Biomedical engineering

Summary

Introduction

In this chapter we describe the ion-channel switch (ICS) biosensor. By a biosensor, we mean a sensor built out of synthetic biological material – in our case, built out of an artificial membrane, ion channels, spacers, and tethers. The purpose of the biosensor is to detect the presence of analyte molecules.

The ICS biosensor provides an example of how devices built out of tethered engineered membranes can be used for high-resolution detection of specific target (analyte) molecules. When it senses the presence of analyte molecules, the ICS biosensor responds by changing its electrical impedance significantly. The amount the impedance changes by depends on the concentration of the analyte molecules. Therefore, by examining the current flowing through the biosensor, one can detect both the presence of the analyte and its concentration. A key feature of the ICS biosensor is that the detection is performed by measuring the time-dependent conduction of the engineered membrane, which is dependent on the ensemble of aqueous pores and a single pair of conducting gramicidin (gA) dimers present. A gramicidin dimer [83, 84, 466] is an example of a conducting ion channel which allows ions to pass through the tethered membrane. Using specific molecular components and excitation potentials, the ICS can be designed to detect specific analyte molecules of interest.

The ICS biosensor provides an interesting example of engineering at the nanoscale. It is remarkable that the functionality of the device depends on approximately 100 lipids, and a pair of ion channels modulating the flow of billions of ions in a typical sensing event of approximately 5 min. Since the gramicidin monomers (each with a conducting pore of diameter 0.4 nm and length 2.8 nm) move randomly in the outer lipid leaflet of the membrane (1.4 nm thick), we can view the biosensor as a fully functioning nanomachine with moving parts. Indeed, each individual gramicidin monomer diffuses randomly over an area of order 1 μm2. Furthermore, the 4-nm-thick lipid bilayer is tethered 2–3 nm away from the gold surface by hydrophilic spacers, thereby allowing ions to diffuse between the membrane and gold. This permits a flux in excess of 106 ions per second to traverse each gramicidin channel comprised of two gA monomers.

Summary

Biochemistry deals with chemical processes occurring in living organisms. An elementary understanding of the biochemistry of cell membranes is important for understanding the key ideas behind engineered artificial membranes and biosensing. This chapter gives a brief description of the key chemical components in cell membranes, namely, lipids, proteins, and peptides. Lipids are the primary component of all cell membranes; and peptides and proteins are the primary component of signaling in living organisms. Additionally, we discuss tethers and spacers, which provide the basis for connecting engineered membranes to a bioelectronic interface; this bioelectronic interface is connected to electrical measurement equipment that allow signals to be acquired from the device.

With the availability of powerful computer-based visualization tools (discussed at the end of this chapter), it is strongly recommended that the reader use these tools to get a better feel for the various types of biochemical molecules described in this chapter. These software tools are also a first step in more sophisticated molecular dynamics simulations that are discussed in Part III of the book.

Bonded and Nonbonded Molecular Interactions

A chemical bond is an attraction between two or more atoms that facilitates the formation of multiatom chemical compounds. There are two main types of chemical bonds: ionic bonds and covalent bonds. In addition there are also van der Waals forces, a type of nonbonded interaction between atoms, which include attraction or repulsion between atoms that do not arise from covalent or ionic bonds. Specifically, van der Waals forces are a consequence of quantum dynamics in which repulsive or attractive forces between atoms result from correlations in the fluctuating polarizations of atoms. In this section we define the covalent and ionic bonds as well as the van der Waals interaction between atoms and their importance in the dynamics of cell membranes. Additionally, we describe their importance when constructing molecular dynamics simulations.

Covalent bond. A covalent bond is a chemical bond formed by sharing electrons between atoms. Covalent bonds are key to the structure of nearly all chemical compounds such as water, lipids, peptides, and proteins. The covalent bond is one of the strongest bonds compared with ionic bonds and van der Waals interactions. Typically the covalent bond has a bond energy in the range of 120 to 1000 kJ/mol (120– 500 kJ/mol for single bonds, and up to 1000 kJ/mol for triple bonds).

