Modeling of Mass Transport Processes in Biological Media

By Dan Zhao

Modeling of Mass Transport Processes in Biological Media focuses on applications of mass transfer relevant to biomedical processes and technology—fields that require quantitative mechanistic descriptions of the delivery of molecules and drugs. This book features recent advances and developments in biomedical therapies with a focus on the associated theoretical and mathematical techniques necessary to predict mass transfer in biological systems.

The book is authored by over 50 established researchers who are internationally recognized as leaders in their fields. Each chapter contains a comprehensive introductory section for those new to the field, followed by recent modeling developments motivated by empirical experimental observation. Offering a unique opportunity for the reader to access recent developments from technical, theoretical, and engineering perspectives, this book is ideal for graduate and postdoctoral researchers in academia as well as experienced researchers in biomedical industries.

• Offers updated information related to advanced techniques and fundamental knowledge, particularly advances in computer-based diagnostics and treatment and numerical simulations
• Provides a bridge between well-established theories and the latest developments in the field
• Coverage includes dialysis, inert solute transport (insulin), electrokinetic transport, cellular molecular uptake, transdermal drug delivery and respiratory therapies

• Language: English
• Publisher: Academic Press
• Release date: Aug 24, 2022
• ISBN: 9780323857413

Book preview

9780323857413_FC

Modeling of Mass Transport Processes in Biological Media

First Edition

Sid Becker

Department of Mechanical Engineering, University of Canterbury, Christchurch, New Zealand

Andrey V. Kuznetsov

Department of Mechanical & Aerospace Engineering, North Carolina State University, Raleigh, NC, United States

Filippo de Monte

Department of Industrial and Information Engineering and Economics, University of L’Aquila, L’Aquila, Italy

Giuseppe Pontrelli

Institute of Applied Mathematics - CNR, Rome, Italy

Dan Zhao

Department of Mechanical Engineering, University of Canterbury, Christchurch, New Zealand

Table of Contents

Cover image

Title page

Copyright

Contributors

Preface

Chapter 1: Applications of porous media in biological transport modeling

Abstract

1.1: Introduction

1.2: Applications of porous media in modeling in modeling transport phenomena in arteries

1.3: Fluid–structure interaction in biomedical applications

1.4: Brain aneurysms

1.5: Magnetic resonance imaging (MRI)

1.6: Concluding remarks

References

Chapter 2: Metabolic consumption of microorganisms

Abstract

2.1: Introduction

2.2: Model formulation and metabolic mass transfer

2.3: Analysis of the equation governing monotonic growth

2.4: Closed form analytical solution of the monotonic growth

2.5: Results and discussion

2.6: Conclusions

References

Further reading

Chapter 3: Numerical simulation of deformability cytometry: Transport of a biological cell through a microfluidic channel

Abstract

Acknowledgments

3.1: Introduction

3.2: Modeling biological cells in an RT-DC channel

3.3: Hydrodynamic stresses on the cell surface

3.4: Cell shapes and cell deformation

3.5: Extraction of the cell viscosity

3.6: Approximation error of the cell volume over the channel

3.7: Conclusions

Appendix: Derivation of the deformation measure from cell contours

References

Chapter 4: Computation of organelle age during axonal transport

Abstract

Acknowledgments

4.1: Introduction

4.2: Governing equations

4.3: Simulation of the DCV concentration in the axon

4.4: Age distribution model of DCVs and mean age of DCVs in boutons

4.5: Parameter value estimation

4.6: Numerical approach

4.7: Results

4.8: Discussion, model constraints, and future directions

References

Chapter 5: Continuum models of drug transport to multiple cell-type population

Abstract

5.1: Introduction

5.2: Formulation of the problem

5.3: Method of solution

5.4: Case study: A 3D rectangular biological tissue

5.5: Results and discussion

5.6: Conclusions

5.A: Appendix A

5.B: Appendix B

5.C: Appendix C

5.D: Appendix D

5.E: Appendix E

References

Chapter 6: Computational investigation of the role of low-density lipoprotein and oxygen transport in atherosclerotic arteries

Abstract

Acknowledgment

6.1: Introduction: Atherosclerosis and the role of mass transport

6.2: Mass transfer of low-density lipoproteins and oxygen in arteries: Theoretical background

6.3: Mass transfer of low-density lipoproteins in arteries: Computational modeling

6.4: Mass transfer of oxygen in arteries: Computational modeling

6.5: Limitations of the current models and future directions

6.6: Conclusions

References

Chapter 7: Fluid dynamics and mass transport in lower limb vessels: Effects on restenosis

Abstract

Acknowledgment

7.1: Introduction

7.2: Lower limb vessels: Anatomy, physiopathology, treatment options, and their failure

7.3: Modeling the hemodynamics of treated lower limb vessels

7.4: State-of-the-art computational mass transport models of lower limb vessels

7.5: Conclusions and future directions

References

Chapter 8: Numerical modeling in support of locoregional drug delivery during transarterial therapies for liver cancer

Abstract

8.1: Introduction

8.2: State of the art of computational techniques

8.3: Clinical parameters

8.4: State of the art of experimental techniques

8.5: Closing remarks

References

Chapter 9: Active gel: A continuum physics perspective

Abstract

Acknowledgments

9.1: Introduction

9.2: A short insight into active gel physics

9.3: A continuum model of active gels

9.4: Worked examples

9.5: Conclusions and future challenges

Appendix: Cylindrical coordinates

References

Chapter 10: Modeling nasal spray droplet deposition and translocation in nasal airway for olfactory delivery

Abstract

Acknowledgment

10.1: Introduction

10.2: Materials and methods

10.3: Results

10.4: Discussions and conclusion

References

Chapter 11: Drug delivery and in vivo absorption

Abstract

11.1: Drug delivery

11.2: State of the art

11.3: Oral administration

11.4: Conclusions

References

Chapter 12: Modeling the physiological phenomena and the effects of therapy on the dynamics of tumor growth

