Computational hydrodynamics of capsules and biological cells
Modeling and simulation of capsules and biological cells

Pozrikidis wrote in the preface of Computational Hydrodynamics of Capsules and Biological Cells that ‘Computational biofluid-dynamics addresses a diverse family of problems involving fluid flow inside and around living organisms, organs and tissue, biological cells, and other biological materials’. Borhan and Gupta phrased it this way in Chapter 6 of Modeling and Simulation of Capsules and Biological Cells: ‘The motion of flexible particles [i.e. capsules and biological cells] in a confined domain, such as a fluid-filled channel or tube, is of interest to a variety of natural and biological processes and industrial applications’. This means that research about the dynamics of biological cells, which are considered ‘natural capsules’, can expand our understanding of the functionality of organisms under physiological, pathological, and therapeutical conditions. These data can be used to maintain or restore functionality of organisms so that they can operate successfully within a population. Artificial capsules, on the other hand, are used or evaluated in many applications, including in the pharmaceutical, cosmetics, and food industries for protecting the special properties and controlling the release of active substances (e.g. insulin, growth hormones, and antibiotics), aromas (e.g. of cheese), and flavours (e.g. of mushrooms) 1. Capsule technology is also explored in many bioengineering applications, such as transplantation of primary or genetically engineered cells for the treatment of many disorders and diseases 11.

Blood cells have been used for many years to study single-cell dynamics and cell–cell interactions in relation to blood flow under various conditions. Blood is a complex biological fluid composed of plasma (proteins and other solutes), deformable cells (erythrocytes and leukocytes), and rigid particles (platelets). Healthy human erythrocytes (red blood cells) have a biconcave shape and are surrounded by a highly deformable viscoelastic membrane. Tumbling, tank-treading, swinging, and stretching have been discussed as the cell's mechanical responses to flow 4. Erythrocytes contribute to blood viscosity through their individual motion, and viscosity of blood has been used as an indicator in the understanding and treatment of disease 3. In malaria, for example, erythrocytes are infected with Plasmodium parasites which causes a gradual stiffening of the red blood cells’ membranes significantly reducing their ability to deform and to perform their physiological function (i.e. atmospheric gas-uptake and -delivery), and to contribute to normal blood flow 2. In sickle cell disease, erythrocytes undergo dehydration and become distorted in shape (sickle cell appearance), show abnormal adhesion properties to the vascular endothelium, and slow down the blood flow in the microcirculation 7. Research about the fluid dynamics, aggregation properties, and paths of leukocytes (white blood cells) under controlled conditions can reveal information about vessel flow, blood cell interaction, and blood wall-adhesion and -penetration important for these cells to carry out their protective (immune) functions within an organism 6. Platelets are recruited in the event of injury and play a central role in the blood-clotting (thrombus) process. Research has concentrated on platelet motion in blood (e.g. shear-dependent flipping) and platelet-blood vessel wall adhesion dynamics and thrombus growth 910. Finally, significant effort has been made to understand the deformation and adhesion dynamics of (circulating) tumor cells that have detached from the primary tumor and entered into the blood stream in order to establish themselves in new locations within an organism (i.e. the process of tumor metastasis) 58.

There are several different ways to study cell dynamics to understand an organism's functionality. For instance, one can conduct laboratory experiments with living cells in vivo and in vitro, or observe the dynamics of artificially created (engineered) capsules and cells in appropriate model systems; yet, another way is to mathematically analyse biological processes, develop sophisticated mathematical models, and numerically simulate biological events in silico with the ultimate goal to quantitatively explain and predict cell behaviour. A combination of different approaches is often the best way to understand biological phenomena. This approach is used frequently in contemporary science because studying a biological system in its complexity can give researchers a deeper understanding of the overall functionality of a single organism from its molecular and cellular building blocks up to its place as a self-functioning unit within a population. In recent decades, it has become a frequent practice that investigators from different fields (e.g. biologists, physicists, chemists, mathematicians, and engineers, as well as physicians and pharmacologists) come together to solve problems in the biological and biomedical sciences in an interdisciplinary fashion.

The two books by Constantine Pozrikidis deal primarily with mathematical evaluations and in silico investigations (modeling and simulations) of particles in motion. I chose to review both books in this manuscript because I found that they complement each other in that information provided in one book is either absent, described in more detail, or expanded upon in the other. While Modeling and Simulation of Capsules and Biological Cells contains six chapters on various aspects of fluid flow, Computational Hydrodynamics of Capsules and Biological Cells has a total of eight chapters and goes beyond the scope of the first, but is not a second edition. Both books contain a collection of chapters contributed by investigators from around the world who provide their expert experiences in fields such as biology and physiology, mathematics, mechanical, and chemical engineering, as well as computer and information science. I will use the chapter layout of the more recently published book to describe its content and then discuss additional information provided by authors of the other book.

