagrangian particle tracking andexamined how the particle deposition efficiency ε converges with decreasing ∆t(Fig. 3.4). The test shown here corresponds to the data point for d = 100 nm(Brownian particle) in Fig. 3.11. As ∆t falls below 0.002 s, ε essentially converges.All Lagrangian results reported here have used a ∆t of 0.002 s or smaller.(a) (b)Figure 3.3: (a) shows the convergence of a velocity profile with increasingmesh refinement toward the analytical solution [3]. (b) shows the linein the air channel along which the velocity profile is taken.29Figure 3.4: Verification of time step resolution in the Brownian force calcu-lation3.2.1 Eulerian model: particle deposition with constantunidirectional in-flow(a) (b)Figure 3.5: (a) A snapshot of the LOAC at the time of its maximum defor-mation when the membrane between the two chambers stretches to 7%strain. The elastic modulus of the walls and the negative pressure in theside chamber are taken from the experiment [1]. The color contour inthe air chamber represents magnitude of the air velocity (left color bar,in m/s) whereas that inside the solid indicates the level of the von Misesstress (right color bar, in Pa). (b) The cyclic stretching of the membraneproduces a mild oscillation of amplitude 2.7% in the surface numberdensity of deposited nanoparticles, with an average slightly above thatfor a rigid membrane.A special feature of the experimental LOAC device is the stretchable elastic mem-brane that can extend and retract according to the breathing cycle. According tophysiologic data, the amplitude should be around 7%, which is what Huh et al. [1]30used. However, we find that such a stretching affects the surface deposition onlyslightly. In our LOAC model, the membrane stretching amounts only to a periodicchange in the substrate area, up to 7%. The surface deposition changes accord-ingly, with a maximum variation of about 5.4% Fig. 3.5. Since the effect is minorand the computational cost of fluid-structure interaction is high, we assumed thatall the walls of the air channel are rigid in the LOAC device. Moreover, We notethat in vivo and in vitro, the membrane stretching has other effects (e.g. openingpores and allowing silica particles and leukocytes to cross)[1]. Since we focus onparticle deposition only, we do not take these effects into account.Figure 3.6 shows the results from Eulerian simulations for fine SPs. Before thesimulation starts, the LOAC is filled with clean air. At t = 0, we impose a constantvelocity V0 = 0.337 mm/s at the inlet with a constant particulate volume concen-tration of c0 = 5.236×10−6 (cf.Table3.1). Hereafter, “concentration” refers to thebulk volume concentration of the SPs unless otherwise stated. As the particle-laden air displaces the clean air inside the LOAC, adsorption of particles occur onthe substrate. Figure 3.6(a) shows five snapshots of the evolving concentration dis-tribution on the mid-plane that cuts the air channel into symmetric halves. A profiledevelops with higher c in the middle due to the higher air velocity and advection.The adsorption at the bottom generates an asymmetry in the c profile at early times(t = 2 s). In time, surface adsorption and desorption reach an equilibrium, and thesubstrate can no longer take up more SPs. Thus the c profile becomes uniformat c = c0 roughly around t = 12 s. Figure 3.6(b) illustrates the same process bythe fraction of occupied sites (θ ) on the substrate. First the particles are depositedat the entrance of the channel until the fraction of occupied sites saturates locally.Note the relative deficit of adsorbed particles at the left and right edges of the plane.These are next to the side walls of the LOAC, with reduced air flow supplying fewernumber of particles. The front of saturation propagates downstream until the frac-tion of occupied sites reaches saturation. Note that the saturation level correspondsto θs = 1.63× 10−3, or a mere 0.16% coverage of the area of the substrate. Thispercentage is determined by equilibrium between adsorption and desorption in theLangmuir kinetics, and is consistent with experimental data [102, 103].The temporal evolution of surface coverage by deposited particles is depictedmore explicitly in Fig. 3.6(c,d). Figure 3.6(c) shows the average number density31(a)(b)(c) (d)Figure 3.6: Deposition of SPs under constant in-flow with the Langmuirmodel. (a) Time evolution of volume concentration distribution c inthe air channel shown on the mid-plane highlighted in the left figure.The height of the rectangles has been stretched by 4 times to facilitatevisualization. After about 12 seconds the channel is full of smoky air atthe constant particle volume concentration c0 = 5.24×10−6. (b) Tem-poral evolution of the distribution of the fraction of occupied sites θ onthe substrate. (c) Time evolution of the averaged areal number densityρs of deposited particles on the substrate. (d) Time evolution of thedeposition efficiency ε .ρs of deposited particles on the substrate, while Fig. 3.6(d) plots the temporal vari-32ation of the “deposition efficiency” ε , defined as the ratio between the number ofdeposited particles and the number of particles that have entered the LOAC up tothe present time [33]. The areal number density of particles ρs increases in timewith a sigmoidal shape. Initially the rate of increase is small since the particle-laden air has not reached much of the substrate. Upon equilibration, ρs saturatesat a level ρ0 = 0.164 µm−2, which corresponds roughly to a sparse coverage of16 SPs of diameter d = 100 nm per 100 µm2 of the substrate. The depositionefficiency ε increases first, and then declines toward 0 as deposition saturates. De-spite the sparse coverage on the substrate, ε reaches a peak value of 0.16, close toexperimental data [102, 103].Having described the general features of the deposition process, we explorethe effects of a few key factors and parameters next. The first is the three adsorp-tion/desorption models of SPs. The model used in Fig. 3.6 is the Langmuir modelof Eq. (3.8). Then we consider a simpler “rapid-uptake” model that correspondsto c = 0 as a boundary condition for the bulk concentration at the solid substrate.This resembles the “rapid reaction” limit in mass transfer [107], and is a reasonableapproximation in the limit of instantaneous adsorption of the particles followed byrapid transport through the epithelium so that no desorption occurs. Finally, wehave the Frumkin model that allows interactions among adsorbed particles on thesubstrate.Figure 3.7(a) compares the predictions of the three adsorption models for theamount of SP deposition. The average number density ρs plateaus for the Lang-muir and Frumkin models, but settles into an unbounded linear growth for therapid uptake model. Interestingly, for a “reasonable” interaction parameter α = 10suggested by literature [108–110], the Frumkin adsorption kinetics only results ina slight reduction in ρs relative to the Langmuir model, which is recovered withα = 0 in the Frumkin model. Due to the sparse surface coverage of SPs, theirrepulsive interaction will only have an observable effect at higher α values. Forexample, if we raise α by tenfold to 100, the steady-state ρs decreases by 13%.The second parameter to be varied is the diffusivity D of the particulates inair. This directly affects the particle flux toward the substrate through Fickiandiffusion. Figure 3.7(b) shows that with larger diffusivity D, the deposition on thesubstrate is initially faster, but falls below that of a lower D as time passes. This33(a) (b)(c)Figure 3.7: (a) Effect of the surface adsorption models on ρs(t), the averagedareal particle number density on the substrate. (b) Effect of diffusivityD on the number density ρs of deposited particles. (c) Effect of slipvelocity vs on the particle deposition on the substrate.crossover is more obvious for the larger D values, but is a general feature of thesurface deposition that stems from the spatial heterogeneity of particle deposition.A larger D produces faster initial deposition because of rapid diffusion in all threedimensions. First, the surface adsorption creates a deficit of particles in the air layernext to the substrate, and a larger D delivers particles more efficiently to that layerin the direction normal to the substrate. Second, streamwise diffusion also deliversparticles farther downstream, thus activating adsorption over longer sections of thesubstrate. Finally, spanwise diffusion also drives the particles toward the side walls,thus shrinking the low-deposition edges that would otherwise arise from ineffectiveadvection (see Fig. 3.6b). A natural consequence of the faster initial adsorptionis more rapid attainment of saturation over a wider area of the substrate. Thus,the saturation front travels more rapidly downstream for the higher D, shrinkingthe downstream portion of the substrate that is available for continued adsorption.This explains the earlier saturation of ρs for higher D and the crossover betweendifferent curves in Fig. 3.7(b).34Finally, we consider the effect of a slip velocity vs on the surface of the substratethat may arise from the presence of a thin film of lubricating liquid. The airwaysurface is covered by a surfactant solution, ranging in thickness from 1.8 µm inbronchioles to 0.1 µm in alveoli [105, 106]. Its viscosity is close to that of water[106]. The slip velocity is calculated by considering the shear flow of air atop theliquid layer, and turns out to be vs = 1.37 µm/s under physiologic conditions (seeTable 3.1). The presence of the slip facilitates the air flow near the substrate, whichin turn enhances the convection transport of the SPs to the near-wall region to beadsorbed. But for vs = 1.37 µm/s, the increase in deposition is minuscule andhardly visible in Fig. 3.7(c). In order to see a considerable difference, we testeda much larger slip velocity vs = 137 µm/s, 100 times the value estimated above.This does enhance particle deposition initially. But as the saturated steady state isapproached, the bulk solid concentration becomes uniform c = c0 everywhere inthe air. Therefore, the slip-enhanced advection ceases to play any role, and does notaffect the saturated surface density of deposited particles on the substrate, which isentirely determined by the Langmuir kinetics.3.2.2 Eulerian model: particle deposition in bidirectional pulsatileflowHaving established a baseline using constant in-flow, we now explore how tidalbreathing affects the deposition of SPs on the substrate. For normal breathing in aresting state, we assume equal duration of 2 s for inhalation and exhalation [72].We start the simulation with the air channel filled with clean air. At t = 0, particle-laden air flows into the chamber with a fixed velocity V0 and solid volume concen-tration c0 at the inlet. At t = 2 s, we instantaneously reverse the flow direction byprescribing V0 on the original outlet as well as a “clean air” condition c1 = 0, to beexplained below. As a result, the original inlet is now an outlet with the stress-freeboundary condition for the air flow and no-flux for the advection-diffusion of thesolid phase. At t = 4 s, we revert to the inhalation setup with V0 and c0 at theoriginal inlet, and the cycle repeats itself. We have typically simulated 4 cycles asthese seem sufficient to reveal all the key physics. Experimental measurements areusually for a similar number of cycles [102, 103].For the exhalation phase, the boundary condition for the solid concentration35(a)(b) (c)Figure 3.8: Deposition in bidirectional pulsatile flow. (a) Particle volumeconcentration c in the mid-plane (cf. Fig. 3.6a) at the end of the first 4inhalations (left) and 4 exhalations (right). The height of the rectangleshas been stretched by 4 times to facilitate visualization. (b) Averagedbulk concentration at the outlet, scaled by the inlet concentration c0, asa function of time during the first 4 cycles. (c) Variation of the averageparticle number density ρs adsorbed onto the substrate over the first 4cycles of breathing.c1 at the new inlet, i.e., the original outlet, requires special attention. Numericalexperimentation shows that for the V0 and c0 values used for the inhalation, hardlyany solid particles will have passed the exit during the first 4 breathing cycles.Figure 3.8(a) shows the bulk SP concentration profiles on the mid-plane at the endof the first 4 inhalations and exhalations. At the end of the first inhalation (t = 2s), essentially no particle has reached the exit of the LOAC. Note that the mid-plane enjoys the greatest flow velocity inside the channel, and advection should beweaker along other parallel planes. The same conclusion is borne out by averaging36the solid concentration at the exit, which amounts to a negligible 2.39×10−8c0 att = 2 s (Fig. 3.8b). Each subsequent inhalation advances the solids further towardthe exit, but even at the end of the fourth inhalation (t = 14 s), the average solidconcentration at the exit is a mere 0.5% of c0 (Fig. 3.8b). Therefore, if we limitourselves to the first few breathing cycles, posing an “inlet” condition of c1 = 0for the exhalation phase is reasonable, as the air downstream of the LOAC (ordeeper inside the alveolar sac) will contain essentially no solids. This assumptionbecomes questionable for faster breathing during physical exercise or for longersimulations involving many cycles. We will adopt an extended channel for suchcases in Sec. 3.2.3.Figure 3.8(c) shows the temporal evolution of the adsorbed particle numberdensity ρs over 4 breathing cycles. The initial inhalation produces a rapid increasein surface adsorption, following the same portion of the curve in Fig. 3.7(a) forconstant-inflow with the Langmuir adsorption model, resulting in the bulk distri-bution at t = 2 s in Fig. 3.8(a). Upon the start of the exhalation phase, clean airinvades the LOAC channel in reverse direction. The Langmuir kinetics on the sub-strate is such that desorption occurs over the entire area of the substrate, causing thedecline in ρs. At the end of the exhalation (t = 4 s), the substrate retains about 52%of the particles that have been deposited during the inhalation phase. In the bulk,the clean air has not displaced all the particle-laden air inside the LOAC (Fig. 3.8a,t = 4 s). The remaining adsorbed particles on the substrate and particles in thebulk ensure that in the second cycle of inhalation, the surface deposition will ex-ceed that of the first cycle, and the same trend continues for later cycles (Fig. 3.8c).Eventually, the assumption of “clean-air exhalation” (c1 = 0) becomes untenable,as the particle-laden air encroaches on the clean air over time and eventually exitsthe LOAC channel. In a physical experiment, continued cycles will lead to a limitwith the channel fully filled with particle-laden air at c0 and a substrate at the sat-urated number density ρ0. With our artificial condition c1 = 0, the ρs(t) curve inFig. 