Abstract

12.1: Introduction

12.2: Formal reaction kinetics

12.3: Creating tumor model with formal reaction kinetics

12.4: Concluding remarks

References

Chapter 13: Mathematical models of water transport across ocular epithelial layers

Abstract

13.1: Introduction to fluid transport in the eye

13.2: Formulation of the problem of water and solute transport

13.3: Mechanisms involved in water transport across cell layers

13.4: Aqueous humor production at the ciliary processes

13.5: Transport across the corneal endothelium

13.6: Transport across the retinal pigment epithelium

13.7: Conclusions

References

Chapter 14: Multidimensional modeling of solid tumor proliferation following drug treatment: Toward computational prognosis as a tool to support oncology

Abstract

14.1: Introduction

14.2: The workflow of the simulation framework

14.3: Mathematical formulation

14.4: Results

14.5: Conclusions

References

Chapter 15: Modeling LDL accumulation within an arterial wall

Abstract

15.1: Introduction

15.2: Wall-free and single-layer models

15.3: Multilayer modeling

15.4: Published results of multilayer models

15.5: Conclusions and future developments

References

Chapter 16: Modeling transport of soluble proteins and metabolites in the brain

Abstract

Acknowledgments

16.1: Introduction

16.2: Blood-brain barrier

16.3: Flow through the parenchyma

16.4: Conclusions

References

Chapter 17: Hybrid-dimensional models for blood flow and mass transport: Sequential and embedded 3D-1D models

Abstract

17.1: Introduction

17.2: 3D-1D geometric sequential multiscale models

17.3: 3D-1D geometric embedded multiscale models

17.4: Conclusions

References

Chapter 18: Chemical thermodynamic principles and computational modeling of NOX2-mediated ROS production on cell membrane

Abstract

Acknowledgment

18.1: Introduction

18.2: Thermodynamic principles for biochemical systems modeling

18.3: Mathematical modeling of NOX2 enzyme function

18.4: Data analysis and estimation of unknown model parameters

18.5: Biological insights into NOX2 enzyme function

18.6: Summary and conclusion

18.7: Model limitations and future directions

References

Index

Copyright

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Image 1

Contributors

Michela Abrami     Department of Engineering and Architecture, Trieste University, Trieste, Italy

Sebastian Aland

HTW Dresden, Dresden

TU Bergakademie Freiberg, Freiberg, Germany

A. Andreozzi     Department of Industrial Engineering, University of Naples Federico II, Naples, Italy

Said H. Audi

Department of Biomedical Engineering, Medical College of Wisconsin

Department of Biomedical Engineering, Marquette University, Milwaukee, WI, United States

Sid Becker     Department of Mechanical Engineering, University of Canterbury, Christchurch, New Zealand

N. Bianco     Department of Industrial Engineering, University of Naples Federico II, Naples, Italy

Tim Bomberna

IBiTech—Biommeda (Biofluid, Tissue and Solid Mechanics for Medical Applications)

Cancer Research Institute Ghent (CRIG), Ghent University, Ghent, Belgium

Christina Chan     Department of Chemical Engineering and Materials Science, Michigan State University, East Lansing, MI, United States

Claudio Chiastra

Laboratory of Biological Structure Mechanics (LaBS), Department of Chemistry, Materials and Chemical Engineering Giulio Natta, Politecnico di Milano, Milan

PoliToBIOMed Lab, Department of Mechanical and Aerospace Engineering, Politecnico di Torino, Turin, Italy

Monika Colombo     Laboratory of Biological Structure Mechanics (LaBS), Department of Chemistry, Materials and Chemical Engineering Giulio Natta, Politecnico di Milano, Milan, Italy

Anna Corti     Laboratory of Biological Structure Mechanics (LaBS), Department of Chemistry, Materials and Chemical Engineering Giulio Natta, Politecnico di Milano, Milan, Italy

Allen W. Cowley, Jr     Department of Physiology, Medical College of Wisconsin, Milwaukee, WI, United States

Michele Curatolo     Department of Architecture, University Roma Tre, Rome, Italy

Giampaolo D’Alessandro     Department of Industrial and Information Engineering and Economics, University of L’Aquila, L’Aquila, Italy

Barbara Dapas     Department of Life Sciences, Trieste University (Cattinara), Trieste, Italy

Ranjan K. Dash

Department of Biomedical Engineering, Medical College of Wisconsin

Department of Biomedical Engineering, Marquette University

Department of Physiology, Medical College of Wisconsin, Milwaukee, WI, United States

Maria Valeria De Bonis     Initiatives for Bio-Materials Behavior Srls c/o University of Basilicata, Potenza, Italy

Filippo de Monte     Department of Industrial and Information Engineering and Economics, University of L’Aquila, L’Aquila, Italy

Giuseppe De Nisco     PoliToBIOMed Lab, Department of Mechanical and Aerospace Engineering, Politecnico di Torino, Turin, Italy

Charlotte Debbaut

IBiTech—Biommeda (Biofluid, Tissue and Solid Mechanics for Medical Applications)

Cancer Research Institute Ghent (CRIG), Ghent University, Ghent, Belgium

Rosario di Vittorio     Department of Engineering and Architecture, Trieste University, Trieste, Italy

Dániel András Drexler

Physiological Controls Research Center, University Research and Innovation Center

Institute of Biomatics and Applied Artificial Intelligence, Óbuda University, Budapest, Hungary

Mariia Dvoriashyna     Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, United Kingdom

Rossella Farra     Department of Medical Sciences, Trieste University (Cattinara), Trieste, Italy

Luca Formaggia     Department of Mathematics, Politecnico Di Milano, Milan, Italy

Alexander J.E. Foss     Department of Ophthalmology, Nottingham University Hospitals NHS Trust, Nottingham, United Kingdom

