Computational Models in Biomedical Engineering: Finite Element Models Based on Smeared Physical Fields: Theory, Solutions, and Software

By Milos Kojic, Miljan Milosevic and Arturas Ziemys

Computational Models in Biomedical Engineering: Finite Element Models Based on Smeared Physical Fields: Theory, Solutions, and Software discusses novel computational methodologies developed by the authors that address a variety of topics in biomedicine, with concepts that rely on the so-called smeared physical field built into the finite element method. A new and straightforward methodology is represented by their Kojic Transport Model (KTM), where a composite smeared finite element (CSFE) as a FE formulation contains different fields (e.g., drug concentration, electrical potential) in a composite medium, such as tissue, which includes the capillary and lymphatic system, different cell groups and organelles.

The continuum domains participate in the overall model according to their volumetric fractions. The governing laws and material parameters are assigned to each of the domains. Furthermore, the continuum fields are coupled at each FE node by connectivity elements which take into account biological barriers such as vessel walls and cells.

• Provides a methodology based on the smeared concept within the finite element method which is simple, straightforward and easy to use
• Enables the modeling of complex physical field problems and the mechanics of biological systems
• Includes features that are illustrated in chapters devoted to applications surrounding tissue, heart and lung
• Includes a methodology that can serve as a basis for further enhancements by including additional phenomena which can be described by relevant relationships, derived theoretically or experimentally observed in laboratories and clinics

• Language: English
• Publisher: Academic Press
• Release date: Sep 11, 2022
• ISBN: 9780323906692

Milos Kojic

Dr. Milos Kojic is one of the leading scientists in the computational mechanics and finite element method and its application in engineering and biomedicine. He is currently Full Member and Professor of Nanomedicine, Department of Nanomedicine, The Methodist Hospital Research Institute, Houston, TX, USA, as well as the Director of the Bioengineering R&D Center, Kragujevac, Serbia. In his long professional carrier, Dr. Kojic has been Professor of Mechanics at University of Kragujevac, Serbia (retired), Visiting Scholar of MIT; Research and Development Engineer of ADINA R&D, Boston; Senior Research Scientist, Harvard School of Public Health; and Member of the Serbian Academy of Science and Arts, from 2009. Dr. Kojic’s research is primarily concerning the finite element method, implementation in engineering and biomedicine; and software development. He has formulated and implemented a number of original concepts and solutions, among which is the Governing Parameter Method for inelastic analysis of solids and structures, and recently the smeared finite element models for field problems and mechanics, also known as the Kojic Transport Model (KTM). He initiated and has been PI of the FE software package PAK for solids and fluids, field and coupled problems, and biomechanics. The PAK software has been developing over decades with participation of several generations; today, it is the basic tool for applications in industry and in research within various domestic and international grants. Dr. Kojic is the lead author of over 10 textbooks in Serbian and two books by world leading publishers: Inelastic Analysis of Solids and Structures, from Springer, and Computer Modeling in Bioengineering, from J. Wiley and Sons.

Book preview

Computational Models in Biomedical Engineering

Finite Element Models Based on Smeared Physical Fields: Theory, Solutions, and Software

