Bachelor's thesis describing a multiscale spatio-temporal model of multiple competing respiratory gases and in particular the effect of nitric oxide in the oxygen deprivation caused during methemoglobin-anemia.
Multiscale Analysis of Hypoxemia in Methemoglobin Anemia
1. Multiscale Analysis of Hypoxemia in Methemoglobin Anemia
Thesis submitted in partial fulfillment of the
requirements for the degree
of
Bachelor of Technology
In
Chemical Engineering
By
Tanmoy Sanyal
Roll No. 08CH3025
UNDER THE SUPERVISION
OF
Dr. Saikat Chakraborty
DEPARTMENT OF CHEMICAL ENGINEERING
INDIAN INSTITUTE OF TECHNOLOGY, KHARAGPUR
2012
2. .
Department of Chemical Engineering,
Indian Institute of Technology, Kharagpur-721302.
________________________________________________________________________
CERTIFICATE
This is to certify that the thesis entitled ―Multiscale Analysis of hypoxemia in
Methemoglobin Anemia‖ submitted by Mr. Tanmoy Sanyal, to the Department of
Chemical Engineering, in partial fulfillment for the award of the degree of Bachelor of
Technology is an authentic record of the work carried out by him under my supervision
and guidance. The thesis has fulfilled all the requirements as per the regulations of this
institute and, in my opinion, has reached the standard needed for submission.
Date: 3rd
May, 2012 ---------------------------------
Dr. Saikat Chakraborty
Department of Chemical Engineering
Indian Institute of Technology, Kharagpur
3. .
ACKNOWLEDGEMENT
This project work would not have been possible without the guidance of Dr. Saikat
Chakraborty. I express my sheer indebtedness to him for the suggestions and stimulating
discussions throughout the year that were vital for shaping this project and for his cordial
treatment and consistent moral support.
I am thankful to all my professors in the department who have been a source of
inspiration. I express my sincere gratitude towards them for helping me reach the state
that I am today.
Finally, I wish to express my heartfelt gratitude to my friends and
classmates, staff members, and others who directly or indirectly helped me in carrying
out this project.
Tanmoy Sanyal
Roll No. 08CH3025
Department of Chemical Engineering
Indian Institute of Technology, Kharagpur
4. 1
Abstract 3
CHAPTER 1: Introduction
1.1. Pathophysiology of Methemoglobinemia 5
1.1.1.Congenital Methemoglobinemia
1.1.2.Aquired Methemoglobinemia
1.1.3 Multiscale nature of the disease
1.2.Treatment of Methemoglobinemia 11
1.3.Objective and Organization of the thesis 12
CHAPTER 2: Literature Review1
2.1. Physiology and Structure of human lung 15
2.2. Role of Nitric Oxide (NO) 17
2.3 Methemoglobin reducing pathways and Methylene blue therapy 19
2.4. Different Approaches of Mathematical Modeling 21
2.5. Diffusion Capacity of the Lung 24
CHAPTER 3: Formulation and Simulation of Multiscale model
3.1 Reaction Chemistry 28
3.2. Micro-scale :RBC 29
3.2.1 Detailed Model
3.2.2 Spatially Averaged Model
3.3. Meso-scale : Capillary 36
3.3.1 Detailed Model
3.3.2 Spatially Averaged Model
3.4. Macro-scale : Lung 44
3.5 Solution of coupled micro-meso model 47
3.6 Results and Discussion 52
5. 2
CHAPTER 4: Therapeutic strategies for Methemoglobinemia
4.1. Oxygen Therapy 69
4.2. Methylene Blue therapy 70
4.2.1.1 Micro-scale :RBC
4..2.1.2 Meso-scale :Capillary
4.2.1.3 Numerical Simulation of coupled-scale model
4.3 Results and Discussion 80
CHAPTER 5: Conclusions & Future Work 88
CHAPTER 6: References 92
CHAPTER 7: Nomenclature 95
CHAPTER 8: List of Figures & Tables 101
6. 3
Abstract
Methemoglobinemia is a disease that results from abnormally high levels of methemoglobin
(MetHb) in the red blood cell (RBC), which is caused by simultaneous uptake of oxygen (O2)
and nitric oxide (NO) in the human lungs. MetHb is produced in the RBC by irreversible NO-
induced oxidation of the oxygen carrying ferrous ion (Fe2+
) present in the heme group of the
hemoglobin (Hb) molecule to its non-oxygen binding ferric state (Fe3+
). This paper studies the
role of NO in the pathophysiology of methemoglobinemia and presents a multiscale quantitative
analysis of the relation between the levels of NO inhaled by the patient and the hypoxemia
resulting from the disease. Reactions of NO occurring in the RBC with both Hb and
oxyhemoglobin are considered in conjunction with the usual reaction between oxygen and Hb to
form oxyhemoglobin. Our dynamic simulations of NO and O2 uptake in the RBC (micro scale),
alveolar capillary (meso scale) and the entire lung (macro scale) under continuous, simultaneous
exposure to both gases, reveal that NO uptake competes with the reactive uptake of O2, thus
suppressing the latter and causing hypoxemia. We also find that the mass transfer resistances
increase from micro through meso to macro scales, thus decreasing O2 saturation as one goes up
the scales from the cellular to the organ (lung) level. We show that NO levels of 166 ppm or
higher while breathing in room air may be considered to be fatal for methemoglobinemia patients
since it causes severe hypoxemia by reducing the O2 saturation below its critical value of 90%.
Further, we develop models for therapeutic procedures such as oxygen therapy and methylene
blue therapy. Simulation of Oxygen therapy enables us to stratify patients into candidates who
will respond to supplemental Oxygen (having ambient NO levels of less than 216 ppm) and
those who should be subjected to antidotal therapies like administration of methylene blue.
Following this, a complete multiscale, unsteady state model is developed for methylene blue
therapy that helps predict the effect of medicine dosage on treatment and optimizes the dosage.
For an ambient NO concentration of 220 ppm, the optimal dosage is predicted as 0.35 , in
terms of plasma concentration of methylene blue.
8. 5
1.1 Pathophysiology of Methemoglobinemia
Methemoglobinemia is a disorder characterized by the presence of above-normal levels of
methemoglobin (MetHb) in the blood. Methemoglobin is a compound formed from hemoglobin
(Hb) by oxidation of iron atom from ferrous to ferric state, which is generated in the human
respiratory system from the co-operative binding of hemoglobin with nitric oxide. MetHb lacks
the electron needed to form a bond with oxygen (O2), and hence, is incapable of oxygen
transport. Thus, methemoglobinemia is characterized by moderate to severe hypoxemia. Case
studies [1,2] reveal that when suffering from the disease, the arterial blood turns brown due to
higher levels of MetHb, which produces a bluish tinge in skin color. Methemoglobinemia can be
seldom linked to heredity and is most often of the acquired type, caused by external sources of
nitric oxide [2].Such Nitric Oxide (NO) sources may accelerate the rate of formation of MetHb
up to one-thousand fold, overwhelming the protective enzyme systems and acutely increasing
susceptibility to the disease. [5] It may be mentioned that the chief agent of this disease,
methemoglobin is not entirely foreign to the human body.
Generally, red blood cells (RBCs) are continuously
exposed to various oxidant stresses [6] and so
MetHb is continually produced in humans. In
addition, the oxy-hemoglobin dissociation curve is
shifted to the left, impairing the delivery of oxygen
at the tissue level as shown in Figure 1.1. The RBCs
have two mechanisms [7] that work to keep the
Fig 1.1 Effect of Methemoglobinemia on oxy-hemoglobin dissociation curve[1].
9. 6
abnormal hemoglobin percent down to generally less than 1%. One reduces the oxidizing
compounds before they can change the hemoglobin; the other changes the abnormal hemoglobin
back into working hemoglobin by way of two enzyme pathways. Under normal conditions, the
ferric iron in MetHb is readily reduced to the ferrous state through the function of enzyme called
as cytochrome b5 oxidase which is present in erythrocytes and other cells. Typically,
methemoglobin levels are maintained at 1% of the total circulating blood hemoglobin in a
healthy adult. However, injury or toxic agents convert a larger proportion of hemoglobin into
MetHb. Increased concentrations of methemoglobin reduces the efficiency to bind with oxygen,
thus reducing oxygen pressure in the blood. Effectively, therefore the patient becomes anemic.
Methemglobinemia can be of two major types-hereditary or congenital, and acquired from
exposure to external sources of toxins.
1.1.1. Congenital methemoglobinemia
The congenital form of methemoglobinemia has an autosomal recessive pattern of inheritance.
Due to a deficiency of the enzyme diaphorase I (NADH methemoglobin reductase), MetHb
levels rise and the blood of patients lose their full functionality for oxygen binding and transport
to tissues. Instead of being red in color, the arterial blood of changes to a brown texture. Change
in the color of blood to a large extent, especially in the subcutaneous vessels in turn produces a
pale blue tinge in the skin. In a bizarre story that‘s been passed down through the ages, an entire
family from isolated Appalachia had blue skin. This is one of the best known examples of
hereditary Methemoglobinemia.
Hereditary methemoglobinemia is caused by a recessive gene. This means, that the
condition will manifest in an offspring, if and only if both the parents are carriers of the genetic
abnormality, [8]. Another cause of congenital methemoglobinemia is seen in patients with
10. 7
abnormal hemoglobin variants such as
hemoglobin M (HbM), or hemoglobin
H (HbH), which are not amenable to
reduction even with the body‘s intrinsic
enzyme systems that specifically serve
this purpose. Methemoglobinemia can
also arise in patients with pyruvate
kinase deficiency due to impaired
production of NADH - the essential
cofactor for diaphorase I. Similarly, patients with Glucose-6-phosphate dehydrogenase (G6PD)
deficiency may have impaired production of another co-factor, NADPH [1]. Once formed,
methemoglobin can be reduced back to hemoglobin either enzymatically or nonenzymatically
via potential pathways which require the usage of above mentioned enzymes and constitute the
body‘s defense mechanism towards the disease.
1.1.2. Acquired methemoglobinemia
Acquired methemoglobinemia is more common than the inherited forms. Exposure to exogenous
oxidizing drugs and their metabolites such as benzocaine, dapsone [3] and nitrate toxicity from
contaminated water or industrial fumes may overwhelm the body‘s protective mechanism and
increase MetHb levels substantially leading to disease symptoms. Frank R. Greer (2005) [4] in
his clinical report on infant methemoglobinemia identified nitrate toxicity from water as one of
the main reasons for infant methemoglobinemia. The current EPA standard of 10 ppm nitrate-
nitrogen for drinking water is specifically designed to protect infants. The widespread use of
Fig 1.2 The Blue family of Kentucky
11. 8
nitrate fertilizers increases the risk of well-water contamination in rural areas. It occurs after
exposure to certain chemicals and drugs as shown in Table 1.1 [4].
It should be noted that methemoglobin is continually generated in the body due to
various oxidative stresses and this, mere presence of MetHb does not produce anemic conditions.