3.8(c) will settle into a perfectly periodic oscillation.373.2.3 Eulerian model: breathing patterns during exercising andsmokingWith the bidirectional pulsatile flow model established, we examine next howchanges in breathing pattern during physical exercise and cigarette and marijuanasmoking affect the deposition of SPs. Ongoing research indicates a 2- to 8-foldincrease in inhalation dosage of ultrafine particles during exercise [111–115]. Thebreathing pattern shows two prominent changes during physical activity: increasedair speed and tidal volume per inhalation and elevated breathing frequency. Com-pared with the resting state, air volume of each inhalation increases from 7 to 40liter/min during moderate to strenuous exercise, which corresponds to an increasein averaged air speed from 0.337 to 1.91 mm/s. In addition, the breathing fre-quency increases from 15 to 20 cycles per minute [112, 116, 117]. Experimentally,it is challenging to determine the individual and combined effect of these two fac-tors on alveolar deposition of air-borne particulates. Limited evidence suggests thatparticle deposition increases with the tidal volume but decreases with the breath-ing frequency [111, 112]. We will examine each factor separately for its effect onparticle deposition in our LOAC model.Figure 3.9(a) shows the particle deposition ρs at the elevated velocity V0 = 1.91mm/s, characteristic of moderate exercising, compared with ρs at V0 = 0.337 mm/sfor the resting state [116, 117]. At the higher V0, the “clean-air exhalation” bound-ary condition of c1 = 0, used in Figs. 3.8, becomes inaccurate as the faster flowdoes carry some particles outside the LOAC channel during the inhalation phase.To resolve this complication, we have added an auxiliary segment downstream ofthe LOAC that has thrice its length, extending the length L from 2 to 8 mm. Nowc ≈ 0 again holds at the exit at the end of the inhalation, and we impose V0 andc1 = 0 at the new inlet for the exhalation phase. To measure the particle deposition,we collect ρs data from the original segment (L = 2 mm) only. All the simulationson the effect of exercising are carried out in this extended LOAC device. As ex-pected, the faster air flow delivers more SPs into the LOAC and produces a higherdeposition, by a factor of about 3 in Fig. 3.9(a).Now we turn to the effect of a higher breathing frequency while keeping theairflow speed at the same V0 = 0.337 mm/s. Figure 3.9(b) shows that per breathingcycle, the higher frequency produces a lower amount of deposited particles. This38(a)(b) (c)(d) (e)Figure 3.9: (a) Effect of the airflow velocity on SP deposition, with the fre-quency kept at the rest-state 15 cycles per minute or 4 s per cycle. (b)Effect of the breathing frequency on SP deposition, with the airflow ve-locity kept at the rest-state V0 = 0.337 mm/s. The number density ofdeposited particles ρs is plotted against the number of breathing cycles.(c) Same as the above, with ρs plotted against real time. (d) Effect ofexercise on SP deposition, with the faster airflow (V0 = 1.91 mm/s) andhigher breathing frequency (20 cycles/min or 3 s per cycle). ρs is plot-ted against the number of breathing cycles. (e) Same as the above, withρs plotted against real time.is easily understood since at the higher frequency, inhalation and exhalation eachlasts 75% of the duration of the base line case. Therefore, on average only 75%of the SPs enter the LOAC for each inhalation, and there is also a shorter time forthe adsorption on the substrate. If we plot the adsorption not against the cycles39but against real time, the shorter breath at higher frequency still produces a lowerparticle deposition if averaged over time (Fig. 3.9c). This is because the shorterbreath draws in fewer SPs in each inhalation before expelling them by the clean airduring exhalation. Effectively a shorter initial segment of the substrate is exposedto the particulates in the air.These results indicate opposite effects of the two factors that accompany phys-ical exercise: faster airflow increases SP deposition whereas higher breathing fre-quency decreases it. Both predictions are consistent with in vivo measurements[111, 112]. Putting both factors together in Fig. 3.9(d,e), we see that the formerdominates the latter, and overall physical exercising increases the amount of par-ticulates deposited in the LOAC by a factor of about 3 relative to the resting state.This conclusion is consistent with prior experimental observations. Measurementsdone on running in urban areas have consistently shown increased inhalation ofparticulates, with the amount of increases ranging from 1.8 to about 8 times [113–115, 118].Finally, we investigate the effect of breathing patterns relevant to smoking.Tobacco and marijuana smokers often adopt characteristic “puff profiles” that differfrom the natural breathing pattern [94]. One that has been widely studied clinicallyis the inhale-hold-exhale pattern, in which the smoker holds the breath for a periodth between inhalation and exhalation. The holding time th ranges from 1 to a fewseconds for tobacco smoking, and can be as long as 20 seconds for marijuanasmoking [90–93]. To investigate this factor, we adopt the Langmuir kinetics alongwith the baseline parameters in the LOAC channel of regular length (i.e., L = 2 mmof Sec. 3.2.2, without the extension used in Fig. 3.9). At the start of the exhalation,we impose c1 = 0 at the inlet as done previously.Figure 3.10 explores the effect of breath-holding, with th = 1 and 10 s, in com-parison with the default breathing pattern without pause (th = 0). Plotted againstbreathing cycles (Fig. 3.10a), our results show that a longer hold gives the air-borneparticles extra time to adsorb onto the substrate per cycle. Averaged over the 3 cy-cles simulated, the amount of deposition per unit substrate area per cycle increasesfrom 0.0307 µm−2 (for th = 0) to 0.0358 µm−2 (th = 1 s) and 0.0483 µm−2 (th = 10s). This can be compared with experimental measurement of the plasma concentra-tion of ∆9-tetrahydro-cannabinol after 10 puffs of marijuana smoking [93], which40(a) (b)Figure 3.10: Effect of breath-holding on particle deposition during smoking.For holding time th = 0, 1 s and 10 s, the number density of depositedparticles ρs is plotted against the number of breathing cycles in (a) andagainst real time in (b).increases by a factor of about 2 as the pause increases from th = 0 to th = 10 s.Assuming the same bioavailability in both cases, the increased drug intake is qual-itatively consistent with our model prediction, but quantitatively underestimated.Figure 3.10(b) plots the deposition in real time. If averaged over a relatively shorttime, say the first 14 s, the rate of deposition is lower for longer holding periods.Our model uses the geometric design of the LOAC device of Huh et al. [1, 9],and the original experiments involved only steady-state flows corresponding to theunidirectional simulations of Sec. 3.2.1. The bidirectional simulations of variousbreathing patterns typical of a resting state, physical exercise and smoking havepredicted interesting trends that are consistent with in vivo observations. NewerLOAC designs have started to incorporate bidirectional flows [119] but we havefound no directly comparable data in the literature. Our bidirectional numericalresults in Sec. 3.2.2 and Sec. 3.2.3 may serve as guidelines for designing futuredevices.3.2.4 Lagrangian model: particle trackingWhile the effective continuum description of the particulate phase is reasonable forfine SPs, larger particles will require a more detailed discrete description as dragand gravity forces on individual particles become important factors. Meanwhile,Brownian force must be explicitly accounted for through the Langevin equation(Eq. 3.6). We have carried out Lagrangian particle tracking for particles in the4150 nm ≤ d ≤ 1 µm range. For this subsection, we ignore desorption of the La-grangian particles, effectively adopting the simplest “rapid uptake” kinetics on thesubstrate. Experiments have shown that irreversible adsorption may arise from theso-called adhesion hysteresis, with a greater energy barrier against desorption thanthe energy initially gained through adsorption [120]. Given the no-slip boundarycondition on the substrate, the hydrodynamic forces will be ineffectual for dis-lodging an adsorbed particle. So will be the Brownian force for relatively largeparticles.Without desorption and surface saturation, both inhalation and exhalation willcontinually deposit particles on the substrate, and it becomes less meaningful tostudy the inhalation and exhalation phases separately. Thus, we have limited thefollowing discussion to a constant-flow condition similar to Sec. 3.2.1. At t = 0,we release 1000 particles at the inlet and track their trajectories downstream. Theparticles are initially randomly distributed across the entry plane, with an initialvelocity of V0. In the simulations, we treat the particles as points with no spatialdimension; they do not interact hydrodynamically nor occupy an excluded volumespatially. Thus, the results are equivalent to the superposition of 1000 simulationsof single particles. Whenever a particle comes into contact with the substrate, it isimmobilized and adsorbed. On the top and side walls, the particles do not adherebut bounce back with a reflected velocity. The deposition efficiency ε is defined asthe fraction of the deposited particles among the total released. The simulations arecarried out till all particles have either settled onto the substrate or passed throughthe LOAC. The duration ranges from 2 s to 50 s, with smaller particles takinglonger time.The main results of the Lagrangian simulations are shown in Fig. 3.11. Thedeposition efficiency ε depends strongly on the particle size. For clarity, we haveplotted separate ε curves with and without Brownian forces. Without Brownianforce, the fate of the particles is determined by the competition between the dragforce and gravity. Larger particles are increasingly dominated by gravity, and thusexhibit a greater ε in Fig. 3.11. Figure 3.12 shows 3 snapshots of non-Brownianparticles of diameter d = 100 nm and 500 nm. At the start, the particles are carrieddownstream by the air flow. For d = 100 nm, gravitational settling is slow and onlyparticles initially near the substrate and the side walls have a long enough residence42Figure 3.11: Deposition efficiency ε for non-Brownian and Brownian parti-cles of different sizes. ε is calculated by dividing the number of ab-sorbed particles by the total number of particles released.Figure 3.12: Snapshots of the distribution of non-Brownian particles at t = 2,4 and 6 s, for particle diameter (a) d = 100 nm, and (b) d = 500 nm.The color indicates the instantaneous particle speed, with blue particles(up = 0) being stationary and adsorbed onto the substrate.time to settle onto the substrate. The rest are afloat in air and being advecteddownstream, eventually producing a very low ε . The settling is more prominentfor the larger particles (d = 500 nm). By t = 6 s, most particles that remain insidethe LOAC have settled onto the substrate. This explains the increasing ε with din Fig. 3.11. For d ≥ 600 nm, all the particles released settle in the LOAC and ε43Figure 3.13: Snapshots of the distribution of Brownian particles of diameterd = 100 nm at t = 2, 4 and 6 s. The color indicates the instantaneousparticle speed, with blue particles (up = 0) being stationary and ad-sorbed onto the substrate. Note the marked enhancement in depositionrelative to the non-Brownian particles of Fig. 3.12(a).approaches unity.Brownian motion modifies the deposition behavior of the relatively fine parti-cles, and causes ε to vary non-monotonically with d. For the finer particles (e.g.,d = 50 nm), gravitational settling is slow and surface adsorption is dominated byBrownian motion. The Brownian force gives the particles a stochastic movementnormal to the substrate and, with a certain probability, drives particles into thesubstrate for adsorption. Figure 3.13 shows three snapshots of Brownian particlesof size d = 100 nm. Comparing these with their counterparts in Fig. 3.12(a), wefirst note that Brownian motion randomly perturbs the instantaneous speed of theparticles against the clear stratification of non-Brownian particles. In fact, the par-ticles are moving in various directions at any instant. The strong Brownian motioncauses a stream-wise spreading of the particles that blurs the traveling front clearlyseen for the non-Brownian particles (see t = 2 s plot of Fig. 3.12). Simultaneously,the Brownian motion agitates the particle motion in the direction normal to thesubstrate, and greatly enhances the adsorption relative to the non-Brownian parti-cles. For the finer particles, therefore, Brownian motion and gravity both promotesurface adsorption, with the former being dominant.44Figure 3.14: Effect of the orientation of gravity on the deposition efficiencyε for Brownian particles of different sizes. The angle θ is betweenthe directions of air flow and gravity (see Fig. 3.1). Gravity is alongthe flow direction at θ = 0, perpendicular to the flow at θ = 90◦ andagainst the flow at θ = 180◦.With increasing particle size, the effect of Brownian motion declines, and sodoes ε . In the meantime, gravitational settling starts to play an increasing rolein enhancing deposition. Eventually, ε levels off and starts to increase for largerparticles (d ≥ 300 nm). Note that the Brownian