The first three chapters of Computational Hydrodynamics of Capsules and Biological Cells introduce the reader to the field of membrane mechanics and the use of boundary-integral formulations to study flow-induced deformations of capsules. More specifically, Pozrikidis describes in Chapter 1 the mathematical framework useful for studying the motion and deformation of erythrocytes in infinite and wall-bounded shear flow. We learn that in the absence of flow, normal erythrocytes, when suspended in an isotonic solution, have a unique biconcave disk shape. When conditions change (e.g. fluid flow and viscosities), we can observe capsule deformation; ‘flexural stiffness’ helps a deformed capsule to return to the unstressed biconcave shape. Pozrikidis discusses shear rates, internal and external fluid viscosities, and capsule motion near a wall. He points to MATLAB^® computer programs useful to simulate flow-induced deformations of two-dimensional biconcave capsules. He discusses the advantages and shortcomings of both two- and three-dimensional capsule models. The second chapter is about flow-induced deformation of artificial capsules. The authors describe how artificial capsules can be produced and emphasize that their mechanical properties depend on the fabrication method. They discuss the mechanics of thin membranes undergoing large displacements and deformations, and evaluate numerical models for fluid–structure interactions by studying spherical capsules that experience deformations under the effect of stresses (e.g. osmotic pressures). They point out that numerical models have sometimes the great advantage of describing variables that are difficult to measure experimentally, such as the elastic tension developing in a membrane. Chapter 3 provides a fascinating discussion of simulations of red blood cell motion in complex geometries (e.g. a model vessel network, as is depicted on the cover of the book) by using a high-resolution fast boundary-integral method for multiple interacting blood cells. They discuss the pros and cons of this method, and provide an outlook for potential future directions for investigation. The authors also mention red blood cell lysis (haemolysis), a phenomenon which can be caused by high shear stress; they point out that the quantitative prediction of haemolysis remains a challenge, but is important because haemolysis has significant implications for the design of various devices used for processing or manipulating blood, such as the heart-lung machines, cardiac mechanical assist devices, and artificial valves.

The following two chapters (Chapters 4 and 5) review immersed-boundary formulations to describe capsule mechanics and deformation under flow conditions. The authors of the fourth chapter introduce the reader to the lattice-Boltzmann method, which is a mesoscopic simulation technique where the fluid is represented by ‘a collection of fictitious particles moving and colliding on a lattice mesh arising from the discretization of the fluid domain’. They emphasize that appropriate boundary conditions are crucial for a meaningful simulation; they discuss periodic boundary conditions, stationary boundaries, and moving boundaries, and then illustrate the immersed-boundary method. The authors explain models of red blood cell mechanics and aggregation by looking at unstressed cell geometry, fluid and membrane properties, and membrane interaction forces. They describe single cells and groups of cells, flow in small channels, ‘Rouleaux formation’ (i.e. cell shapes developed under the influence of attractive intercellular surface forces) and the dissociation of rouleaux shapes in shear flow (break-up of cell aggregates). Furthermore, they illustrate the simulation of cell suspension flow in microchannels, which is used to mimic blood flow in microvessels. Chapter 5 is about a front-tracking immersed-boundary method for simulating cell motion for several flow configurations. The author uses this method to study deformation of spherical and ellipsoidal capsules, vesicles, and red blood cells in simple shear flow. Vesicles are defined as ‘liquid capsules enclosed by an incompressible membrane endowed with flexural stiffness but lacking resistance against shearing deformation’. The author also considers intercellular interactions (e.g. the interception of spherical particles), capsule motion near a wall, and suspension flow of over 1000 capsules in a channel. Finally, the rolling of a deformable capsule on an adhesive substrate under shear flow is illustrated. The author mentioned that adhesive rolling plays an important role in the metastasis of cancer cells and in the scavenging of biomass, bacteria, and surface-bound particles, as well as in targeted drug delivery (e.g. the binding of a drug carrier to the vascular wall or diseased cell).

Chapters 6 and 7 deal with discrete models. More specifically, the authors of Chapter 6 look at discrete models of the membrane to study the motion of the internal and external fluids with regard to cell deformation dynamics. They point out that in order to accurately capture statics and dynamics, mechanical membrane models must incorporate many characteristics, including shear elasticity, bending rigidity, and membrane viscosity. They study the membrane of a red blood cell as a ‘network of interconnected nonlinear springs emulating the cytoskeleton spectrin network, [where] dissipative forces in the network mimic the effect of the lipid bilayer’. The authors describe that in shear flow, red blood cells show tumbling at low shear rates and tank-treading at high shear rates. Furthermore, they describe that the simulated cells show strong deformation near the tumbling-to-tank-treading transition. The seventh chapter is devoted to the simulation of the motion of red and white blood cells in microvessels. The author describes a two-dimensional model in which red blood cells are represented by a two-dimensional assembly of viscoelastic elements. Various cell behaviours are simulated, including transverse cell migration (i.e. the migration of erythrocytes away from the wall towards the centreline leading to the formation of a cell-free or cell-depleted layer at the wall), partition of cells in diverging bifurcations (cells encounter a region of strongly varying flow), and the motion of multiple cells (e.g. the interaction of red and white blood cells).