It is the percentage saturation of MetHb i.e. the total fraction of blood hemoglobin in the form of
MetHb, that decides the onset and spread of Methemoglobinemia. The clinical consequences of
methemoglobinemia are related to the blood level of MetHb as shown in Table 1.2. ABG
(Arterial Blood Gas) determination will show a normal arterial Oxygen partial pressure (PaO2),
with an arterial fractional O2 saturation in hemoglobin (SaO2) lower than expected for the given
PaO2. The arterial blood will take on a characteristic "chocolate brown" color. The blood gas lab
will further report the makeup of the other possible forms of hemoglobin: reduced,
carboxyhemoglobin & methemoglobin. A pulse oximeter cannot distinguish between normal &
abnormal hemoglobin and will report a combination of saturated hemoglobin,
carboxyhemoglobin & MetHb.
13. 10
Local anesthetics Benzocaine; lidocaine; Propitocaine; Prilocaine
Table 1.1--Reported Inducers of Methemoglobinemia
Signs & Symptoms
10-20% Mild cyanosis
30-40% Headache, fatigue, tachycardia, weakness, dizziness
>35% Dyspnea, lethargy
50-60% Acidosis, arrhythmias, coma,seizures, badycardia, hypoxia
>70% Fatal
Table 1.2. Clinical consequences of methaemoglobinaemia
1.1.3 Multiscale Nature of the disease
Thus, we see that this disease involves physical processes of gaseous exchange at the lung
alveoli, diffusion and convection of dissolved gases in the alveolar capillaries and simultaneous
chemical reaction and diffusion inside the red blood cells. Clearly, the phenomenon spans
several varying length scales and may well be termed as a ―multiscale disease‖. The scales
involved are the micro-scale constituting the scale of the red blood cells, the meso-scale
constituting capillary scale and the macro scale representing the scale of the organ, in this case,
the whole lung. Transport at each of these levels, is coupled nonlinearly through the processes of
14. 11
diffusion, convection and reaction, the nonlinearity arising from the reaction kinetics of the
participating gaseous components (in this case O2 and NO) with hemoglobin. The smaller length
scales are embedded hierarchically in the next larger length scale. Because of the coupling and
hierarchical embedding of scales, the processes at smaller length scales significantly affect and
alter those at larger scales. Clearly, any attempt to construct mathematical models of
methemoglobinemia must be multiscale in nature and must take into account this intricate scale-
coupling.
1.2. Treatment of Methemoglobinemia
Methemoglobinemia can be treated either by supplying with oxygen at hyperbaric conditions or
by methylene blue 1% solution (10mg/ml) 1-2mg/kg administered intravenously slowly followed
by IV flush with normal saline[10]. Methylene blue restores the iron in hemoglobin to its normal
(reduced) oxygen-carrying state. This is achieved through the enzyme inducing effect of
methylene blue on levels of diaphorase II (NADPH methemoglobin reductase). Diaphorase II
normally contributes only a small percentage of the red blood cells reducing capacity but is
pharmacologically activated by exogenous cofactors, such as methylene blue to 5 times its
normal level of activity. Genetically induced chronic low-level methemoglobinemia may be
treated with oral methylene blue daily. Also, vitamin C can occasionally reduce cyanosis
associated with chronic methemoglobinemia but has no role in treatment of acute acquired
methemoglobinemia.
15. 12
1.3. Objective and Organization of the thesis
In this thesis, we attempt to quantify NO induced hypoxemia in methemoglobin anemia, with a
particular focus on the more common acquired type of the disease. We start with the basic
reaction scheme of coupled O2 and NO uptake by blood hemoglobin and this provides us with
the kinetics, which together with fundamental transport equations, constitute the entire model. As
stated before, methemoglobinemia is essentially a multiscale disease with coupled transport and
reaction spanning disparate length scales. In this work, we develop a rigorous multiscale model
that consists of L-S averaged low dimensional forms of the fundamental CDR equations. The
low-dimensional model for each scale is embedded hierarchically in the next larger scale and the
model equations at all the three (micro, meso, macro) scales are solved not sequentially but
simultaneously so as to allow coupling between the scales. We use our model to study the
interdependent O2 and NO dynamics in patients suffering from methemoglobinemia. Our model
simulations allow us to quantify the increased methemoglobin levels inside the RBC in
methemoglobinemia, as well as estimate the overall pulmonary saturations of oxygen and nitric
oxide, that result in moderate to severe hypoxemia in the patient. We also estimate the levels of
inhaled NO that could be fatal for the patient. This discussion on the quantitative study of the
disease is well supplemented by a similar multi-scale formulation of the standard therapeutic
procedures. We use the model developed with minor modifications to simulate both Oxygen
Therapy and Methylene Blue therapy. We perform analyses of the sensitivity of Methylene Blue
16. 13
under different conditions and show how the data generated can be used as a quick way to
estimate optimal methylene blue concentrations to achieve treatement.
This thesis is organized as follows Chapter 2 mentions about the Literature referred for
pursuing this project. Section 2.1 gives us a brief idea about the architecture of lung.. Section 2.2
describes the NO generation in human body and its role in maintaining many biological
activities. Section 2.3 illustrates the mechanism of methemoglobin reduction pathways in the
human body and briefly outlines their reaction schematics Section 2.4 talks about the traditional
and modern approaches of mathematical modeling. Section 2.5 explains the diffusion capacity in
the lung and its significance. Chapter 3 presents the details of the multiscale model employed.
Section 3.1 delineates the reaction chemistry which is the basis of our mathematical modeling.
Sections 3.2, 3.3 and 3.4 constitute the formulation of the model across three different scales-
micro, meso and macro, respectively followed by relevant results obtained and their analysis in
Section 3.5. Section 4 describes the pathological treatment for methemoglobinemia and is
divided into two parts— a short discussion on oxygen therapy and its simulation followed by a
complete multiscale model formulation and its simulation for methylene blue therapy. Chapter 5
draws out conclusions from the quantification of the disease and the results obtained during the
process of the same and also talks about scope for future work. Chapter 6 shows all the
nomenclature used in the thesis. Chapter 7 contains all the references used while carrying out the
project. Chapters 8 and 9 include the list of all the figures and tables present in the thesis.
18. 15
2.1. Physiology and Structure of Human Lungs
Respiration is a complicated multiscale phenomenon which requires the gas molecules to travel
through the bronchioles, alveoli, capillaries to finally reach the red blood cell. The path of
transport of gases from the lungs to red blood cell can be divided into three representative length
scales and transport at these three scales are intensely coupled with each other. The smallest
length scale termed as micro scale ( is the red blood cell (RBC).
Figure 2.1 shows the schematic diagram of flow
gases diffusing into the red blood cell from alveolus
through the capillary membrane. The human red blood
cell is a biconcave disc, which alters shape and
orientation as it drifts along blood vessels of different
diameters. Average dimensions of this discoid are 8 mm
in diameter and 1.6 mm in thickness [11]. Gases that
diffuse into the RBC from the bloodstream will react
with the hemoglobin and bind to it to form different complexes, e.g. oxygen binds to form the
extremely important oxyhemoglobin that functions as the oxygen transporter to tissues; Carbon
dioxide produced in the process of respiration binds with hemoglobin to produce carbo-
aminoglobin which serves as a carrier of CO2 for its ultimate disposal from the system. The next
higher length scale which comprises of the capillaries and alveoli is called meso-scale
( . Figure 2.2 presents a view of structure of capillaries and alveoli in the human lung.
Fig 2.1 Schematic of O2 transport in the
micro-scale [20]
19. 16
The capillaries form a dense network on the
walls of the alveoli which provides huge
surface area for efficient gaseous exchange.
The diameter of the capillary segment is about
(10 microns), just large enough for a red blood
cell. The lung is regarded as a collection of
300 million alveoli each 0.3 mm in diameter,
which is equivalent to roughly about 2 billion
capillaries!
The lung in its entirety represents the macro-scale ( . It is thus clear that when
a pulmonary process happens parallel over all the three scales and there exists coupling between
sub-processes at each of these three scales, the combined effect that is manifested can only be
measured in terms of macro-scale properties. Hence any quantitative analysis of such coupled
scale phenomena must always be reported in terms of macro-scale quantities (saturations of
species and partial pressures of gaseous components, for instance).
The lung is divided into two main networks, perfusion network which involves the flow
of blood in the capillaries and the ventilation network, which involves the air-flow pathways.
Each is an intricate network of capillaries. Figure 2.3 shows a schematic of ventilation- perfusion
network in the human lung.
Fig 2.2 Barium angiogram showing dense network of
capillaries within the lung [12]
20. 17
(a) (b)
Fig. 2.3. (a) The gas exchange between air in an alveolus and blood circulating through pulmonary capillaries.
(b) Schematic showing the Alveoli and Capillaries surrounding the Alveoli. (Red denotes oxygenated blood,
blue denotes deoxygenated blood) [21].
.
2.2. Role of Nitric Oxide (NO)
Inhaled Nitric oxide (NO) reacts with deoxygenated blood to form Methemoglobin (MetHb)
which is the crux of the disease. NO is a colorless, odorless gas that is only slightly soluble in
water. Its main sources of emissions are combustion processes. Moreover, Fossil fuel power
stations, motor vehicles and domestic combustion appliances emit nitrogen oxides mainly
constituting of nitric oxide. NO is readily oxidized to nitrogen dioxide and peroxidation then
occurs. Following are the reactions involved:
Hb (Fe2+
) + NO MetHb (Fe3+
) + NO3
2NO + O2 2NO2
21. 18
NO + O2 ONOO-
The rates of uptake and release of NO from hemoglobin is 6.25x105
fold greater than those of
oxygen [11]. At certain levels, inhaled nitric oxide concentrations can cause vasodilation in the
pulmonary circulation without affecting the systemic circulation and hence is important from a
physiological perspective. However, Nitric Oxide always contains some contamination of
Nitrogen Dioxide whenever found in air. Endogenous nitric oxide synthesis occurs by nitric
oxide formation from physiological substrate in cells of many of the organ systems such as nerve
tissue, blood vessels and the immune system. One such important internal source is the complex
L-Arginine. Nitric oxide may be more potent than nitrogen dioxide in introducing certain
changes in lung morphology. Once NO is inhaled, it rapidly diffuses across the alveolar-capillary
membrane into the smooth muscle of pulmonary vessels to activate soluble guanylate cyclase
(Figure 2.4). This enzyme is responsible for the conversion of GTP to cGMP which in turn
relaxes smooth muscle via several mechanisms.
Fig. 2.4. NO signaling pathway in the lung. PKG
indicates cGMP-dependent protein kinases;
NOS, NO synthase; L-arg, L-arginine; sGC,
soluble guanylate cyclases; and RSNO, S-
nitrosothiol
22. 19
These effects are probably responsible for vasodilation in the pulmonary circulation and acute
bronchodilator effect of inhaled nitric oxide (as has been recognized since 1987) [13]. Nitric
oxide has an affinity for haem-bound iron which leads to the formation of methemoglobin and
the stimulation of guanylate cyclase. Additionally, it can also deaminate DNA, evoke DNA chain
breaks, and inhibit DNA polymerase and ribonucleotide reductase. It might be antimitogenic and
inhibit T cell proliferation in spleen cells [10]. Although early studies of inhaled NO in the
treatment of pulmonary hypertension used concentrations of 5 to 80 ppm, it has since been
realized that concentrations greater than 20 ppm provide little additional hemodynamic benefit in
most patients.
