Computational Blood Cell Mechanics:

Road towards Models and Biomedical Applications

Ivan Cimrák and Iveta Jančigová

About this book

In this work, we build computational models step by step. We expect these models to give us correct answers to questions about reality. The models do not have to correspond to reality in all aspects, only to such extent that they give us answers to the questions of interest. For us, a red blood cell is the object that we always have in mind. So, in this book, we:

Identify

the shape and structure of this object.

Determine

the mechanical properties of the materials, of which this object consists.

Implement

into a numerical model the behavior of the object using the fundamental laws of physics.

Test

the model, by creating a code that is capable of simulating the cell behavior.

Calibrate

the model parameters using experimental measurements of basic cell manipulation.

Validate

the computational model against more complex physiological experiments.

Use

the validated model to predict outcomes of other experiments and to formulate hypotheses.

Table of contents

Here is a short desription of the individual chapters. The full table of contents is provided below.

Illustrative simulation example

Chapter 2 describes an introductory simulation that presents a computer script. It explains some of the possibilities and available features step by step. If you are interested mainly in building a model or if you have previous experience with cell modeling, this chapter may be skipped. It is primarily meant for readers not familiar with these topics to illustrate basic outcomes and to make it a bit clearer what we are trying to achieve.

Cell model

Chapter 3 contains the modeling core of the book. Here, we establish the cell model, first with some hidden flaws but then revisited and improved. The modeling part is complemented with section, which discusses the fluid solver and section, which addresses the cell-cell interactions. The cell model is then discussed within the context of other approaches.

Model vs bioreality

To link the biological reality with the developed model, we first review and derive theoretical foundations and then we use the theory as a starting point for direct comparison of model and real-world experiments with cells. Data from experiments need to be approached carefully and we point out several issues that should be considered.

Practical issues

Now the model is ready for practical use, but before we actually use it, we point out some issues that may arise. These include strategies for seeding of dense simulations or suitable discretisations.

Applications

We describe some - with absolutely no intention of being exhaustive - applications of the presented model. We show how this model may be used for computing capture rates of circulating tumor cells in microfluidic devices or for evaluating collision rates during deterministic lateral displacement in periodic obstacle arrays. The knowledge of individual cell deformation during the passage of the device may be used for evaluation of the blood damage index and for design optimisation of ventricular assist devices. The user-friendly open-source approach in computer implementation of the model has also led to its use for optimisation of micro-roughness of channel surfaces.

Ideas for extension

The model, or more precisely, the force-based approach to cell modeling, is flexible enough that several other biological phenomena, such as cell adhesion, inclusion of a cell’s inner structure or modeling of stiff cells can be tried.

Dreaming up the future

Chapter 8 does not contain either Conclusions or Summary. In it we discuss, where the modeling may lead some time in the future. While writing this chapter, we have unleashed our imagination a little bit and we have replaced the sci-entific context with a sci-fiction context.

Preface (.pdf)
Acknowledgements
Symbols and Abbreviations

1 Introduction (.pdf)
1.1 Computational modeling as a tool for understanding
1.2 Compass over map
1.3 How to read this book

2 Illustrative simulation example
2.1 Why simulation before modeling?
2.2 Basic setup
2.3 Adding more complexity
2.4 Overview of the simulation module in ESPResSo
2.5 A few more words about simulations

3 Cell model
3.1 Introduction
3.1.1 Biological background of the model
3.1.2 Relaxed shape of the red blood cell
3.1.3 Mechanics defined by biology
3.1.4 Components of the fluid-cell model
3.1.5 Classification of elastic forces
3.2 Local membrane mechanics
3.3 Global membrane mechanics
3.4 Membrane mechanics revisited (.pdf)
3.4.1 How to make bending torque-free?
3.4.2 Triangle annihilation and its implications for global area force
3.4.3 Competing in-plane and not-out-of-plane moduli and implications for volume force
3.5 Fluid-object coupling
3.5.1 Fluid solver
3.5.2 Fluid-structure interaction
3.5.3 Computational setup
3.5.4 Identification of the friction coefficient
3.5.5 Sensitivity analysis
3.5.6 Flow through membrane
3.6 Cell-cell interaction
3.7 Brief overview of other approaches
3.7.1 Categorisation of red cell models
3.7.2 Comparison of models and solvers