Miloš Kojić

Department of Nanomedicine, Houston Methodist Research Institute, United States

Bioengineering Research and Development Center, Kragujevac, Serbia

Serbian Academy of Sciences and Arts, Belgrade, Serbia

Miljan Milošević

Bioengineering Research and Development Center, Kragujevac, Serbia

Institute for Information Technologies, University of Kragujevac, Serbia

Belgrade Metropolitan University, Serbia

Arturas Ziemys

Department of Nanomedicine, Houston Methodist Research Institute, United States

Table of Contents

Cover image

Title page

Copyright

1. Basic processes in living organisms

1.1. Introduction: mass transport as a vital process in living organisms

1.2. Circulatory system

1.3. Tissue

1.4. Cells

1.5. Specificities of the body organs with respect to transport

1.6. Tissue microenvironment within organs and physiological barriers to transport

2. Fundamental laws for physical fields and mechanics

2.1. Diffusion

2.2. Heat conduction

2.3. Flow through porous media

2.4. Electrostatics

2.5. Fluid flow

2.6. Mechanics of solids

3. Kojic transport model (KTM) for physical fields

4. Smeared finite element formulation for mechanics

4.1. FE modeling of 3D solid deformation

4.2. Shell deformation

4.3. Large strain FE formulation

4.4. Fluid mechanics

4.5. Solid–fluid and solid–solid interaction

4.6. Composite smeared finite element for mechanics (CSFEM)

4.7. Numerical examples

5. Multiscale hierarchical models for diffusion in composite media and tissue

5.1. Introduction

5.2. Multiscale diffusion and numerical homogenization

5.3. Coupled convective and diffusive transport within vessels and tissue

5.4. Examples

6. Application of Kojic transport model (KTM) to convective-diffusive transport and electrophysiology in tissue and capillaries

6.1. Introduction—mass transport in living organisms

6.2. KTM for convective and diffusive transport

6.3. Application of KTM in electrophysiology

6.4. KTM for drug release from nanofibers

6.5. Examples

7. Heart electrophysiology and mechanics

7.1. Heart physiology

7.2. Electrophysiology

7.3. Heart mechanics

7.4. Computational models for the heart tissue passive mechanical response

7.5. Finite element models of the left ventricle—wall deformation and blood flow

8. Description of the software accompanying the book

8.1. Introduction

8.2. General structure of graphical user interface (GUI) software accompanying the book

8.3. Procedures for running examples and visualization of results

Index

Copyright

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1: Basic processes in living organisms

Abstract

A brief overview of the basic processes in living organisms is presented in this introductory chapter. The goal is to provide the reader with information about fundamental physiological processes, which are modeled by computational methods in subsequent chapters. The text gives an insight into the complexity of biological systems and background for understanding the approximations used further in the mathematical interpretations and generation of numerical results. Mass transport is first described, as the vital process mostly governed by diffusion and then is given basic information about the transport of matter within the circulatory system, tissue, and cells. Finally, the role of the heart, lung, liver, and brain as body organs is outlined, with reference to specificities of the tissue microenvironment of these organs.

Keywords

Body organs; Cells; Circulatory system; Diffusion; Mass transport; Microenvironment; Tissue

1.1. Introduction: mass transport as a vital process in living organisms

The life of biological organisms is fundamentally associated with growth, motion, multiplication, evolution, interactions with their environment and other living organisms. All these biological fundamental events to occur need energy. Usually, the energy in living systems is accumulated and used in chemical and electrochemical forms. To maintain the integrity of living systems and the energy balance, biological organisms need to absorb the energy of physical or chemical nature through complex processes of mass transfer and conversion. Thus, mass transfer (or mass transport) is one of the fundamental manifestations of living organisms that enable all their functions. Would it be the slow movement of a single cell organism ameba on the surface of the water in a lake or humans undergoing strenuous exercises in the Olympic Games, the mass transfer phenomena are present all the time. Even in the event of death, living organisms decay, and their accumulated mass is transformed and may dissipate into the environment according to the same principles of mass transfer that sustained life before. The mass transfer phenomena involved in living organisms may be described and understood by universal laws of physics connecting mass transfer efficiency with properties of transported matter itself and boundary conditions of a particular system.

This book describes applications of computational methods and physical laws, which rely on mass transfer phenomena, ultimately leading to improved therapeutics in clinics. The novel computational methodology presented and implemented in the book is developed by the first author and supported in implementation by the book coauthors and a number of collaborators in Serbia (R&D Center for Bioengineering BioIRC and the University of Kragujevac) and at the Houston Methodist Research Institute. The material of the book covers many complex biological systems and problems of biomedical and bioengineering nature. The remaining portion of this chapter is devoted to refreshing readers with the key properties of biological tissues at the levels of a single cell, organ tissues, and partially at the systemic level of the whole body. This short review is considered helpful for presentation in subsequent chapters, but also it provides an insight into the fundamentals of the computational approaches discussed later.