2.3 Methemoglobin reducing pathways and Methylene Blue therapy
Finally, we must also discuss the pathways by which the body manages to keep
methemoglobin levels at their allowed values. The two most important pathways are the cyt-b5
pathway and the NAPDH-Flavin Pathway. Kinoshita et.al in their seminal work [7] on the
simulation of these pathways show that at different concentrations of ambient methemoglobin in
the system, the human body switches from one pathway to another. Figure 2.5 presents a
schematic of the reactions that occur during methemoglobin reduction through these pathways. It
must be mentioned that pathway II is more common one and accounts for over 95% of the
methemoglobin reduction. Kinoshita et.al showed that initially, under low levels of oxidative
stress, Pathway I is adopted as a defense mechanism and slowly with increasing concentration of
Methemoglobin, the body switches over to Pathway II. Pathway I though not a major contributor
23. 20
in normal cases, is the one that is activated by external administration of Methylene blue and
thus is of interest when we simulate Methylene blue therapy. It is evident from this figure that
G6PD is most essential for the NADPH-Flavin Pathway to activate. Thus, methylene blue
therapy in disease conditions will only work when this pathway receives proper activation. That
is to say cases that present themselves with an intrinsic G6PD deficiency e.g. in congential
methemoglobinemia, cannot be treated effectively by Methylene blue therapy [7].
Fig 2.5 The cyt-b5 and the NADPH-Favin Pathway
24. 21
In methylene blue therapy the NADPH-Flavin Pathway is activated to a much larger extent than
in normal situations. Methylene blue is oxidized into leukomethylene blue by accepting an
electron from NADPH in the presence of NADPH-methemoglobin reductase. Leukomethylene
blue then donates this electron to MetHb resulting in its conversion back to Hb
2.4. Different approaches of Mathematical Modeling
The human body is an organizational miracle. It is made up of wide range of length scales
which function in unison to regulate the various physiochemical processes that take place in our
body. Proteins and nucleic acids constitute one of the smallest length scales of the body (10-8
m), followed by organelles( 10-7
m), cells (10-6
to 10 -5
m ), capillary spacing (10-4
m) and
organs(10-1
m) where each is organized hierarchically inside the whole body whose length scale
is of 1m [14]. Thus we can say that the whole physiological system is a multiscale system
composed of different scales. In the respiration process different levels of organizations are
coupled so as to promote the transfer of gases from the air we breathe to the red blood cell. Each
level of organization exerts a different resistance to the transport of gases. Modeling of this
Fig 2.6 Mechanism of action of Methylene Blue
25. 22
multiscale physiological transport and reaction structure will allow us to quantify each scale and
hence to study in detail the physics and chemistry of each scale. Modeling efforts in this area
have progressed in two parallel paths, namely the bottom-up approach and the top-down
approach. This is illustrated in Figure 2.7 for the case of transport of oxygen saturation.
The bottom-up approach [15, 16, 17] consists of describing the physical system by
solving the detail three dimensional convection-diffusion reaction equation (CDR) and coarse
graining of the solution to obtain the macro description which is the oxygen saturation in the
blood. This approach requires use of numerical techniques to solve the complex three
dimensional CDR equations. Though high on accuracy, this method is computationally
formidable and is very expensive in terms of computer-run-time. Furthermore, often in bio-
engineering problems, it is not at all required to obtain extremely detailed point to point profiles,
which also diminishes the need for extremely precise three dimensional data at every point of the
geometrical domain in question.
Fig 2.7.Schematic showing the different approaches of modelling pulmonary oxygen uptake [20]
26. 23
The complexity increases many folds when we consider simultaneous transfer of oxygen and
carbon dioxide (Bohr-Haldane effect) where each of them will encounter different resistences at
different scales and both compete for the hemoglobin in the blood. This disadvantage of the
bottom-up approach is subjugated in the top-down approach [18, 19]. Here the physical system is
simplified under a- priori assumptions regarding the convective and diffusive time scales of the
system and also, cross coupling between the scales is neglected. Thus the top-down approach
completely abolishes the idea of scale coupling, so inherent in biological processes, and lacks the
rigor of the bottom up approach. Hence mulstiscale problems cannot be effectively described by
a top-down approach. The numerical complexity of the Bottom-up approach and the a-priori
assumptions to simplify the structure done in the top-down approach can be eliminated with the
help of recently developed averaging model. This approach as shown in Figure 2.7 consists of
spatially averaging the complex three dimensional CDR equations to reduce the dimensionality
of the problem and obtaining the solution analytically. The spatial averaging of CDR equations is
done with the help of Liapunov-Schmidt (L-S) method of classical bifurcation theory. The L-S
method results in a low dimensional model which, however, retains all the small parameters and
hence all the important physics of the lower scales embedded in it i.e. the micro and meso. For
problems of multiscale pulmonary transport, such an approach has been shown [20] to retain
information about the lower scales like the size and shape of the red blood cell, its shear rate and
the surrounding plasma boundary layer thickness, as well the solubility and diffusivity of oxygen
in the red blood cell and in the unstirred layer. This method has been used by Chakraborty et al
[20] in modeling the diffusion of oxygen uptake in the lung. The analytical expression of the
diffusion capacity of the RBC developed by Chakraborty et al in [20] retains information about
the size and shape of the red blood cell, its shear rate and the surrounding plasma boundary layer
27. 24
thickness, as well the solubility and diffusivity of oxygen in the red blood cell and in the
unstirred layer. Similarly, the model at meso scale retains the information about the radius of the
capillary, thickness of the alveolar epithelium and the capillary endothelium membranes, the
flow profile in the plasma, the slip velocity between the RBC and the plasma, hematocrit
(volume fraction of RBC in the plasma) and the diffusivity and solubility of gases in the plasma
and the membranes. Macro scale also retains the complexities resulting from the ventilation-
perfusion heterogeneities and right to left shunts in the lung. Direct spatial averaging of the CDR
equation without using the LS method results in a low-dimensional model but it doesn‘t retain
the above parameters. Thus, in this type of approach functional as well as morphological changes
at the red blood cell or the capillary level could be easily accounted in the low-dimensional
model. Hence, these low dimensional models could be used as tools to analyze the process of
pulmonary exchange under both normal as well as pathophysiological conditions.
2.5. Diffusion Capacity of the lung
The overall diffusing capacity of the lung (DL) is defined as the volume of gas
transferred in milliliters per minute per mm of Hg of alveolar partial pressure. Diffusion capacity
is a measurement of the lung's ability to transfer gases. Term ‗Θ‘ denotes the diffusion capacity
of RBC. In 1957, Roughton and Forster proposed in their classic paper [21] that the resistances
offered by the capillary membrane (1/DM) and the reaction rate in the RBC (1/ ΘVc) are in series.
They can be summed up to give the resistence to gas transfer (1/DL) between the alveolar gas in
the RBC given as:
28. 25
, where
DL Overall diffusing capacity
DM Diffusing Capacity of the membrane separating the alveolar air from blood
Vc Total volume in milliliters of blood in the lung capillaries exposed to alveolar air
Θ Diffusing Capacity of RBC
The Diffusion Capacity of the RBC, (Θ) is the effective (mass transfer disguised)
reaction rate between a reactive gas and the hemoglobin in the RBC. It depends on three physical
and chemical processes, namely, internal mass transfer in the RBC, external mass transfer (due
to diffusional gradients of the gas) in the stagnant plasma layer surrounding the RBC, and the
actual rate of reaction between the dissolved gas and the hemoglobin inside the RBC.
Hughes and Bates [22] summarized the experimental efforts for evaluating ‗Θ‘ (for
O2and CO) in the last 50 years. According to this paper, Holland [23], Forster [24], Reeves and
Park [25], and Borland and Cox [26] used direct and indirect ways of measuring diffusing
capacity of the RBC.
The numerical values of the two components of the overall diffusing capacity are
different for different gases. The transport of gases can be reaction limited or diffusion limited
depending upon the relative values. In reaction limited case, the diffusional gradients are
established faster compared to the reaction inside the RBC. In diffusion limited case, reaction
time is less than the diffusion time. Here, we have concentrated in microscale modeling i.e.,
reactive uptake of NO in the RBC for the quantification of methemoglobinemia. When a person
1 1 1
L M cD D V
29. 26
is exposed continuously to NO, the reactions will attain equilibrium and the whole transport of
NO through the lung will be diffusion controlled. Several attempts were made previously to
quantify the uptake process of NO in lungs via simple analytical means as well as computational
models. But, computational models were difficult to solve because of its non-linear complexity.
Hence, spatially averaged models were used in this thesis to reduce the computational efforts of
the bottom-up approach immoderately. Moreover they are also free from the a priori
assumptions made in the top down approach. Its expediency lies in a way that it reduces the
dimensionality of the problem. Hence, the low dimensional model for each scale is embedded
hierarchically in the next larger scale, and solves for all the three scales i.e., micro, meso and
macro simultaneously so as to allow interaction between the scales. This method was used by
Chakraborty et al, [20] for the quantification of O2 in human lungs. However, effect of O2 on
NO‘s diffusion process or the vice-versa was not accounted. Also, gas diffusion process of one
gas was assumed to be independent of other gas presence which is not true. In this thesis
simultaneous uptake of both NO and O2 is considered. Computation models for diffusion of both
the gases are obtained where dependency of NO on the diffusion of oxygen in RBCs is clearly
evinced. We have used Lipunov-Schmidt Spatial Averaging approach to obtain expressions for
diffusing capacities (Θij) of the red blood cell for each of NO and O2 in terms of size and shape
of the RBC, thickness of the unstirred plasma layer surrounding the RBC, diffusivities and
solubilities of the gases in RBC and boundary layer, hematocrit, and the slope of the dissociation
curve. Two components of diffusion capacities were observed for each of NO and O2 caused by
pressure difference of the other gas.
31. 28
3.1 Reaction Chemistry
We being with presenting the scheme of coupled reactions that take place at the lowest
scale, i.e. the RBC. The chemical reaction system of NO with the RBC hemoglobin in the
presence of O2, is given by
(1)
(2)
(3)
Reaction (1) shows that Oxygen-Hb equilibrium is assumed to follow the Hill equation with n
being the Hill constant (=2.34), and the equilibrium rate constant ( ) being given by
(4)
where Torr and Torr. [27] The reaction rate constant for NO induced
oxidation of oxyhemoglobin to MetHb (given by Eq. (2)) was calculated by Eich and co-workers
(1996) [28]. Sheele et al., (1999) [29] reported the reaction rate constant for the reaction between
NO and deoxyhemoglobin (Eq. (3)) as , and also measured the dissociation rate
of Nitrosylhemoglobin. The dissociation rate constant of MetHb (in Eq. 2) is obtained to be
from Power et al. (2007) [30].