Bingmei M. Fu     Department of Biomedical Engineering, The City College of the City University of New York, New York, NY, United States

Eamonn A. Gaffney     Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford, United Kingdom

Gabriele Grassi     Department of Life Sciences, Trieste University (Cattinara), Trieste, Italy

Lucia Grassi     Department of Engineering and Architecture, Trieste University, Trieste, Italy

Mario Grassi     Department of Engineering and Architecture, Trieste University, Trieste, Italy

M. Iasiello     Department of Industrial Engineering, University of Naples Federico II, Naples, Italy

Khalil Khanafer     Mechanical Engineering Department, University of Michigan, Flint, MI, United States

Levente Kovács

Physiological Controls Research Center, University Research and Innovation Center

Institute of Biomatics and Applied Artificial Intelligence, Óbuda University, Budapest, Hungary

Andrey V. Kuznetsov     Department of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC, United States

Ivan A. Kuznetsov

Perelman School of Medicine

Department of Bioengineering, University of Pennsylvania, Philadelphia, PA, United States

Shay Ladd     Department of Chemical Engineering and Materials Science, Michigan State University, East Lansing, MI, United States

Laura Lagonigro     Modeling and Prototyping Laboratory, College of Engineering, University of Basilicata, Potenza, Italy

Francesco Marra

Initiatives for Bio-Materials Behavior Srls c/o University of Basilicata, Potenza

Department of Industrial Engineering, University of Salerno, Fisciano, SA, Italy

Sean McGinty     Division of Biomedical Engineering, University of Glasgow, Glasgow, United Kingdom

Francesco Migliavacca     Laboratory of Biological Structure Mechanics (LaBS), Department of Chemistry, Materials and Chemical Engineering Giulio Natta, Politecnico di Milano, Milan, Italy

Gesmi Milcovich     School of Chemical Sciences, Dublin City University, Dublin, Ireland

Paola Nardinocchi     Department of Structural and Geotechnical Engineering, Sapienza University, Rome, Italy

Giuseppe Pontrelli     Institute of Applied Mathematics - CNR, Rome, Italy

Felix Reichel     MPL & MPZPM Erlangen, Erlangen, Germany

Rodolfo Repetto     Department of Civil, Chemical and Environmental Engineering, University of Genoa, Genoa, Italy

Jose Felix Rodriguez Matas     Laboratory of Biological Structure Mechanics (LaBS), Department of Chemistry, Materials and Chemical Engineering Giulio Natta, Politecnico di Milano, Milan, Italy

Gianpaolo Ruocco

Initiatives for Bio-Materials Behavior Srls c/o University of Basilicata

Modeling and Prototyping Laboratory, College of Engineering, University of Basilicata, Potenza, Italy

Shima Sadri     Department of Biomedical Engineering, Medical College of Wisconsin, Milwaukee, WI, United States

Xiuhua April Si     Department of Aerospace, Industrial, and Mechanical Engineering, California Baptist University, Riverside, CA, United States

Luciano Teresi     Mathematics and Physics, University Roma Tre, Rome, Italy

Namrata Tomar     Department of Biomedical Engineering, Medical College of Wisconsin, Milwaukee, WI, United States

C. Tucci     Department of Medicine and Health Sciences Vincenzo Tiberio, University of Molise, Campobasso, Italy

Alisa S. Vadasz     College of Engineering, Informatics, and Applied Sciences, Northern Arizona University, Flagstaff, AZ, United States

Peter Vadasz     College of Engineering, Informatics, and Applied Sciences, Northern Arizona University, Flagstaff, AZ, United States

Kambiz Vafai     Department of Mechanical Engineering, University of California, Riverside, CA, United States

Lucas Daniel Wittwer

HTW Dresden, Dresden

MPL & MPZPM Erlangen, Erlangen

TU Bergakademie Freiberg, Freiberg, Germany

Neil T. Wright     Department of Mechanical Engineering, Michigan State University, East Lansing, MI, United States

Jinxiang Xi     Department of Biomedical Engineering, University of Massachusetts, Lowell, MA, United States

Paolo Zunino     Department of Mathematics, Politecnico Di Milano, Milan, Italy

Preface

This book presents a collection of 18 chapters, each of which considers the modeling of mass transfer involved in different biological processes. This collection brings together more than 50 internationally recognized researchers who are leading the development of transport models in their respective fields. The chapters provide detailed introductions so that the reader new to the field is not alienated by the technical diction. In fact, it is anticipated that both the advanced academic clinician and the emerging researcher will find the material presented here clear and informative.

Chapter 1 presents a critical review of the porous media representations of transport phenomena in biological systems. The chapter begins with a summary of the literature covering modeling transport phenomena in arteries and continues with porous media in modeling of endovascular coiling in the treatment of cerebral aneurysms. Finally, the chapter presents a summary of diffusive transport models of brain tissues with a focus on the application of molecular dynamics models in the simulation of transport through biological membranes.

Chapter 2 develops a novel theory of the metabolic consumption by microorganisms that are compared both quantitatively and qualitatively to existing theoretical models and to empirical observations. The chapter provides the reader with a comprehensive historical review of the existing models of predictive population dynamics and critically reviews the current mathematical expressions of this field. Different mathematical approaches are presented that are directly linked to the underlying physical phenomena.

Chapter 3 addresses recent developments and challenges associated with deformability cytometry with special emphasis on the area of real-time deformability cytometry. Theoretical and numerical models of the physical characterization of the mechanical properties of large populations of cells are reviewed and the numerical handling of the fluid-solid interaction observed in microfluidic chip technology are described. A model is presented in this chapter that represents a significant step toward the field’s need to develop a methodology that captures the full viscoelastic characterization of biological cells in high-throughput experiments.

Chapter 4 reviews the modeling of the transport of organelles in the axon. The field’s current theoretical understanding of the prediction of the age of organelles as they are transported along the axon is reviewed. The chapter provides a computational technique to simulate the age of dense core vesicles during their transport to active terminal sites that accumulate and release neurotransmitters.