Mass transport phenomena can be grouped into passive and active transport, and both are equally important in biological systems. However, their dominance and importance may depend on the scale of the consideration. Passive and active transport mechanisms usually are happening together, while the mass fluxes may not occur in parallel. The active transport has several different mechanisms, which can be specific to a certain biological scale, e.g., cell or systemic level; thus, the active transport will be presented below separately—in describing specificities of the transport within tissue and cells (Section 1.4), once we discuss the basic organization of living systems.

Diffusion is the most fundamental process, which represents passive mass transport. The presence of a transported mass within a physical space is measured by its concentration c [mass/volume]. Diffusion occurs from regions or compartments with higher concentrations to those of lower concentrations. The net movement of mass follows a concentration gradient (with the negative sign) and is mathematically described by phenomenological Ficks's law used extensively throughout the book,

(1.1.1)

where D is diffusion coefficient or diffusivity, and q is the mass flux [mass per unit area and unit time]; and ∂c/∂x is the concentration gradient in direction x (partial derivative is used because concentration is also a function of time). It can be seen that the diffusion flux is proportional to the concentration gradient and diffusivity. While the concentration gradient depends on the current state of the system in which diffusion occurs, the diffusivity is the property that depends on the diffusing matter and the environment. The value of D depends on the size of diffusing particulates (e.g., atoms, molecules) and their interaction with the environment. The D value in pure water usually is in the order of 10 −⁹ m²/s, but it can be reduced to 10 −¹⁰ m²/s or less in the biological environment. There can be other factors that may influence diffusivity, such as temperature. Diffusion according to the relation Eq. (1.1.1) is called Fickian diffusion. It gives a macroscopic phenomenological relationship. However, molecules individually are displaced according to Brownian motion or random displacements, meaning that their displacements may happen at certain times against the concentration gradient. Diffusion is one of the most simple and fundamental mass transfer mechanisms in living systems, and it is present at all scales of a biological organization.

The distribution of molecules in a biological environment, as in any other environment, is a spatiotemporal problem meaning that the concentration of the substance is a function of time and specific location. Change of concentration over time is frequently referred to as biodistribution within biological systems. Pharmacokinetics, abbreviated by PK, is the branch of pharmacology science that exclusively deals with the biodistribution of substances in biological tissues. Drug biodistribution problems are highly important in the field of drug delivery and medicine and will be addressed in several subsequent chapters.

In the remaining text of this chapter, we follow mass transport from vessels, which represent the circulatory system within the body, to tissue and cells, occurring bidirectionally, and summarize the main characteristic of these biological entities. Additionally, basic information is presented regarding organs and their role in the process of mass exchange.

1.2. Circulatory system

The living body is the highest physiological hierarchical system incorporating multiple organs and cells and maintaining the homeostasis and viability of all body functions (Hall and Hall, 2020). Because many organs and tissues are tightly integrated within a body, the body is not just a collection of tissues. For example, neural and cardiovascular systems connect all organs and tissues ensuring that organs work properly. The blood vessel system serves to supply the tissue with various vital ingredients, such as nutrients or oxygen, and takes back the products generated by the processes within living cells. The supply is carried by the arterial branch, which spans from large vessels to capillaries, while the mass flow back goes within the venous system, from capillaries to large veins. Examples of blood vessel systems are shown in Fig. 1.2.1 where the geometrical complexity of the structure can be seen, particularly in the case of a tumor.

For the completeness of this introduction, we give basic data also used in Chapter 6. Regarding mass transfer within blood vessels, there exist transport of biological fluid and particulate transport for which the fluid is the carrying medium. In engineering terms, fluid flow can also be called hydraulic transport. In this book, we are focused on the transport of matter, which goes from vessels to tissue and vice versa. The overall transport in the vessel-tissue system depends, first of all, on blood vessel properties, which include hydraulic and diffusive components within the vessels and the vessel walls.