In normal RBCs, typical levels of methemoglobin produced as shown above are less than
1%. Its generation is controlled through two MetHb reducing pathways. One of these systems is
the redox cycle consisting of cytochrome b5 (cytb5) and cytochrome b5-metHb reductase (b5R),
which uses NADH for electron transfer to cytb5 (―cytb5-NADH system‖). The other pathway
uses flavin as an electron carrier for the reduction of MetHb coupled with NADPH oxidation,
catalyzed by NADPH-dependent flavin reductase (FR) (―flavin NADPH system‖). Both these
32. 29
major pathways have been discussed in great detail by Kinoshita et. al (2003) [11]. This
enzymatic reduction of methemoglobin maybe written in a simplified way as
. (5)
It may be noted that Eq. (5) illustrates the conversion of MetHb back to normal hemoglobin and
can thus be used in understanding the therapeutic strategies for the disease. This will be taken up
in greater detail in Chapter 5 where we discuss Methylene Blue therapy.
The multiscale models for quantification of hypoxemia in Nitric Oxide induced
Methemoglobinema consist of developing CDR equations at the micro (RBC) scale and coupling
it with similar transport models at the meso (capillary) and macro (lung) scales. In the
subsequent sections, the detailed governing equations at each scale are presented and the scale-
coupling is demonstrated.
3.2 Micro Scale - RBC
Transport of Oxygen from the alveolus (where partial pressure of oxygen is denoted by ) to
the red blood cell occurs across the membranes
(alveolar epithelium and capillary endothelium) and
the plasma. Human erythrocytes are biconcave discs
with average dimensions of 8 mm in diameter and 1.6
mm in thickness. Figure 3.1 presents a schematic
representation of oxygen uptake in pulmonary
capillaries.Fig 3.1 Gas Transport at micro and meso
scales
33. 30
3.2.1. Detailed Model
The convection diffusion-reaction equations in Lagrangian coordinates for a single RBC of any
arbitrary shape with volume and external surface area are given as
(6)
(7)
where is the substantial derivative and is the three dimensional Laplacian operator.
is the partial pressure of the gaseous component dissolved in the RBC, is the (fractional)
saturation of the species as obtained from the reaction scheme in Eqs. ((1)-(3)), and
are the diffusion coefficients of the gaseous component j and Hemoglobin inside the erythrocyte
respectively, is the total intra-cellular hemoglobin concentration and is the solubility of
the corresponding gaseous component in the RBC. The rates of reaction of the component gases
with the hemoglobin in the RBC are functions of both partial pressure and saturation and are
shown by in Eqs. (6) and (7). Considering a symmetric geometry, the appropriate
boundary conditions for these equations are given by:
and are finite at the center of the cell , (8)
(9)
(10)
Equations (6) and (7) are also subject to initial conditions given by
(11)
(12)
34. 31
where is the outward unit normal to the surface, is the mixed venous partial pressure of
oxygen, is the spatially averaged partial pressure of the gas component in the
plasma. It has been shown ([31]) that a thin unstirred plasma layer is formed around the surface
of the red cell and retards oxygen transfer from the plasma. in Eq.(9) represents the mass
transfer coefficient in the unstirred layer, which quantifies the mass transfer resistance between
the RBC and the plasma. Solution of the coupled CDR equation of the RBC with the unstirred
(boundary) layer surrounding it ([11]), gives
(13)
where is the thickness of the boundary layer, and are the solubility and the diffusion
coefficients of the corresponding component in the stagnant layer, respectively.
We proceed to write the system of equations in their detailed form where j=1 and j=2
refer to O2 and NO, respectively, and i=1, 2 and 3 indicate the species HbO2, MetHb and HbNO,
respectively. We note here that in the rest of the development, we shall use the subscript i (j)
interchangeably with the actual name of the species (gaseous component). In their complete
form, the CDR equations for the micro scale, thus, are
(14)
(15)
(16)
(17)
(18)
The spatially averaged partial pressure of the gas in the plasma that appears in the boundary
conditions for this system represents the coupling that exists with the next higher scale, viz. the
35. 32
meso scale. Thus, to establish the complete low-dimensional scale coupled model, we spatially
average Eqs. (14) to (18).
3.2.2. Spatially Averaged Model
We perform spatial averaging of the above equations over the volume of a single RBC using
the Liapunov-Schmidt based spatial-averaging method ([11, 20]).To simplify the notation, we
use the subscripts 1, 2 and 3 to denote saturations of species HbO2, MethHb and HbNO,
respectively, and and to denote and , respectively. Treating the reaction scheme
in Eqs. (1), (2) and (3) as elementary, the equilibrium relations in terms of fractional saturations
can be written as
, (19a)
(19b)
, (19c)
Also, at any instant the total hemoglobin in the system (free + bound form) is constant. The total
hemoglobin saturation in the RBC is 100 %, i.e.,
(20)
Solving these above equations, we have the fractional saturations as
(21a)
, (21b)
36. 33
. (21c)
The dissociation constants, as mentioned in section 2.1.2, are obtained by differentiation and are
given as
(22a)
(22b)
(22c)
(22d)
(22e)
. (22f)
Time derivatives in Eqs. (14)–(18) are written in terms of as
, (23)
and then the reaction rate terms are eliminated from Eqs. (14)–(18) to obtain
(24a)
(24b)
where the coefficients are given as
37. 34
11 1 1 2 1 1 22 32 2 11 21
2
11 22 11 32 21 32 21 12 31 12 31 22
12 2 12 22 2
21 1 21 31 1
22 2 1 2 1 22 3
[ [ ] ( ) ( )
[ ] ]/ ;
[ ] ( )( ) / ;
[ ] ( )( ) / ;
[ [ ] (
T Hb
Hb T
T Hb
T Hb
T Hb
G D Hb D D
D Hb Deno
G Hb D D Deno
G Hb D D Deno
G D Hb D
2 2 2 11 21
2
11 22 11 32 21 32 21 12 31 12 31 22
1 2 1 22 32 2 11 21
2
11 22 11 32 21 32 21 12 31 12 31 22
) ( )
[ ] ]/ ;
[ ] ( ) ( )
[ ] ;
Hb T
T
T
D
D Hb Deno
Deno Hb
Hb
. (25)
Equations (24a) and (24b) are spatially averaged over the volume of the RBC, following the
method outlined in Appendix C of [11], to obtain the averaged equations given by
, (26)
(27)
where the coefficients are given as
,
,
,
. (28)
Multiplying both sides of Eq. (26) by and that of Eq. (27) by
we obtain the final spatially averaged forms
38. 35
Θ Θ (29)
Θ Θ (30)
which represent the micro-scale model for the simultaneous reactive uptake of O2 and NO in
RBC in Eulerian co-ordinates, that we eventually use for computational purpose.
Here, is the spatially averaged partial pressure within the RBC, and is the slope of the
equilibrium dissociation curve given by . The quantity Θ is called the ‗Diffusing
Capacity‘ [11, 20], which, in a multi-component reaction-diffusion system, represents the
reaction-altered mass transport coefficient of gas ‗i’ under the driving force of gas ‗j’. For a
single gaseous component, the diffusing capacity includes the effects of diffusional gradients
within the RBC as well as that in the stagnant plasma layer, and the effect of facilitated transport
of gas due to reactive coupling with hemoglobin. The expressions for the diffusing capacities are
obtained from Eq (28) as
Θ (31)
Θ (32)
Θ (33)
Θ (34)
where, is a mass transfer coefficient of gas ‗i’ in the stagnant plasma layer surrounding the
RBC [11], is the internal Sherwood number of the RBC, and is a function of . An
analytical expression for Shi for a discoid-shaped RBC as a function of its ratio of height (H) to
39. 36
diameter (D) has been given in Eq. (124) in Appendix C of [11]. Cellular deformations often
alter H/D and the corresponding change in the Sherwood number may be calculated using the
reference stated above. In general, erythrocytes have dimensions of 8 mm in diameter and 1.6
mm in thickness i.e. have a H/D of 0.2, and therefore, have Shi =2 [11], which is the value used
in our analysis.
3.3 Meso Scale : Capillary
A pulmonary capillary consists of a continuous plasma phase in which the red cells are
suspended. Our meso-scale model assumes the plasma phase can be considered a continuum in
which suspended RBCs behave like sources/ sinks of O2 and NO.
3.3.1. Detailed Model
The meso scale model equations are given by
For Oxygen:
Θ
Θ (35)
For Nitric Oxide:
Θ
Θ (36)
where h is the hematocrit i.e. volume fraction of RBC in the blood, is the concentration of
gas j (j=1 or 2) in the plasma phase expressed in L (STPD) of O2 per L blood, is the
40. 37
effective diffusivity of gas in the plasma (effective diffusivity is used instead of molecular
diffusivity, since it is known that the red cell rotation and other motions in the shear field tend to
promote radial transport), s are given by Eqs. (21) to (24), is obtained by solving the
averaged equations in the micro scale (Eqs. (19) and (20)), and and are related linearly
as
(37)
where is the solubility of gas j in the plasma. We may also note here, that is not
equal to the velocity at which the red cells are carried along by the plasma, and there exists slip
between the RBC and the plasma phase, which our model accounts for. Equations (35) and (36)
are subject to the boundary conditions
(38)
(39)
where is the alveolar partial pressure, DM,j is the membrane diffusing capacity of the
component under consideration and has units of L (STPD) of O2/ (mmHg s), a is the capillary
radius and VC is the total capillary bed volume. Of the two boundary conditions, Eq. (39) ensures
centerline symmetry, while Eq. (38) accounts for mass transfer across the membrane using the
concept of membrane diffusing capacity and also couples the transport inside the capillary with
that in the macro scale, i.e. the alveolus.
The velocity of the red cells ( and the plasma ( are not constants but are given by
([15, 31])
41. 38
,
while, the hematocrit, i.e., the volume fraction of the RBC at any radius is expressed as ([15,
31])
where, = radius of the RBC rich region, = radius of the capillary, B=Blunting factor, that
represents deviation from parabolic flow profile ( , and slip=relative slip between
plasma and RBC ( . However, for our analysis, we consider the hematocrit to be
constant along the transverse direction (=h). We further assume a fully developed parabolic
velocity profile for the plasma, given by
(40)
and a constant velocity profile for the RBC which retains the effect of slippage and is given by
(41)
where, = average velocity of plasma in the capillary. Typical values used later in this analysis
are slip = 0.1 and h = 0.45. It may be further noted that since all reactions are essentially
confined to the RBC and none occur in the plasma phase, = 0 in both Eqs. (35) and
(36).
3.3.2. Non Dimensionalsation
In this section, we non-dimensionalize the meso-scale governing equations and their boundary
conditions, in order to compare the magnitude of the various time scales involved in the transport
42. 39
process, and accordingly simplify the governing equations. The non-dimensional forms of the
different variables are
(42)
(43)
while the different dimensionless numbers are given by
, (44)
Θ
Θ
Θ (45)
and the constituent time scales are given by
Θ
(46)
The above expressions show that the dimensionless numbers are ratios of different time-scales in
the system, e.g. is the ratio of radial diffusion time to transit time of blood in the capillary
( , is the ratio of axial diffusion time to capillary transit time, Θ is the ratio of capillary
transit time to mass transfer time between RBC and plasma, while Θ is the ratio of blood
transit time to mass transfer time across the alveolar membrane . Using these dimensionless
variables, and using the fact that, , the governing Eqs. (35) and (36) along with their
boundary conditions can be written in the dimensionless form
For O2 (j = 1)
Θ
1, + Θ ,12 2, 2, , (47)
For NO (j = 2)
43. 40
Θ
1, + Θ ,12 2, 2, , (48)
with boundary conditions given by
Θ
(49)
(for j = 1 and 2) . (50)
An order of magnitude analysis of the convection, radial and axial diffusion times in a capillary
shows that the radial diffusion time is the smallest, the convection time comes next, and the axial
diffusion time is the largest. Since axial diffusion is very slow as compared to convection
(i.e. is very large ~103
), we neglect the axial diffusion term in the above model. We may also
note here that the low value (~10-4
) of the transverse Peclet number facilitates radial averaging of
the above model. Similar to the process carried out in the micro-scale (RBC), the averaging at
the meso-scale is also performed using Liapunov– Schmidt technique, which allows us to reduce
the radial dimension of the model.