motion is directionally isotropic;it drives the particles toward and away from the substrate with equal probability.As a result, for an intermediate size range (500 nm ≤ d ≤ 700 nm), the Brow-nian motion hinders gravitational settling more than it drives the particles to thesubstrate, and so the Brownian particles exhibit a lower desorption ratio than thenon-Brownian particles (Fig. 3.11). For the largest particles (d ≥ 700 nm), Brow-nian motion becomes negligible, and the two ε curves merge. It is interestingto note that in vivo experimental measurements also show ε to vary with d non-monotonically (Figs. 16, 17 of Rostami [100]), consistent with our prediction forBrownian particles.So far, we have only considered gravity acting toward the substrate. In viewof the macroscopic structure of the lungs, however, gravity may act in a range ofdirections relative to the alveolar surface and air flow. For example, it may be par-allel to the epithelium and along the flow direction in the lower lobes, and against45the flow in the upper lobes. Supine or prone postures will place more bronchiolarand alveolar surfaces perpendicular to gravity than an upright one. The LOAC canbe inclined to test such effects. We have carried out simulations with gravity actingin co-flow (θ = 0) and counter-flow (θ = 180◦) directions as well as along a 45◦angle with the flow direction (Fig. 3.14). For the finer particles (d = 100 nm), grav-itational settling is small relative to Brownian motion, and so essentially the sameε prevails regardless of the direction of gravity. For larger d, gravity parallel to thesubstrate (θ = 0 and 180◦) yields lower ε than gravity pointing toward the substrate(θ = 90◦) for the lack of gravitational settling. Curiously, there is essentially nodifference between the two cases with gravity along (θ = 0) or against (θ = 180◦)the air flow, and both agree with the curve of no gravity at all. Counter-flow grav-ity should prolong the residence time of the particles within the LOAC relative toco-flow gravity. But a quick estimation shows that for d = 500 nm, for example,the Stokes drag is about three orders of magnitude greater than the buoyant weightof the particle. Thus, the difference in the residence time is negligibly small. The45◦-inclined gravity has a component toward the substrate, and produces an in-termediate ε between the perpendicular and parallel cases discussed above. Forthe largest particles, even this partial gravity results in nearly perfect settling asε approaches unity. Therefore, for the parameter values tested here, gravitationalsettling plays a significant role only if it has a non-zero component directed to-ward the substrate. One may infer that more particulates are deposited with thebody horizontal than upright (e.g., during night versus day times), and more in themiddle sections of the lungs than in the upper and lower lobes. These inferencesremain to be tested experimentally.3.3 Summary and discussionThe main motivation for this work is to understand the transport of submicron parti-cles (SPs) in a lung-on-a-chip (LOAC) device and show that computer simulationscan inform and potentially accelerate the design and application of these devicesfor analyzing particulate- and microbe-alveolar interaction. We have presented twocomputational models: an Eulerian model for fine SPs and a Lagrangian model forcoarser SPs. Using LOAC as a surrogate for the human alveolus, these models were46applied to examine the deposition and adsorption of SPs (a) under different exer-cise and breath-holding patterns and (b) for a range of particle sizes with varyingrelative importance of Brownian motion and gravitational settling. The first set ofresults are for fine SPs for which gravitational settling is negligible and an Euleriancontinuum approach is suitable, with Brownian motion represented by a diffusivityparameter. For larger particles, we employ Lagrangian particle tracking to probehow gravitational settling and Brownian effect cooperate or counter-act each other.Special attention is given to elucidating the underlying physical mechanisms byadopting parameter values relevant to human breathing and LOAC experiments.The key findings are summarized below.(a) Using physiologically relevant values for the bulk SP concentration and theadsorption and desorption coefficients in the Langmuir and Frumkin models,we find that the surface coverage by adsorbed particles saturates at a lowlevel of 0.16%.(b) Tidal breathing produces a cyclic change in the amount of surface deposi-tion, with a gradual increase over the cycles. The exhalation phase typicallyfeatures a reduction in deposited SPs as a result of desorption.(c) Physical exercise, characterized by increased air-flow volume in each breathand higher breathing frequency, increases the amount of particulate deposi-tion by a factor of about 3 relative to the state of rest. This is thanks to thedeeper breath with greater air-flow volume, as higher breathing frequencycauses a moderate decrease in deposition.(d) Breath-holding between inhalation and exhalation, characteristic of cigaretteand marijuana smoking, increases the deposition by an amount that increaseswith the duration of the holding period.(e) Lagrangian particle tracking finds the deposition efficiency first to decreasewith particle size for diminishing effects of Brownian motion, and then to in-crease for more pronounced gravitational settling. Thus the deposition varieswith particle size non-monotonically. Spatial orientation of the epitheliumrelative to gravity also affects the deposition efficiency.47(f ) Where comparisons can be made with experimental data in the literature, theabove results are in qualitative and sometimes semi-quantitative agreement.These findings may have several clinical implications. First, because physi-cal exercise leads to increased particle deposition in the airways, it is advisablefor individuals to avoid or curtail outdoor exercise on days with poor air quality.Second, smokers should be warned against prolonged breath-holds during puffs.In contrast, for individuals with asthma or chronic obstructive pulmonary diseasewho use inhalers for a clinical indication, prolonged breath-holding should be en-couraged.We must note that as in any modeling work, certain simplifications and assump-tions have to be made to make the problem tractable. One such assumption is theLangmuir and Frumkin kinetics for surface adsorption and desorption, which areboth valid for a monolayer of adsorbed particles. In the current parameter range,where the surface areal coverage is sparse (∼0.16% in Fig. 3.6c), the monolayermodels are suitable. For greater amount of deposition, a multilayer adsorptionmodel such as the Brunauer-Emmett-Teller (BET) model may be more appropriate[121]. In addition, the estimation of parameter values is subject to a degree of un-certainty, as discussed in section 3.1.4. Finally, in Lagrangian particle tracking, weare unable to include sufficient number of particles to reach the appropriate bulkconcentration. This is just a matter of linearly increasing the amount of computa-tions, since the particles do not interact, and should not have affected the qualitativetrend in the results.48Chapter 4Case study II: Mesenchymaltransmigration of cancer cellsIn this chapter1, we aim to model the extravasation of a tumor cell (TC) throughthe gap between endothelial cells (ECs). To achieve this purpose, we set up afluid-structure interaction problem, with partial differential equations describingthe cytosol fluid flow and the elastic deformation of various elastic components—the TC membrane, the TC nucleus, the ECs and the extracellular matrix (ECM).Besides, the stress fiber (SF) and focal adhesion (FA) assembly evolves in timeaccording to a set of differential and algebraic equations that describe the activa-tion of actomyosin in the SF coupled with the growth of the FA. Thus, the modelalso comprises interconnected chemo-mechanical factors. They can be graphicallyrepresented by the schematic in Fig. 4.1.The equations governing SF and FA dynamics, cytosolic flow and elastic de-formation of various components were described in the chapter. 2. In the followingsections, we first describe the problem setup and then represent the results anddiscussion. Finally we wrap up the chapter in the summary.1The results of this chapter were published in:”Arefi, SM Amin, Daria Tsvirkun, Claude Verdier,and James J. Feng. ”A biomechanical model for the transendothelial migration of cancer cells.”Physical Biology 17, no. 3 (2020): 036004.”49Contractile forceFA: high-affinity integrin SF: myosin activationECM deformationTC nuclear deformationEndothelial deformationTC extravasationCytosolic flowFigure 4.1: Schematic showing the various components of the model. Apointed arrowhead means “activate” or “promote” whereas a flat arrow-head means “impede” or “inhibit”.4.1 Problem setupI this section, we first describe the geometric setup and the boundary conditions forthe wet-contact simulation. The dry-contact simulation is set up in a similar way,with the cytosolic flow and TC membrane omitted. Then we summarize all themodel parameters that we used in our simulations. We will end this section with abrief discussion of the numerical techniques used in the simulations.4.1.1 Model descriptionTo the best of our knowledge, there has been only one prior model [59] for thetransmigration of tumor cells through endothelium, and will be the mostly imme-diate inspiration for our own modeling. Cao et al. [59] built a mechanical modelon the transmigration of a hyperelastic TC nucleus through small constrictions.By specifying a constant force that pulls on the nucleus, an elastic equilibrium iscomputed where elastic resistance due to the nuclear and endothelial deformationsbalances the pulling force. In our simulations, the geometric setup and the hy-perelasticity of the TC and EC are modeled after Cao et al. [59], with the tumorcell being pulled through an axisymmetric opening in hyperelastic endothelium(Fig. 4.3). Our model goes beyond Cao et al.’s model in several aspects. First,we use finite elements to track the transmigration dynamically, with cell and tis-sue deformation, dynamic remodeling of the stress fibers and the focal adhesionsand the displacement of the tumor cell occurring simultaneously in time. Sec-ond, the pulling force in the TC invadopodia is modeled by adopting Deshpande50et al.’s models [81–83] for the stress fibers and the focal adhesions, both controlledby biochemical signaling with a mechanical feedback. For simplicity, the modelaccounts for a single invadopodium, even though in vitro images suggest the pos-sibility of multiple protrusions or branches [5]. In addition, we have neglectedany plasma flow inside the capillary, because the transmigration essays typicallydo not have ongoing perfusion of the vascular network [5], or because the cancercells tend to occlude the capillary to prevent flow [11]. The opening that allowsthe passage appears to be small in comparison to the dimension of the TC nucleus[5, 59]. Therefore, it is reasonable to assume tight contact between the TC and EC.Given the multiple cellular components involved in the process, including the TCnucleus, cytosolic fluid and cell membrane, as well as the EC, the mechanics ofcontact requires a careful treatment.In their model, Cao et al. [59] have ignored the TC cytosol and membrane, andassumed direct solid-solid contact between the TC nucleus and the EC. Conceiv-ably, the lubrication between the nucleus and the membrane [122] and between themembrane and the constriction [123] can play significant roles. In this work, there-fore, we will test two cases: the “dry-contact” case of Cao et al. [59] with directsolid contact between the TC nucleus and the EC (Fig. 4.3a), and the “wet-contact”case that includes the cytosol and TC cell membrane (Fig. 4.3b). In the dry-contactsimulation, the contact force is handled by a penalty method commonly used insolid-contact models [124]. In the wet-contact case, the cytosol flow within thethin gap between the TC and EC provides the lubricating pressure that preventssolid-solid contact. Including the cytosol narrows the opening available for thenucleus, and introduces additional viscous friction. So the wet contact should pro-vide a greater resistance to the passage of the TC, and the dry contact is expectedto provide the lower bound for the pulling force needed for passage. In both cases,the extravasation hinges on nuclear deformation, which is known to be a key factorin cell migration [125]. The contractile apparatus of the invadopodium consistsof stress fibers and focal adhesions; the two are connected in series with the nu-cleus at one end and the ECM at the other. In the wet-contact simulations, weassume that the cavity is filled with the cytosol of the TC, its membrane being incontact with the ECM. We have in general a fluid-structure interaction problem,with partial differential equations to be solved for the cytosol fluid flow and the51(a) (b)Figure 4.2: (a) Schematic showing the geometric setup for the “dry-contact”simulation. Stress fibers AB and focal adhesions BC link the nucleus tothe roof of the ECM cavity. The TC membrane and cytosol are ignored,and the critical condition of the breakthrough of the TC nucleus is deter-mined from the direct but frictionless contact between the nucleus andthe endothelium. (b) Schematic illustrating the geometric setup for the“wet-contact” simulation, with the TC membrane and cytosol explicitlyaccounted for.elastic deformation of various elastic components—the ECM, EC and TC nucleusand membrane. Besides, the SF-FA assembly evolves temporally according to aset of ordinary differential equations. The dry-contact case lacks the fluid flow andfluid-structure interaction, and is therefore simpler.4.1.2 Geometric setupFor the The geometric dimensions listed in Table 4.1, we have largely modeledafter the prior simulation of Cao et al. [59]. The axisymmetric geometry of thesimulation is shown in Figure 4.3. The nucleus of the cancer cell is initially a sphere52Figure 4.3: The geometric setup for the so-called “wet-contact” simulation,with the cytosol and the membrane of the tumor cell explicitly repre-sented.of radius rn = 3 µm. It is made of a soft core of radius rc = 2.7 µm and a stiffershell of thickness δ = 0.3. The endothelium is modeled as a flat elastic layer with acircular hole of radius rg = 1.5 µm in the middle, which is smaller than the nuclearradius rn. The EC thickness is taken to be h= 6 µm. Experimental literature reportssmaller values of 3–4 µm [126]. Our larger value is an effective thickness forthe endothelium that also includes the basement membrane outside the EC [127].Above the endothelial cells the ECM was treated as an elastic material with a cavityin the middle. The radius of the cavity is Rcavity = 7.5 µm. The rationale forthe cavity is that cancer cells are known to create a space in the ECM either bydegrading it [128] or by exploiting the tissue’s plasticity [129]. At the start, theTC nucleus is centered 5 µm below the centerline of the EC and 13 µm below53the center of the ECM cavity. The upper part of the TC membrane is not treatedseparately since it is in contact with the ECM. The lower part is a spherical shellthat is initially concentric with the nucleus, with an inner radius of R = 4.5 µm anda thickness of ε = 10 nm. The upper end of the TC membrane is attached onto theendothelium at the intersection of the round membrane and endothelial layer.4.1.3 Governing equationsIn this section we briefly bring up the governing equations that required for thesimulations in this chapter. Although these equations were described in chapter 2,they are represented in here to make this chapter more self-contained.The SF dynamics is governed by a kinetic equation for actomyosin activationand a Hill-like constitutive equation for the SF:dηdt= (1−η)kae− tθ − kd(η− ττmax), (4.1)τ =0, v fv0 <−ηkv,ητmax(1+ kvηv fv0), − ηkv ≤v fv0≤ 0,ητmax,v fv0> 0.