The cardiovascular system takes an especially critical role in transport because it redistributes oxygen, nutrients, metabolites, hormones, as well as drugs through all organs. The main components of the cardiovascular system are the heart and major vessels responsible for blood perfusion and its distribution across all other vessels. It takes 1–2min on average to reperfuse all blood volume in an adult human body. It is natural to represent all vessels as a tree: large vessels such as the aorta move a bulk volume of oxygenated blood into smaller arteries that branch into smaller and smaller ones and reach organ capillaries. Deoxygenated blood from capillaries gets collected into veins gradually increasing in size, to be finally pumped by the heart to the lung. The mass exchange with tissues in organs only happens in capillaries that have a diameter of around 10–20μm and a capillary wall of ∼1μm. Larger vessels have thick and mechanically strong walls preventing any mass exchange across them. The systematic circulation that redistributes blood volume through organs is driven by perfusion and pressure gradients created by the heart. However, diffusion becomes an important transport mechanism at the capillary level only, where perfusion velocity drops from 10cm/s in the aorta to 100–500μm/s in capillaries (illustrated in Fig. 1.2.2).

Figure 1.2.1  Blood vessel system. (A) Blood vessels within mouse brain (Kojic et al., 2015); (B) Blood vessels within mouse tumor. (A) Lab of The Singleton Department of Pediatric Radiology, Texas Children's Hospital, USA. (B) Rita Serda, Baylor College of Medicine, Houston, TX, private communication.

The body also has the lymphatic branch, which is a part of the circulatory system. This system provides the moving of interstitial fluids drained from tissues under the pressure created by the cardiovascular system. Lymphatic flow is much slower than in other vessels, and it merges with the venous system returning fluid to the cardiovascular system.

Perfusion transport efficiently mixes and redistributes blood in a body. However, concentration profiles of the transported molecules in the systemic blood pool depend on how mass reaches the blood volume. The O2 levels in the blood can be taken as steady under normal conditions because of the constant gas exchange in the lung. Food, water, and drugs usually are consumed orally, hence the mass has to pass to the digestive tract to be absorbed and processed. Such a process will create an onset of concentration in blood of specific material taking minutes or hours after the consumption. Furthermore, a fraction of the consumed dose of nutrients or drugs may be lost due to digestion and excretion, which may reduce concentration in blood. All organs, including those of the cardiovascular and digestive systems, will consume available nutrients or other substances, contributing to the decline of concentration in the blood. In some cases, medications are administered intravenously, which allows drugs to bypass the digestive system at first and increase drug delivery to tissues due to larger concentrations in blood.

Figure 1.2.2  Characteristics of the cardiovascular system of a 13kg dog. (A) Cross-sectional area of the blood vessel for arteries, veins, and capillaries; (B) velocity distribution; (C) pressure distribution; (D) blood volume in arteries, veins, and capillaries. According to Rushmer, R., 1976. Cardiovascular Dynamics, fourth ed. Saunders, WB, Philadelphia, with permission.

In order to gain insight into the [blood vessel]-[vessel wall]-[tissue] system, Fig. 1.2.2 shows the cross-sectional area, velocity, pressure, and blood volume distributions in the blood vessel system of a dog. It can be seen that there are differences in the order of magnitude in the cardiovascular system (arteries, veins, and capillaries). These physical characteristics are fundamental for transport within the cardiovascular system. In large vessels, convective transport dominates while on the capillary level, diffusion prevails. When studying transport through vessel walls, the striking fact is that practically the entire blood vessel wall surface area belongs to capillaries. This follows from Fig. 1.2.2A: for a straight circular vessel, the wall surface area is proportional to the cross-sectional area, with the coefficient of proportionality of 4/d (d being the vessel diameter) (Rushmer, 1976). It has been estimated that there are 40 billion capillaries in the average human body. If all the capillaries in the human body were lined up in a line, the line would have a length over 100,000 miles. Therefore, in studying the supply of nutrients or drugs to cells within tissues, it is essential to achieve the desired transport through capillary walls, as they constitute the major biological barrier to gradient-driven transport.