3.3.3. Spatially Averaged Model
Here, we present a brief outline of the Liapunov-Schmidt (LS) reduction of the Convection
Diffusion equations of the meso-scale [19, 32]. For the sake of mathematical simplicity, we
consider equations for only O2, while those for NO follow a similar treatment. From Eq. (47), we
have
Θ
Θ
44. 41
along with boundary conditions,
Θ
,
where .
We introduce the following definitions of average partial pressure and cup mixing partial
pressure, given by
, (51)
, (52)
respectively, where the non-dimensional velocity has the usual parabolic profile for fully
developed flow, as mentioned in Eq. (42). We choose to write both pressure and velocity as
, (53)
, (54)
where and are spatial averages and and are fluctuations about the average. The
fluctuations satisfy the solvability criterion given by
, (55a)
. (55b)
Plugging these into Eq. (52), and using the solvability criterion gives
, (56)
We note that , and then proceed to spatially average Eq. (47) over the cross section of the
capillary, which gives
45. 42
Θ Θ
1, + Θ ,12 2 2, . (57)
Using Eqs. (53) - (56) in Eq. (47), and subtracting the result from Eq. (57), we obtain
Θ
Θ ,11 1 1, + Θ ,12 2 2, . (58)
Assuming steady state and considerably high axial Peclet number (~104
, refer to section
3.3.2) allows us to neglect the axial diffusion terms and both Eqs. (57) and (58) can be further
simplified. The simplified form of Eq. (57) gives
Θ Θ Θ
(59)
This is known as a Global Evolution Equation. The simplified form of Eq. (58) is given as
Θ Θ
1, + Θ ,12 2 2, . (60)
The Liapunov-Schmidt expansion technique then consists of solving Eq. (B.10) for
by expanding it in powers of the transverse Peclet Number , ([33]), as
, (61)
and retaining only the first order terms, which have been shown to be sufficient for qualitative
and even quantitative accuracy [33]. Once we calculate , we make use of Eq. (56) to obtain
the relationship between the modes as
Θ
, (62)
46. 43
Similarly, using Eq. (49), we can solve for . The equations thus obtained are called the Local
Equations. An in-depth development of and discussion on the above may be found in [33].
The averaged meso-scale model is thus represented by three modes, namely the averaged partial
pressure of gas in the blood plasma , the cup-mixing partial pressure of the gas, and the
pressure inside the RBC, – the micro-scale variable that affects the meso-scale due to
scale-coupling. Post-averaging, we obtain a set of Differential Algebraic Equations (DAEs),
consisting of two global evolution equations:
Θ Θ Θ (63)
Θ Θ Θ (64)
and four local equations:
Θ
(65)
Θ
(66)
Θ (67)
Θ (68)
where is the steady-state partial pressure at the capillary-membrane interface, and the
parameter k is the ratio between the alveolar partial pressures of O2 and NO, whose value is ~10-6
in this analysis. It may be mentioned that the effects of scale-coupling between the micro and the
meso levels are manifested in the equations through the magnitude of k. The above system of
DAEs must be solved in conjunction with the non-dimensional Eulerian forms of the micro scale
Eqs. (29) and (30). The Eulerian forms of these equations are given as
Θ Θ (69)
47. 44
Θ Θ (70)
3.4 Macro Scale : Lung
In this section, we develop relations
governing gas ventilation ( ) and blood
perfusion ( ) in the lung, the ratio of which,
known as the ventilation to perfusion ration
( / controls the alveolar partial pressure
( , that in turn affects the meso and
micro scales through scale coupling. For an
―ideal‖ lung, this ratio is known to be 1 [34],
but in a ―normal‖ lung, it varies spatially
across the different lung compartments, from 0.3 to 3.3. In addition, we also consider
intrapulmonary right to left shunts ( / = 0), constituting about 3-5% of the blood flow. The
spatial variation of ventilation and perfusion rates at the macro scale (i.e., lung) is accounted for
by using a simple compartment model, which is illustrated in Fig. 3.2.
As may be noted from Figure 3.2, compartment 1 of our six compartment model
corresponds to the space in the lung between ribs ―Top‖ and rib number 2, compartment 2
corresponds to that between ribs 2 and 3, and so on, for the first five compartments. The fraction
of blood flow through the sixth compartment corresponds to the shunt fraction . For a
―normal lung‖, the distribution of the ventilation to perfusion ratio with rib position has been
Fig. 3.2 Schematic of human the lung showing positions of
the ribs and the compartments
48. 45
obtained by fitting polynomials to data reported by West [34] (Figure 6A in [20]), which is given
as
(71)
where y is the normalized distance along the ribs ( , given by
(72)
where has been described in Figure 3.2 and . For the sixth compartment, i.e.,
the shunt, / = 0. For the shunt, blood flow i.e. perfusion rate is given by ,
while the remaining volume of flow is distributed across the other five compartments following a
log-normal distribution, given by
(73)
where =1, 2, 3, 4, 5 is the compartment number and is the cumulative distribution function
of log-normal distribution, given by
(74)
and (75)
As mentioned earlier, the alveolar partial pressure of O2, is different for each compartment
and is determined by the local / ratio of that compartment. The relationship between and
the venous partial pressure of O2 denoted by , has been fitted as a sigmoidal function of /
to the data presented by West (1977) [34] and the relationship is given by
. (76)
49. 46
For a ―normal lung‖ breathing in room air, = 40 mmHg and we obtain from the data, =
122.5 mmHg and = 1.28. It may be mentioned here, that the value used in Eq. (76) is an
average value in that particular compartment, obtained from integrating Eq. (71) as
, (77)
where the integral is taken over the whole compartment . Writing out the limits appropriately,
the above quantity becomes
. (78)
For the sixth compartment, as mentioned earlier the ventilation-perfusion ratio is simply zero.
Using the values of and / for each compartment obtained by the above method, we
calculate the partial pressure of the gas (where O2 is of chief interest but we also calculate for
NO) and the saturation of blood exiting each compartment ( and , respectively)
using the steady-state coupled micro-meso scale model (Eqs. (63)–(70)). Blood from the six
compartments is allowed to mix in the pulmonary vein, and this mixing process is modeled as
one in a perfectly stirred tank. The final gas saturation of mixed blood in the pulmonary vein,
( determines the total amount of uptake of the gaseous component j (=1 for O2 and 2 for
NO) in the lung, which is given by
(79)
where is the venous saturation of the gas. For O2, this is calculated from the venous partial
pressure, using the relation proposed by Severinghaus [35],
, (80)
50. 47
while for NO,
(81)
Thus, we have established a hierarchy of scales where the macro scale embeds the meso and the
meso scale embeds the micro scale, and the entire system must be solved simultaneously (not
sequentially) to capture the effect of the transport-reaction coupling across the scales.
3.5 Solution of coupled micro, meso and macro scales
Considering the macro scale compartmental model with the micro and meso scale ODEs, we
have a system of coupled DAEs. However, numerical solution of this system in its present form
results in an ill-conditioned mass matrix that is hard to handle, even with standard tools such as
MATLAB. So, we exploit the linear interdependency of the modes in the meso-scale local
equations (Eqs (63)-(70)) to convert the system of DAEs into a system of 4 coupled ODEs that
describe the coupling at the two lower scales in terms of the partial pressure in the RBC (which
is representative of the micro-scale) and the cup-mixing partial pressure in the plasma (which is
representative of the meso-scale). The macro scale algebraic equations supply the compartmental
alveolar partial pressure (Eq.(76)) as a boundary condition to this micro-meso system of ODEs.
For each compartment, the ODE system computes the saturation and partial pressure profiles
which are then combined together by way of Eq. (79) to obtain macro scale saturations. The
entire schematic is presented in the form of an information flow diagram in Figure 3.3 that
reflects the numerical coupling between and hierarchical embedding of scales.
51. 48
It may be mentioned that the primary objective of solving the macro scale and obtaining the
results in terms of the macro-scale variables is that these are most easily measurable and allows
us to perform a host of parametric studies, which bring out interesting dependencies, as shall be
seen later.
The derivation of the final ODE system fit for numerical solution, involves only elementary
algebra and the final form is
(82)
(83)
(84)
Fig. 3.3 Information flow diagram, demonstrating how variables communicate across different scales, and
how the simulation algorithm functions
52. 49
(85)
which can be handled easily by available ODE solver algorithms in softwares such as MATLAB.
Algebraic simplification gives the coefficients in Eqs. (82)–(85) as
2
,1 ,1 1, ,1
1 1, ,11
1 1 1
2
,2 2, ,2
1 ,12
2 2
1, 1, ,1 1, ,1
,11 ,1
11, 1, 1, ,1
1
1
,1
3
1 ,
8 8 192
,
8 192
3 (8 )
46
1
(1 )
T T m T
m rbc
T m T
rbc
m m T m T
rbc rbc
m m m T
rbc
Pe Pe Pe
X h
B B B
Pe Pe
Y h
B B
p Pe Pe
p
Bp Pe
Z
h B
2, 2, ,2 2, ,2
2 ,2
2
,
3 (8 )
4
m m T m T
rbc
p Pe k Pe
p
D
(86a)
2
,1 1, ,1
2 ,21
1 1
2
,2 ,2 2, ,2
2 2, ,22
2 2 2
1, 1, ,1 1, ,1
,21 1,
12, 2, 2, ,2
2
2
,
,
8 192
3
1 ,
8 8 192
3 (8 )
( )
46
(1 ) 1
T m T
rbc
T T m T
m rbc
m m T m T
rbc rbc
m m m T
rbc
Pe Pe
X h
B B
Pe Pe Pe
Y h
B B B
p Pe Pe
p
Bp k Pe h
Z k
h B h
2, 2, ,2 2, ,2
22 2,
2
,
3 (8 )
( )
4
m m T m T
rbc
p Pe k Pe
p
B
(86b)
2
,1 1, ,1
1 11
1 11 1 1
2
,2 2, ,2
1 12
1 11 2 2
1, 1, ,1 1, ,1
11 1,
1
1
1 11
(1 )
,
( [ ] ) 8 192
(1 )
,
( [ ] ) 8 192
3 (8 )
4(1 )
( [ ] )
T m T
rbc T
T m T
rbc T
m m T m T
rbc
rbc T
Pe Peh
X
u Hb B B
Pe Peh
Y
u Hb B B
p Pe Pe
p
Bh
Z
u Hb
2, 2, ,2 2, ,2
12 2,
2
,
3 (8 )
4
m m T m T
rbc
p Pe k Pe
p
B
(86c)
53. 50
2
,1 1, ,1
2 21
1 22 32 1 1
2
,2 2, ,2
2 22
1 22 32 2 2
1, 1, ,1 1, ,
21
2
1 22 32
(1 )
,
{ [ ] ( )} 8 192
(1 )
,
{ [ ] ( )} 8 192
3 (8 )
(1 )
{ [ ] ( )}
T m T
rbc T
T m T
rbc T
m m T m T
rbc T
Pe Peh
X
u Hb B B
Pe Peh
Y
u Hb B B
p Pe Pe
h
Z
u Hb
1
1,
1
2, 2, ,2 2, ,2
22 2,
2
4
.