(4.2)Our FA model is based on the thermodynamic equilibrium between two con-formational states of low- and high-affinity integrins, subject to the conservationof the total number of integrins: ξH + ξL = ξ0. Only the high-affinity integrinscontribute to the formation and growth of the FA; the tensile force τ in the SF issustained by the FA and shared by all the high-affinity integrins in the FA:kBT lnξHξL= ∆µ−Φ+Fb db, (4.3)τ = ξHFbAb, (4.4)where kB is the Boltzmann constant, T is temperature, and ∆µ = µL−µH is the freeenergy difference between the low- and high-affinity integrins. The stretch energyΦ is expressed as a piecewise quadratic potential in the integrin bond length db,54and the force Fb = ∂Φ/∂db on each high-affinity integrin is piecewise linear:Φ=12λd2b , db ≤ dm,−λd2m +2λdmdb− 12λd2b , dm < db ≤ 2dm,λd2m, db > 2dm,(4.5)Fb =λdb, db ≤ dm,2λdm−λdb, dm < db ≤ 2dm,0, db > 2dm.(4.6)where λ is the stiffness of the bond and dm is the maximum bond length.The elastic components of the model, including the TC membrane, TC nucleus,the EC and the ECM, all obey the neo-Hookean constitutive equation, and the localforce balance is expressed simply as the stress tensor having zero divergence:σ = GJ−53(FFT − I3(FFT ))−K(J−1)I, (4.7)∇ ·σ = 0. (4.8)The elastic moduli differ among the various components and are listed in Sec-tion 4.1.5.The cytosolic flow is governed by the Navier-Stokes equations in the laminarflow regime:∇ · v = 0, (4.9)ρ(∂v∂ t+ v ·∇v)=−∇p+µ∇2v. (4.10)Finally, the displacement of the TC nucleus, the SF and FA stretching and themovement of the ECM at the FA are coupled by a kinematic condition, whichcompletes the set of governing equations:vA + v f + vb = vC. (4.11)55Figure 4.4: Illustration of the boundary conditions. The green lines desig-nate fixed boundaries and the red lines show boundaries with distributedforces.4.1.4 Boundary conditionsAround the outline of the computational domain, three types of boundary condi-tions are imposed. Along the axis of symmetry (r = 0), of course, all radial deriva-tives vanish. The outer boundaries of ECM and the endothelial layer, marked bythe green lines in Fig. 4.4, are fixed in space with zero displacement. Finally, onthe outside of the TC membrane we impose a constant ambient pressure p = 0.On all the fluid-solid interfaces, including that between the cytosol and theECM, the EC, the TC nucleus and the TC membrane, we impose continuity in the56velocity v and traction:vsolid = v f luid , (4.12)n ·σsolid = n ·σ f luid , (4.13)n being the local unit normal vector. The tension inside the SF-FA assembly re-quires some special treatment. To avoid local singularities, we distribute the force τin the stress fibers over a small area marked in red on top of the nucleus in Fig. 4.4,of area An. A uniform traction of magnitude τ/An is applied in the vertical direc-tion over this area. Similarly, on the roof of the cavity where the FA meets theECM, we distribute τ over the FA area Ab.The dry-contact setup is similar and simpler, in that the TC cell membrane andcytosol are neglected. On solid surfaces exposed to the cytosol in the wet-contactsetup, i.e. the inner surfaces of the ECM and the EC, as well as the TC nucleus, wenow impose the zero-stress condition: n ·σsolid = 0.4.1.5 Parameters estimationThe model parameters fall into 3 categories: geometric, physical and biochemical,and they are tabulated in Tables 4.1–4.3 in that order together with the sources fortheir values. Where possible, the parameters are evaluated by using experimentalmeasurements from the literature or our own laboratory. Otherwise the values arechosen based on similar modeling and computational studies in the literature. Inthe following, we highlight several notable parameters.57Symbol Description Value Sourcesrn TC nuclear radius 3µm [59, 130]δ TC nuclear shell thickness 0.3µm [59]ε TC membrane thickness 10 nm [65, 131]rm radius of remaining TC body 4.5µm this workR ECM Cavity radius 7.5µm [59]h endothelial thickness 6µm [59, 126, 127]rg endothelial gap radius 1.5µm [5, 59, 130]Ab focal adhesion area 4µm2 [86, 132]Table 4.1: Geometric parameters used in our model. Additional lengths aremarked and given in Fig. 4.3.The physical parameters consist of the properties of the various elastic compo-nents and the cytosolic fluid (Table 4.2). To allow strain hardening in the elasticcomponents at large strain, we adopt the neo-Hookean hyperelastic constitutive re-lationship for all the elastic components, i.e. the ECM, EC, TC nuclear shell andcore and the TC membrane. The same Poisson ratio ν = 0.3 is assumed for allthese components, but their shear moduli differ, as given in Table 4.2. Note thatthe TC membrane is much more rigid than the ECM and EC; this is chosen basedon reported elastic modulus for lipid bilayers [133].Symbol Description Value SourcesGEC EC shear modulus 1 kPa [59, 134, 135]GECM ECM shear modulus 5 kPa [136]Gc TC nuclear core shear modulus 5 kPa [59, 137]Gs TC nuclear shell shear modulus 50 kPa [59]Gm TC membrane shear modulus 2 MPa [133]ν Poisson’s ratio for all elastic components 0.3 [59]µ TC cytosol viscosity 10−3 Pa·s [138]ρ TC cytosol density 1000kgm−2 [131]κ TC nuclear drag coefficient 0.1 kg/s this workTable 4.2: Physical parameters used in our model.58The TC nucleus is the limiting factor in extravasation, and deserves a few morewords here. Following Cao et al. [59], we consider the nucleus as made of a hy-perelastic shell that encases a softer core of chromatin and other structures, bothlayers being governed by neo-Hookean constitutive relationships. The shell con-sists mainly of a matrix of Lamin A/C, which affords it an higher stiffness thanthe inside of the nucleus. We have adopted a shear modulus of Gs = 10Gc fol-lowing Cao et al. [59]. Experiments showed that the viscosity and density of thecytosol are not significantly different from those of water [131, 138], and so wehave chosen µ = 10−3 Pa·s and ρ = 1000 kg/m3.Symbol Description Value Sourcesθ decay time for calcium signal 100 s [139, 140]ka myosin activation rate 0.1 s−1 [139]kd myosin deactivation rate 0.01 s−1 [139]τmax maximum tension in SF 100 nN [86, 132, 141]kv coefficient in SF constitutive equation 10 [139, 140]v0 reference shortening rate of SF 1 µm/s [86, 142]ξ0 total number of integrins per unit area 5000 µm−2 [139, 143]∆µ free-energy difference between low/high-affinity integrins 5kBT [82]λ stiffness of integrin bond 0.15 nN/µm [139]dm maximum strain of integrin bond 130 nm [139]Table 4.3: Biochemical parameters used in our model. In evaluating ∆µ , wehave taken kB = 1.381×10−23 m2 kg/(s2 K) and T = 310 K.Finally, the biochemical parameters of the model are tabulated in Table 4.3.Most of these are adopted from Deshpande et al. [82, 139]. The total number ofhigh and low affinity integrins (ξ0) in an isolated cell is taken to be 5000 µm−2[139, 143].4.1.6 Numerical techniquesThe numerical problem formulated in the above is solved by COMSOL Multi-physics, a finite-element package. The dry-contact simulation consists in solvingfor the elastic deformation of the various components under the driving force thatdevelops inside the SF-FA complex. A prominent feature is the contact force be-tween the nucleus and the EC. This is handled by the penalty method, a common59technique in solving solid-contact problems [124].The wet-contact simulations consist essentially in fluid-structure interactions,with coupled solutions of the cytosolic flow and the elastic deformation of theelastic components (ECM, EC and TC components). The cytosol in the thin gapbetween the TC nucleus and EC provides a lubricating pressure. In addition, wehave found it necessary to introduce a repulsive pressure to prevent very closeapproach between the TC nucleus and the EC surface, which could cause numericalblowup. Such a repulsion is commonly used in the literature for preventing solid-solid contact across a narrow fluid gap [144]. This repulsive pressure is imposedon the nuclear and endothelial boundary nodes, and is defined using the shortestdistance d from a node to the opposite surface:Prep = P0tanh(d−1−d−10 )+12, (4.14)where P0 = 100 kPa, d0 = 0.588 µm and d is measured in micrometers. The valueof d0 determines how steeply Prep decays with the separation d, and the minimumseparation is about 0.25 µm during the transmigration under the standard set ofparameters (see Section 4.1.5).The P0 value is chosen to ensure the integrity of thelubricating film against TC-EC contact for the conditions tested in this study. Wehave tested P0 values down to 50 and 10 kPa. Results showed that in this range, P0does not strongly affect the threshold for transmigration.In our simulations, COMSOL does implicit time stepping, and adaptively re-fines or remeshes an unstructured triangular grid. We have done numerical exper-imentation to ensure adequate temporal and spatial resolutions. As an example,Fig. 4.5 shows the effect of mesh refinement on the contractile force in the stressfibers in the dry-contact simulation. All the results reported have used the so-called“normal resolution” with roughly 10,000 elements for the dry-contact setup, and27,000 elements for the wet-contact setup, with the smallest and largest grid sizesbeing 0.007 µm and 1.98 µm, respectively. The maximum time step is ∆t = 0.05s, which is adequate for resolving the transients in the simulations.60Figure 4.5: The effect of different mesh resolution on the contractile force inthe stress fibers. The solutions using the coarse, normal and fine reso-lutions are sufficiently close, and we have used the normal resolution inall results reported later.4.2 Results and discussionTo test the invadopodium-based hypothesis of cancer cell transmigration, we seekto answer the following question: what conditions determine whether a tumor cellsucceeds in extravasation? This depends on whether the active driving force gen-erated in the SF-FA machinery is able to deform the TC and the EC sufficientlyto allow the passage of the TC. We will first test the dry-contact scenario and thenthe wet-contact scenario. The results presented below correspond to the baselineparameter set described in the section. 4.1.5.4.2.1 Dry contact simulationFigure 4.6 shows a time sequence of snapshots illustrating how the nucleus ispulled through the narrow opening in the endothelium by the SF. The endothe-lial opening has a radius rg = 1.5 µm in the middle, while the TC nucleus has anundeformed radius rn = 3 µm, twice the size of the opening. First, we initiate thecalcium signal to trigger the myosin activation. This produces tension in the SFthat pulls the TC nucleus toward the gap in the TC. Since the nucleus has an un-deformed size that is bigger than the gap size, it stalls and comes into solid-solid61Figure 4.6: Snapshots showing the progression of transmigration in the dry-contact setup of the simulation. The color contours show the local strainin the vertical direction.contact with the surface of the EC (t = 50 s). In the contact zone, there is a nor-mal contact force but no tangential friction. Under the pulling force by the SF, thenucleus, EC and the ECM continue to deform elastically until t = 90 s, when thenucleus is approaching the narrowest part of the EC constriction. Since the nuclearshell is much more rigid than the EC [59], it deforms much less in this process.Then a breakthrough happens as the nucleus transmigrates successfully (t = 100s). Note that at this instant, the normal force from the EC propels the TC nucleusforward. The transmigration is completed by t = 130 s.Figure 4.7 shows the evolution of the velocity vn of the centroid of the nucleusduring its transmigration. Upon start of the simulation, the velocity rises till t ≈ 20s. This acceleration is owing to the geometric setup that puts an initial clearancebetween the tumor nucleus and the endothelium, and is not an intrinsic part of thephysics. Afterwards, the nuclear velocity exhibits four distinct phases as demar-cated in the graph. First, the nucleus movement is impeded after it makes contactwith the endothelial surface, and the velocity declines. Second, the nucleus and theEC deform as the SF contracts more, and vn slowly accelerates. Third, the nucleussnaps through the gap, with a sharp rise in vn followed by a sharp decline. The ac-celeration is partly due to the release of stored elastic energy in the nucleus and theEC after the breakthrough; this is visible