Hydraulic and diffusive transport through capillary walls occurs through permeable walls. Fig. 1.2.3A shows a capillary with epithelial cells at the capillary outer surface, which is in contact with the surrounding tissue filled with interstitial fluid. The transport of fluid and other particulate occurs through pores (fenestrae) shown in figure. A collagen sleeve can surround a capillary (schematics shown in Fig. 1.2.3B) as an additional biological barrier for mass transport (Yokoi et al., 2014; Kojic et al., 2015).

Since the body holds a diverse hierarchy of tissues, cells, and organs, various transport mechanisms are present at the same time. The mass exchange at the systemic level and body can be tackled by a multiphysics and multiscale computational model. The spatial scales span practically from 10 −⁹ (molecules) to 10⁰ m (a body). The associated times of the processes at different scales also are different. Thus, proper awareness should be exercised in analyzing human body mass exchange. Furthermore, transport problems can be significantly complicated by the need to incorporate tissue biomechanics, such as in the case of the heart or lung.

Figure 1.2.3  (A) Capillary structure. Left: capillary interior filled with flowing fluid (plasma and blood cells), epithelial cells at the internal capillary surface, and interstitial tissue fluid surrounding capillary. Right: Capillary wall with basal membrane and pores (fenestrae). (B) Schematics of collagen sleeve surrounding capillary as a barrier for transport from capillary to tissue (Yokoi et al., 2014; Kojic et al., 2015).

Modeling of transport within capillaries and further in the tissue will be shown in Chapters 5 and 6. It is important here to note that pressure gradient or concentration gradient is the driving factor causing mass exchange. The fluid pressure difference across the capillary walls dominates in the aortic branch, while in the veins osmotic pressure has an important role in mass transfer from tissue to the capillary system. The same governing physical laws are applicable to the lymphatic system within the body, as well.

1.3. Tissue

Biological tissue is the medium connected to the vessel system for mass exchange. Tissue is composed of extracellular medium and cells, Fig. 1.3.1. In this section, we describe the main properties of the extracellular space since the mass exchange between vessels and tissue occurs via this domain of the tissue; detailed analysis is given in numerous references, e.g, (Halper and Kjaer, 2014; Kular et al., 2014).

Extracellular space is a complex porous medium filled with a biological fluid. The solid phase is composed of various types of proteins forming the extracellular matrix (ECM) schematically shown in Fig. 1.3.1. Composition of the ECM includes: collagen of different types (I, II, III, V, and XI), which is formed as fibrils of size 10–12nm in diameter in connective tissue, affecting adhesion and migration; elastin, which provides tensile strength and recover/recoil; fibronectin, arranged into a mesh of fibrils and linked to cell surface receptors (integrins), located in the basement membrane of the ECM, important in cell adhesion and wound healing; laminins, tenascins, and others.

Figure 1.3.1  Tissue composition. An enlarged region with cells and extracellular space (middle panel) and enlarged extracellular space (right panel). Image form E.J. Koay, CT imaging, data recorded at MD Anderson Cancer Institute, Houston, under approved Institutional Review Board protocol PA14-0646.

Extracellular space represents the tissue microenvironment and physiologic barrier to transport. The mass transport within extracellular space is a very complex biophysical process that follows from the composition of this medium. It can be considered, in a simplified approach, that fluid flow within the extracellular space is driven by a pressure gradient. The proportionality coefficient, called the Darcy coefficient, depends on the pore size within the medium and must be determined experimentally for a considered tissue. On the other hand, particulate transport usually involves biochemical interactions with the solid phase, and diffusivity in Eq. (1.1.1) depends on the pore size and current concentration. These interactions on the molecular level were evaluated in Sections 5.2.1 and 5.2.2 using molecular dynamics (MD), where appropriate scaling functions were introduced for the correction of diffusivity.