3 (8 )
4
rbc
m m T m T
rbc
p
B
p Pe k Pe
p
B
(86d)
The above coefficients are plugged into Eqs. (82)-(85), and solved numerically. The solution of
the system of ODEs (Eqs (60)-(63)) was performed using MATLAB. The mass matrix being
extremely stiff and ill-conditioned due to the large difference in variable values across different
scales, appropriate stiff solvers were employed (ode15s and ode23s routines).
55. 52
3.6 Results and Discussions
In this section, we present the O2 and NO saturation (in hemoglobin) profiles that
quantify gaseous uptake in methemoglobinemia, as well as study the effects of different
parameters (including NO concentration) on the onset of the disease by simulating the multiscale
model derived in the previous section. Table 1 lists the standard data set used for the simulation,
along with the equivalent of the units used, in S.I. It may be mentioned here, that the diffusing
capacity for O2 transport through the capillary membrane (DM,O2) is known from literature [11],
while that for NO is estimated using the formula
(87)
where is the membrane diffusivity of the gas j in the RBC. The membrane
diffusivities for both O2 and NO are obtained from available literature [36, 37], and are listed
below.
Variable Value
57. 54
We use deoxygenated blood with an O2 partial pressure of 40 Torr and = 75 % as initial
conditions for our simulation.
In the plots that come next, we analyze the diffusing capacities as well as the saturation
of various species in the system both as functions of the ambient levels of Nitric Oxide in the
system as well as their steady state profiles in the capillary, against different parameters. The
Table 3.1 – Data used in simulation (other parameter values used in this thesis are list in Table 4.1 on pg.
81)
58. 55
Fig.3.4 Plots of Micro scale diffusing capacities against partial pressure of Oxygen in the RBC, with alveolar
partial pressure of NO as a parameter
59. 56
Figure 3.4 shows the variation of the micro scale diffusing capacities with the O2 partial pressure
in the RBC, where the NO concentration in the RBC is a parameter. As was stated in section 2.5
these diffusing capacities represent the reaction-enhanced effective diffusion coefficients of one
gas in the presence of another. Figure 3.4(a) shows the diffusing capacity of O2 plotted against
the partial pressure of O2 in the RBC. The biphasic nature of the plot shows that O2 uptake
decreases for low or very high partial pressures and maximizes at around 25 torr. Figures 4(b)
and (c) show and , which are indicative of the effect of gas 1 on 2 and vice versa and
reveal that they have nearly the same magnitude. This shows that the change O2 and NO enforce
upon the diffusion dynamics of each other is almost similar in extent. However, as seen in Fig
ure 3.4(d), is several orders of magnitude higher than the rest of the capacities, which easily
shows why NO is so potent. The reaction altered diffusion of NO, thus, is extremely high and
any effect O2 has in stemming it is considerably small. This is where we can begin to infer that
even small quantities of NO, in finite time, must lead to the onset of methemoglobinemia if not
removed or treated properly.
60. 57
Figure 3.5 presents the partial pressure profiles of O2 and NO, in the RBC and the plasma that
are obtained directly by solving the micro-meso coupled scale system (Eq. (63-70). Figures
3.5(a) and (b) present the variation of the partial pressures of Oxygen in the RBC and the plasma,
respectively, along the capillary length for different values of alveolar NO concentration. We
note that for both the micro- and meso-scales, steady states are attained at around 20% of the
capillary distance for an alveolar pressure of 110 torr. It may also be noted that the partial
pressure of Oxygen inside the RBC always remains slightly lower than that inside the plasma so
that a diffusional gradient exists until equilibrium is attained. Though intuitively obvious, we
shall soon see that this does not hold true for Nitric Oxide. Figures 3.5(c) and 4(d) show the
Fig 3.5 Partial pressure profiles of O2 and NO in the plasma and RBC
61. 58
profiles of the partial pressures of Nitric Oxide at the meso (capillary) and micro (RBC) scale. A
close look at the magnitudes at which equilibration occurs shows that the pressure inside the
RBC, though initially lower steadies at a slightly higher value than that in the plasma. This could
be attributed to the interesting dynamics of convection, diffusion and reaction that the two scales
with widely differing time-scales are locked into. The fact that the plots equilibrate very soon
inside the capillary, demonstrates that the partial pressure of O2 inside the capillary is not
significantly affected by the NO environment. However, the same cannot be said for saturation
as we shall see in the next set of results.
.
Fig. 3.6 Profiles of saturation of O2 and NO, along the capillary for different ambient levels of NO.
62. 59
Figure 3.6(a) presents the simulated profile for oxygen saturation in the rbc (SHbO2) along the
capillary for different values of alveolar partial pressure of Nitric Oxide. The figure shows that
when the value of alveolar NO is nearly zero, oxygen saturation attains equilibrium at 97% and
the oxygen tension increases to 110 Torr. Oxygen diffuses through the membrane separating the
air and the blood from the high partial pressure in the alveoli (110 Torr) to the area of lower
partial pressure in the pulmonary capillaries (40-42 Torr). For this case, the oxygen saturation
reaches a steady state within 15-20 % of the total capillary length of 0.1 cm. It may be mentioned
that SHbO2=90% in the arterial (oxygenated) blood is considered to be the critical oxygen
saturation required for the patient to survive. Thus, we note from Figure 3.6(a) that in the
presence of alveolar NO in methemoglobinemia patients, this critical oxygen saturation (~90%)
occurs when the alveolar NO concentration is 150ppm or above, when considering the effect of
only micro and meso scales. Figure 3.6(b) shows the profile for the total NO saturation in the
RBC, SNO (=SMetHb + SHbNO) along the length of the pulmonary capillary. The steady state is
attained at 20% of the total capillary length. Figures 3.6(c) and (d) show the individual
saturations (along the capillary) of the two components of reacted NO in the hemoglobin, namely
Nitrosylhemoglobin (HbNO) and Methemoglobin (MetHb), respectively. As discussed before,
the level of MetHb in normal hemoglobin is known to be 1%. However, it may observed from
Figure3.6(c) that as the concentration of NO in the RBC increases, MetHb saturation increases
significantly to about 6% (in the critical case), leading to the onset of methemoglobinemia.
63. 60
Fig 3.7 Saturation profiles for (a) Oxygen, (b) Nitric Oxide, (c) Methemoglobin and (d) Nitrosyl Hemoglobin in the
capillary (meso-scale) for different values of alveolar partial pressure of Oxygen.
64. 61
Figure 3.7 illustrates the results of the meso scale simulation. The saturations of the species
involved, O2, NO, Methemoglobin (MetHb) and Nitrosyl-hemoglobin (HbNO) are plotted vs. the
alveolar NO concentrations. Since this set of plots is obtained at the meso-scale, the
compartmental variations of do not affect the values significantly. Hence, has been
kept as a parameter and values assigned to it are nearly equal to the actual compartmental values
that result from a macro-scale calculation. As is intuitively obvious, increasing increases
the critical NO concentration above which O2 saturation falls below 90%, leading to severe
hypoxemia. Also, the variation in NO saturation is significantly less than that for O2, which
bolsters our claim about the inability of the O2 environment to stem NO dynamics. This effect is
more pronounced at the macro scale as we shall see later. We may also observe from Figures 3.7
(b), (c) and (d), that the principal contribution to the NO saturation is from the generated
methemoglobin and only very little from HbNO. This can be interpreted as an indirect indication
of the greater role of MetHb in causing metheoglobinemia (and not Nitrosylhemoglobin-anemia
for instance). Indeed, NO concentrations of just 60 ppm or higher in the lung can cause
methemoglobin concentrations to spike (at values of 80 to 90 Torr). Methmoglobin levels
steadily rise from 1% (in disease-free condition) to as high as 10%, with increase in NO
concentration, leading to the onset of methemoglobinemia (as corroborated by others [5]).
65. 62
Figure 3.8 presents an interesting feature that was observed during the simulation studies carried
out with the micro-meso coupled scale model. The ambient Nitric oxide in the system, can be
considered to be either generated internally, which is possible since NO is produced internally
from L-arginine (refer Chapter 2), or exogenously by inhalation. The difference in source leads
to differences in the distribution of ambient NO across the compartments (refer section 3.4) of
the lung. An endogenous source leads to a constant ambient NO level for each compartment,
while continuous inhalation of NO, as occurs in the exogenous case, will cause NO ppm to vary.
It was computationally predicted that an exogenous source leads to greater variation in NO
distribution across the different compartments, as shown in Figure 3.8(b). Since under disease
conditions the NO saturation varies significantly across the compartments as observed in Figure
3.7, we can hypothesize that the main source of NO in methemoglobin anemia and consequently
the whole model may switch over from being endogenous initially to purely exogenous. It is
interesting to verify if this actually happens and if there exists a threshold value of the NO ppm
that represents transition from one model to another. However, this shall not be pursued here.
Fig 3.8 Comparison of NO saturation for endogenous and exogenous sources of NO
66. 63
Fig 3.9. Saturation profiles for (a) Oxygen, (b) Nitric Oxide, (c) Methemoglobin and (d) Nitrosyl Hemoglobin in the
lung (macro-scale) for different values of venous partial pressure of Oxygen
67. 64
Figure 3.9 presents the O2 and NO saturation profiles in the macro-scale. The results obtained at
this scale are the final outputs of the entire multi-scale simulation, spanning the micro, meso and
macro scales, and are reported in terms of measurable physiological quantities at the macro-
scale. Here, the parameter chosen is the venous partial pressure of O2, which is varied from 20
Torr to 40 Torr. It may be noted that the macro-scale saturations of both O2 and NO have
diminished significantly from those reported at the meso-scale (Figure 3.7) This happens
because as we embed smaller scales hierarchically in larger ones their mass transfer resistances
add up (not necessarily linearly). The increase in total resistance is observed to increase
significantly when the macro scale is added to the coupled micro-meso scales. We also find that
the variation in NO saturation in the macro scale is lesser than the corresponding variation in the
meso scale while O2 saturation gets considerably affected by alveolar NO concentrations in both
these scales. Since macro scale results represent directly measurable quantities, we can
conclusively claim that the physical conditions of O2 do not have significant influence on the NO
dynamics, while NO deposition in the alveolus considerably alters O2 dynamics. Our present
conclusion, in conjunction with the effect of the very high diffusing capacity (Figure
3.4(d))shows why Methemoglobinemia occurs; -- NO keeps accumulating in the alveolar
epithelium undergoing unhindered reactive uptake, while it keeps competing with O2 uptake,
causing subsequent hypoxemia.