Summary

Biological membranes such as the cell membrane play an essential and ubiquitous role in cellular communication, energy storage, structural support, and protecting the contents of living cells. Biological membranes are dynamic structures with remarkable properties: they have moving parts, they act as electrical circuits with long-range memory, they contain protein macromolecules (ion channels) that selectively open and close to admit ions, thereby regulating the electrical activities of cells, and they spontaneously form pores (electroporation) which is crucial for the operation of antimicrobial drugs.

This book focuses on building and mathematically modeling artificial membranes that mimic biological membranes. Recent advances in bioengineering and biochemistry allow us to build artificial membranes precisely to mimic the remarkable properties of biological membranes and also to build synthetic biological devices out of these membranes. Also advances in computational molecular biology allow us to construct largescale computer simulation models at the atomic-spatial scale to gain deep insight into the dynamic properties and structure-function relationship of such artificial membranes and devices. This dual approach of synthesis (building membranes and novel devices) and analysis (mathematical modeling) is vital for the future development of sensing devices, drug delivery mechanisms, and synthetic biological devices.

Summary. This book provides a comprehensive description of artificial membranes aimed at advanced undergraduate and graduate students, and researchers in electrical engineering, biological and chemical engineering, biophysics, and applied mathematics. Construction of artificial membranes and devices is intimately linked with careful mathematical modeling so as to predict the performance and improve the design. As a result this book is organized into three interrelated parts:

Part I gives an overview of the book along with an elementary primer on biochemistry for engineers and applied mathematicians.

Part II deals precisely with engineering and building artificial membranes, building novel synthetic biological devices out of these artificial membranes, and evaluating how such devices perform in experimental and clinical studies. The devices studied include a super-resolution biosensor (which can be viewed as a fully operational nanomachine), an electroporation platform (for studying how membranes spontaneously form waterfilled pores), and an electrophysiological platform (for noninvasive measurements of cells).

Summary

Introduction

Part II gave a detailed description of fabricating synthetic biological devices using engineered membranes and also discussed methods to perform experimental measurements. Constructing mathematical models that accurately predict the behavior of engineered membranes is important, both for the design of devices built out of the membrane and for estimating parameters such as analyte concentration and membrane conductance. This chapter focuses on constructing reaction-rate models for engineered membranes. Such models comprise a system of nonlinear ordinary differential equations and can be viewed as a black-box phenomenological model for the underlying synthetic biological device. Reaction-rate models account for the dynamics of tethered membranes that are excited by voltages less than 50 mV; therefore, the process of electroporation is not present. Specifically these models in combination with experimental measurements can be used to estimate the structural integrity of the tethered membrane and the disulfide bonds that anchor the tethers to the gold bioelectronic interface.

When an electrical potential is established across a tethered membrane, the macroscopic reaction-rate model for the current flowing through the membrane is equivalent to a lumped electrical circuit composed of resistors and capacitors as shown in Figure 8.1. In other words, the simple resistor–capacitor (RC) circuit of Figure 8.1 models the electrical response of the engineered membrane at a macroscopic level. The resulting current voltage response is a second-order linear system of ordinary differential equations. Though widely used to study the dynamics of tethered membranes, this reaction-rate model cannot account for the restricted diffusion dynamics that occur at the electrode surface. Modeling these effects requires introducing fractional-order operators. The fractional-order macroscopic model of the tethered membrane accounts for both the charging dynamics of the membrane and the restricted diffusion dynamics at the electrode surface. This chapter also presents a fractional-order macroscopic model for tethered membranes that contain sterol molecules (e.g., cholesterol). All the reaction-rate models in this chapter are validated using real-world experimental data from engineered membranes with different tether densities, lipid compositions, and analyte solutions.

Remark: Digital data acquisition, i.e., how the measured current is time sampled and digitized for processing on a computer, is not described in this book. Excellent resources for the design of analog-to-digital converters, time discretization, and antialiasing filters are [52, 102].