in the snapshot for t = 100 s in Fig. 4.6.62Figure 4.7: The nucleus velocity vn during the passage through endothelium.The deceleration after breakthrough reflects the decline in the tensile force on arapidly shortening SF. Finally, the nucleus settles down to a small velocity, as theSF gradually disassembles following the loss of tension.We can compare the model prediction with experimental observations of David-son et al. [145]. The simulation has captured the qualitative features of the exper-iment, including the four phases in the variation of the nuclear velocity. However,the time scale and the peak velocity are quite different. Our model predicts a transittime of roughly 2 min, compared with the experimental value of over 100 min. Apossible explanation for the difference is that the experiment is based on the crawl-ing of a fibroblast through narrow constrictions due to the gradient of chemoattrac-tant, which is a slow process determined by the biochemical rates of chemotaxis.In contrast, the breakthrough of nucleus during extravasation is governed by themuch faster dynamics of SF contraction.During the transmigration, the active myosin level η , the fraction of high-affinity integrins ξH/ξ0 in the FA and the contractile force τ in the SF followsimilar trajectories of temporal evolution (Fig. 4.8). According to Eq. (4.1), thecalcium signal prompts the rapid assembly of the actomyosin stress fibers. In themeantime, the myosin deactivation also rises with η , the balance between the tworesulting in a plateauing of η around t = 50 s. After η plateaus, the SF tensionτ continues to grow because of the “viscous” v f /v0 term in the constitutive equa-tion (Eq. 4.2). The nucleus coming into contact with the EC, its motion slows63(a) (b)Figure 4.8: Dynamics of transmigration illustrated by the evolution of (a) theactive myosin level in the SF η and the fraction of high-affinity integrinsξH/ξ0 in the FA; (b) the tension τ in the SF.down as does the rate of SF contraction v f , thus raising τ . The rising tension alsodrives the continual rise of the high-affinity integrins ξH (Eq. 4.3), which amountsto a growth of the FA in a well-documented positive feedback [87]. As the nucleussnaps through the passage (90< t < 100 s in Fig. 4.6), its sudden acceleration leadsto a large negative v f , and consequently a sudden drop in the contractile force τfrom a peak of about 20 nN toward zero according to the SF constitutive equation(Eq. 4.2). As a consequence, the actomyosin SF disassembles quickly and η de-clines, and the FA weakens with declining high-affinity integrins ξH . Later, as thenucleus decelerates quickly, the SF contraction speed v f declines and the tensionτ recovers somewhat (cf. Eq. 4.2) before decaying slowly over a long time. Thesame pattern is mirrored by ξH .In Section 4.3, we have presented a parametric study that shows how the suc-cess of the extravasation depends on model parameters such as the size of the ECopening, ECM modulus, and duration of the decaying signal that stimulates myosinactivation.4.2.2 Wet-contact simulationThe wet-contact simulations include the effects of the cytosol and the TC cell mem-brane. The cytosol affects the translocation of the nucleus both geometrically andhydrodynamically. Geometrically, a film of cytosol that cushions the nucleus fromthe endothelium effectively narrows the space available for the nucleus, and thushinders extravasation. Hydrodynamically, the cytosol film acts like a lubricating64layer, producing a viscous friction and a lubricating pressure between the nucleusand the endothelium, in place of the solid contact force in the dry-contact situa-tion. Since the dry-contact model is frictionless, the viscous friction probably alsoincreases the resistance to extravasation for the wet-contact model. Finally, havingthe cytosol inside the membrane also allows the model to account for the “backpressure”, i.e. the pressure difference between the fluid ahead of the nucleus andbehind it, a potentially important effect [59]. Should the contractile force fail topull the nucleus through the gap, it will slowly pull the nucleus against the mem-brane toward a static state where the nucleus, TC membrane and the EC outsideare in solid contact. In this sense, the dry-contact force computed in the previoussubsection should provide a lower bound for the contractile force that is requiredfor extravasation.Figure 4.9 shows snapshots during passage of the nucleus through the openingin the endothelium. The color scale in the solid indicates the strain component inthe vertical direction, while that in the cytosol marks the magnitude of the velocity.As the stress fibers pull the nucleus into the narrow gap (t = 60 s), not only does thenucleus deform and stretch in the vertical direction, so do the EC and the ECM. Thedeformation in the ECM directly above the nucleus is caused by the pulling forcein the SF and the FA. Meanwhile the cytosol is squeezed in the upper chamber,where the pressure rises and pumps the cytosol downward through the gap. Later(t = 72.5 s), the upward movement of the nucleus has generated such a low pressureunderneath that the cell membrane caves in at the bottom. This concave shape maybe partly due to the model’s neglecting the cortical tension. It is unclear whethersuch change of shape occurs during extravasation in vitro or in vivo. The greatestamount of nuclear deformation occurs at t = 180 s. Once the centroid of the nucleusis past the narrowest point of the gap (t = 183 s), the elastic resistance on thenucleus drops suddenly. In fact, the relaxation of the deformed EC now providesan upward pushing force on the nucleus. As a result, the tension on the stressfibers drops quickly, while the ceiling of the ECM relaxes as well. The passageis complete by t = 196.5 s. Ultimately, the back pressure and viscous friction dueto the cytosolic flow are negligibly small. The cytosol influences the passage ofthe TC nucleus mostly by the lubrication film occupying space and transmittingpressure between the nucleus and the EC.65Figure 4.9: Snapshots illustrating the extravasation of a tumor-cell nucleusin a wet-contact simulation. The left color bar shows the level of thevertical strain component in the solids while the right color bar showsthe velocity magnitude (m/s) in the cytosol.Similar to the dry-contact case, the wet-contact transmigration can be analyzedmore quantitatively through the temporal evolution of the nuclear velocity vn, themyosin activation η and tension τ in the SF, and the high-affinity integrins ξH inthe FA. The nuclear velocity vn, depicted in Fig. 4.10, exhibits the same 4 phasesas in Fig. 4.7. But the wet-contact case takes much longer in phase I; there is a longperiod (roughly from t = 20 s to 120 s) in which the nucleus decelerates slowly.This is evidently due to the lubricating flow in the thin film between nucleus andthe EC. Subsequently, the acceleration phase II and breakthrough phase III take66Figure 4.10: Dynamics of transmigration illustrated by the temporal evolu-tion of the velocity of the centroid of the nucleus in time. The dry-contact result is also shown for comparison.(a) (b)Figure 4.11: The temporal evolution of (a) the active myosin level η on thestress fibers and the fraction of high-affinity integrin ξH/ξ0 in the focaladhesions; (b) the tension τ on the stress fibers. The force for the dry-contact case is also shown for comparison. In later times (t > 200 s),τ and ξH recover somewhat as in the dry-contact case, but this portionof the evolution is omitted for a better view.roughly the same time as in the dry-contact case. During the breakthrough, thenucleus attains a much higher velocity (maximum of 45 µm/min, almost 4 timesthat for the dry-contact case).The evolution of η , τ and ξH in Fig. 4.11 resembles that of the dry-contact casein Fig. 4.8, with some interesting quantitative differences. Although the myosinactivation η saturates at more or less the same level, the tension τ rises to much67larger magnitude in the wet-contact simulation. The presence of the cytosol nar-rows the space available for the nucleus to pass, and also adds a viscous frictionagainst the passage. Thus the nucleus encounters stronger initial resistance andmoves up more slowly as the nucleus deforms gradually. This explains the longdeceleration phase II of Fig. 4.10. Through the constitutive equation of the stressfiber (Eq. 4.2), the decelerating SF contraction rate allows τ to continue to rise af-ter η has saturated. This is also the cause of the upturn in τ at t = 20 s when thenucleus initially encounters resistance in the gap and decelerates sharply. Throughthe stress-feedback mechanism in the FA, the rising τ raises ξH . In fact, the tensionis so strong that the fraction ξH/ξ0 reaches 1 at t ≈ 50 s, signifying complete con-version to high-affinity integrins, and stays at 1 until the extravasation completes.The breakthrough around t = 180 s is accompanied by a sharp peak in the nuclearvelocity and a precipitous decline in the SF tension and high-affinity integrin level.The actomyosin stress fibers also start to disassemble. Afterwards, the nuclear ve-locity is largely determined by the “Darcy-like” drag coefficient. The maximumtension in the SF over the entire process is about 70 nN.In comparing the wet-contact and dry-contact simulations, one notes similartemporal dynamics in the SF-FA assembly, based on two positive feedbacks be-tween the tension τ and actomyosin activation η on the one hand, and betweenτ and the FA strength ξH on the other. As anticipated in Section 4.1.1, the wet-contact mode incurs more intense dynamics. The resistance to nuclear passage isgreater, and the SF develops a stronger tension τ , accompanied by a higher levelof myosin activation η and FA development ξH . The wet-contact model predictsa transit time of roughly 4 min, on the same order as in vitro observations of TCextravasation (10–15 min) [5, 57].Our model has predicted successful extravasation in both the dry-contact andthe wet-contact setup using a set of realistic parameters. The contractile devicein the SF-FA assembly is able to generate the force required to deform the tumorcell and the endothelium sufficiently to allow transmigration. This supports thehypothesis of Chen et al. [2, 5] that the tumor cell uses contractile forces insidean invadopodium to pull the nucleus through the narrow gap between neighboringendothelial cells. Most interestingly, the required tension for the nuclear transmi-gration is approximately 70 nN for wet contact and 20 nN for dry contact. These68two values straddle that of 38 nN predicted by the earlier model of Cao et al. [59].Which more closely approximates reality will be tested by experimental measure-ments.4.3 Parametric studiesThe ”baseline” set of parameters tabulated in the section. 4.1.5 is used in all thesimulation results so far. Besides, we have also done some limited exploration ofthe parametric space, in particular to determine how the outcome of TC transmigra-tion depends on key parameters, including the size of the gap (rg) in the endothe-lium, the ECM rigidity and the time scale of the chemical signal that stimulatesmyosin activation. These explorations have been done in the dry-contact setup forits lower computational cost.Effect of the gap size.Figure 4.12 represents the impact of gap size on the driving force. A successfulextravasation is characterized by a tension force τ that rises in time as the nucleusdeforms and squeezes slowly through the gap, and falls precipitously upon break-through. Such is the case for the larger gaps (rg ≥ 1.2 µm). Note that as the gapshrinks in size, the force rises to larger values as the nucleus deforms more severelybefore transmigration occurs. It also takes longer time. For a gap that is too narrow(rg = 0.9 µm), the nucleus becomes stuck and fails to extravasate, suggesting a crit-ical ratio rg/rn = 0.3 ∼ 0.33 for passage. Incidentally, the previous model of Caoet al. [59] predicts a critical ratio of rg/rn = 0.3 for the dry-contact setup, close toour predictions. This is also consistent with experimental observations that arresthappens at pores whose cross-sectional area is 10% of that of the nucleus [130].Effect of ECM rigidity.Figure 4.13(a) shows the effect of shear modulus GECM of the ECM on the drivingforce τ due to SF contractility. The key observation is that a stiffer ECM facilitatestransmigration. For GECM ≥ 500 Pa, the ECM is sufficiently stiff to provide secureanchoring for the SF-FA machinery such that the latter produces a sharply rising τto pull the nucleus through. Increasing ECM modulus makes the passage faster. On69Figure 4.12: Effect of the EC gap radius rg on extravasation of the TC nucleusin the dry-contact simulations. The outcome is depicted by the tensionin the SF-FA assembly, and extravasation fails for too small a gap (rg≤0.9 µm).the other hand, softening the ECM below 500 Pa has a more qualitative effect andtransmigration eventually fails for GECM ≤ 100 Pa. As illustrated by the snapshotin Fig. 4.13(b), the soft ECM deforms greatly in response to the tension in the stressfibers, and the nucleus remains stuck at the endothelial gap. This again illustratesthe importance of having a secure anchoring pad for the contractile machinery.The effect the ECM rigidity can be compared with the effect of the cantileverspeed in measuring the extravasation force. A higher cantilever speed leaves theECs less time to relax and adapt to the deformation brought on by the TC, and thusthe ECs will appear more rigid. As a result, the force rises more rapidly in Fig. 11,similar to the trend in Fig. 4.13(a). Conversely, a slower cantilever speed leads toeffectively softer ECs and a slower increase in the force. The two situations differ,however, in that softer ECs favor extravasation in the experiment while a softerECM impedes extravasation here.Effects of duration of the activating signal.In our biochemical model, the actomyosin activation is driven by a chemical signal,e.g. an ionic influx of Ca2+. In Eq. (4.1), this signal decays exponentially in time70(a) (b)Figure 4.13: Effect of the ECM modulus GECM on extravasation of the TCnucleus in the dry-contact simulations. (a) Temporal