68. 65
Fig 3.10. Saturation profiles for (a) Oxygen, (b) Nitric Oxide, (c) Methemoglobin and (d) Nitrosyl Hemoglobin in
the lung (macro-scale) for different values of shunt at venous partial pressure of Oxygen of 40 Torr
69. 66
In Figure 3.10, the species saturations vs. NO levels are presented again, but this time with
intrapulmonary left to right shunt fraction as a parameter. Two values are considered: a zero
shunt and a standard 5% shunt. As is expected, saturations are higher for the zero shunt case
since this allows more blood supply to oxygenated alveolar regions. Once again we find that NO
saturations do not change much for different shunt fractions due to the very high reactivity (
of NO. As mentioned before (section 3.4), the macro-scale has been modeled as a series of
compartments with all the intrapulmonary shunt lumped together into the 6th
(and last)
compartment. When these compartments, with their different s are solved in the macro-
scale computations, high reactivity of NO ensures significant total NO uptake even before the
last compartment, i.e. the shunt, is reached. Thus, whether we simulate with a 5% shunt or a zero
shunt, when the saturations obtained from each compartment are combined, the final result has
little to do with the shunt used, as is obvious from Figures. 3.10(a) and (b). Also, From Figure
3.10(a) we find that even with zero shunt, the O2 saturation tends to dip below the critical value
of 90% at NO levels slightly higher than
150 ppm.
Finally, in Figure 3.11 we present
the variation of critical NO concentrations
plotted against the venous partial pressure
of O2 at the macro scale. As mentioned
before, competition between the reactive
uptakes of the two gases, suppresses O2 use,
and leads to hypoxemia. 90% O2 saturation
Fig. 3.11 Critical Nitric Oxide levels (in ppm) for different values of venous partial pressure of Oxygen simulated
at zero and 5 % shunt fractions
70. 67
is usually termed as the critical O2 saturation since any drop in saturation below 90% could be
fatal to the patient due to the severe hypoxemia it causes. Our simulations reveal that for the
standard case of 40 torr and 5% shunt, the critical concentration of NO is around 166 ppm.
Also, we see that the critical ppm increases with and shunt fraction. A higher critical
concentration of NO in a methemoglobinemia patient implies less susceptibility to extreme
hypoxemia
72. 69
4.1 Oxygen Therapy
So far, we have discussed about the mechanism of methemoglobin anemia, i.e. how the disease
conditions manifest themselves and performed a detailed quantitative analysis of the same. In
this chapter we discuss the therapeutic approaches towards treatment and curing of the
symptoms. The first line of antidotal therapy that is used is administration of Methylene Blue.
Though the most standard practice, their exists alternate approaches towards treatment that
produce reasonably good results within a certain cut-off level of Nitric oxide concentration. The
most common of these alternate, initial methods is Oxygen Therapy. The quantitative basis for
this therapy has already been established from the results obtained in the previous chapter. From
Figure 3.11, we concluded that Nitric Oxide levels over 166 ppm are fatal for the patient since
they decrease the Oxygen saturation to less than its critical value of 90%. However, the final
end-capillary saturation of oxygen is dependent not only on the ambient level of NO but also on
the compartmental distribution of Oxygen in the different lung compartments. In oxygen therapy
we exploit this dependence of the O2 distribution and manipulate the distribution to ensure O2
saturations of over 90% at critical and even post-critical concentrations of Nitric Oxide.
The therapy is carried out under hyperbaric conditions and O2 at high pressure of around
650 torr is introduced externally. This washes away any existent distribution, since the maximum
value occurs at around 106 ppm in the somewhere in the 4th
lung compartment, which is
about six times lesser than the hyperbaric condition. Oxygen therapy then, is essentially
equivalent to simulating the coupled three scale model with values of all compartments
equal to the constant pressure of the external introduction. We carry out this simulation using
73. 70
shunt fraction as a parameter. The end-capillary oxygen saturation is plotted for different levels
of ambient Nitric Oxide for zero shunt and 5% shunt.
In Figure 4.1, the simulation of oxygen
therapy is demonstrated. The most significant
information that we obtain from this plot is
that, above a NO concentration of around 216
ppm, even supply of 100% pure oxygen, fails
to maintain O2 saturation above 90%. 216
ppm, thus represents the critical or cut-off
NO concentration above which oxygen therapy
will no longer work. Thus now we have stratified methemoglobinemia patients into two distinct
categories-
Patients with NO levels below 216 ppm, for whom Oxygen therapy will suffice.
Patients with NO levels above this threshold, who are therefore candidates for Methylene
Blue therapy.
4.2 Methylene Blue Therapy
Methylene blue is the first-line antidotal therapy. Methylene blue works as a cofactor for the
enzyme NADPH-methemoglobin reductase as shown in Figure 2.6 . Methylene blue gets
oxidized into leukomethylene blue by accepting an electron from NADPH in the presence of
Fig 4.1 O2 saturation in Oxygen therapy
74. 71
NADPH-methemoglobin reductase. Leukomethylene blue then donates this electron to MetHb
resulting in its conversion back to Hb. The standard dose, used by medicine practitioners is 1-2
mg/kg and is administered over a certain time after which its effects start manifesting. This
might be treated as some form of dead time needed by the diffusing Methylene blue to overcome
various mass transfer resistances throughout the different scales.
It must be pointed out at this juncture that sometimes lack of response to methylene blue
suggests the congenital form of methemoglobinemia in which a deficiency of the intrinsic
mechanisms of glucose-6-phosphate dehydrogenase or NADPH (the reduced form of
nicotinamide-adenine dinucleotide phosphate) methemoglobin reductase exists. In these patients,
or any patient not responding to methylene blue, transfusions of packed red blood cells are
necessary to increase the amount of nonionized hemoglobin, thereby allowing increased oxygen
binding and transportation [38].
4.2.1 Mathematical Modeling
As discussed earlier towards the end of section 3.1, the modeling of Methylene blue therapy
simply consists of plugging an extra reaction into the 3-reaction scheme (Eqs (1), (2) and (3)),
and solving the multiscale model developed earlier. However certain obvious modifications need
to be made. In an analysis of drug delivery, two questions are essential-
In what extent is the drug required i.e. dosage?
How fast does the drug work, i.e. system response time?
75. 72
This directly shows that an unsteady state analysis needs to be performed and correct information
can only be extracted from the complete spatio-temporal profiles of the saturation of different
species under action of Methylene blue Certain important assumptions need to be clearly stated
at the outset-
The body‘s own mechanisms of restraining methemoglobin generated due to various
continuous oxidative stresses on the RBC are the cyt-b5 pathways and the flavin-NADPH
pathways (refer Chapter 2 for a detailed discussion). If we maintain that 1 mol of
Methylene blue produces 1 mole of Leukomethylene blue, which initiates and sustains
the Flavin-NADPH pathway in the same 1:1 ratio, then, the remaining part of the
mechanism being completely elementary, we can directly use the rate constants for the
Flavin-NADPH reduction but keep the enzyme concentration equal to administered
Methylene Blue concentration. This directly lets us monitor the effect of dosage. This
assumption, is valid under low levels of ambient Nitric Oxide concentration when the
coupled reactions of the pathway under consideration [7] are all approximately first order
(low concentration asymptote of Michaelis Menten kinetics).
It has been shown by Kinoshita et.al [7], that there is a switch in pathway from the
Flavin-NADPH route to the cyt-b5 route with steadily increasing concentration of
Methemoglobin. Our analysis does not incorporate this transition.
The model developed does not take into account the effect of the dead time discussed
earlier. Its significance thus has not been quantitatively established and we estimate its
value such that the overall response time of the patient to the therapy is minimized.
4.2.1.1 Micro-Scale: RBC
76. 73
The modified reaction schematic which will be an extension of that presented in section 3.1 is –
(88)
(89)
(90)
. (91)
The nomenclature for the model remains entirely similar to that described in Chapter 3, so we
shall not go into it again. The set of reaction diffusion equations in the micro-scale will be
similar to Eqs ((14)-(18)) with unsteady components and addition of an extra term which
represents the consumption of Methemglobin by enzyme (E(red))
(92)
(93)
(94)
(95)
(96)
Adopting an approach similar to the one presented in section 3.2 of Chapter 3, we eliminate all
reaction rate terms except between Eqs. (92)-(96), and use the definition of to write
the system of equations in (92)-(96) as
(97a)
77. 74
(97b)
where the coefficients are given by Eq. (25) and the rest are as –
(98a)
(98b)
and,
1 1 11
2 21 31
1 12 22
2 2 22 32
[ ]
[ ] ( )
[ ] ( )
[ ] ( )
T
T
T
T
a Hb
a Hb
b Hb
b Hb
(99)
The spatial averaging of this system of equations is exactly similar to the one presented in
section 3.2 and so we will directly present the final spatially averaged micro-scale equations as
Θ Θ (100)
Θ Θ
(101)
where all symbols have their usual meanings and the reaction term is given as
(102)
and the enzyme concentration [E], satisfies the constraint,
78. 75
(103)
Usually, is the standard concentration of the drug (Methylene blue) as available in IV
drips whereas is determined through suitable experimentation and or modeling of the
different pathways discussed earlier [ref].
4.2.1.2 Meso-scale : Capillary
The meso-scale formulation also, will be exactly similar as before (section 3.3), with the addition
of an extra reaction term in the CDR equations, which will cause it to be retained in the final
Global Evolution equations but not in the Local Equations. The process of spatial averaging, and
all details remain the same, so we present here directly the meso-scale CDR equations and their
spatially averaged forms.
The CDR equations now containing the extra term due to reactive consumption of
Methemoglobin in the micro-scale are given by—
For Oxygen:-
Θ Θ
(104)
For Nitric Oxide:
79. 76
Θ
Θ
, 1 , + 2 , 2 ,
(105)
where all symbols have their usual meanings.
The spatially averaged equations in their non-dimensional forms are modified as –
Θ Θ Θ
2, + ,1 4 ] (106)
Θ Θ Θ
2, + ,2 4 ] (107)
and four local equations:
Θ
(108)
Θ
(109)
Θ (110)
Θ (111)
where is the non dimensionalized reaction rate and is the Dam Kohler Number in the
plasma of constituent i and is given as
(112)
being a reference reaction rate taken at some reference values of the partial
pressures of gases in the RBC.
80. 77
The micro-scale equations developed in the previous section are non-dimensionalised as –
Θ Θ
(113)
Θ Θ
2, + . 2 2+ 22+ 32 4 1, , 2, (114)
The macro-scale formulation remains entirely the same as in section 3.4 with no modification,
except for the obvious fact that the final output of the macro-scale model are summed up across
different lung compartments, so that they will no longer be functions of space, but only of time.
Hence we skip the macro-scale formulation here, since it will essentially become a repetition of
section 3.4.
4.2.1.3 Numerical Solution of the micro-meso coupled scale problem
The micro meso coupled scale equations now represent a system of coupled PDEs of 1st
order.