4 Model vs. bioreality
4.1 Is the model correct?
4.1.1 Verification
4.1.2 Validation
4.2 What does the theory of membrane mechanics tell us?
4.2.1 Mechanics of the continuum model for a membrane
4.2.2 Pressure in the spring network system
4.2.3 Microscopic pressure around one point
4.2.4 Relation between model and system parameters
4.3 Calibration and validation experiments
4.3.1 Stretching red blood cells
4.3.2 Couette/shear flow
4.3.3 Poiseuille flow and parachute shapes
4.3.4 Other types of experiments in flow in confined regions
4.3.5 Experimental data for elasticity of other blood cells
4.4 Issues with biological data
4.4.1 Limitations of single-cell calibration experiments
4.4.2 Replicating experimental setup in simulation
4.4.3 Need for image processing
4.4.4 Validation of flow of many cells on a single-cell scale

5 Practical issues
5.1 Introduction
5.2 Periodicity of the simulation domain
5.3 Seeding of dense suspensions
5.3.1 Random seeding
5.3.2 Cell growth
5.3.3 Free fall
5.4 Discretisation
5.4.1 Triangular mesh vs. fluid lattice
5.4.2 Fluid boundary discretisation
5.4.3 Creating triangulations of objects
5.5 Computational complexity and parallelisation

6 Applications
6.1 Microfluidic cell manipulation
6.2 Designing a simulation study
6.3 Periodic obstacle arrays (POAs)
6.4 Capture rates for rare cells in POAs
6.4.1 Models for calculating capture rates
6.4.2 Performing a simulation study
6.4.3 Parameter ranges
6.4.4 Comparability of simulation runs - Fixed flow rate
6.4.5 Initial conditions and randomness
6.4.6 Observed characteristics
6.4.7 Interpretation of results
6.5 Collision rates and deterministic lateral displacement in POAs
6.5.1 Performing a simulation study
6.5.2 Parameter ranges and observed characteristics
6.5.3 Interpretation of results
6.6 Blood damage index
6.7 Ventricular assist devices
6.8 Analysis of surface micro-roughness

7 Ideas for extension
7.1 Advantage of modular approaches or isolate cleverness
7.2 Rolling and adhesion of cells
7.2.1 Adhesion models
7.2.2 Calibration of model parameters
7.3 Cell with a nucleus
7.4 Solid objects
7.5 Movement of cell clusters (emboli)

8 Dreaming up the future (.pdf)

A Force- and torque-free bending modulus
A.1 Four-point interaction
A.2 Comparison to other approaches
A.3 Computationally friendly expressions

B Comparison of area interactions to other approaches
B.1 Local area interaction
B.2 Global area interaction

C Force- and torque-free volume modulus
C.1 Global volume interaction
C.2 Comparison to other approaches
C.3 In-plane or out-of-plane volume forces
C.4 Search for the volume energy

D Calculus of spring network deformations
D.1 Shear modulus
D.2 Area expansion modulus
D.3 Comment on differences in implementations of local area forces

E Complete example script

F Simulation setup
F.1 Units
F.2 Calculation of membrane coefficients
F.3 Determining friction coefficient and mesh densities
F.4 Setting up interactions
F.5 Fluid parameters

Bibliography
Index

Resources

Buy the book

Book Computational Blood Cell Mechanics: Road towards models and biomedical applications is available at Amazon and CRC Press. It is part of the Chapman and Hall Mathematical and Computational Biology Series. While it is not a textbook, certain portions of it are suitable as a supplementary material for a course on computational blood cell modeling.

Install ESPResSo

sample-image ESPResSo is an open-source Extensible Simulation Package for Research of Soft Matter. The 4.0 release (September 2018) is available here. The computational core is developed mainly at the Institute of Computational Physics in Stuttgart, Germany. It contains a lattice-Boltzmann solver for fluid.

Use Object-in-fluid

red blood cellObject-in-fluid is a computational framework developed in Žilina, Slovakia as part of ESPResSo. It lets the user model and simulate closed elastic objects using python scripts. The documentation and current information is available at Cell-in-fluid Research Group website and the latest version can be downloaded from here.

Watch Youtube channel

red blood cellHere you can watch animations from simulations that use the model described in the book. Some of the videos are proof-of-concept works for various applications.