evolution of thetension τ for 4 values of the ECM modulus. Extravasation fails ifthe ECM is too soft (GECM ≤ 100 Pa). (b) A snapshot of the ECMdeformation at t = 210 s for GECM = 100 Pa shows great deformationof the soft ECM that accompanies failure of extravasation. The colorcontours indicate the vertical component of the stretching.with a time constant θ . Figure 4.14 plots the time evolution of the contractile forceτ for four values of θ . If we lengthen θ from the baseline value of θ = 100 s,the contractile force τ grows more slowly initially, but in time reaches a similarmaximum before declining. The passage takes a longer time to complete. As weshorten θ , however, the decay of the stimulating signal is faster, and τ achieves arelatively low maximum force. For θ ≤ 10 s, this maximum falls below what isneeded to deform the nucleus sufficiently and transmigration fails.4.4 Summary and discussionIn this chapter, we proposed a biomechanical model for the trans-endothelial mi-gration of a tumor cell (TC) during extravasation, with the aim of testing a hy-pothesis established by Chen et al. [2, 5] that the transmigration is driven by thecontractile force in the invadopodium protruding from the cancer cell to the ECMoutside the endothelium. The model integrates several components into a coher-ent framework: the elastic deformation of the tumor and endothelial cells and theECM, the fluid flow inside the TC cytosol, and the coupled dynamics of the stress71Figure 4.14: Effect of time scale θ of the activation signal (see Eq. 4.1).Transmigration requires a minimum θ , which is between 10 and 50s.fibers (SF) inside the invadopodium and the focal adhesions (FA). Using a set ofparameters chosen to approximate a realistic extravasation, the model predicts suc-cessful transmigration of the TC. The main results of the work can be summarizedas follows:• Triggered by an external signal such as an influx of calcium ions, the SFassembles and grows simultaneously with the FA, thanks to a positive feed-back between the tension and actomyosin activation, and another betweenthe tension and growth of the focal adhesions through growing high-affinityintegrins.• Driven by the tension in the SF-FA assembly, the TC nucleus deforms asit enters a prescribed gap representing an endothelial junction. In the meantime, the endothelial cells (ECs) and the ECM also deform until a criticalpoint when the TC passes through the gap to extravasate.• The measured resistance force, in the range of 70∼100 nN [146], is con-sistent with model predictions of a required driving force of 70 nN. A sim-pler “dry-contact” model, which ignores the TC cytosol and membrane andassumes solid-solid contact between the TC nucleus and the endothelium,72under-predicts this force grossly.Moreover, these results support the invadopodium-based mechanism for theextravasation hypothesized by Chen et al. [2, 5]. Quantitatively, the invadopodiumis able to generate the force required to pull the TC through a narrow openingin the endothelial layer. Besides, the comparison between the model predictions,using both the dry-contact and the wet-contact setups, with the experimental datademonstrates the need to account for the cytosol and membrane of the tumor cells.Otherwise, a dry-contact model would incorrectly predict transmigration at drivingforces that are too low by a factor of about 3.5.We must emphasize the limitations of this work as well. Extravasation of tu-mor cells is a highly complex process, with multiple biochemical and mechanicalfactors at work. Of necessity some of these are neglected in the modeling. Per-haps the most important of these is the remodeling of the endothelium, which canbe through mechanical [147] and biochemical pathways [50]. Stretching and fluid-induced shear stresses not only remodel the cytoskeleton of ECs [147, 148] but alsothe cell-cell junctions [149]. Biochemically, cancer cells promote stress fiber as-sembly and actomyosin contraction inside ECs through the Rho pathway, causingretraction of the ECs and opening of their junctions, both facilitating the passageof the tumor cell [55, 150, 151]. They can also directly attack the EC junctions bydisrupting the VE-cadherin complex [50, 56, 152].Another potentially important omission is the biochemical pathways for ac-tomyosin activation. This has been lumped into a simple “activation signal” thatdecays exponentially in time. In reality, there exist multiple pathways for such ac-tivation, e.g. for actin polymerization through the Rac1 pathway and for myosinassembly by the RhoA pathway [83], which may add nuances to the developmentof the tensile force in the invadopodium.Finally and more technically, evaluation of the model parameters has been adifficult task. For one, certain model parameters, such as the magnitudes of theactivation and deactivation rates and the duration of the external signal, are notexplicitly available from prior measurements, and we have adopted values fromprior modeling. In view of the above limitations, one should consider the modelpredictions in this study a tentative validation of the hypothesis that we have set out73to test, subject to refinement of the various model assumptions and more accurateevaluation of the parameters.74Chapter 5Case study III: Amoeboidtransmigration of leukocytes andtenertaxisTo explore the tissue level of immune response in the long-on-a-chip device, Huhet al. [1] stimulated the epithelium on the air channel with a pro-inflammatory cy-tokine and released immune cells to the fluid perfusing through the blood channel.This immune reaction requires passage of leukocytes through endothelium. Byusing real-time fluorescent microscopy, they observed that leukocytes migrate onthe apical membrane of the endothelium until they reach a junction between twoadjacent endothelial cells and then they transmigrate through the junction.In general, leukocytes can directly transmigrate through the body of endothe-lial cells or through the gap between adjacent endothelial cells. These are known,respectively, as the transcellular and paracellular route of diapedesis. What reg-ulates the usage of one route over the other is still ambiguous. Martinelli et al.[63] recently proposed the so-called ”tenertaxis” hypothesis, according to which,leukocytes find and traverse the path of least mechanical resistance [63].Martinelli et al. [63] conducted several experiments to examine the correla-tion between junctional integrity and the route of diapedesis. They did in vitroexperiments of diapedesis using various types of endothelial cells (ECs) such asrat brain and heart endothelium. The brain endothelium has stronger junctional75integrity than heart endothelium. They observed that in rat brain endothelium thenumber of transcellular diapedesis is higher than the paracellular route, while inrat heart endothelium the usage of paracellular route is dominant. They also en-hanced or disrupted endothelial junctions using pharmacological agents and hor-mones. They noticed that disrupting junctional integrity leads to a remarkableincrease (about two folds) in paracellular diapedesis accompanied by decrease intranscellular transmigration. They also used a shear flow on the endothelium asa mechanical modifying agent to promote the junctional strength and remodelingof the cytoskeleton. This alteration causes a significant increase in transcellulardiapedesis. They concluded that strong junctional integrity is correlated with dom-inant transcellular route of transmigration, and the EC junction tightness and localstiffness are the major determinants of the route of diapedesis. As a result, theyhypothesized that leukocytes choose a path with the least mechanical resistanceduring the transmigration.In this chapter, we aim to examine this hypothesis using a simplified model forendothelium and leukocytes with invadosome-like protrusions. In the followingwe first describe the details of our model. Then we present the results of resistanceforce during the penetration of the protrusions through the trancellular and para-cellular routes, both for the control endothelium with standard properties and forendothelia with altered junctional and cellular properties.5.1 Problem setupLeukocytes extend protrusions into the EC to explore and sample the mechanicalresistance of the EC surface and find the least resistance route. In fact, when aprotrusion reaches the basal surface of the EC, an opening forms in the endotheliumthrough which diapedesis of leukocytes can occur [66]. Essentially, finding thepath of least resistance for leukocytes is an exercise in comparing between themechanical resistance due to the deformation of the EC body to form a transcellulartunnel and the mechanical resistance against a protrusion through the endothelialjunctional opening. Therefore, our model is designed to simulate the mechanicalresistance of the ECs monolayer against the protrusive force of the leukocytes ineither route.76Figure 5.1: Schematic of a leukocyte extending protrusions on the endothe-lium.As shown in Figure 5.1, our model involves three elements: (i) elastic ECs andthe basement membrane underneath, (ii) leukocytes protrusions and (iii) adhesionmolecules at the endothelial junction.(i) We treat the ECs and the basement membrane as hyperelastic solid compo-nents obeying the neo-Hookean constitutive equation and ignored the cytosol andother structures inside the cell [75]. Besides, in our model we can set the elasticmodulus of the EC near the edges and the EC body separately. It is worth mention-ing that the EC nuclues is very stiff and diapedesis hardly happens in the nucleusarea [66]. Therefore, we do not consider a protrusion on the nucleus region in ourmodel.(ii) We considered the leukocytes protrusion as a rigid rod with a diametersimilar to the average diameter of protrusions observed in experiments. The rodpenetrates the EC cell body or the EC junction with prescribed displacement.(iii) Cell-cell adhesion molecules, e.g., vascular endothelial cadherins (VE-cadherins) are modeled as a distributed linear spring between two adjacent ECs[153, 154].5.1.1 Geometric setupThe micro-environment of the protrusions is composed of a monolayer of ECs andthe basement membrane underneath. The length and depth of a spread EC are 13.2µm and 25.8 µm [155] and its height is between 1∼3 µm depending on the locationof the nucleus [126]. The shape of the spread EC is like a Gaussian function [156–158]. The nuclear bulge has a maximum height of 3 µm, [63, 159]. The basement77(a) (b)Figure 5.2: Geometric setup for the simulations of (a) the paracellular pro-trusion and (b) the transcellular protrusion.membrane underneath the endothelium has a thickness of 2 µm [127]. This layerprovides a biologically relevant environment for the modeling of endothelium. Theequilibrium gap size between two neighbouring ECs ranges between 10 ∼ 100nm [160, 161] and we chose 10 nm for our simulations. The average size of theprotrusions is 340 nm in diameter [66].Figure 5.2 shows the geometries that used for the simulations of the protru-sions. Without loss of generality, we used a portion or subdomain of the endothe-lium for the simulations to reduce the computational cost. The subdomain usedfor the modeling of the paracellular protrusion is a three-dimensional cylinder withthe diameter of 3 µm and includes two adjacent ECs, the junction and the base-ment membrane underneath (Fig. 5.2a). For the transcellular protrusion, we used atwo-dimensional axisymmetric domain with the radius of 1.5 µm which representsa part of the EC body (away from the nucleus and the junction) and the basementmembrane beneath (Fig. 5.2b). The protrusion is modeled by a rod with diameterof 340 nm similar to the average diameter of protrusions reported in experiments.The rod has a cylindrical body and a hemispherical head.785.1.2 Parameters estimationIn this section the baseline parameters that are required for the simulation of theextending protrusions are presented. These parameters can be categorized into twogroups: parameters related to the EC body and parameters related to the endothelialjunction. Important baseline parameters were listed in Table 5.1.Endothelial cellsEndothelial cells contain different organelles and structures that can withstandexternal stresses. The nucleus is the stiffest organelle inside the cell and cy-toskeleton is the major structure that determines the response of the cell to in-dentation [64]. Many studies have been done to measure the elastic modulusof an EC body and in general the reported values vary between 200 ∼ 5600 Pa[59, 64, 135, 137, 158, 162]. For example, Caille et al. [137] measured the elasticmodulus of the cell body and nucleus to be 500 Pa and 5000 Pa, respectively, usingcompression of an endothelial cell between two microplates. To find the elasticmodulus of endothelial cells at different regions of the cell, Mathur et al. [135]conducted AFM indentation experiments on the human umbilical vein endothelialcells (HUVECs) monolayer at different location of the cell. The results of line fit-ting on the force-indentation curve showed that the elastic modulus is 6.8 kPa onthe top of the nucleus, 3.3 kPa near the nucleus and 1.4 kPa away from the nucleusand in the proximity of the edges. Costa et al. [162] carried out the measurementof pointwise elastic modulus of human aortic endothelial cells using the AFM in-dentation technique. They measured an average elastic modulus of 1.5 kPa for thecell body away from stress fibers and 5.6 kPa for a location above stress fibers. Wehave adopted the value of 500 Pa (corresponds to the elastic modulus of heart ECs)for the elastic modulus of the EC body (including the EC edges) when simulatingthe paracellular protrusion. Also the elastic modulus of the basement membrane is5 kPa similar to the elastic modulus of extracellular matrix [59].During transcellular diapedesis, a leukocyte adheres to the apical surface of anEC, and actively remodels the EC cortex underneath [163, 164]. Thus, the effectiveEC modulus is much reduced in comparison to those measured by passive mechan-ical deformation. To quantify this effect, Isac et al. [163] conducted the following79experiment. They removed an attached