1.4. Cells

Cells are the smallest and fundamental units of life. Cells consist of cytosol, which is a complex water-based mixture of biomolecules, salts, and water; organelles dispersed in the cytosol; and a cell membrane enclosing all of those components. There are prokaryotic and eukaryotic cell types, where the first type does not have a nucleus, while the other has it. Bacteria are the best-known example of prokaryote cells, which are one of the oldest living organisms on the planet, and are responsible for many infectious diseases. Cell size is usually within a few micrometers, with shapes that can range from spheres to spirals.

Eukaryote cells are plant and animal or human cells. We further focus on typical cells of this type according to the scope of the book. These cells are usually 10–20μm in size with spherical-like shapes. A cell, taken from the image in Fig. 1.3.1, is shown in Fig. 1.4.1A. Cell interior with a nucleus, organelles, and cytoskeleton is schematically shown. In Fig. 1.4.1B, the basic composition of a cell membrane is displayed. In the text that follows, the role of some organelles and characteristics of the cell membrane are described. This information serves as the basis for the development of computational models in subsequent chapters.

The size and shape of cells can be significantly different across different cell types, and they may change according to their biological state. Each cell has different organelles in singular or multiple copies-that govern various biological functions. Each organelle, e.g., nucleus, endoplasmic reticulum, Golgi apparatus, mitochondria, or lysosomes, represents an individual compartment within a cell, separated by its membrane. The interior of an organelle may possess chemical composition and physical properties different from the cell cytosol. The compartmentalization aspects are especially important in view of the mass transport because the entire cell interior is divided physically into domains with specific transport and material properties. Furthermore, membranes separating cell cytosol and cell surrounding, or cell cytosol and organelle interior, are made of lipids and have different physiochemical nature compared to the water-based environment of cell cytosol or the interior of organelles. Thus, the analysis and computer simulations of transport across various intracellular structures should incorporate physical reality, which may become challenging for numerical solutions. Here we review a few important organelles.

Figure 1.4.1  (A) Cell with a schematically shown nucleus, organelles, and cytoskeleton. (B) Schematics of the cell membrane as a lipid bilayer, with ionic channel and transport protein.

Cell membrane. The cell membrane (schematics in Fig. 1.4.1B) is the outer shell of a cell that holds the cell interior and is responsible for the communications and mechanical interactions of the cell with its surrounding. The membrane is made of lipid bilayer where polar portions of lipid molecules are exposed to a water-based environment. Phospholipids, glycolipids, and sterols are the most common types of lipids in the membrane. Cholesterol is the most important component of sterols regarding membrane mechanical properties, such as fluidity. The lipid bilayer thickness is approximately 10–15nm, but it can vary because of other structural elements attached to the membrane, such as proteins or polysaccharides. The lipid-made membrane also implies that molecules well soluble in water will hardly pass the lipid phase and need specific transport mechanisms to get inside the cell. And in the opposite, lipophilic molecules may preferentially occupy the lipid phase avoiding water-based media of cytosol.

Integral proteins are the key part of the cell membrane (in Fig. 1.4.1B shown as transport protein) because they perform multiple functions such as signaling and molecular transport. Biological signaling is the ability to sense the presence of specific molecules upon their interaction with specific proteins, called receptors. Those receptors are proteins that are located on the outer side of the membrane. Once they have formed a molecular complex with ligand molecules, they undergo conformational changes triggering signal transduction across the membrane into the cell in a form of conformational changes of proteins. The cell membrane also contains proteins forming various porous structures in the membrane to regulate ion and water balance inside cells. Many of those proteins govern the active transport of molecules across the membrane, during which energy, stored in the form of adenosine triphosphate (ATP) molecules inside cells, is consumed. However, some protein channels may be passive (in Fig. 1.4.1B shown as ionic channel), i.e., transport may be driven by a concentration gradient, and may be semipermeable so that molecules or ions can be transferred only in one direction across a cell membrane. Having such a diversity of transport mechanisms within the membrane, it is important to establish correct transport models in the analysis of mass transport within cells.