Summary

This section details the precise ambient conditions and hardware specifications under which all experiments reported in this book were conducted. §C.1 provides the parameters for the ion-channel switch (ICS) biosensor experiments. §C.2 provides the experimental parameters estimated for the pore formation measurement platform (PFMP) for the dynamics of peptidyl-glycine-leucine-carboxyamide (PGLa) pore formation. §C.3 provides the experimental parameters of the electroporation measurement platform (EMP) model. Finally, §C.4 through §C.5.1 discuss coarse-grained molecular dynamics (CGMD) and molecular dynamics (MD) simulation parameters.

Channel Switch Biosensor

All experimental measurements, unless otherwise stated, were conducted at 27◦C in a phosphate-buffered solution with a pH of 7.2, and a 0.15Msaline solution composed of Na+, K+, and Cl−. At this temperature the tethered membrane is in the liquid phase. A pH of 7.2 was selected to match that typically found in the cellular cytosol of real cells. The forward and reverse reaction rates in Table C.1 are obtained from [220, 221, 283]. In Table C.1, notice that the computed diffusion coefficient for ferritin is 80 μm2/s and that for hCG is 250 μm2/s; this is an expected result because the ferritin has a molecular weight of 450 kDa and hCG a molecular weight of 25.7 kDa.

The governing equations (10.10), (10.12), (10.13), and (10.15) with the boundary conditions (10.11) are solved numerically with the commercially available finiteelement solver COMSOL 4.3a (Comsol Multiphysics, Burlington, MA). To solve the advection-diffusion equations (10.10) and (10.15) the COMSOL module Transport of Diluted Species is used. The surface reaction diffusion equation (10.13) is solved using the Weak Form Boundary PDE module. The simulation domain is meshed with approximately 46,000 triangular elements constructed using an advancing-front meshing algorithm. The generalized Poisson–Nernst–Planck and Poisson–Nernst–Planck are numerically solved using the multifrontal massively parallel sparse direct solver [14] with a variable-order variable-step-size backward-differential formula [53]. The Levenberg– Marquardt algorithm (10.17) is implemented using the MATLAB function lsqnonlin with (10.16) computed from the results of the COMSOL simulations.

Summary

Introduction

In this chapter (and the following three chapters) we study mesoscopic mathematical models for the dynamics of engineered membranes. This chapter constructs mesoscopic models for the following two synthetic biological devices built out of artificial membranes:

Summary

Introduction

Recall that the ion-channel switch (ICS) biosensor is a synthetic biological device built out of an artificial membrane, ion channels, spacers, and tethers. Its purpose is to detect the presence of analyte molecules in fluid. When it senses the presence of analyte molecules, the biosensor responds by changing its electrical impedance significantly. The amount the impedance changes by depends on the concentration of the analyte molecules in the fluid. Therefore, by examining the current flowing through the biosensor, one can detect both the presence of analyte and its concentration.

This chapter studies how the current flowing through the ICS biosensor can be modeled using a two–time scale nonlinear system of ordinary differential equations (for chemical reactions) coupled with the fractional-order macroscopic model of Chapter 8 (for electrical dynamics). The fractional-order macroscopic model of the tethered membrane accounts for both the electrical dynamics of the membrane and the restricted diffusion dynamics at the electrode surface (as detailed in Chapter 8), while the two–time scale nonlinear system of differential equations models the chemical reaction dynamics that occur on the membrane surface of the ICS bisensor. We call the resulting system a reaction-rate-constrained model. Such reaction-rate models are useful for modeling the dynamics of tethered membranes that are excited by potentials less than 50 mV (so that electroporation does not occur).

After developing reaction-rate-constrained models for the ICS biosensor (with μm2 electrode size), we consider biosensors with very small electrodes (area on the order of 1 × 10−12 m2 = μm2).We construct a stochastic model for the current response of such microelectrode ICS (mICS) biosensors as a noisy finite-state Markov chain observed in noise, i.e., a hidden Markov model (HMM), where each state is associated with the number of conducting gramicidin A (gA) dimers.When the electrode size is very small, individual dimer events are recorded – which is a finite-state process – as opposed to the ensemble average of millions of such events in a large electrode, which gives an ordinary differential equation.