leukocyte from the endothelium under-neath using a specific nano-surgery at the initial stages of transcellular diapedesis.By using fluorescence microscopy they observed that actin filaments were depoly-merized at the interaction site and the cytoskeleton were remodeled. Moreover, byusing scanning electron microscope (SEM), Barzilai et al. [164] clearly showedthat the actin filaments are disassembled underneath the leukocyte protrusion. Isacet al. [163] also measured the elastic modulus of the EC beneath the leukocyte atthe interaction site using AFM indentation. They showed that the interaction siteis maximum 10 folds softer than its surrounding (∼ 300 Pa versus ∼ 3000 Pa).They used human umbilical vein endothelial cells for their experiments while Mar-tinelli’s experiments were done using rat heart or brain endothelial cells [63]. Sincewe wanted to model Martinelli’s experiments, we adopted the fact of maximum 10folds softening at the interaction site from Isac’s experiments and we did not usetheir reported value for the elastic modulus of the EC body. we used Ebody= 50Pa for the elastic modulus of the EC body during the transcellular protrusion (inaccordance with our choice of the elastic modulus of the Ebody= 500 Pa near theedges for the paracellular penetration).Endothelial junctionsEndothelial cells are mechanically connected to the neighboring cells mainly bytheir VE-cadherin proteins. VE-cadherins are force sensitive proteins and theirlifetime increases with increasing tension on VE-cadherin/VE-cadherin bonds be-fore rupture (catch-bond behavior) [153, 165]. Force dependence of the bond canbe modeled by a linear elastic spring [65, 153, 154]:fbond = kadh (Lbond−L0) (5.1)Where fbond is the force of each bond, kadh is the spring constant, Lbond is the bondlength and L0 is the equilibrium bond length. The spring constant kadh was reportedto be between 10−5 ∼ 10−2 N/m for cell-cell adhesion bonds [153]. The equilib-rium bond length was reported to be between 10 ∼ 100 nm [65, 154]. Moreover,each bond can sustain 50 pN for high loading rate (1000 pN/s) and 32 pN for lowloading rate (100 pN/s) [166]. Liu et al. [167] used microfabricated force sensors80to measure the force at the endothelilal cell-cell junction. They also measured thearea of cell-cell contact and reported the stress of 1 nN/µm2 at the endothelial cell-cell junction due to VE-cadherins. The normal stress, i.e. the total force betweentwo neighboring ECs per unit area, can be related to the bond force by:Fbonds = Nb fbond (5.2)Where Nb is the bond density per unit area. Nb can be modeled by a kinetic equationfor formation and dissociation of bonds in which the rate constants are functions oftension in the bonds [65, 154]. We ignored the kinetics of Nb and used the equilib-rium value of 150 µm−2 for the density number of VE-cadheins bonds [154]. Thecontact area of the two adjacent ECs in our 3D simulations has a height of 1 µmand depth of 3 µm. Considering all the aforementioned constraints and the valueof stress (1 nN/µm2) due to VE-cadherins, we used the value of 0.01 N/m for theequivalent spring constant (Keq = Nbkadh).Symbol Description Value SourcesEEC(P) Elastic modulus of the EC body (Paracellular case) 500 Pa [137]EEC(T ) Elastic modulus of the EC body (Transcellular case) 50 Pa [137, 163]Eedge Elastic modulus of the EC edge 500 Pa [137]EBM Elastic modulus of the basement membrane 5000 Pa [59]ν Poisson’s ratio 0.3 [59]Keq Equivalent VE-cadherins spring constant 0.01 N/m CalculatedL0 Equilibrium VE-cadherin bond length 10 nm [154]Table 5.1: Important baseline parameters used in our model.5.1.3 Numerical setupWe treated the ECs and the basement membrane as Neo-Hookean hyperelstic ma-terials. The constitutive equation for these material isσ = GJ−53(FFT − I3(FFT ))−K(J−1)I, (5.3)81where Fi j = ∂xi/∂X j is the deformation gradient tensor, with X and x being the un-deformed and current positions of a material point, and J = detF . The coefficientsG and K are the shear and bulk modulus, respectively, connected via the Poissonratio ν : K = 2G31+ν1−2ν . Finally, the governing equation of solid deformation is givenby:∇ ·σ = 0. (5.4)We used the augmented Lagrange method and the penalty method to model thenormal contact force between the rod and the EC surface during transcellular andparacellular penetration, respectively [124]. Also we assumed a frictionless con-tact between the protrusion and the ECs. The sets of equations were solved usingCOMSOL Multiphysics.Boundary conditionsThe basal surface of the basement membrane has a zero displacement boundarycondition. The contact surface of the two adjacent ECs has a spring foundationboundary condition in order to model the cell-cell adhesion force because of VE-cadhein bonds. To model the protrusion, we used a rod similar to the average sizeof protrusions above the junction or the cell body and prescribe the kinematics of itsmovement into the cell, in terms of the displacement drod(t). We then compute howthe resistance force on the rod, Fr(drod), varies with the depth drod . Because theelastic deformation happens instantaneously, without viscous damping, essentiallywe are computing a series of quasi-static states with the rod at different positionsof insertion. Thus, the speed of motion d′rod(t) plays no role in the result.The simulation ends when the rod displacement drod reaches a threshold db forbreakthrough. This threshold is set to the thickness of the EC monolayer (db =1µm) for the paracellular protrusion and db =0.98 µm for the transcellular protru-sion. The threshold of 0.98 µm for the transcellular protrusion was chosen becausethe thickness of the EC membrane is 10 nm [65, 131], and at this point the apicalmembrane reaches the basal membrane so they may open up a transcellular tunnel[66]. All the other surfaces are free to deform with no loads or constraints.82(a) (b)Figure 5.3: (a) Contour plot of the von-Mises stress (in Pa) when the rodreaches the basement membrane during the paracellular protrusion. Tohave a better view, half of the EC domain was removed in the figure. (b)Mid-plane of the domain.5.2 Results and discussionIn this section we first present the results using baseline parameters and then in-vestigate the effects of manipulating the elastic modulus of the endothelium. Fur-thermore, we focus on the results of the contact resistance force against extendingthe transcellular or paracellular protrusions. This resistance force can be used asa measure to compare the mechanical resistance of the transcellular route with theparacellular route, and to predict which is the preferred route with smaller resis-tance.5.2.1 Baseline resultsParacellular routeFigure 5.3(a) shows the level of von-Mises stress in 3D at the end of paracellularpenetration. Figure 5.4 shows the snapshots of the von-Mises stress at the mid-plane during penetration of the rod through the endothelial junction and Figure 5.5is the plots of vertical contact force as a function of displacement of the rod. Be-83Figure 5.4: Progression of the rod in the paracellular route at the mid-plane(refer to Fig. 5.2). The color bar shows the level of von-Mises stress inPa.cause of the geometry of the problem, the rod travels downward by 0.1 µm beforeit makes contact with the EC surfaces at the top of the junctional gap. Initially, therod deforms the upper edges of the ECs at the junction and the resistance force islarge. The first peak of the resistance force is about 10 pN, and it happens when therod pushes the edges of the ECs apart and opens up a gap. Then the contact forcedecrease to about 6 pN and slightly decreases when the rod proceeds in the gap.Again the force increases when the rod starts to deform the basement membranewhich is much stiffer than the ECs (with a elastic modulus ten times as large). Thesecond peak of the resistance force occurs at the end of the penetration when therod reaches the basement membrane. During the paracellular protrusion, the max-imum resistance force is about 15 pN and it occurs at the end of the penetration.It was shown in earlier studies that the protrusive force is mainly provided bythe actin filaments polymerization in the core of the protrusion and myosin is ex-cluded from the core [67]. Direct measurement of the generated force by an actinfilament polymerization revealed a value of 1.5 pN [168]. Each protrusion con-tains at least 10 actin filaments [169] so actin polymerization inside the core ofthe protrusion can easily provide 15 pN to accomplish the paracellular penetration.Because of the quasi-static setup of the problem, in which we prescribe the dis-placement of the rod, the force-displacement curve of Figure 5.5 has interestingimplications for a “dynamic simulation” where we prescribe the pushing force Fr84Figure 5.5: Plot of the vertical contact force versus the rod displacement. Thedashed arrow suggests how the rod will pass over an unstable portion ofthe curve in a dynamic simulation with a prescribed force.on the rod, and compute its displacement drod(t). If the force Fr is below 10 pN,the rod will stop on the first up-slope of the curve in Figure 5.5. With larger forces,the rod passes the peak, and then jumps over an unstable portion of the curve toland on a larger displacement on the second up-slop of the curve. In fact, we havedone such dynamic simulations to confirm the above scenarios. Of course, therewould be a jump in the opposite direction if we start with an initial state of fullpenetration and then gradually decrease the protrusion force. But such a scenariois not relevant to the diapedesis process.Transcellulr routeFigure 5.6 shows the level of von-Mises stress during the transcellular penetrationof the EC and Figure 5.7 is the plot of the vertical contact force during progressionof the rod. The protrusive force on the EC produces a transcellular tunnel similarto the finger-like protrusions of leukocytes [63]. The force increases non-linearlywith the depth of penetration due to the hyperelastic behaviour of the EC and alsothe geometric non-linearity of the deformation. Therefore, when the protrusionreaches the threshold of db=0.98 µm, the resistance force reaches to its maximum85Figure 5.6: Snapshots of the level of von-Mises stress (in Pa) during transcel-lular tunnel formationof 22 pN which corresponds to the average normal stress of 242 Pa at the tip of therod.Generally, in vitro and in vivo experimental studies [63, 170, 171] demon-strated that paracellular transmigration is the preferred mode that happens most ofthe time. Based on our baseline simulation results, the required force to overcomethe mechanical resistance in paracellular and trancellular routes are 15 pN and 22pN, respectively. This shows that the penetration is easier using the paracellularroute, in good agreement with experimental observations [63, 161]. To probe thetenertaxis hypothesis, however, we need to manipulate the relative level of resis-tance in the EC body and junctional areas. This was achieved in prior experimentsby drugs or hormones treatment, shear flow effect, and comparing different celltypes that offer different levels of junctional resistance. In the following, we willsimulate such changes in the endothelial resistances.Effect of the elastic modulus of the EC bodyAs we mentioned in section 5.1.2, the maximum softening of the EC body duringtranscellular penetration is about 10 folds. In this section, we present the resultsfor the parametric study on the lower level of the softening of the EC body. wechanged the elastic modulus of the EC body from Ebody=50 Pa to Ebody=100 and86Figure 5.7: Plot of resistance force versus rod displacement500 Pa for the transcellular penetration. Figure 5.8 shows the plots of resistanceforce during transcellular penetration. The peak forces for the penetration to the ECbody with Ebody=100 Pa and Ebody=500 Pa are 44 pN and 199 pN, respectively. Asexpected, these values are much higher than that of 22 pN predicted for the baselinemodulus of Ebody = 50 Pa. Interestingly, Ng et al. [172] used the tip of a cantileverin an atomic force microscope to exert a mechanical force on the cell body ofHUVECs. They measured the mechanical force varying between 5∼ 100 nN. Thisindentation induced a transcellular tunnel with a diameter about 2 µm. Hence,the normal stress for the formation of this tunnel should be between 1600 Pa and31800 Pa. The average normal stress based on our simulation to penetrate the cellbody for Ebody=500 Pa is about 2200 Pa which is within the range reported by Nget al. [172]. Obviously, their AFM cantilever does not disrupt the EC cortex as anattached leukocyte would [163]. Thus, a much greater force or pressure is neededon the AFM cantilever to produce the transcellular tunnel than the protrusive forcerequired of a leukocyte in vivo.5.2.2 Effects of manipulating of the endotheliumTo simulate the experimental manipulations of the EC resistance in the junctions,we did parametric studies on the elastic modulus of the EC near the edge for para-87Figure 5.8: Plots of resistance force vs displacement during transcellular pen-etration for the various elastic modulus of the EC body.cellular route.Studies done by Martinelli et al.