The final chapter (Chapter 8) is entitled ‘Multiscale modeling of transport and receptor-mediated adhesion of platelets in the bloodstream’. More precisely, the authors simulate the motion dynamics and adhesive phenomena of platelets near a wall, mimicking an injured vascular endothelial cell wall. Platelets, which are the smallest formed elements in blood, are defined as ‘microscopic oblate spheroid-shaped particles whose transport and haemostatic functions are strongly affected by the haemodynamic flow environment’. The authors describe that the motion of platelets is significantly different than that of leukocytes and spherical cells in linear shear flow in that platelets spend most of their time in the horizontal orientation. They describe different types of platelet motion near a surface (wall), discuss the role of Brownian motion in regard to platelet trajectory, orientation, and frequency of contact with a surface, characterize particle collision in the proximity of a wall, and investigate adhesion and aggregation dynamics.

The authors of Modeling and Simulation of Capsules and Biological Cells also provide an excellent review of many aspects of flow-induced capsule deformation. In fact, some topics are described in greater detail and with better-selected illustrations. To provide a few specific examples, I liked the introduction of the various types of membrane elasticities, such as linear-, rubber-, and visco-elasticity. Also, the descriptions of the compression dynamics of capsules between parallel plates and in spinning rheometers are excellent because the author uses a combination of mathematical equations, diagrams, schematics, and fascinating photographic images to explain the membrane deformations during the plate-squeezing process and at different rotation speeds, respectively. Another example is the very helpful schematic illustration of cell biomechanics and blood flow which demonstrates that these processes involve a broad range of disparate length scales; the authors also provide an illustration of the multi-scale nature of haemodynamics modeling (e.g. a model of cell adhesion, a lumped flow model of the circulation, and a three-dimensional model of the circulation). I also liked the discussion about leukocyte deformation and recovery. More specifically, the authors show illustrations comparing laboratory observation of leukocyte (lymphocyte) recovery, visualized with fluorescent illumination, and corresponding numerical simulation. What is missing in this book, however, are the comprehensive descriptions and quantitative analyses of flow-induced deformation of artificial capsules, the multiscale modelling of adhesion and aggregation characteristics of platelets, and the simulation of the migration dynamics of large numbers of biological capsules in microvessels of straight tubes and in diverging bifurcations. Also, this book does not provide in silico application options (e.g. MATLAB^®). Finally, it is important to mention that there is a short list of corrections available on the following website: http://dehesa.freeshell.org/CC/errata.html.

I found both books well written and structured, and the sequence of topics presented in the chapters is appropriate. However, the more recently published book appears to be a bit more uniform when it comes to the organization of each chapter (i.e. introduction, main text, discussion/summary, references). The indexes for key-word searches are relatively short (three pages in the Computational Dynamics book and five pages in the Modeling and Simulation book). A glossary is not provided in either book and would have been a welcome addition. The illustrations are well selected and of high quality. Using a combination of presentation tools, such as tables, diagrams, photographs, schematics, and mathematical equations in the book and a referral to internet sites where animations (e.g. the tank-treading of red blood cells), computer codes, and other information are provided, help the reader understand the complexity of hydrodynamic phenomena of capsules and biological cells. The authors successfully managed to demonstrate what it takes to quantitatively describe, model, and simulate events that occur in microvessels under various conditions.

The reader will notice that some of the physical problems described and analysed in different chapters are sometimes similar or even identical (e.g. cell behaviour in simple shear flow or capsule motion near a wall), but the editor explains that ‘the repetition is desirable so that solutions produced by different numerical approaches can be compared and the efficiency of alternative formulations can be assessed’. He emphasized that it is ‘the consistency of the results [that] validates the procedures and offers several alternatives’.

Both books are part of a different series: The more recently published book is part of the Mathematical and Computational Biology Series, and the other book is part of the Mathematical Biology and Medicine Series. According to the publisher, these books are meant ‘to appeal to students, researchers, and professionals in the mathematical, statistical and computational sciences, fundamental biology and bioengineering, as well as interdisciplinary researchers involved in the field’. Pozrikidis adds that his books can serve either as a research reference or as an instructional text for all those who have a general interest in the mathematical and computational sciences, and a specific interest to capsule and cell dynamics and biomechanics. I agree with the editor and publisher that these two books will attract readers who have different educational backgrounds and who are at different stages in their careers. Both books are fascinating and, I believe, a welcome addition to the growing number of publications in the fast-advancing field of biological dynamics.