Chapter 5 presents a model drug of delivery in a tissue composed of a multiple cell type population. A review of the most common components used in the three compartment continuum models of drug delivery to single cell type populations is presented. This is followed by a review of models describing transport in domains composed of multiple cell type populations. The chapter presents an analytical solution to the problem of the conservation of drug mass in a tissue composed of three cell types.

Chapter 6 considers the computational modeling of the cardiovascular transport of low-density lipoproteins (LDLs) and oxygen and their influence on atherosclerosis. The chapter presents a very comprehensive review of the theoretical and mathematical understanding of the physiology underlying transport in the vessel wall. The chapter considers existing computational models of mass transport of LDL and oxygen in different vascular segments, including coronary, carotid, femoral arteries, and aortas and special attention is given to their atherogenic role. The potential and challenges posed by the application of existing theoretical models to patient-specific fluid dynamics simulations are presented as well.

Chapter 7 reviews the modeling of transport associated with the cardiovascular system of lower limbs. The chapter presents a thorough description of the field that details the theory underlying computational studies of fluid dynamics and mass transport in lower limb vessels and focuses on the relationship between local hemodynamics and restenosis in both the surgical and endovascular treatment options for peripheral artery disease. A clear link is made between the physiology and physics of these treatments and the theoretical models used to represent them.

Chapter 8 presents a detailed review of the state of the art of in silico modeling of the mass transport phenomena related to drug delivery in the hepatic arterial tree. The important physiological considerations associated with the response of the cardiovascular system to treatments are presented. The chapter discusses the challenges and advantages offered by computational modeling for improving the efficacy of transarterial embolization therapies for unresectable liver-confined hepatocellular carcinoma. An informed list of specific recommendations to better predict patient-specific injection parameters for better treatment efficacy is presented.

Chapter 9 discusses a chemomechanical model of active gels within the context of a stress-diffusion theory, augmented with a theory of remodeling. A chemomechanical model of active gels is developed by using the perspective of continuum physics; the activation of the polymer network is viewed within the context of a stress-diffusion theory that is coupled to a model of growth and remodeling. A model is developed and a thorough discussion of the theoretical foundations of the model is presented.

Chapter 10 describes the transport of drugs in the route from the nose to the brain in therapies for olfactory dysfunctions. The chapter also describes the physiology and the physics associated with olfactory drug delivery and presents a study of the feasibility of gravitationally driven droplet translocation in order to enhance the nasal spray dosages in the olfactory region. A computational nasal spray testing platform is developed that includes a nasal spray releasing model, an airflow-droplet transport model, and a Eulerian wall film formation/translocation model.

Chapter 11 presents a very comprehensive review of the general field of drug delivery and of the models of drug absorption. The models presented in the chapter are set in their historical context, and to this end, a very comprehensive description of the evolution of drug delivery and therapies is presented. The chapter explains that the challenges associated with using mathematical modeling to aid researchers in overcoming biological barriers require a vision that must extend beyond drug release kinetics to include processes related to adsorption, distribution, and elimination.

Chapter 12 addresses the mathematical modeling of tumor growth and response to chemotherapeutic therapies. The chapter considers many specific physiological characteristics of the patients that affect the efficacy of the treatment and subsequent tumor evolution. A mathematical model is developed that captures the observed fundamental physiological phenomena: the tumor proliferation and necrosis, the tumor’s response to the drug, tumor cell washout, and the pharmacokinetics of the drug. Parametric fitting from experimental observations is also discussed.

Chapter 13 examines the fluid dynamics and transport associated with physiological processes in the eye. The physics underlying fluid transport across ocular epithelial layers are discussed including osmotic and mechanical pressure differences across the layer, electroosmosis along the clefts separating adjacent cells, and local osmosis. In a detailed review of the mathematical representations of these phenomena, the chapter focuses on models of transport in the ciliary epithelium, the corneal endothelium, and the retinal pigment epithelium.

Chapter 14 provides an engineering perspective on the development of virtualized oncological prognosis, including all the fundamental steps needed to develop an oncological digital support tool. The model is applied to diffuse large B-cell (non-Hodgkin’s) lymphoma proliferation in personalized organ volumes, and its validation and output are discussed with a few clinical examples. The tool uses digital images taken of patients’ tumors to provide data of tumor geometry that can be used in a personalized model to predict the growth of the tumor that is treated and compare this to the predicted growth in the absence of treatment.

Chapter 15 considers the transport of LDLs through the arterial wall and their association with cardiovascular diseases. This chapter reviews existing mathematical models of LDL transport and categorizes these into groups: the wall-free models, the single-layer models, and the multilayer models. Emphasis is given to the mathematical descriptions of the models in the multilayer category. The applications of these models to nonsimplified geometries that are encountered in patient-specific cardiovascular physiology are presented.

Chapter 16 discusses mathematical models of the molecular transport within the brain and across the blood-brain barrier (BBB). A detailed description of the underlying physiology associated with the transport and delivery of molecules to the brain is presented. A review of the field is presented that outlines models of filtration across the BBB and cerebrospinal fluid flow through the parenchyma.

Chapter 17 concerns the coupling of models that capture cardiovascular transport phenomena occurring at the macroscale and at the microscale. The field’s approach to overcoming the challenge associated with the coupling between the three-dimensional and one-dimensional representations of transport is discussed. Sequential and embedded coupling strategies are reviewed and the mathematical expressions of the models are presented both explicitly and in detail.

Chapter 18 provides a comprehensive overview of the thermodynamic principles and relationships for general biochemical reactions and special oxidation-reduction (redox) reactions involving electron transfer. These principles and relationships are linked to the thermodynamically constrained mathematical models of enzymatic and redox reactions associated with reactive oxygen species production and scavenging and other biochemical processes. The chapter provides a detailed review of the mathematical and computational approaches to develop, parameterize, and validate kinetic models describing the different aspects of NOX2 enzymatic function on the cell membrane.