The typical current response of the ICS biosensor and mICS biosensor is illustrated schematically in Figure 9.1. All the models in this chapter are validated using realworld experimental data from engineered membranes with different tether densities, lipid compositions, and analyte solutions.

Summary

Introduction

The goal of the present chapter is to describe a procedure to formally determine solutions for solid mechanics problems, fluid mechanics problems and problems with filtration and diffusion. Mechanical problems in biomechanics can be very diverse, and most problems are so complex that it is impossible to derive analytical solutions and often very complicated to determine numerical solutions. Fortunately, in most cases it is not necessary to describe all phenomena related to the problem in full detail; simplifying assumptions can be made, thus reducing the complexity of the set of equations that have to be solved. The present chapter deals with formulating problems and solution strategies, starting from the most general set of equations and gradually reducing the generality by imposing simplifying assumptions. In Section 13.2 this will be done for solids. Section 13.3 is devoted to the solution of fluid mechanics problems. The last section of this chapter discusses diffusion and filtration.

Solution Strategies for Deforming Solids

In this section, it is assumed that the material (or material fraction) to be considered can be modelled as a deforming solid continuum. This implies that it is possible and significant to define a reference configuration or reference state. With respect to the reference state, the displacement field as a function of time supplies a full description of the deformation process to which the continuum is subjected (at least under the restrictions given in previous chapters, such as a constant temperature). After all, for a displacement field that is known as a function of time, it is possible to directly calculate the local deformation history (applying the kinematics: see Chapter 10) and subsequently, the stress state as a function of time (using the constitutive equations: see Chapter 12). The relevant fields, from a mechanical point of view, that are obtained in this way for the continuum have to satisfy the balance equations (see Chapter 11). In addition, the initial conditions have to be fulfilled: at the beginning of the process, the positions of the material points in space have to be prescribed, usually in accordance with the reference configuration, and also the initial velocities of the material points have to be in agreement with the specification of the initial state.

Summary

Introduction

Up to this point, all treated problems have been in some way one-dimensional. In Chapter 3 we have discussed equilibrium of two- and three-dimensional bodies and in Chapter 4 the fibres were allowed to have some arbitrary orientation in three-dimensional space. But, when deformations were involved, the focus was on fibres and bars, dealing with one-dimensional force–strain relationships. Only one-dimensional equations were solved. In this chapter and the ones that follow, the theory will be extended to the description of three-dimensional bodies, and it is opportune to spend some time looking at the concept of a continuum.

Consider a certain amount of solid and/or fluid material in a three-dimensional space. Although in reality for neighbouring points in space the (physical) character and behaviour of the residing material may be completely different (because of discontinuities at the microscopic level, which become clearer on reducing the scale of observation), it is common practice that a less detailed description (at a macroscopic level) with a more gradual change of physical properties is used. The discontinuous heterogeneous reality is homogenized and modelled as a continuum. To make this clearer, consider the bone in Fig. 7.1. Although one might conceive the bone at a macroscopic level, as depicted in Fig. 7.1(a), as a massive structure filling all the volume that it occupies in space, it is clear from Fig. 7.1(b) that at a smaller scale the bone is a discrete structure with open spaces in between (although the spaces can be filled with a softer material or a liquid). At an even smaller scale, at the level of one single trabecula, individual cells can be recognized. This phenomenon is typical for biological materials at any scale, whether it is at the organ level, the tissue level or cellular level, or even smaller. At each level, a very complex structure can be observed, and it is not feasible to take into account every detail of this structure. That is why a homogenization of the structure is necessary, and, depending on the objectives of the modelling (which is the question that has to be answered in the context of the selection of the theoretical model), a certain level of homogenization is chosen.

Summary

Introduction

The two- and three-dimensional convection–diffusion equations play an important role in many applications in biomedical engineering. One typical example is the analysis of the effectiveness of different types of bioreactors for tissue engineering. Tissue engineering is a rapidly evolving interdisciplinary research area aiming at the replacement or restoration of diseased or damaged tissue. In many cases, devices made of artificial materials are only capable of partially restoring the original function of native tissues, and may not last for the full lifetime of a patient. In addition, there is no artificial replacement for a large number of tissues and organs. In tissue engineering, new, autologous tissues are grown. The tissue proliferation and differentiation process is strongly affected by mechanical stimuli and transport of oxygen, minerals, nutrients and growth factors. To optimize bioreactor systems, it is necessary to analyse how these systems behave. The convection–diffusion equation plays an important role in this kind of simulating analysis.