Because of the semipermeable features of membranes and lipids acting as an electrical insulator, the imbalance of ion concentrations across the membrane creates the electrical membrane potential. The interior of a cell usually has a negative charge with respect to the outside domain. The potential inside the cell ranges from −40 to −80mV for most cells. The membrane potential has the critical role of cell homeostasis by powering cells for all operations with help of the ATP synthesis, which serves as a universal energy equivalent for all biochemical processes; it also enables electrical communications among some cells, e.g., neurons, muscle cells.

The potential is generated by many ions, where the monovalent sodium and potassium cations, Na+ and K+, are the most important. However, calcium Ca²+, magnesium Mg²+, and chlorine Cl − ions also may contribute to the potential. The semipermeable nature of membrane and active transport—i.e., integral membrane proteins acting as ion pumps, create ion fluxes across the membrane and polarize the membrane. At the same time, diffusion and electrical field counterbalance the active transport. The resting potential of a cell is achieved once those fluxes balance each other, and it is maintained as long as the physiology of a cell is not compromised. This aspect of cell membrane behavior is particularly important for muscle cells and will be modeled in Chapter 7 (related to heart modeling). It is worth mentioning that each ion has its own specific protein pumps, and those pumps may work in one direction only. For electrically active or excitable cells such as neurons, the changes in membrane potential are part of normal physiology for signal transduction through propagating changes in membrane potential (and ion concentration imbalance) along the cell body. Electrical, biochemical, and mechanical triggers may induce traveling depolarization of membranes. This process is complex, but the basis for it is the voltage-gated ion channels embedded in the membrane that can change their specific ionic permeability.

Cytoskeleton. The cytoskeleton of cells (schematically shown in Fig. 1.4.1A) is a dynamic network of protein filaments spanning within the interior of cells, immersed in the cytosol, and tightly interacting with the cell membrane. The cytoskeleton provides the cell with mechanical resistance and maintenance of the cell shape. It is a very dynamic structure and also contributes to cell movement, as well as to some transport phenomena in cells such as endocytosis—interacting with the cell membrane. The cytoskeleton also serves as a backbone of a cell by connecting and organizing many cellular components, such as organelles. Furthermore, cytoskeleton filaments serve as an intracellular transport highway, where different cellular components and organelles can be transported actively by consuming ATP energy.

There are many types of filaments and some are more specific to certain cell types. All filaments are polymers made of proteins, but they are specific for the type of filaments. Microtubules are the most important components of the cytoskeleton. The network of microtubules is composed of individual dynamic filaments made of tubulin proteins that may rearrange in response to the needs of a cell. The outer diameter of filaments is around 25nm, and the inner diameter is around 11nm. The microtubule network is especially important for intracellular transport, cell mechanical resistance, or cell division. Other types of filaments, such as microfilaments, may be critical to muscle contraction. These filaments are approximately 10nm in diameter and are made of actin proteins so that they can generate contractile or pushing forces within the cell and beyond.

Nucleus. The nucleus contains the genetic material of cells. Usually, cells have a single nucleus, but certain cell types may have no nucleus at all or may have few of them. The nucleus is wrapped by a double membrane separating its interior from the cytoplasm and contains filaments that provide mechanical support and organization of genetic material. The space between inner and outer membranes is about 20–40nm. The inner nucleus membrane is integrated with nucleus filaments, while the external membrane is connected to the cytosol cytoskeleton and other organelles. The nucleus double membrane is impermeable to larger biological molecules; therefore, it has nuclear pore structures, while the membranes have an internal gap of approximately 9nm. These membranes form channels enabling an exchange of large biological molecules, ions, and other molecules between the nucleus and the rest of a cell. The nucleus may be broken down during cell division, and it is reformed back after new cells are created. It may be the largest organelle of a cell, and it can have different mechanical properties from the cytosol.