This book presents the physics, physiology, and mathematics involved in the modeling of transport phenomena occurring in biological media. It is anticipated that the reader will find the book very helpful in bridging the gap between transport modeling and biology.

Sid Becker, Mechanical Engineering Department, University of Canterbury, New Zealand

Chapter 1: Applications of porous media in biological transport modeling

Khalil Khanafera; Kambiz Vafaib    a Mechanical Engineering Department, University of Michigan, Flint, MI, United States

b Department of Mechanical Engineering, University of California, Riverside, CA, United States

Abstract

A critical review of the applications of porous media in a biological system is conducted in this study. Transport phenomena in porous media are receiving a great deal of attention from many investigators due to their importance in many biomedical applications. Such applications include drug delivery, medical imaging, transport in biological tissues, and porous scaffold for tissue engineering. This chapter is structured in three sections: the first part summarizes pertinent studies in modeling transport phenomena in arteries. The second part focuses on the applications of porous media in modeling endovascular coiling in the treatment of cerebral aneurysms. The third part summarizes the available diffusion models and effective diffusion coefficients associated with brain tissues. Finally, future studies on the applications of porous media are recommended in this review.

Keywords

Porous media; Biological systems; Critical review; Transport phenomena

1.1: Introduction

Transport phenomena in porous media have received substantial interest by many researchers in the last few decades due to their importance in various engineering and medical fields. Such applications include solar panels, thermal management of electronics, thermal insulation in buildings, oil recovery, drug delivery, tissue engineering, medical imaging, and diffusion process in brain tissues (Vafai, 2000; Vafai, 2005; Vafai and Hadim, 2000; Vafai and Sozen, 1990; Vafai et al., 2006; Vafai and Tien, 1981, 1982; Khanafer and Vafai, 2006; Hadim and Vafai, 2000). Many pertinent parameters associated with transport phenomena in porous media were analyzed by many authors in the literature. Such parameters include investigating the influence of non-Darcy, porosity variation, anisotropic, and local nonequilibrium thermal between fluid and solid phases. For example, Alazmi and Vafai (2000) conducted a comprehensive study to analyze the changes within flow and heat transfer models in porous media. The authors categorized these models using four groups, namely, constant and variable porosity, thermal dispersion, and local thermal nonequilibrium. The same authors investigated the influence of varying several parameters such as Reynolds number, Darcy number, porosity, inertia, and slip coefficients on transport phenomena in porous media through testing different models of interfacial conditions between porous and fluid layers (Alazmi and Vafai, 2001, 2002). Another important topic studied in the literature is turbulent flow in porous media. Vafai et al. (2006) conducted a study on different turbulent models for transport phenomena in porous media. This chapter aims to provide a comprehensive review of porous media interpretations of fluid flow and heat transfer in biological systems.

1.2: Applications of porous media in modeling in modeling transport phenomena in arteries

Computational modeling is considered an important tool in understanding the macromolecular transport process in arteries. Many numerical and mathematical models were developed and utilized by investigators to study the transport of low-density lipoprotein (LDL) through the arterial walls (Ai and Vafai, 2006; Yang and Vafai, 2006; Stangeby and Ethier, 2002a,b; Zunino, 2002; Prosi et al., 2005; Wada and Karino, 2000). Stangeby and Ethier (2002a) studied numerically the transport of macromolecules such as LDL across the artery wall of a stenosed artery. Coupled analysis of lumen blood flow and transmural fluid flow was attained through the solution of Brinkman’s model. Their results indicated that the concentration of LDL increased downstream of the stenosed artery. Prosi et al. (2005) introduced and discussed several models to study the transport of macromolecules in the bloodstream and the arterial walls. The authors showed that the concentration of LDL at the luminal side of a stenotic segment wall was about 10% higher than for the undisturbed segment. These models were applied by Wada and Karino (2000) and Rappitsch and Perktold (1996) for the analysis of the macromolecular transport in the arterial wall, which coupled the transport within the lumen and the wall.

The most realistic models to investigate the transport of LDL in arteries are the multilayer models, which break the arterial wall down into several layers and model the transport within the wall (Prosi et al., 2005; Huang et al., 1994; Huang and Tarbell, 1997; Tada and Tarbell, 2004; Fry, 1985, 1987; Karner et al., 2001). Yang and Vafai (2006) developed a robust four-layer model to investigate the transport of LDL in the arterial wall coupled with the transport of blood flow in the lumen as depicted in Fig. 1.1.

Fig. 1.1

Fig. 1.1 Physical description of the model. Reprinted with permission from Yang, N., Vafai, K., 2006. Modeling of low density lipoprotein (LDL) transport in the artery-effects of hypertension, Int. J. Heat Mass Transfer 49, 850–867, Elsevier.

The endothelium, intima, internal elastic lamina (IEL), and media were all modeled as macroscopically homogeneous porous media and the volume-averaged porous media equations were utilized to model the transport phenomena in various layers. The effect of Staverman filtration and osmotic reflection coefficients were introduced in their study to account for the selective permeability of each porous layer to certain solutes. The effects of hypertension and boundary conditions were analyzed by the authors in detail. It was found that the filtration velocity and the LDL concentration profiles in the arterial wall were substantially dependent on the various types of boundary conditions (Fig. 1.2).

Fig. 1.2

Fig. 1.2 Comparison of filtration velocity profiles at the lumen endothelium interface for various boundary conditions. Reprinted with permission from Yang, N., Vafai, K., 2006. Modeling of low density lipoprotein (LDL) transport in the artery-effects of hypertension, Int. J. Heat Mass Transfer 49, 850–867, Elsevier.