[63] showed that enhancing the junctional in-tegrity by drugs increases the level of cortical F-actin near the EC edges. Fur-thermore, to test the effects of altering junctional integrity by drugs or hormonestreatment on the mechanical resistance of the gap, they did AFM nanoindentationexperiments on the junctions. They observed that by disrupting the junctional in-tegrity in rat heart endothelium the transendothelial electrical resistance decreases20% and the elastic modulus near the junction decreases to 0.75% of its initialvalue. Besides, the transendothelial electrical resistance increases 25% and theelastic modulus increased to 175% of its initial value when they enhanced the junc-tional integrity. Moreover, Costa et al. [162] showed that the existence of stressfibers underneath the tip of cantilever in the AFM indentation experiment on theEC cell body increases the measured elastic modulus by about three folds. It canbe concluded that the increase in the junctional strength is mainly due to the in-crease in the level of F-actin filaments and can be translated to an increase in theelastic modulus of the EC near the edges (Fig. 5.2) in our simulations. Consideringthese experimental data, we chose increases by a factor of 1.75 and 4, respectively,in the elastic modulus of the ECs near the edges with respect to its baseline value88Figure 5.9: Plots of resistance force vs rod displacement during paracellularpenetration for various elastic modulus of the ECs near the junction(E=500 Pa). Therefore, we have tested Eedge=875 Pa and Eedge=2000 Pa to seehow the junctional strength affects the required force to overcome the mechanicalresistance of the the paracellular penetration.Figure 5.9 plots the resistance force during rod penetration for various elasticmodulus near the ECs junction. The maximum resistance force for the strength-ened junction endothelium is 24 pN and 49 pN for Eedge=875 Pa and Eedge=2000Pa, respectively. Therefore the maximum resistance force for the strengthenedjunction has increased markedly from that value of 15 pN corresponding to thewide-type EC modulus. The model shows that by strengthening the EC junctions,the resistance of paracellular transmigration can surpass that of the transcellularroute. This will switch the preferred route of leukocyte diapedesis from paracellu-lar to transcellular, as observed by [63], in support of the tenertaxis hypothesis.Comparison of the resistance force from the simulation results for the softestEC body (Fresistance = 22 pN) and the stiffest junction (Fresistance = 49 pN) showsthat there might exist situations that the transcellular route is the favorable path andthe protrusion requires less force to penetrate and reach the basal surface of the EC.Martinelli’s experiments [63] have demonstrated that manipulation of the relativestrength of endothelial resistance can make the transcellular route preferable. Our89simulations have confirmed this shift in usage of the route of diapedesis, and thustentatively confirmed the hypothesis of tenertaxis.5.3 Summary and discussionImmune response to infections requires frequent crossing of leukocytes throughthe endothelium. Leukocytes transmigrate either directly through an individual EC(transcellular route) or from the junction between ECs (paracellular route). It isunclear what determines the usage of one route over the other. To address thisquestion, Martinelli et al. [63] proposed the idea of tenertaxis. According to thishypothesis, leukocytes choose the path with least mechanical resistance againstprotruding. Leukocytes extend and retract protrusions into the apical surface of theendothelium. Thus, they sense the stiffness of the surface of ECs or the strength ofthe cell junctions to find the path of minimum mechanical resistance.We tested this hypothesis using numerical simulation of hyperelastic deforma-tion during the extension of paracellular and transcellular protrusions. Based on theliterature, paracellular route is the prevalent route. Moreover, experimental stud-ies demonstrated that the elastic modulus of the EC body beneath the leukocytemay decreases drastically due to depolymerization of the actin filaments and thecytoskeleton remodeling. By considering the baseline parameters, our simulationresults showed that the required force to penetrate the endothelium from transcel-lular route is greater than the paracellular route (15 pN vs 22 pN). This rationalizesthe preference of leukocytes to use the paracellular route most of the time.By enhancing the junctional strength of the wild type endothelium, the elasticmodulus of the EC near the edges increases due to increase in the level of corticalactin filaments. Experiments have demonstrated that manipulation of the relativestrength of endothelial resistance can make the transcellular route preferable. Weshowed that there might be situations that the mechanical resistance of the tran-scelluar route is less than the paracellular route, which would leads to a switchin the preferred route. Therefore, our simulations have tentatively confirmed thehypothesis of tenertaxis. It should be noted that the confirmation of the tenertaxishypothesis might have clinical implications for anti-inflammatory drugs develop-ment [63].90Chapter 6ConclusionsBio-mimicking microfluidic devices have been established to provide a biologi-cally relevant framework for drug development and toxicological studies. In thisdissertation, we have described three case studies in the area of particle and celltransport in bio-mimicking microfluidic devices:• Submicron particle transport and deposition in the alveolar channel of a lung-on-a-chip (LOAC) device• Mesenchymal transmigration of cancer cells through endothelial monolayerin a microvasculature networks (µVNs) device.• Amoeboid transmigaration of leukocytes through endothelial monolayer invascular channel of a LOAC device.In the following, we will present the concluding remarks and the potential prospec-tive works for each case study.Transport and deposition of submicron particles in a LOAC deviceThe effect of air-borne submicron particles (SPs) on human health is an active areaof research, with clinical relevance evidenced by the current COVID-19 pandemic.As in vitro models for such studies, LOAC devices can represent key physical andphysiological aspects of alveolar tissues. However, wide-spread adoption of the91LOAC device for SP testing has been hampered by low intra-laboratory and inter-laboratory reproducibility. To complement ongoing experimental work, we carriedout finite-element simulations of the deposition of SPs on the epithelial layer of awell-established LOAC design.We solved the Navier-Stokes equations for the fluid flow in a three-dimensionaldomain, and studied the particle transport using Eulerian advection-diffusion forfine SPs and Lagrangian particle tracking for coarse SPs. Using Langmuir andFrumkin kinetics for surface adsorption and desorption, we investigated SP ad-sorption under different exercise and breath-holding patterns. Conditions mimick-ing physical exercise, through changes in air-flow volume and breathing frequency,enhance particle deposition. Puff profiles typical of smoking, with breath-holdingbetween inhalation and exhalation, also increase particle deposition per breath-ing cycle. Lagrangian particle tracking shows Brownian motion and gravitationalsettling to be two key factors, which may cooperate or compete with each otherfor different particle sizes. Comparisons are made with experimental data wherepossible, and show qualitative and semi-quantitative agreement. These results sug-gest that computer simulations can potentially inform and accelerate the designand application of LOAC devices for analyzing particulate- and microbe-alveolarinteractions.The significance of the results lies mainly in identifying the principal factorsthat determine deposition in an LOAC device, and delineating their effects indi-vidually and collectively on the outcome. These factors include surface adsorptionand desorption kinetics, breathing patterns and puff profiles in smoking, Brown-ian force on fine particles and gravity on larger ones. Besides, our results provideguidelines for designing future LOAC devices and measurement protocols in termsof the pattern of depositions to anticipate and the key data to record. Moreover,these findings may have several clinical implications. First, because physical ex-ercise leads to increased particle deposition in the airways, it is advisable for indi-viduals to avoid or curtail outdoor exercise on days with poor air quality. Second,smokers should be warned against prolonged breath-holds during puffs. In con-trast, for individuals with asthma or chronic obstructive pulmonary disease whouse inhalers for a clinical indication, prolonged breath-holding should be encour-aged. Future studies might involve extending the model to incorporate more com-92plex biological processes, e.g., transport of the deposited SPs through the mucuslayer atop the epithelium, and transmigration of particles through blood-air barrier.We must note that as in any modeling work, certain simplifications and as-sumptions have to be made to make the problem tractable. One such assumption isthe Langmuir and Frumkin kinetics for surface adsorption and desorption, whichare both valid for a monolayer of adsorbed particles. In the current parameterrange, where the surface areal coverage is sparse, the monolayer models are suit-able. For greater amount of deposition, a multilayer adsorption model such as theBrunauer-Emmett-Teller (BET) model may be more appropriate [121]. In addi-tion, the estimation of parameter values is subject to a degree of uncertainty, asdiscussed in section 3.1.4. Finally, in Lagrangian particle tracking, we are unableto include sufficient number of particles to reach the appropriate bulk concentra-tion. This is just a matter of linearly increasing the amount of computations, sincethe particles do not interact, and should not have affected the qualitative trend inthe results.Mesenchymal transmigration of cancer cellsIn this study, we modeled the chemo-mechanics of the stress fibers and the focaladhesions by following the kinetics of the active myosin motors and high-affinityintegrins, subject to mechanical feedback. This is incorporated into a finite-elementsimulation of the extravasation process, with the contractile force pulling the nu-cleus of the tumor cell against elastic resistance of the ECs. To account for theinteraction between the TC nucleus and the endothelium, we consider two scenar-ios: solid-solid contact and lubrication by cytosol. The former gives a lower boundfor the required contractile force to realize transmigration, while the latter providesa more realistic representation of the process. Using physiologically reasonableparameters, our model shows that the stress-fiber and focal-adhesion ensemble canproduce a contractile force on the order of 70 nN, which is sufficient to deform theECs and enable transmigration.Our results have provided insights into the role of mechanical properties ofthe tumor cell and its micro-environment in a successful extravasation. This studycan be extended by incorporating the active remodeling and junction opening of93the endothelial cells in response to initiation of extravasation by TCs and also bio-chemical signaling pathways that regulate the level of myosin activity in stressfibers.We must emphasize the limitations of this work as well. Extravasation of tu-mor cells is a highly complex process, with multiple biochemical and mechanicalfactors at work. Of necessity some of these are neglected in the modeling. Per-haps the most important of these is the remodeling of the endothelium, which canbe through mechanical [147] and biochemical pathways [50]. Stretching and fluid-induced shear stresses not only remodel the cytoskeleton of ECs [147, 148] but alsothe cell-cell junctions [149]. Biochemically, cancer cells promote stress fiber as-sembly and actomyosin contraction inside ECs through the Rho pathway, causingretraction of the ECs and opening of their junctions, both facilitating the passageof the tumor cell [55, 150, 151].Another potentially important omission is the biochemical pathways for ac-tomyosin activation. This has been lumped into a simple “activation signal” thatdecays exponentially in time. In reality, there exist multiple pathways for such ac-tivation, e.g. for actin polymerization through the Rac1 pathway and for myosinassembly by the RhoA pathway [83], which may add nuances to the developmentof the tensile force in the invadopodium.Amoeboid transmigration of leukocytes and tenertaxisLeukocytes require trafficking through endothelium during immune reaction. Leuko-cytes transmigration occurs either directly through the body of an EC (transcellu-lar route) or from the junction between adjacent ECs (paracellular route). It isunknown what determines the usage of one route over the other. Martinelli etal. [63] hypothesized that leukocytes seek the path of least mechanical resistance(tenertaxis). In the third case study, we examined this hypothesis using numericalsimulation of mechanical resistance during paracellular and transcellular penetra-tion. Model predictions show that normally the paracellular route is the prevalentroute. By considering the baseline parameters, our simulation results showed thatthe required force to penetrate the endothelium from transcellular route is greaterthan the paracellular route (15 pN vs 22 pN). This rationalizes the preference of94leukocytes to use the paracellular route most of the time.By enhancing the junctional integrity of endothelium using modifying agents,One may increase the elastic modulus of the EC near the edges. Moreover, theelastic modulus of the EC body beneath the leukocyte may decreases significantly.We showed that by increasing the elastic modulus of the EC near the junction, thereexist situations that the mechanical resistance of the transcelluar route becomesless than the paracellular route, which will make the transcellular route preferable.Our simulations have confirmed this reversal, and thus tentatively confirmed thehypothesis of tenertaxis.Our model can be extended in several directions. First, we have ignored theforce-generation mechanisms inside the protruding invadopodia. A more completemodel should integrate such mechanisms with the elastic deformation of the EC,much like in our Chapter 4. Second, it was observed that unsuccessful protru-sions tend to last a much shorter time [170]. Possibly there is a positive feedbackmechanism that prolongs the successful protrusions and we can incorporate thisfact to our model. Finally, endothelium actively remodel through mechanical andchemical signaling pathways [50, 147] during leukocyte diapedesis, and this activeresponse can be included in a future model.95Bibliography[1] Dongeun Huh, Benjamin D. Matthews, Akiko Mammoto, Martı´nMontoya-Zavala, Hong Yuan Hsin, and Donald E. Ingber. Reconstitutingorgan-level lung functions on a chip. Science, 328(5986):1662–1668, 2010.→ pages x, xi, xii, 1, 4, 11, 20, 21, 26, 30, 31, 41, 75[2] Michelle B. Chen, John M. Lamar, Ran Li, Richard O. Hynes, andRoger D. Kamm. 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