We solve them using the standard Method of Lines approach [39]. In this method, the spatial
components are discretised but not the time derivatives, so that we end up with a set of ODEs in
time at different points in the spatial domain. These ODEs when solved in a coupled fashion
produces the entire spatio-temporal profiles of the required quantities.
Let and denote the values of x(z,t) at points z and in space.
With this notation, we discretise the equation system (106)-(114) , and put it in the following
compact form that is well suited to solution using standard routines in MATLAB.
81. 78
(115a)
(115b)
Θ Θ Θ
2, + ,1 4 +1 + [ . ]
(115c)
Θ Θ
. 2 4 1, , 2, Δ ( , , −Δ )
(115d)
where for i=1 and for i=2; ‗k’ being the ratio between the compartmental
pressures of NO and O2 ( . Also for i = 1 and for i = 2.
for i = 1, and for i = 2.
The other symbols are as –
(116a)
82. 79
(116b)
At this point, a few things must be mentioned about the algorithm adopted for the solution of this
formidable set of PDEs. The Method of Lines, once applied reduces the system to a set of ODEs
as shown above. The ODEs can be solved in a coupled fashion by considering the mass matrix
formed and this approach was adopted while solving the ODEs under steady state conditions in
Chapter 3. However, given the tremendous stiffness of the system, such an approach will
invariably be computationally very expensive. So, we adopt an explicit space marching method.
With the initial conditions given, we solve for a particular point in space say z, and use the
solution as an initial value for the ODE at the next point in space . Though some accuracy is
lost, this is fairly reliable and cuts down hugely on runtime. Further note, that the Eqs. (115) and
(116) contain derivative terms , for which initial guesses must be made to provide initial
conditions to the solver. The initial guesses are taken to be zero for both the derivatives, since
when we switch on the Methylene blue therapy, the system will have attained steady state and
the spatial profiles too will have equilibrated (refer Figures3.5 and 3.6).
84. 81
4.2.2 Results and Discussions
Simulation for the Methylene blue therapy was carried out under standard conditions of a
venous partial pressure of oxygen around 40 torr. In the meso-scale simulations, the alveolar
partial pressure of O2 is maintained at 110 torr. In the previous section, we have established that
patients with NO levels above 216 ppm are candidates for Methylene blue therapy. So, here we
use ambient NO levels of 220 ppm and higher.
The data used in the simulation are –
Variable Value
Between to
55 torr
Partial pressure of NO, corresponding to
the ambient ppm
The total concentration of enzyme was calculated as the equivalent plasma concentration of
dosage available in standard IV drips (1.2 mg/Kg body weight, and assuming an average 60 Kg
body weight).
Table 4.1 - Data used in therapy simulation
85. 82
Figure 4.2 illustrates the result of the micro-meso coupled scale computation. The spatio
temporal profiles of the saturations of the different species are presented at an ambient Nitric
Oxide concentration of 220 ppm. The dosage was taken to be , and the relaxation
time, referred to as dead time earlier in the previous section was approximated to 1 sec. The
value is decided from the steady state saturation profiles (Figure 3.5) which show that the time
required to attain steady state under disease conditions is of the order of 1 sec, which is also the
pulmonary transit time. The surface plots show that during treatment, the oxygen saturation rises
Fig 4.2 Spatio temporal profiles of meso-scale saturations of (a) Oxygen, (b) Methemoglobin, (c) Nitric
Oxide and (d) Hemoglobin
86. 83
while methemoglobin plummets sharply, within an approximate response time of 1.6 secs. The
oxygen saturation climbs steadily from around 85% to 95% under the effect of therapy.
87. 84
Fig 4.3 Dynamic profiles of Macro-scale saturations of (a) Oxygen, (b) Methemoglobin, (c) Nitrosyl-hemoglobin and (d)
Hemoglobin, under Methylene blue therapy with dose as parameter.
88. 85
In Figure 4.3, we show the time evolution of the saturation of the constituent species as a result
of the therapy, the simulation being carried out over all three length scales present in the system-
micro, meso and macro, and the results being reported in the macro scale. The active dose of
Methylene blue that reduces methemoglobin, is taken here as a parameter. The ambient Nitric
oxide was taken at 220 ppm, and the dose was continuously varied from to . The
results show that after 1 sec of real time, when the therapy is applied, Oxygen saturation (Figure
4.3 (a)) climbs sharply upto higher levels of 90% to 95% depending on the dosage administered,
while Methemoglobin levels (Figure 4.3 (b)) drop below 1%. The hemoglobin levels, as shown
in Figure 4.3(d) however remain fairly constant. This is expected since the increase in
hemoglobin due to enzymatic reduction of methemoglobin is offset by its rapid binding with
oxygen to produce oxyhemoglobin, thereby contributing to the increasing trend of the oxygen
saturation. The response time of the therapeutic procuedure can be estimated to be around 2 secs
and this rather short time can be attributed to not considering several mass transfer resistances
that appear across the different scales.
It is interesting to note that the relative change in saturation of different species, from
their steady state values on application of the therapy remains constant as we keep on increasing
the dose of methylene blue. This gives us direct insight into the sensitivity of the system to the
medicine and at a first glance it tells us that sensitivity decreases with increasing dosage. This
can be better understood when we plot system sensitivity against dosage for different
environments of Nitric Oxide.
89. 86
In this plot, we perform a formal sensitivity analysis of the change in saturation of oxygen due to
the change in the dosage for different levels of NO. Sensitivity in this case can be formally
defined as . It is not only a measure of how responsive the system is to perturbation
in the administered dosage, but also signifies, albeit indirectly, how effective a particular
concentration of methylene blue in reducing methemoglobin. The analysis reveals that with
increase in concentration of dose, sensitivity goes down, which shows that it is meaningless to
provide very high dosages. Thus there exists a tradeoff between the increase in O2 saturation (as
evident from Figure 4.3) and the sensitivity of the medicine. This can be used to make a quick
estimate of the optimum drug dosage for a known NO environment which is shown next.
Fig 4.4 Sensitivity analysis of methylene blue dose
90. 87
In Figure 4.4 we make a
rough estimate of optimum
drug dosage by comparing
the two conflicting trends—
increasing O2 saturation and
decreasing sensitivity of the
dose. For 220 ppm of NO, the
plot provides us an
approximate optimum dose
. This extremelyof
simplistic method can be used to obtain approximately the optimum drug dose for any given NO
environment.
Finally we come to a rather subtle point about the numerical algorithm used for solution
of the scale coupled PDEs. As stated in section 4.2.1.3, the Method of Lines was used to
discretise the problem and render it into a form easily solvable in MATLAB. In the expressions
that result due to discretization, as shown in Eq. (115) the step size in space denoted by
appears. Apparently, the Method of Lines is supposed to increase in accuracy [39] with
decreasing spatial step size. However, depending on the problem at hand, very low step sizes
may lead to numerical instability. In our simulations we used a rather high value of the step size,
i.e. to say a few grid points along z. Here, we give a heuristic sketch of why such stability
problems are to be expected—
The discretised forms in Eqs (115b) and (115c) may be written as
Fig 4.5 Optimization of methylene blue dose
91. 88
Thus, and . That is to say, for high values of
, one of the derivatives remains fairly constant while the other becomes lower and lower. This ensures
stability as it does not allow any unbounded increase of either of the two partial pressures. However,
notice that in the other extreme limit, we have, and
. Hence, one derivative in the system now grows unboundedly with decreasing step
size and at some step size such unbounded growth offsets the slowly increasing nature of the other
derivative and the entire system heads towards numerical instability and ultimately reaches what is called
a blow up. The value at which this occurs
can be obtained through trial and error and
should always be kept above this
value, even if it means working with a few
grid points.
We obtained the critical value of
(or h, as in Figure 4.5) through trial and
error. Figure 4.5 reports the growing
instability with decreasing step size and at 15 grid
points, complete blow up occurs as shown by the oscillation. The maximum allowed number of grid
points was estimated to be 12.
Fig 4.5 Changes in stability with step-size
93. 90
5.1 Conclusions
In this analysis, we develop a complete multi-scale model of methemoglobin anemia and
provide quantitative estimates of different important quantities that might improve our
understanding of the disease and its possible therapeutic strategies. We start with the
fundamental convection-diffusion-reaction equations and derive a coupled three (micro-meso-
macro) scale model for the reactive transport of nitric oxide and oxygen in the red blood cell, the
pulmonary capillary and the lung. The multi-scale model is subjected to a rigorous Liapunov-
Schmidt reduction (of the classical Bifurcation Theory) which reduces the dimensionality of the
model, thus reducing the computational effort for solution while retaining all the essential
physics of the original CDR model. The low dimensional model of each scale is embedded
hierarchically in the next larger scale and the models at the three scales are solved
simultaneously. Numerical simulation of our coupled multiscale model provides us with
quantities of physiological significance and direct insights into the mechanism of
methemoglobinemia. By way of comparing the saturation profiles of O2 and NO at different
length scales, we conclusively show that while the presence of NO significantly influences
reactive uptake of O2, the O2 partial pressure hardly affects NO deposition in the alveolar
epithelium. This is central to the understanding of the why and the how of the disease.
The multi-scale nature of the disease is reflected in the reduction of overall gas
saturations (both O2 and NO) as we pass from one scale to the next higher one. Mathematically,
it illustrates the intricate scale coupling, while physically it reflects the increased mass transfer
resistance across the scales. One of the key findings of this paper is the critical or the safe limit
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of NO in the lung alveoli. Our simulations show that for NO concentration above 166 ppm, O2
saturation falls below the critical limit of 90% required for the human body to barely function.
Thus, we find that an alveolar NO concentration of 166 ppm or higher in methemoglobinemia
patients would be fatal due to severe hypoxemia and requires immediate therapy.
As a natural extension, we go on to discuss the commonly-used therapeutic approach for
methemoglobinemia. We present a discussion on Oxygen therapy and applying the model
developed, we are able to classify patients into those who would respond to Oxygen therapy and
those who are candidates for Methylene blue therapy. Our simulations show that the threshold
level of NO in the system which decides this aforementioned stratification is around 216 ppm.
Finally, we present a multiscale simulation of Methylene blue therapy using an unsteady state
version of the model developed previously. The therapy is shown to be effective in raising O2
saturation from 88% in disease conditions to about 95 %. The effects of different magnitudes of
dose of medicine are computationally predicted and sensitivity and optimization analyses are
outlined; for an ambient NO level of 220 ppm, the optimal dose plasma concentration of the dose
is estimated to be 0.35 .
5.2 Future Work
There exists scope for future work in extending and validating (or eliminating) some of the
hypotheses that were made at various points of the work, based on model predictions. The
disparity in NO saturations for endogenous and exogenous environments of Oxygen, as observed
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in section 3.5 needs to be further probed into. If such a transition can be verified, either from
literature or from direct experiments, then a more accurate biphasic model can be formulated that
takes into account this switch-over. Further, the multiscale simulation of methylene blue therapy
predicts rather short response times, which has been attributed to neglecting mass transfer
resistances in the lung during the construction of the compartmental macro-scale model. It is
interesting to see how the effect of such resistances can be formulated to provide slower response
times, thus making the simulation more realistic and closer to real life.