Ai and Vafai (2006) investigated numerically the coupled analysis of the transport of macromolecules such as LDL inside a stenosed artery. Different layers of the arterial wall were modeled as a homogeneous porous medium. The advection–diffusion equations in porous media were used to model the species field in the arterial wall layers. The physical parameters needed are computed based on the available data from in vivo and in vitro measurements. The benefit of the model used in that study was that its setup was based on in vivo/vitro measurements and the computed exact solutions of the concentration field, leading to more reliable results. The effects of hypertension and geometrical variation on the LDL accumulation within the wall were analyzed. Figs. 1.3 and 1.4 illustrated the effect of varying the transmural pressure on the filtration velocity and concentration. It can be seen from these figures that the filtration velocity and concentration increased significantly with hypertension.

Fig. 1.3

Fig. 1.3 Effect of varying the transmural pressure on the filtration velocity at different interfaces. Reprinted with permission from Ai, L., Vafai, K., 2006. A coupling model for macromolecule transport in a stenosed arterial wall, Int. J. Heat Mass Transfer 49, 1568–1591, Elsevier.

Fig. 1.4

Fig. 1.4 Effect of varying the transmural pressure on the concentration profile at the lumen/endothelium interface. Reprinted with permission from Ai, L., Vafai, K., 2006. A coupling model for macromolecule transport in a stenosed arterial wall, Int. J. Heat Mass Transfer 49, 1568–1591, Elsevier.

Yang and Vafai (2008) developed an analytical solution for the transport of LDL in the arterial wall coupled with blood flow inside the lumen. The authors utilized a robust four-layer porous model. The analytical results were found to be in excellent agreement with the numerical data for different physiological conditions (Fig. 1.5). The distribution of LDL concentration in each layer of the curved artery wall was studied analytically by Wang and Vafai (2015). The influence of curvature on the growth of atherosclerosis within the arterial wall was also analyzed in their investigation. Their results showed that the average concentration in the circumferential direction was found to decrease in the axial direction for a curved artery compared with a straight artery. The increase in concentration at the lumen/endothelium interface in the axial direction was found to have an insignificant effect on the concentration profile at the other wall interface layers.

Fig. 1.5

Fig. 1.5 Comparison of the species profiles across the intima between the numerical and analytical results. Reprinted with permission from Yang, N., Vafai, K., 2008. Low-density lipoprotein (LDL) transport in an artery—a simplified analytical solution, Int. J. Heat Mass Transfer 51, 497–505, Elsevier.

Iasiello et al. (2018) used COMSOL Multiphysics software to numerically study the boundary layer effects on the concentration of LDL in a multilayer artery model for geometries of both a straight artery and an aortic-iliac bifurcation. Their results illustrated that hypertension increased the concentration of LDL in the proximity of the wall for a straight artery. Moreover, it was found that an increase in the diameter of the lumen caused a slight increase in LDL concentration near the wall, and this was attributed to small velocity gradients near the wall. For the aorta-iliac bifurcation, the concentration boundary layer was found to grow with the Reynolds number, especially when recirculation occurred. The effects of hypo-and hyperthermia, as well as the curvature of the artery on LDL deposition in a multilayer curved artery model, were studied numerically by Iasiello et al. (2019). The heat source/sink was applied from the interior side of the lumen. The heterogeneity of various layers was considered by the authors in the multilayer model. The Darcy-Brinkman equation, the Staverman-Kedem-Katchalsky equation with a reaction term, and the energy equation were utilized by the authors to analyze the wall layers. The results presented by the authors demonstrated that the curvature of the artery had an insignificant effect on LDL deposition. Mahjoob and Vafai (2009) obtained an analytical solution for bioheat transport through tissue/organ by utilizing a local thermal nonequilibrium model in porous media (Fig. 1.6). The biological media was modeled as a blood saturated tissue represented by a porous matrix. The analytical results obtained by the authors were found to be in excellent agreement with the numerical results (Fig. 1.7).

Fig. 1.6

Fig. 1.6 Schematic diagram of the tissue-vascular system. Reprinted with permission from Mahjoob, S., Vafai, K., 2009. Analytical characterization of heat transfer through biological media incorporating hyperthermia treatment, Int. J. Heat Mass Transfer 52, 1608–1618, Elsevier.

Fig. 1.7

Fig. 1.7 Comparison of the temperature profile between exact and numerical results assuming blood-tissue local thermal equilibrium. Reprinted with permission from Mahjoob, S., Vafai, K., 2009. Analytical characterization of heat transfer through biological media incorporating hyperthermia treatment, Int. J. Heat Mass Transfer 52, 1608–1618, Elsevier.

Most of the studies in biomedical applications treated the arterial walls as a solid nonelastic medium, which does not represent the physiological conditions. In the cardiovascular systems, the flow of blood is under continuous interaction with arterial walls and therefore they constitute an intrinsically coupled system. The interactions between blood flow and wall deformation may alter flow patterns in pathological conditions.

1.3: Fluid–structure interaction in biomedical applications

Fluid–structure interaction (FSI) is receiving great interest in the literature due to its significant applications in different fields such as aerospace, biomedical, civil, mechanical, etc. (Alamiri and Khanafer, 2011; Khanafer et al., 2015, 2016; Khanafer, 2013, 2014). Most of the investigations in biomedical applications assumed rigid artery wall which does not mimic the physiological conditions. The interactions between blood flow and wall deformation may change flow patterns in pathological conditions. To the best knowledge of the present authors, one study was found in the literature that dealt with FSI in porous media as related to the biological application. Al-Amiri et al. (2014) analyzed numerically turbulent flow in a flexible wall artery during hyperthermia treatment. The wall of the artery was modeled as an elastic porous medium. Different heating protocols were examined in their study to illustrate its effect on the temperature variations in both the blood vessel and the tumor. The results presented in that investigation demonstrated that the flexible wall model had a significant effect on the heat flux variation along the bottom surface of the tumor tissue at different flow conditions.