Mitochondria. Mitochondria are double-membrane organelles like nuclei. These organelles are power plants of cells responsible for energy production by supplying cells with adenosine triphosphate (ATP) molecules. ATP molecule is the universal unit with biochemical energy fueling practically all active cellular processes, such as cell movement, division, intracellular transport, transmembrane potential, and all other cellular processes. ATP stores chemical energy, which is released through biochemical ATP conversion into adenosine diphosphate (ADP). Besides energy production, mitochondria participate in cell proliferation and death as well. The size and number of mitochondria can vary significantly among cells.

Mitochondria are multicompartment organelles with different physicochemical properties in their compartments. The outer mitochondria membranes are freely permeable to small molecules such as ions or water and have protein transporters to move larger biomolecules or special molecules across them. The intermembrane space (perimitochondrial space) separates inner and outer membranes. While the chemical composition of this space is similar to the cytosol, this compartment contains a specific protein, cytochrome c, which can be released outside it during processes of cell death. The inner membrane is very different from the outer membrane and other cell membranes. It has numerous folds (cristae) that increase its surface area many times compared to the outer membrane. The inner membrane also has a different chemical composition with respect to the outer membrane and contains numerous complexes of integral proteins important for transmembrane potential, electron redox reactions, ATP synthesis, and molecular transport. The inner membrane is practically impermeable to molecules so that property helps to develop its transmembrane potential. This potential fuels electron redox reactions within integral proteins of the membrane in synthesizing ATP molecules inside the mitochondria matrix enclosed by the inner membrane. The matrix contains also a mitochondrial genome making this organelle different from the others.

Endoplasmic reticulum (ER). ER is a large network of the membrane-enclosed sac with cisterna or tubular shapes dispersed through the cytosol. ER connects directly to the nucleus outer membrane. The ER is primarily responsible for the synthesis of biomolecules. There are two types of ERs: rough and smooth ER, depending on the presence of ribosomes on the surface of the ER. Rough ER membrane contains many ribosomes that perform protein synthesis. Some of those proteins are excreted from a cell, while the smooth ER specializes in the synthesis and transport of other molecules. Smooth and rough ERs are in a dynamic state, and they change according to the needs.

The sarcoplasmic reticulum can be viewed as a variation of the smooth ER, and it is found in muscle cells with various biochemical compositions and functions. Its primary role is calcium ion storage and the release during muscle contraction.

1.5. Specificities of the body organs with respect to transport

In this section, we summarize the main characteristics of the body organs with respect to mass transport and mass exchange (Hoehn and Marieb, 2010; Tortora and Derrickson, 2018).

Different types of cells may compose tissue giving the tissue-specific biological features that are needed for the functioning of the body organs. As described in the previous section, cells within the tissue are embedded into the extracellular matrix, which is a complex medium made of many different salts, organic molecules, water, enzymes, polysaccharides, and other biomolecules. Extracellular matrix composition, chemistry, and mechanics can be specific to organ tissue and cells within it. Many organs in the human body are organized into systems, such as digestive, nervous, respiratory, reproductive, skeleton-muscular, circulatory, and others. Each system can be composed of numerous organs. For example, the digestive system includes the mouth, esophagus, stomach, small intestines, large intestines, liver, pancreas, etc. The biological tissues composing these organs can be even more diverse. This diversity, or heterogeneity, leads to the high complexity of the biological environment. But it is common that in order to function, all organs need oxygen and nutrients, as well as waste removal. Here, the circulatory systems interweave with the tissues of organs and serve as the main mass transport highway.

Heart, lung, liver, brain, kidney, guts, pancreas, muscle, blood, and skin are frequently analyzed entities according to their biomedical importance. We here review a few of them briefly, presenting their anatomical