Download files

File Type Description
rbc374nodes.dat mesh first of two files needed for red blood cell with 374 mesh nodes, radius = 1
rbc374triangles.dat mesh second of two files needed for red blood cell with 374 mesh nodes
sphere393nodes.dat mesh first of two files needed for a sphere with 393 mesh nodes, radius = 1
sphere393triangles.dat mesh second of two files needed for a sphere with 393 mesh nodes
motivation_first.py python script simple script with one cell, p. 9 in the book
motivation_second.py python script more complex script with two cells and boundaries, p. 15 in the book
check_flow.py python script script for checking fluid flow through the membrane, p. 61 in the book
capture_rates.py python script script for computing capture rates based on cell-obstacle contacts, p. 157 in the book
nucleus_bonded.py python script one possibility how to model a cell with nucleus, p. 185 in the book; meshes for this script available upon request
nucleus_non_bonded.py python script another possibility how to model a cell with nucleus, p. 185 in the book; meshes for this script available upon request
seeding_regular.py python script script that shows regular seeding, p. 130 in the book
seeding_random.py python script script that shows random seeding, p. 131 in the book
seeding_random_overlap.py python script random seeding modified to allow for initial overlap of membranes that is resolved during warm-up period, p. 132 in the book
seeding_freefall_fall.py python script first stage of freefall seeding - cells fall in a larger simulation box, p. 134 in the book
seeding_freefall_shake.py python script second stage of freefall seeding - fallen cells are shaken in a desired simulation box, p. 134 in the book
appendix_cell_wall.py python script example script that can be used for calibration of cell-wall interaction, p. 238 in the book
appendix_cell_cell.py python script example script that can be used for calibration of cell-cell interaction, p. 239 in the book
appendix_simulation.py python script more complete version of the motivation example script, p. 229 in the book

Selected Papers

Shear‐induced hemolysis is a major concern in the design and optimization of blood‐contacting devices. Even with a small amount of mechanical stress, inflammatory reactions can be triggered in the cells. Blood damage is typically estimated using continuum fluid dynamics simulations. In this study, we report a novel cell damage index (CDI) obtained by simulations on the single‐cell level in a lattice Boltzmann fluid flow. The change of the cell surface area gives important information on mechanical stress of individual cells as well as for whole blood. We are using predefined basic channel designs to analyze and compare the newly developed CDI to the conventional blood damage calculations in very weak shear stress scenarios. The CDI can incorporate both volume fraction and channel geometry information into a single quantitative value for the characterization of flow in artificial chambers.
Cell Damage Index as Computational Indicator
for Blood Cell Activation and Damage
Markus Gusenbauer, Renáta Tóthová, Giulia Mazza, Martin Brandl, Thomas Schrefl, Iveta Jančigová, Ivan Cimrák Artificial Organs, 2018
We review the Lattice-Boltzmann (LB) method coupled with the immersed boundary (IB) method for the description of combined flow of particulate suspensions with immersed elastic objects. We describe the implementation of the combined LB–IB method into the open-source package ESPResSo. We present easy-to-use structures used to model a closed object in a simulation package, the definition of its elastic properties, and the interaction between the fluid and the immersed object. We also present the test cases with short examples of the code explaining the functionality of the new package.
An ESPResSo implementation of elastic objects immersed in a fluid
Ivan Cimrák, Markus Gusenbauer, Iveta Jančigová
Computer Physics Communications, 2014
Recently, computational modelling has been successfully used for determination of collision rates for rare cell capture in periodic obstacle arrays. The models were based on particle advection simulations where the cells were advected according to velocity field computed from two dimensional Navier–Stokes equations. This approach may be used under the assumption of very dilute cell suspensions where no mutual cell collisions occur. We use the object-in-fluid framework to demonstrate that even with low cell-to-fluid ratio, the optimal geometry of the obstacle array significantly changes. We show computational simulations for ratios of 3.5, 6.9 and 10.4% determining the optimal geometry of the periodic obstacle arrays. It was already previously demonstrated that cells in periodic obstacle arrays follow trajectories in two modes: the colliding mode and the zig–zag mode. The colliding mode maximizes the cell-obstacle collision frequency. Our simulations reveal that for dilute suspensions and for suspensions with cell-to-fluid ratio 3.5%, there is a range of column shifts for which the cells follow colliding trajectories. However we showed, that for 6.9 and 10.4%, the cells never follow colliding trajectories.
Collision rates for rare cell capture in periodic obstacle arrays
strongly depend on density of cell suspension
Ivan Cimrák
Computer Methods in Biomechanics and Biomedical Engineering, 2016

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