1.4: Brain aneurysms

A cerebral or brain aneurysm is an abnormal bulge in the wall of an artery in the brain. The rupture of the brain aneurysm causes bleeding into the brain, which leads to a hemorrhagic stroke, brain damage, and consequently death. The endovascular coiling technique has been widely used to treat intracranial aneurysms. This technique involves blocking blood flow into the sac of the aneurysm through the deployment of tiny coils. Several experimental studies were conducted in the literature to investigate flow characteristics of cerebral aneurysms after endovascular treatment using coils or stents (Lieber et al., 2002; Szikora et al., 1994; Turjman et al., 1994). A better understanding of the blood flow and hemodynamics characteristics inside brain aneurysms is a very challenging topic in clinical research. Consequently, computational fluid dynamics is considered a crucial tool in the assessment and treatment of cerebral aneurysms using stents and coils. The accuracy of CFD models has been significantly enhanced by the introduction of patient-specific geometries and physiological boundary conditions. However, computational modeling of coil embolization in the treatment of cerebral aneurysms has received less attention in the literature due to the high irregularly shaped geometry of the coil. To overcome this problem, the embedded coil was modeled using porous media theory (Khanafer et al., 2009, 2010; Wiśniewski et al., 2021; Cha et al., 2007).

Khanafer et al. (2009) developed a numerical model using porous media theory to determine the reduction in the blood flow velocity and pressure resulting from the deployment of the endovascular coil within the brain aneurysm. Physiological waveform velocity was used as a boundary condition. The results presented by the authors illustrated that the magnitude of the velocity within the sac of the aneurysm was significantly affected by the presence of the coil. By utilizing the definition of porosity, the authors showed that a volume density of 20% was adequate to arrest blood flow in the aneurysm. Further, the authors presented a simple formula to estimate the required length of the coil for arresting the blood flow within the aneurismal sac. Wiśniewski et al. (2021) conducted a numerical study to simulate the hemodynamics within the aneurysmal sac after coiling. The dome of the aneurysm was modeled as a porous medium in which, due to its low Reynolds number, the flow was governed by Darcy’s law. The blood flow was modeled as an incompressible and non-Newtonian. Its shear-thinning was modeled using a modified power law viscosity. To assess hemodynamic variations, CFD was utilized for a case without coiling and another case for a wide range of porosities to resemble 1%–30% volume packing density (VPD) of coiling. Their results illustrated that both pressures at the aneurysm wall and residual flow within the aneurysmal sac decreased when VPD exceeded 10% (Fig. 1.8). Cha et al. (2007) proposed a new approach that allowed a better understanding of the complex interaction between the endovascular coil and the local blood flow. Semiheuristic porous media sets of equations were used in that investigation to describe the intraaneurysmal flow. The results suggested that the lower permeability of the coil mass at a given packing density resulted in a faster thrombosis formation within the aneurysmal sac.

Fig. 1.8

Fig. 1.8 Velocity distributions at aneurysm neck for selected VPD cases ( Wiśniewski et al., 2021 ).

Umeda et al. (2017) conducted a numerical study to predict aneurysm recurrence after coil embolization using porous media theory. Different parameters were analyzed in their study including morphological parameters, coil packing density, and hemodynamic variables to assess their relations with aneurysm recurrence. The flow in the coiled regimes was modeled using Darcy’s law. This study illustrated that the aneurysm recurrence was prone to happen in aneurysms with a larger dome or neck size, or in a coil with a lower coil packing density. As such, the coil packing density must be > 25% by residual flow volume (RFV) values in the preoperative modeling to avoid aneurysm recurrence (Fig. 1.9).

Fig. 1.9

Fig. 1.9 Preoperative simulation of a right internal carotid artery aneurysm ( Umeda et al., 2017 ).

Beppu et al. (2020) conducted a numerical study to evaluate the occlusion of placing flow-diverter (FD) with CFD using porous media theory for the decision-making in treating wide-neck aneurysms. The flow in the stent region was modeled using Darcy’s law and the pressure was locally balanced with resistance forces. The authors illustrated that the hemodynamic parameters using both control CFD and porous media CFD could predict the angiographic obstruction condition at 6 months after the FD treatment.

1.5: Magnetic resonance imaging (MRI)

Magnetic resonance imaging is considered to be an important tool in various applications such as diagnostic medicine, biomedical research, and porous material characterization (Barrie, 1995; Khaled and Vafai, 2003). Most of the studies in the literature were focused on the applications of MRI in the detection of brain strokes and brain diseases such as neurodegenerative and metabolic conditions, infections, and tumors. Further, MRI was also used for the in vivo measurement of the diffusion of water and intracellular metabolites. The diffusion of water within the brain is a vital topic because of its role in diffusion-weighted magnetic resonance imaging (DW-MRI) which illustrated superior capabilities compared with other imaging techniques in ischemic brain tissue diagnosis (Hotter et al., 2019). DW-MRI technique provides important information about the water exchange between brain tissue regions in normal and diseased states (Bose et al., 1988). The diffusion process in the brain tissue is evaluated in terms of the apparent diffusion coefficient (ADC). The water ADC represented a crucial variable in the evaluation of stroke patients (Stejskal and Tanner, 1965; Taylor and Bushell, 1985; Gelderen et al., 1994). It was reported by many studies that there was a substantial reduction in the apparent diffusion coefficient several minutes after the onset of the stroke. Therefore, a thorough understanding of all the parameters that influence the ADC of water in tissue is vital. For example, Norris et al. (1993) and Latour et al. (1991) claimed that the drop in the ADC after the onset of the stroke was due to a substantial increase in the tortuosity of the existing pathways for the fast diffusion process within the extracellular space. Other studies (Moseley et al., 1990; Mintorovitch et al., 1991; Benveniste et al., 1992) related reduction in ADC to the swelling of the cells, which caused water molecules to move from extracellular space to the intracellular space consequently slower the diffusion process and overall reduction in ADC. Helpern et al. (1992) concluded that the decrease in the permeability of the cell membrane resulted in a substantial decrease in the ADC after acute injury.