Figure 16.1 shows two different bioreactor configurations, both of which have been used in the past to tissue engineer articular cartilage. The work was especially focussed on glucose, oxygen and lactate, because these metabolites play a major role in the biosynthesis and survival of chondrocyte. Questions ranged from: ‘Does significant nutrient depletion occur at the high cells concentrations required for chondrogenesis?’ to ‘Do increasing transport limitations due to matrix accumulation significantly affect metabolite distributions?’ Figure 16.2 shows a typical result for the calculated oxygen distributions in the two bioreactor configurations.

This chapter explains the discretization of the convection-diffusion equation in two or three dimensions. First, the diffusion equation is discussed; thereafter, the convection–diffusion equation is elaborated. The spatial discretization of the weighting function is based on the Galerkin method.

Diffusion Equation

Consider a two- or three-dimensional domain with boundary. As in the onedimensional model problem, the boundary is split into a part along which the essential boundary conditions are specified, and a part along which the natural boundary conditions may be specified. The generic form of the diffusion equation is given by

Summary

In September 1997, an educational programme in Biomedical Engineering, unique in the Netherlands, started at the Eindhoven University of Technology, together with the University of Maastricht, as a logical step after almost two decades of research collaboration between both universities. This development culminated in the foundation of the Department of Biomedical Engineering in April 1999 and the creation of a graduate programme (MSc) in Biomedical Engineering in 2000 and Medical Engineering in 2002.

Already at the start of this educational programme, it was decided that a comprehensive course in biomechanics had to be part of the curriculum and that this course had to start right at the beginning of the Bachelor phase. A search for suitable material for this purpose showed that excellent biomechanics textbooks exist. But many of these books are very specialized to certain aspects of biomechanics. The more general textbooks address mechanical or civil engineers or physicists who wish to specialize in biomechanics, so these books include chapters or sections on biology and physiology. Almost all books that were found are at Masters or post-graduate level, requiring basic to sophisticated knowledge of mechanics and mathematics. At a more fundamental level, only books could be found that were written for mechanical and civil engineers.

We decided to write our own course material for the basic training in mechanics appropriate for our candidate biomedical engineers at Bachelor level, starting with the basic concepts of mechanics and ending with numerical solution procedures, based on the finite element method. The course material assembled in the current book comprises three courses for our biomedical engineering curriculum, distributed over the three years of their Bachelor studies. Chapters 1 to 6 mostly treat the basic concepts of forces, moments and equilibrium in a discrete context in the first year. Chapters 7 to 13 in the second year discuss the basis of continuum mechanics, and Chapters 14 to 18 in the third year are focussed on solving the field equations of mechanics using the finite element method.

Biomedical engineering

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Index

Contents

5 - Ion-Channel Switch (ICS) Biosensor

Summary

2 - Biochemistry for Engineers: A Short Primer

Summary

List of Abbreviations

Preface

Summary

8 - Reaction-Rate-Constrained Models for Engineered Membranes

Summary

Part I - Introduction and Background

Bibliography

Appendix C - Experimental Setup and Numerical Methods

Summary

Part III - Dynamic Models for Artificial Membranes: From Atoms to Device

10 - Diffusion-Constrained Continuum Models of Engineered Membranes

Summary

9 - Reaction-Rate-Constrained Models for the ICS Biosensor

Summary

11 - Local Balance of Mass, Momentum and Energy

13 - Solution Strategies for Solid and Fluid Mechanics Problems

Summary

9 - Motion: Time as an Extra Dimension

7 - Biological Materials and Continuum Mechanics

Summary

16 - Solution of the Three-Dimensional Convection–Diffusion Equation by Means of the Finite Element Method

Summary

10 - Deformation and Rotation, Deformation Rate and Spin

Preface to the First Edition

Summary

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