Theoretical models for the diffusion process in brain tissues were received less interest from researchers. The majority of studies in the literature were experimentally based. Khanafer and Vafai (2006) conducted a comprehensive review on the role of porous media in biomedical engineering as related to magnetic resonance imaging and drug delivery. The authors summarized most of the available diffusion models and diffusion coefficients in the literature. Tables 1.1 and 1.2 summarized these models. Further, Khanafer and Vafai (2006) proposed new models based on porous media theory for the diffusion process in the brain tissue as well as the effective diffusion coefficient.

Table 1.1

Table 1.2

1.6: Concluding remarks

This chapter provides an overview of the role of porous media in applications related to transport phenomena in arteries, brain aneurysms, and magnetic resonance imaging. This review cited the pertinent studies associated with modeling the wall of the arteries as multilayered model. This model was found to determine correctly the flow velocity and mass transfer within different layers of the artery wall. This review also showed the lack of studies on the application of FSI in porous media as related to the biomedical field. Therefore, more studies should be conducted in biological systems using the combination of FSI and porous medium. The wall of the blood vessel is flexible and the rigid assumption was not correct. Thus, FSI should be employed for precise predictions of flow and hemodynamic stresses. Diffusive transport models were found to play an important role in the transport of drugs and nutrients to brain tissues. Different definitions for the effective diffusivity coefficient as well as diffusion models were summarized in this review for various conditions.

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Chapter 2: Metabolic consumption of microorganisms

Peter Vadasz; Alisa S. Vadasz    College of Engineering, Informatics, and Applied Sciences, Northern Arizona University, Flagstaff, AZ, United States

Abstract

This review chapter presents a proposed theory that accounts for the metabolic consumption of microorganisms, which is analyzed and compared with experimental data both qualitatively and quantitatively. As distinct from other models that traditionally disregarded the growth and reproduction from the metabolic process, the proposed model links the latter directly to growth and reproduction in microorganism. The major reason for the latter is that the other models caused inconsistencies between the modeling results and experimental data. A major discrepancy was related to the experimentally observed LAG phase in the growth process. Associating the LAG phase with delay processes, as frequently proposed, has been proven incorrect and will be shown as such in this chapter. The present model shows that the origin of the LAG phase is the existence of unstable stationary states resulting from the explicit inclusion of the metabolic consumption process via a resource consumption and utilization term.

Keywords

Metabolic mass transfer; Monotonic growth; Microorganisms; LAG

2.1: Introduction

Despite the fact that 222 years have passed since the first attempt by Malthus (1798) to propose a predictive model for population dynamics, 182 years since Verhulst (1838) introduced the Logistic Growth Model (LGM), and 93 years since Pearl (1927) showed that the LGM can reproduce accurately experimental growth data in some circumstances, the problem of predictive growth in microbiology (as well as other fields such as ecology) is still not resolved. In addition, it is widely agreed (Baty and Delignette-Muller, 2004; Augustin and Carlier, 2000; Baranyi, 2002) that the LAG Phase, a typical phase in the growth of microorganisms, cannot be estimated accurately by any existing model. We counted at least 20 different models that are being used in "Predictive Microbiology" to describe population growth (14 of them being consistently used in the last few decades). Furthermore, all these models use extensive curve fitting as a really poor alternative to True Predictive Modeling. The 20 models we refer to are: Malthus (Malthus, 1798), Logistic Growth Model (Pearl, 1927; Verhulst, 1838; Baty and Delignette-Muller, 2004), Gompertz model (Gompertz, 1825), Allee model (Alee, 1931), Richards model (Richards, 1959), Baranyi and Roberts’s model (Baranyi and Roberts, 1994; Baranyi et al., 1993; Baty and Delignette-Muller, 2004), Smith model (Smith, 1963), Model with Delay (May, 1973, 1978, 1981; Hutchinson, 1948), Structured Models, age or otherwise (e.g., Hills and Wright, 1995; Baty and Delignette-Muller, 2004; among many others), Bi-Logistic model (Meyer, 1994), Generalized Logistic model (Tsoularis and Wallace, 2002), Generalized Gompertz model (Tsoularis and Wallace, 2002; Farber et al., 1996), Modified Logistic model (Messen et al., 2002), Varying Carrying Capacity Logistic model (Meyer and Ausubel, 1999), Modified Gompertz model (McClure et al., 1994; Baranyi and Roberts, 1994; Gibson et al., 1988; Zwietering et al., 1990; Baty and Delignette-Muller, 2004), Three-Phase Linear model (Buchanan et al., 1997), Monod model (Monod, 1942), Ginzburg model (Ginzburg, 1986; Akçakaya et al., 1988), von Bertalanffy model (von Bertalanffy, 1957), and Unified Generic Growth model (Turner et al., 1976). All the models listed above share at least one common attribute that a large number of these models are not really predictive. One can use some of them to fit a particular set of data, but for the next set of data one needs to fit them again. Therefore, one may raise the question: what is the use of predictive modeling if it cannot predict? No wonder that given this gloomy state of affairs, Buchanan et al. (1997) proposed an ingeniously simple solution, a three-phase linear model. If prediction is anyway not accomplished, why use complicated models at all? This state of affairs led to a profound controversy regarding scientific approaches and the methodology used in biological sciences (see Murray, 1992; Turchin, 2001, to list only a few).

By using one of the most popular models in predictive microbiology, i.e., Baranyi and Roberts’s model, Baranyi and Roberts (1994) attempt to fit 19 data points with 12 model coefficients and obtain results which show some fit. The correct way of using a predictive model with some appropriate curve fitting is linked to the model’s parameter estimation. However, for monotonic growth, a model that generically has three parameter values to estimate needs about 27 data